ABSTRACT
Title of dissertation: CHARACTERIZATION OF
JOSEPHSON DEVICES FOR USE IN
QUANTUM COMPUTATION
Sudeep K. Dutta, Doctor of Philosophy, 2006
Dissertation directed by: Professor Frederick C. Wellstood
Department of Physics
This thesis examines Josephson tunnel junctions as candidate qubits for quan-
tum computation. A large area current-biased junction, known as a phase qubit,
uses the two lowest energy levels in a tilted washboard potential as the qubit states
j0i and j1i. I performed experiments with 10 ? 10 ?m2 Nb/AlOx/Nb qubit junc-
tions, with critical currents of roughly 30 ?A. The state of a device was initialized
by cooling below 50 mK in a dilution refrigerator. In order for quantum mechanical
superpositions to be long-lived, it is necessary to isolate the junction from noisy bias
leads that originate at room temperature. I studied two types of isolation: an LC
fllter, and a broadband scheme that used an auxiliary junction, resulting in a dc
SQUID.
One of the main goals of this work was to determine how well a simple Hamilto-
nian, derived assuming just a few lumped elements, describes the observed behavior
of a macroscopic Josephson device, including coherent dynamics such as Rabi oscil-
lations. I did this by comparing results to the expected behavior of ideal two-level
systems and with more detailed master equation and density matrix simulations.
I performed state manipulation by applying dc bias currents and resonant
microwave currents, and through temperature control. The tunneling escape rate
of the junction from the states j0i and j1i (zero voltage) to the running state (flnite
voltage) depends on the occupation probability of the energy levels and served as
state readout.
Experiments to measure the relaxation time T1 between j1i and j0i were per-
formed by examining the dependence of the escape rate with temperature, yielding
a maximum T1 ? 15 ns. Measuring the decay to the ground state after applying
a microwave pulse revealed at least two time constants, one of about 10 ns and
another as long as 50 ns. The spectroscopic coherence time T?2 was estimated to
be roughly 5 ns by measuring resonance widths and the decay envelope of coherent
Rabi oscillations was found to have a time constant T0 ? 10 ns over a wide range of
conditions.
CHARACTERIZATION OF JOSEPHSON DEVICES
FOR USE IN QUANTUM COMPUTATION
by
Sudeep K. Dutta
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulflllment
of the requirements for the degree of
Doctor of Philosophy
2006
Advisory Committee:
Professor Frederick C. Wellstood, Advisor / Chair
Professor J. Robert Anderson, Advisor
Professor Christopher J. Lobb, Advisor
Professor Steven M. Anlage
Professor Donald R. Perlis
c Copyright by
Sudeep K. Dutta
2006
Preface
While the Acknowledgements recognizes the people that helped me throughout
gradschool, Iwantedtoreservethis page forresources Iused speciflcally inpreparing
this document.
The previous theses from my group are outstanding [1{3]. In the interest of
saving trees, I could have replaced about half of my thesis with references to those
works. Especially in recent months, Fred Strauch has been an invaluable resource.
It would have been easier on both of us if he had just written a few sections of
Chapter 3 himself. Intriguing points that he has made over the years appear in
several other places.
Naturally, many of the explanations I have provided come straight from the
professors. While I was writing this thesis, Dr. Lobb foolishly agreed to play Twenty
Questions with me every Wednesday and many of his answers (occasionally more
than \yes" or \no") appear throughout the thesis. He and John Matthews are re-
sponsible for big chunks of Chapter 2. Over the past few years, Dr. Wellstood has
produced a sizeable stack of notes and chalkboards full of diagrams and derivations.
These covered topics ranging from dilution refrigerator techniques to quantum me-
chanical simulations. I have taken quite a bit from these and chosen to lump all of
the documents under Ref. [4], as I am unable to read his handwriting and decipher
the titles he gave them.
I had the great pleasure of sitting in on PHYS 711: Atomic Physics in the
Fall 2005 semester, taught by Drs. Bill Phillips and Trey Porto. Seeing two-level
systems presented from a difierent point of view was enormously beneflcial. They
cleared up many concepts and I hope they don?t mind that I have reproduced some
ii
of their arguments in Chapter 3 [5].
I don?t think I?ve ever typed anything longer than about 12 pages, so I was
terrifled when I started writing. I started with the LATEX template created by
Dorothea Brosius of the University?s Institute for Research in Electronics and Ap-
plied Physics and only had to make minor changes. The document was compiled
with the open-source MiKTEX 2.4 packages and I used Aleksander Simonic?s wonder-
ful shareware editor WinEdt v5.4. Essentially all of the data analysis was performed
with MATLAB R12 and R14. I drew graphs with OriginPro 7.5 and layout for the
flgures was done in Adobe Illustrator CS2.
iii
In loving memory of my father,
who taught me by example,
\If you have to do something, do it right."
iv
Acknowledgements
I?ve always been amused at how sappy this section of the average thesis is.
Now that I?ve made it through grad school, I have a better idea of where all the
emotion comes from. I hope my efiort, below, doesn?t disappoint future generations.
I will never be able to repay the debt I owe to my advisors, Profs. Fred Well-
stood, Chris Lobb, and Bob Anderson, who restored my faith is physics, mankind,
and myself in just a few years. If and when I grow up, I want to be like Dr. Anderson,
who has an irrepressible wit and is always able to ask the surprisingly penetrating
question. Dr. Chris, a fellow New Jerseyan, has uncommonly clear vision on a range
of matters, which I relied on to get out of many sticky situations.
As for Dr. Wellstood, I just can?t imagine enduring the terrors of grad school
under anyone else?s direction. In times of relative peace, he often let me call the
experimental shots, all the while ofiering a critical eye, encouragement, and endless
notes and derivations. Even though all of my hot leads ended up as incoherent
goose chases, I think he understood that the process was enormously satisfying. As
deadlines approached, I saw how he made sound decisions (on matters of \politics"
and science), and I consider that an invaluable part of my education.
In case my colleagues are reading this and can distinctly picture me screaming,
I don?t mean to suggest that things were always rosy. At all. In fact, I probably
had a violent disagreement with one of the bosses every 17 hours. What sets these
gentlemen apart is that when they told me I was wrong, they usually told me why
I was wrong. By actually bothering to *gasp* advise me, my advisors made me a
better scientist and lowered my blood pressure to record lows.
Allow me to sum up. The profs gave me an interesting and timely project, a
v
lab full of equipment, great people to work with, and good advice whether I asked
for it or not. In the end, though, the success of a chunk of the experiments was left
to me. Now that?s upper management at its flnest. I hope they?re satisfled with
what I chose to do with the pieces of the puzzle.
I should also thank the rest of my thesis committee, Profs. Don Perlis and Steve
Anlage, who put up with (nearly) endless delays. In addition, Dr. Anlage somehow
tolerated me for two years, when I worked for him as an unruly undergrad; it?s fair
to say that a day doesn?t go by in lab that I don?t use something I learned in his
group.
I must also acknowledge the members of the experimental nuclear physics
group. Prof. Phil Roos was my undergrad research and academic advisor and is
occasionally pressed into service to this day. Prof. Betsy Beise taught me about
electronics and how to solder well; as far as I can tell, this remains my only mar-
ketable skill. I learned just about all of the LabVIEW I know from Fraser Duncan.
Prof. Jim Kelly taught me FORTRAN and Mathematica and, perhaps more impor-
tantly, the beneflts of attacking a problem with all you?ve got.
A famous biophysicist once compared grad school unfavorably to a deep, dark
tunnel. Given the location of our lab, I?ll not disagree with that, but I?m sure
it?s much worse if you?re down there alone. It is not possible to overstate the
contribution that Huizhong Xu, Roberto Ramos, Andrew Berkley, Mark Gubrud,
and a slew of undergrads (including Sam Reed, Bill Parsons, and Joe Foley) made
to this thesis. They had the experiment and refrigerator running smoothly when I
joined the project, leaving me in the incredibly enviable position of just having to
concentrate on taking data. I was particularly fortunate to overlap with the flrst
two guys for 18 months, every day of which I learned something new. Although
Huizhong left the project over two years ago, his flngerprints can still be clearly
seen on the detection electronics, acquisition and simulation code, and my insatiable
vi
cookie jones. Roberto, who often let me in on specials at ramos.com, taught me
a great deal about low temperature physics and the philosophy of doing efiective
experiments.
I owe a tremendous amount to everyone currently working on the project
as well. I was very lucky to witness Tauno Palomaki?s seminally flducial year. I
cleverly wrote this thesis very slowly, so that he could do the experiments that
explained many of my results. Rupert Lewis was somehow able to herd a sub-
basement full of students, while still managing to teach me about microwave and
low temperature techniques. I mimicked several experiments that Hanhee Paik flrst
performed; she also taught me half of the Korean I know. It was great fun to watch
as Ben \Coop" Cooper and Tony \Tony" Przybysz brought energy and fresh ideas to
the project. I?m most impressed with Hyeokshin Kwon for withstanding my Korean.
John Wyrwas and Bjorn Van Bael accomplished more than I thought possible in the
chaos of our lab. I don?t know where to put Ben Palmer, but he was a dependable
source of information and insights.
Perhaps the most pleasant surprise about working on this project was our
close collaboration with Prof. Alex Dragt and his group, including Phil Johnson,
Fred Strauch, Kaushik Mitra, Momahed Abutaleb, and Bill Parsons. These are no
ordinary theorists. Not only were they able to turn our confusing experimental data
into publishable results, but they also often made staggeringly reasonable proposals
for new experiments to try.
I am grateful to the rest of the research group for creating such a pleasant place
to work and putting up with all of my quirks. I?ve known Gus Vlahacos longer than
I care to mention; I wonder if either of us will be able to work above sea level.
John Matthews was my one-stop-shop for questions about SQUIDs and LabVIEW.
Crazy David Tobias was always willing to lend a hand or engage in a little high
stakes wagering. The corner of the lab was never quite as friendly after Su-Young
vii
Lee left; she taught me the other half of the Korean I know. Matt Sullivan and
Monica Lilly defy description; I?ll just say that lab would?ve been far duller without
them. Hua Xu and Zhengli Li were as helpful as they could be, even as we slowly
pushed them out of lab. Satheesh Angaiah was my faithful basketball buddy.
The Center for Superconductivity Research has been a terriflc place to work.
I?ll start at the top with Admiral Rick Greene, who runs a tight ship. I?d never admit
this to him, but I learned a great deal (about physics and giving talks) in his seminar.
Doug Bensen and Brian Straughn not only are capable of solving any of the random
problems constantly thrown at them, but can also make you laugh an unhealthy
amount while doing so. I?m glad that I don?t know how much bureaucratic red tape
Belta Pollard, Cleopatra White, Grace Sewlall, and Brian Barnaby cut through so
we could get our work done.
I was often suspicious of how friendly and eager to help the students and post-
docs in the Center were. I had a number of microwave emergencies that Mike Ricci,
Dragos Mircea, Sameer Hemmady, and Nate Orlofi handled cooly; they, for instance,
helped me do the network analyzer measurements in Chapter 5. I could always go
to Yuanzhen Chen and Samir Garzon to ask about low temperature methods or to
\borrow" liquid helium. I eased the misery of writing this thesis by commiserating
with Josh Higgins. I?m proud to say that, by flnishing three months after him,
I won our race. Incredibly, Clippity always seemed to know just when I wanted
to make a list. You?d be hard pressed to flnd anyone more enthusiastic or with a
better name than Diane Elizabeth Pugel. Maria Aronova has the amazing ability of
raising the spirits of any room she walks into; she also was quick to point out how
little Korean I actually knew. My memory is failing me (due to a lack of oxygen
in lab), so I can only give a partial list of the people that I either stole equipment
from or who entertained me in the darker times: Pengcheng Li, Enrique Cobas,
Gokhan Esen, Tarek Ghanem, Dan Lenski (my nemesis), Adrian Southard, Todd
viii
Brintlinger, Yung-Fu Chen, Jim Ayari, and Tobias D?urkop.
Next up is the rest of Department of Physics. Prof. Nick Chant couldn?t have
made it easier for me to meet the program?s requirements, for which he has my
profound gratitude. Jane Hessing?s ability to remember every form that I needed to
flll out was truly remarkable. I?m sure I would have ended up in the wrong building
on the day of my defense if it wasn?t for her. Bernie Kozlowski, Linda O?Hara,
and Tom Gleason never turned me away, whether I came to beg for help or just to
chat. Jesse Anderson and Bob Dahms dealt with all of my purchasing requests with
freakish e?ciency. Pauline Rirksopa, Russ Wood, Bob Woodworth, Al Godinez,
and George Butler always went out of their way to make my life a little bit easier.
Finally, I wish to thank my family: my cousins, Sompa, Chino, Munu, and
Ashim, who did more to make this thesis possible than I could have ever asked;
Subrata, my big brother and flrst physics professor; and Kabita, who has deflned
nearly every aspect of her son.
I am well aware that second chances of the quality that Maryland has given
me don?t come along often in life. I could try to describe what this school means to
me, but I?ve already gone on far too long. I?ll instead end with a cheer. Go Terps!
ix
Table of Contents
List of Tables xiii
List of Figures xiv
List of Abbreviations xviii
1 Introduction 1
1.1 A Brief Review of Quantum Computation . . . . . . . . . . . . . . . 1
1.2 Superconducting Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Josephson Junctions and SQUIDs 11
2.1 Josephson Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 RCSJ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 RCSJ Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 The Tilted Washboard Potential . . . . . . . . . . . . . . . . . 20
2.2.3 Junction IV Curve . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Quantum Mechanical Properties of the Josephson Junction . . . . . . 25
2.3.1 Harmonic Oscillator Approximation . . . . . . . . . . . . . . . 27
2.3.2 Cubic Approximation . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.3 Full Tilted Washboard . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Asymmetric dc SQUID Hamiltonian . . . . . . . . . . . . . . . . . . . 36
2.5 Classical SQUID Behavior . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Current-Flux Characteristics . . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Capacitively-Coupled Junction Qubits . . . . . . . . . . . . . . . . . 54
3 Dynamics of Quantum Systems 59
3.1 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Two-Level Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Three-Level Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.1 The Reduced Density Matrix . . . . . . . . . . . . . . . . . . 73
3.7 Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8 Multi-Level Density Matrix . . . . . . . . . . . . . . . . . . . . . . . 82
4 Qubit Design and Fabrication 84
4.1 Hypres Fabrication Process . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 LC-Isolated Phase Qubit . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.1 Device LC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 dc SQUID Phase Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Device DS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
x
4.3.2 Device DS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Instrumentation and Experimental Apparatus 115
5.1 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.1 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Current and Flux Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 Refrigerator Wiring . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.2 Biasing of DS2 . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4 Voltage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.6 Current Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Device Characterization and Measurement Techniques 160
6.1 IV Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2 Escape Rate Measurement . . . . . . . . . . . . . . . . . . . . . . . . 170
6.3 Current-Flux Characteristics . . . . . . . . . . . . . . . . . . . . . . . 174
6.4 Simultaneous Biasing . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.5 Flux Shaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.6 State Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.6.1 Direct Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.6.2 Microwave Pulse . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.6.3 dc Bias Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7 Tunneling Escape Rate Measurements 209
7.1 Thermal Activation and Macroscopic Quantum Tunneling . . . . . . 211
7.1.1 LC-Isolated Phase Qubits . . . . . . . . . . . . . . . . . . . . 212
7.1.2 dc SQUID Phase Qubits . . . . . . . . . . . . . . . . . . . . . 221
7.2 Low Temperature Escape Rate . . . . . . . . . . . . . . . . . . . . . 226
7.2.1 LC-Isolated Qubits . . . . . . . . . . . . . . . . . . . . . . . . 227
7.2.2 dc SQUID Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.3 Master Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.4 Determination of T1 With a Slow Bias Sweep . . . . . . . . . . . . . . 240
7.4.1 LC-Isolated Phase Qubits . . . . . . . . . . . . . . . . . . . . 245
7.4.2 dc SQUID Phase Qubits . . . . . . . . . . . . . . . . . . . . . 251
7.5 Fast Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8 Spectroscopy and Non-Coherent Dynamics 260
8.1 Spectroscopy of LC-Isolated Phase Qubits . . . . . . . . . . . . . . . 260
8.2 Spectroscopy of dc SQUID Phase Qubits . . . . . . . . . . . . . . . . 265
8.3 Spectroscopic Coherence Time . . . . . . . . . . . . . . . . . . . . . . 272
8.4 Multi-Level and Multi-Photon Transitions . . . . . . . . . . . . . . . 282
xi
8.4.1 Power Dependence . . . . . . . . . . . . . . . . . . . . . . . . 287
8.5 Spurious Junction Resonances . . . . . . . . . . . . . . . . . . . . . . 290
8.6 Spectroscopy of Coupled Qubits . . . . . . . . . . . . . . . . . . . . . 304
8.7 Time-Domain Measurement of T1 . . . . . . . . . . . . . . . . . . . . 312
8.7.1 Microwave Pulse Readout . . . . . . . . . . . . . . . . . . . . 313
8.7.2 Direct Tunneling Readout . . . . . . . . . . . . . . . . . . . . 316
8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9 Coherent Rabi Oscillations 324
9.1 Power Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
9.2 Detuning and Strong Field Efiects . . . . . . . . . . . . . . . . . . . . 334
9.3 Density Matrix Simulations . . . . . . . . . . . . . . . . . . . . . . . 344
9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10 Conclusions 361
10.1 The DiVincenzo Criteria Revisited . . . . . . . . . . . . . . . . . . . 361
10.2 Summary of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 363
A Circuit Hamiltonians 367
B MATLAB Code 369
B.1 Solutions of the Junction Hamiltonian . . . . . . . . . . . . . . . . . 369
B.2 Solutions of the Coupled-Junction Hamiltonian . . . . . . . . . . . . 378
C Three-Level Rotating Wave Approximation 383
D Dynamical Evolution Matrices 385
D.1 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
D.2 Master Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Bibliography 390
xii
List of Tables
2.1 Flux state properties for a low fl dc SQUID . . . . . . . . . . . . . . 45
4.1 ParametersofdeviceLC2 (twocapacitively-coupledLC-isolatedphase
qubits) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Parameters of device DS1 (dc SQUID phase qubit) . . . . . . . . . . 106
4.3 Parameters of device DS2 (two capacitively-coupled dc SQUID phase
qubits) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.1 Commercial electronics used in the escape rate measurement . . . . . 117
6.1 Current- ux characteristic properties of SQUID DS1 . . . . . . . . . 181
7.1 Master equation flt parameters for junction LC2B . . . . . . . . . . . 249
8.1 T1 flt results for SQUID DS1 . . . . . . . . . . . . . . . . . . . . . . . 316
10.1 Summary of characteristic times . . . . . . . . . . . . . . . . . . . . . 364
xiii
List of Figures
1.1 Yearly quantum computation publications . . . . . . . . . . . . . . . 2
2.1 Schematic of a Josephson junction . . . . . . . . . . . . . . . . . . . . 12
2.2 RCSJ model circuit diagram . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 The tilted washboard potential . . . . . . . . . . . . . . . . . . . . . 18
2.4 Plasma oscillations of a capacitively shunted junction . . . . . . . . . 22
2.5 IV curve of a current-biased junction . . . . . . . . . . . . . . . . . . 23
2.6 Quantum states of the tilted washboard . . . . . . . . . . . . . . . . 25
2.7 Number of levels of the tilted washboard potential . . . . . . . . . . . 32
2.8 Energy level transitions of the tilted washboard potential . . . . . . . 32
2.9 Tunneling rates of the tilted washboard potential . . . . . . . . . . . 33
2.10 Ground state escape rate . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.11 Matrix elements of the tilted washboard potential . . . . . . . . . . . 35
2.12 dc SQUID circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . 37
2.13 Potential of a symmetric dc SQUID with respect to symmetry axes . 42
2.14 Potential of a symmetric dc SQUID with respect to junction axes . . 43
2.15 Current- ux characteristics of a symmetric SQUID . . . . . . . . . . 49
2.16 Current- ux characteristics of an asymmetric dc SQUID . . . . . . . 51
2.17 Slopes of the current- ux characteristics of an asymmetric dc SQUID 52
2.18 Circuit diagram for two LC-coupled junctions . . . . . . . . . . . . . 55
3.1 Bloch sphere representation of a two-level system . . . . . . . . . . . 60
3.2 Three-level rotating wave approximation . . . . . . . . . . . . . . . . 66
3.3 Power broadening in a two-level system . . . . . . . . . . . . . . . . . 80
4.1 Hypres fabrication process . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 LC isolation of a current-biased junction . . . . . . . . . . . . . . . . 87
4.3 Efiective impedances of an LC isolation network . . . . . . . . . . . . 90
4.4 Photographs of device LC2 . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 dc SQUID phase qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 96
xiv
4.6 Isolation of a dc SQUID phase qubit from its bias line . . . . . . . . . 99
4.7 Flux line coupling to SQUID . . . . . . . . . . . . . . . . . . . . . . . 101
4.8 Isolation of a dc SQUID phase qubit from its  ux line . . . . . . . . . 103
4.9 Photographs of SQUID DS1 . . . . . . . . . . . . . . . . . . . . . . . 105
4.10 Photographs of coupled SQUIDs DS2 . . . . . . . . . . . . . . . . . . 109
5.1 Overview of experimental set-up . . . . . . . . . . . . . . . . . . . . . 116
5.2 Timing diagram for escape rate measurement . . . . . . . . . . . . . 118
5.3 Refrigerator photographs . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 Sample box photograph . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5 Mixing chamber thermometer calibration . . . . . . . . . . . . . . . . 125
5.6 Schematic of refrigerator wiring . . . . . . . . . . . . . . . . . . . . . 129
5.7 High frequency properties of Thermocoax . . . . . . . . . . . . . . . . 130
5.8 Discrete LC bias fllter . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.9 High frequency properties of an LC fllter . . . . . . . . . . . . . . . . 134
5.10 Photograph of a long copper powder fllter . . . . . . . . . . . . . . . 135
5.11 High frequency properties of copper powder fllters . . . . . . . . . . . 136
5.12 Low frequency properties of the bias fllters . . . . . . . . . . . . . . . 138
5.13 Avoided crossing biasing of device DS2 . . . . . . . . . . . . . . . . . 143
5.14 Gain and phase shift of the homemade voltage ampliflers . . . . . . . 147
5.15 Charging of the voltage line . . . . . . . . . . . . . . . . . . . . . . . 152
6.1 IV curves of device LC2 . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2 IV curves of junction LC2B as a function of fleld . . . . . . . . . . . 165
6.3 Difiraction patterns of device LC2 . . . . . . . . . . . . . . . . . . . . 167
6.4 IV curves of device DS2 . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.5 Switching experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.6 Ib vs. 'A characteristic of SQUID DS1 . . . . . . . . . . . . . . . . . 175
6.7 Bias trajectories for a dc SQUID . . . . . . . . . . . . . . . . . . . . 176
6.8 Swept- ux Ib vs. 'A characteristics of SQUID DS1 . . . . . . . . . . 179
6.9 Simultaneous biasing of a dc SQUID phase qubit . . . . . . . . . . . 186
xv
6.10 Schematic of the  ux shaking procedure. . . . . . . . . . . . . . . . . 190
6.11 Experimental determination of shaking amplitude and ofiset . . . . . 194
6.12 Flux dependence of the number of allowed states . . . . . . . . . . . . 197
6.13 Flux shaking with a variable number of oscillations . . . . . . . . . . 198
6.14 Flux shaking with a large number of states . . . . . . . . . . . . . . . 200
6.15 Schemes for state readout . . . . . . . . . . . . . . . . . . . . . . . . 203
7.1 Temperature-dependent escape rates of junction LC2B . . . . . . . . 210
7.2 Thermal activation in junction LC2B . . . . . . . . . . . . . . . . . . 213
7.3 Efiective escape temperature for junction LC2B . . . . . . . . . . . . 215
7.4 Efiective escape temperature for junction LC2A . . . . . . . . . . . . 219
7.5 Efiective escape temperature for SQUID DS1 . . . . . . . . . . . . . 222
7.6 Efiective escape temperature for SQUID DS2B . . . . . . . . . . . . 224
7.7 Low temperature escape rate of junction LC2B . . . . . . . . . . . . 228
7.8 Low temperature escape rate of SQUID DS1 . . . . . . . . . . . . . . 231
7.9 Flux state dependence of ? for SQUID DS1 . . . . . . . . . . . . . . 234
7.10 Flux state dependence of ? for SQUID DS2B . . . . . . . . . . . . . 236
7.11 Transitions of a two-level system . . . . . . . . . . . . . . . . . . . . . 243
7.12 Master equation simulation of junction LC2B . . . . . . . . . . . . . 246
7.13 Master equation simulation of SQUID DS1 . . . . . . . . . . . . . . . 252
7.14 The efiect of sweep rate on ? . . . . . . . . . . . . . . . . . . . . . . 254
7.15 \Evaporative cooling" of a Josephson junction . . . . . . . . . . . . . 257
8.1 Transition spectra of junction LC2B . . . . . . . . . . . . . . . . . . 262
8.2 Spectra of SQUID DS1 . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.3 Spectral flt parameters for SQUID DS1 . . . . . . . . . . . . . . . . . 268
8.4 Shallow ramp spectrum of SQUID DS1 . . . . . . . . . . . . . . . . . 271
8.5 Resonance widths of LC2B and DS1 . . . . . . . . . . . . . . . . . . 275
8.6 Resonance widths of SQUIDs DS2A and DS2B with difierent ramp
rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.7 Resonance widths of SQUID DS1 at elevated temperature . . . . . . 281
xvi
8.8 Low frequency spectrum of SQUID DS2B . . . . . . . . . . . . . . . 283
8.9 High frequency spectrum of SQUID DS2B . . . . . . . . . . . . . . . 285
8.10 Power dependence of spectral peaks in SQUID DS1 . . . . . . . . . . 288
8.11 Fine spectrum of SQUID DS2B . . . . . . . . . . . . . . . . . . . . . 292
8.12 Resonance shape near an avoided crossing . . . . . . . . . . . . . . . 294
8.13 Three flne spectra of SQUID DS2A . . . . . . . . . . . . . . . . . . . 296
8.14 Model of a spurious junction resonator . . . . . . . . . . . . . . . . . 298
8.15 Fine spectra of SQUID DS1 at base and elevated temperatures . . . . 300
8.16 Fine spectrum of SQUID DS2B at elevated temperature . . . . . . . 302
8.17 Coupled junction spectra . . . . . . . . . . . . . . . . . . . . . . . . . 305
8.18 Avoided crossing spectra of DS2 . . . . . . . . . . . . . . . . . . . . . 308
8.19 Relevant energy levels for a simple controlled-NOT gate . . . . . . . . 311
8.20 T1 measurement of SQUID DS1 with pulsed readout . . . . . . . . . 314
8.21 T1 measurement of SQUIDs with a direct tunneling readout . . . . . 318
9.1 Rabi oscillations in SQUID DS1 . . . . . . . . . . . . . . . . . . . . . 325
9.2 Power dependence of Rabi oscillations in SQUID DS1 . . . . . . . . . 327
9.3 Phenomenological flts of Rabi oscillations in SQUID DS1 . . . . . . . 329
9.4 Power broadening in SQUID DS1 . . . . . . . . . . . . . . . . . . . . 333
9.5 Rabi detuning map of SQUID DS2B at elevated temperature . . . . . 335
9.6 Rabi detuning map of SQUID DS2B at !rf=2? = 5:9 GHz . . . . . . 339
9.7 Strong fleld efiects in SQUID DS2B . . . . . . . . . . . . . . . . . . . 341
9.8 Rabi detuning map of SQUID DS2B at !rf=2? = 2:95 GHz . . . . . . 343
9.9 Power dependence of ?eq in SQUID DS1 . . . . . . . . . . . . . . . . 347
9.10 Loss of visibility in Rabi oscillations . . . . . . . . . . . . . . . . . . . 350
9.11 Density matrix simulations of Rabi oscillations in SQUID DS1 . . . . 354
9.12 Escape rate and current pulse measurements of Rabi oscillations in
SQUID DS2B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
9.13 Efiect of j2i on 0 ! 1 Rabi oscillations. . . . . . . . . . . . . . . . . . 359
xvii
List of Abbreviations
Symbols that are used through the thesis are listed below. Several symbols
that are only used in a few sections within the same chapter have been omitted.
Fundamental constants
h Planck?s constant, 6:626?10?34 J?s
~ h=2?, 1:054?10?34 J?s
e absolute value of the electron charge, 1:602?10?19 C
'0 superconducting  ux quantum, h=2e = 2:068?10?15 T?m2
kB Boltzmann?s constant, 1:381?10?23 J=K
Notation
v vector
M matrix
_a derivative of a with respect to time
Junction parameters and controls
 phase difierence across junction
VJ voltage across junction
Vb measured junction voltage on bias line,
due to VJ and stray series resistance
Ib current bias
I0 critical current
Ir reduced current bias, Ib=I0
CJ shunting capacitance
RJ shunting resistance
mJ efiective mass of \phase particle," CJ ('0=2?)2
LJ Josephson inductance, '0=2?I0 cos 
EJ Josephson coupling energy, '0I0=2?
EC charging energy, e2=2CJ
!p plasma frequency, 1=pLJCJ
!p0 zero-bias plasma frequency, p8ECEJ=~
flC Stewart-McCumber hysteresis parameter, (!p0 RJCJ)2
xviii
Q quality factor, !p RJCJ
Bk in-plane suppression fleld
I?w microwave activation current
PS nominal power of I?w, as set at the source
!rf angular frequency of I?w
C?w capacitance that couples I?w to qubit
SQUID parameters: In addition to the quantities listed below, each of the SQUID
junctions is described by the parameters above. A subscript ?1? indicates the qubit
junction; ?2? indicates the isolation junction. For example, I01 is the qubit junction?s
critical current.
Ib total current bias
I1 current through qubit branch
I2 current through isolation branch
Mb mutual inductance between the bias line and SQUID loop
If  ux current
Mf mutual inductance between the  ux line and SQUID loop
J circulating current
'A applied  ux due to  ux current, MfIf
'T total  ux due to Ib, If, any applied magnetic fleld, and circulating current
L1 geometrical inductance of qubit branch of the SQUID loop
L2 geometrical inductance of isolation branch of the SQUID loop
fl modulation parameter, L(I01 +I02)='0
N'  ux state label
Coupled qubits
CC coupling capacitor
LC stray inductance from coupling lines
?0 purely capacitive coupling constant, CC=(CJ +CC)
? LC coupling constant, pCC=(CJ +2CC)
!C renormalized LC frequency, 1=??pLCCJ?
? (!p) efiective capacitive coupling constant, ?2=?1??2 ?!2p=!2C?
xix
Device names: For the coupled devices, a superscript ?A? or ?B? indicates the indi-
vidual device. For example, IB01 is the qubit junction?s critical current for DS2B.
LC2 two capacitively-coupled LC-isolated phase qubits, LC2A and LC2B;
see x4.2.1
DS1 single dc SQUID phase qubit; see x4.3.1
DS2 two capacitively-coupled dc SQUID phase qubits, DS2A and DS2B;
see x4.3.2
Dissipation
Y shunting admittance
Reff efiective parallel resistance, 1=Re(Y)
Xeff efiective parallel reactance, ?1=Im(Y)
Leff efiective parallel inductance, Xeff=!
Ceff efiective parallel capacitance, ?1=!Xeff
Rbn resistance that characterizes noise on the current bias line
Rfn resistance that characterizes noise on the  ux bias line
Quantum properties and simulation
jni n-th metastable eigenstate of the system; j0i is the ground state
En energy of state jni
!nm angular frequency spacing between jni and jmi, jEm ?Enj=~
? density matrix
?nm element of the density matrix
?n diagonal element of the density matrix, ?nn
?tot probability of being in the zero-voltage state, P?n
Pn normalized occupation probability, ?n=?tot
?n tunneling escape rate from state jni
? total escape rate, PPn?n
Wnm inter-level transition rate from jmi to jni
Wtnm thermal transition rate from jmi to jni
 n;m matrix element of the phase operator ^ that characterizes transitions
^x modifled phase operator, pmJ!p=~?^ ?sin?1 Ir?
T1 dissipation or relaxation time
T` dephasing time
xx
T2 coherence time, (1=2T1 +1=T`)?1
?! full width at half maximum of a resonance,
in terms of angular frequency
T?2 spectroscopic coherence time, 2=?! for 0 ! 1 transition
?nm bare Rabi angular frequency between jni and jmi
?nm efiective Rabi frequency; includes detuning
?R;nm minimal Rabi oscillation frequency in a multi-level system;
deflnes resonance (!rf need not equal !nm)
?R;nm efiective Rabi oscillation frequency with respect to ?R;nm
T0 decay time constant of Rabi oscillation envelope
Experimental data
h(ti) number of times the junction switched to the voltage state
in the histogram bin centered at time ti
? measured escape rate, calculated from h
?bg escape rate in the absence of a microwave current
??=? escape rate enhancement due to a microwave current, (???bg)=?bg
xxi
Chapter 1
Introduction
1.1 A Brief Review of Quantum Computation
In the early 1980?s, Feynman pointed out the di?culties of exactly simulating,
or perhaps imitating, a quantum system using a classical computer, even one that
was probabilistic in nature [6]. In particular, he argued that the size of the classical
computer will grow exponentially with the size of the system being studied. As
an alternative, he ofiered the possibility of using a computer that took advantage
of the quantum mechanical nature of its own elements to perform simulations and
faithfully include the unpredictability of quantum physics.
The building block of a quantum computer is the quantum bit, or qubit, which
has two states j0i and j1i. Unlike a classical bit, which must be deflnitively in either
state, a qubit can be in an arbitrary superposition aj0i+ bj1i, where a and b are
complex coe?cients. For this reason, a quantum computer may simplistically be
thought of as a parallel array of classical processors. A limitation is that, at the end
of a computation, only classical information can be extracted. A quantum computer
would have access to an additional resource with no classical analog, entanglement,
where the states of multiple qubits are inextricably linked. More precisely, the
collective state cannot be expressed as the direct product of single qubit states.
Figure 1.1 shows a plot of the number of papers per year that had quantum
computation terms in their abstracts. It is evident that there was minimal interest
in the fleld at the time of the original proposals, in the 1980?s. While the ideas were
intriguing, it was not thought possible to physically realize a quantum computer.
Also it was known that the potential power of classical analog computing was largely
1
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s115
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s80
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s98
s108
s105
s115
s104
s101
s100
s89s101s97s114
Figure 1.1: Yearly quantum computation publications. The number of papers re-
lated to quantum computation was found using INSPEC with the following search
string: AB qubit Or AB \quantum comput*" Or AB \quantum information". This
by no means gives an accurate total count. For example, the search did not return
Feynman?s well-known paper, Ref. [6]. Some of the early papers went on to become
highly cited, while others have nothing to do with the subject at all.
lost in any physical implementation that was susceptible to noise, and the suspicion
was that the same problem would harm a quantum computer.
The explosion of interest in quantum computation in the 1990?s can be partly
attributed to the development of error correcting algorithms [7{9]. The basic idea
is to encode a single \logical" qubit with two orthogonal entangled states of several
\physical" qubits. In this way, only a small subset of the possible physical qubit
states will be considered valid logical states. During an intentional measurement, no
information about the logical state can be gained by only measuring a single physical
qubit. In the same way, if a single qubit is disturbed through an interaction with the
environment in the form of decoherence, the logical state is not completely destroyed.
A specifled set of errors induced on the physical qubits can be detected with classical
measurements and corrected with a discrete set of quantum gate operations.
2
Concrete algorithms also emerged around the same time and spurred interest.
Shor?s algorithm is the most well-known. It provides an e?cient way of flnding
the prime factors of a large number, based on the quantum Fourier transform [10].
Grover?s algorithm, developed shortly afterwards, ofiers a moderately more e?cient
way of searching an unstructured database [11]. While a classical computer could
in principle solve these same problems, it would take an inordinate amount of time
to do so as the number or database becomes large.
It may well turn out that the most useful quantum computation application
will be quantum simulation, as Feynman originally suggested. Even with the most
powerful classical computers, it is only possible to carry out full simulations of
Schr?odinger?s equation for diatomic and triatomic molecules. With a quantum sim-
ulator of a few hundred qubits, it would be possible to determine the ground state
conflguration of a dozen or more atoms [12]. Or perhaps the potential of adiabatic
quantum computation will be realized [13]. In this approach, a system is constructed
such that the slow evolution of it to its ground state gives a solution of interest. In
the related flelds of quantum communication and cryptography, products are cur-
rently commercially available that allow secure key distribution over distances as
large as 100 km [14].
It is perhaps remarkable that experimentalists in several branches of physics
were in a position to investigate a wide variety of schemes around the same time the
theory gained flrmer footing. Approaches that are currently being pursued include
using the hyperflne levels of individual ions [15], electronic spin states of a quantum
dot [16], and nuclear spin states of individual impurities in semiconductors [17] and
of molecules in a more traditional NMR sense [18]. As I will discuss in more detail
in the next section, several types of superconducting qubits have been proposed and
implemented. One such device, the Josephson phase qubit, is the subject of this
thesis. The superconducting approaches are quite difierent than the previous ones
3
I listed, as they require quantum information to be stored in macroscopic objects.
Aside from the speciflc tools and techniques that each scheme requires, David
DiVincenzo has proposed a list of generic requirements that any viable qubit must
meet [19{21]. They are as follows.
Hilbert space control: The system must be conflned to a precisely known set
of quantum states and methods should be in place to prevent \leakage" to unde-
sired states. The state space should be expandable with the addition of (typically)
particles with a \spin" of 1/2.
State preparation: It should be possible to initialize the system to a well-
deflned state. As the qubit states are often energy eigenstates, this usually means
cooling the system to its ground state. In addition, it may be required to reinitialize
auxiliary qubits throughout a computation.
Low decoherence: The system should weakly interact with its environment,
so that quantum superpositions are not disturbed. The amount of time needed
to perform a gate operation (and perhaps the time needed to perform readout)
determines the quality of isolation needed. A reliable error correction algorithm is
needed to satisfy this requirement; current theoretical estimates vary widely, but the
decoherence time may need to be 104 longer than the time to perform a quantum
gate operation for error correction to work.
Controlled unitary transformations: A universal set of (typically) one- and
two-qubit gates is needed to perform accurate manipulations of the qubit states.
The great challenge in quantum computing is balancing this requirement with the
previous one. Systems that are easy to control also typically interact with their
environments strongly, leading to decoherence.
State-speciflc quantum measurements: It must be possible to readout a spe-
ciflc subsystem of the Hilbert space, to obtain classical information. The simplest
example is a projective measurement of individual qubits, which does not afiect the
4
rest of the system (not possible if all the qubits are entangled).
Quantum communication: \Stationary" qubits (whose properties are given by
the previous requirements) must be converted to \ ying" qubits, which can be used
to faithfully transmit quantum information to a specifled location.
1.2 Superconducting Circuits
In the 1980?s, there was a tremendous amount of theoretical and experimental
work done on the quantum nature of Josephson tunnel junction circuits [22{24].
Experimentally, the quality of tunnel barriers had reached the point where very
sensitive experiments could be performed. The fundamental physics of the devices
was studied and some understanding of the concepts of coherence and dissipation
was established. Twenty years later, there is a variety of superconducting qubits
being pursued, based on this early work.
As I will discuss inx2.1, a Josephson junction can be thought of as an inductor,
in that the voltage across it is proportional to the time derivative of the current
 owing through it. By placing a capacitor in parallel with an ideal junction, a
resonator is formed whose two lowest quantum energy levels can be used as the
states of a qubit. What makes this a feasible approach is that the value of the
Josephson inductance depends on the current  owing through it. This nonlinearity
leads to an anharmonic potential that governs the junction dynamics. Because of
the resulting unequal energy level spacings, transitions between any two states can
be uniquely addressed by applying a properly tuned external high frequency drive
to the system.
The potential energy function that describes many junction circuits resembles
that of a single atom and in fact much of the terminology is borrowed from NMR
and atomic physics. Unlike their natural counterparts, the energy level structure of
these artiflcial atoms can be easily tuned either in the fabrication process or during
5
operation by adjusting, for example, a bias current or magnetic  ux. This tuning
can be performed on selected qubits and the interaction between qubits can often be
adjusted in the course of an experiment. With this flne control, though, comes the
likelihood that the circuits will interact with their environment rather strongly. The
challenge with these devices is to retain some level of control, while isolating them
su?ciently to preserve quantum mechanical phase coherence of the superconducting
state. What makes this feasible is that the relevant energy scales, such as the
Josephson coupling energy, can be fairly large so that thermal  uctuations can be
made negligible. Another concern is that unlike two rubidium atoms, for example,
no two junctions are identical. Some variation in the fabrication may be tolerable,
but the solid state approaches to quantum computation will require some strategy
for ensuring that the relevant properties are extremely uniform, particularly when
scaling to a large number of qubits [25].
Four general classes of superconducting qubits have emerged [26{28]. They
are largely distinguished by the ratio of the Josephson energy EJ (which gives the
maximum energy stored by the nonlinear inductor) to the charging energy EC (which
gives the energy required to add an electron to the junction). The superconducting
phase difierence across a junction  and the number of Cooper pairs NCP form a
conjugate pair and thus obey a Heisenberg uncertainty relation, ?NCP? ? 1 (see
x2.2 of Ref. [2]). The EJ=EC ratio determines the term that dominates the relation
and also gives a name to each approach. I will brie y describe each of the qubit
classes, in increasing order of EJ=EC.
In the charge qubit, EC ? 5 EJ and the number of Cooper pairs on a small
superconducting island distinguishes the qubit states; thus ?NCP is typically small
in this case [29,30]. A gate voltage is used to adjust the electrostatic energy of the
states and tunneling of pairs onto the island is controlled by a small Josephson junc-
tion. State readout can be performed with a very sensitive electrometer, the single
6
electron transistor, among several other techniques that have been demonstrated.
With EC ? EJ, the qubit states of the charge- ux qubit are superpositions of
several number states. In the quantronium conflguration [31,32], two small junc-
tions are placed in a superconducting loop with an additional large junction. The
qubit states difier by the circulating current in the loop and state readout can
be performed, for example, by measuring the tunneling characteristics of the large
junction.
One form of the  ux qubit uses a large area junction (EJ > 10 EC) in a
superconducting loop [33]. The dynamics of the resulting rf SQUID are governed
by a double well potential, where localization in one of the wells corresponds to
circulating current in either the clockwise or counter-clockwise direction. The qubit
can be readout using a nearby SQUID magnetometer. In the persistent current
conflguration, a superconducting loop is interrupted by three junctions, which allows
for a smaller loop inductance and makes the device less susceptible to  ux noise
[34,35]. This is an example of how the potential of the Josephson artiflcial atom
can be engineered to improve qubit performance.
Finally, the focus of this work is the phase qubit, which in its simplest form is
a single current-biased Josephson junction, with EJ  EC [36,37]. The energy level
structure of the junction is controlled by a large current, typically on the order of
10 ?A. This corresponds to a relatively well-deflned phase, with a large ?NCP.
There are two features of the energy levels of the phase qubit that set it apart
from the charge and  ux qubits. For one, while all Josephson devices have a second
excited state j2i, in the phase qubit, the energy difierence between j1i and j2i is
quite close to the splitting between j0i and j1i for all values of the bias current. For
this reason, the phase qubit is more susceptible to leakage out of the qubit space.
In addition, in the other qubit types, there is an accessible \sweet spot," where the
energy levels are (to flrst order) independent of the control parameter (for example,
7
a voltage or  ux). By operating at this value, the efiects of noise can be minimized.
While there is such a point for the phase qubit (at zero current), it is not easy to use
the device there. Nevertheless, a single junction is quite insensitive to charge and
 ux noise, so it only needs to be protected from current noise. In our group, we have
attempted to do this either with an LC fllter [38] or a broadband inductive current
divider [39]. Also, the freedom in choosing an operating point may actually make
certain tasks, such as dynamic coupling of multiple qubits, easier to perform [40].
Apart from difierent conflgurations and isolation techniques, phase qubits1
have been made using several difierent fabrication methods and from a variety of
materials, including NbN/AlN/NbN [37], Nb/AlOx/Nb [39, 41,42], Al/AlOx/Al
[38,43,44], epi-Re/epi-Al2O3=Al [45], YBa2Cu3O7?? [46], and Bi2Sr2CaCu2O8+? [47]
junctions.
1.3 Summary of Thesis
One of the primary goals of this work is to understand the quantum behav-
ior of the phase qubit. Therefore, Chapter 2 contains an overview of Josephson
junctions, the derivation of the Hamiltonian for a simple current-biased junction,
and an overview of the quantum properties that we tried to experimentally examine.
The inductive isolation technique mentioned above uses an auxiliary junction, which
results in a dc SQUID, whose properties I will also describe. Chapter 3 contains
information about the time evolution of simple quantum systems. I will mostly
quote generic analytical results, which predict the type of behavior we should see in
the experiments, and outline the numerical techniques used in simulating the actual
experiments.
Chapters 4 and 5 contain experimental details. The flrst focuses on the de-
1While current-biased junctions have been studied for decades, the work I have cited was re-
cently done speciflcally with quantum information processing in mind.
8
sign and fabrication of the three devices that I studied: two capacitively coupled
LC-isolated junctions (referred to as device LC2), a single dc SQUID (DS1), and
two capacitively-coupled dc SQUIDs (DS2). The latter chapter mostly concerns
the experimental apparatus, including the dilution refrigerator in which all of the
experiments were performed. I will also give an overview of the electronics used in
the measurements.
Starting with Chapter 6, I will discuss experimental results. This chapter
contains details of the measurement of the tunneling escape rate, which is how we
obtained nearly all of the information we have about the qubits. I will also describe
some of the experimental methods we used to characterize the difierent devices.
Our approach to investigating the junction Hamiltonian was to turn as many
experimental knobs as we had access to and attempt to explain the results. This
process begins in Chapter 7, which describes experiments where the temperature,
isolation, and bias sweep rates were varied. The second key goal of my work was
to evaluate, in as many ways as possible, the potential for using these phase qubits
in quantum computation. Therefore, I will also describe how the relaxation time
T1 (which characterizes the speed at which the qubit can dissipate energy to its
environment) can be extracted from these experiments.
The flnal two experimental chapters contain results of microwave activation,
which was our primary method of state manipulation. Most of Chapter 8 is devoted
to spectroscopy, which demonstrates the quantum nature of the qubit in a dramatic
way. I will also describe time domain measurements of T1, will yielded puzzling
results. Coherent Rabi oscillations, which are the prototypical single qubit gate
operation for quantum computation, are the subject of Chapter 9. The oscillations
serve as a very strong test of our understanding of the coherent dynamics in the
junction. Both spectroscopy and Rabi oscillations provide measures of the phase
coherence time T2, an important indicator of the viability of a qubit.
9
Finally, in Chapter 10, I will summarize the key results from the experiments.
10
Chapter 2
Josephson Junctions and SQUIDs
In 1962, Brian Josephson proposed a phenomenon that now bears his name
[48]. The 22-year-old student of Pippard was interested in flnding consequences
of the spontaneous symmetry breaking arguments of P. W. Anderson, who was
visiting Cambridge at the time, and how the phase of a superconductor might be
measured [49]. He suggested that a current of superconducting particles could tunnel
through a thin insulating barrier with no voltage drop. Although it seems that even
he was not certain of the result (given that the paper is entitled \Possible new
efiects in superconductive tunnelling"), several experiments quickly conflrmed his
ideas [50,51]. He was awarded the Nobel prize in 1973 along with Giaever and Esaki,
just a year after BCS theory was recognized.
This chapter opens with a discussion of the basic equations that govern the
behavior of Josephson junctions. Next, the Hamiltonian of a current-biased junction
will be derived and a summary of the classical dynamics presented. In preparation
for treating the device as a qubit, I will review some of the techniques for flnding
quantum mechanical solutions of the Hamiltonian and show results for a typical
device. Although all of our qubits act efiectively as current-biased junctions, one
form of isolation that we use results in a dc SQUID. Therefore, I will also discuss
some basics of these devices, largely focusing on the phenomenology of the allowed
 ux states, because they have a signiflcant experimental impact.
2.1 Josephson Relations
The Josephson efiect occurs for any weak link between two superconducting
banks. The weak link could be a thin normal metal, a constriction in the super-
11
?1 ?2
S1 I S2
(a) (b)
Figure 2.1: Schematic of a Josephson junction. (a) The Josephson efiect involves
superconducting Cooper pairs from one electrode (S1) tunneling through a thin
insulating barrier (I) to another electrode (S2). (b) For this to occur, the barrier
must be thin enough for the wavefunctions of each side to overlap.
conductor, or a grain boundary, for example. Our devices resemble the cartoon
shown in Fig. 2.1(a), where the weak link is created by an insulating barrier. The
Cooper pairs on each side of the junction can be described by an order parameter or
wavefunction ? = p?ei , where ? is the number density of pairs and  is the quan-
tum mechanical phase. Although the wavefunction on one side will exponentially
decay within the barrier, it may be non-zero at the other superconducting lead if
the barrier is thin enough.
The wavefunctions on the left (?1) and right (?2) sides are connected and can
be described by efiective Schr?odinger equations [52]1
i~@?1@t = E1?1 +T?2 and i~@?2@t = E2?2 +T?1: (2.1)
Here, E1 and E2 are the energies of the pairs on the left and right sides and T rep-
resents the coupling between the superconductors; it will be small for thick barriers.
Imagine also that side 1 is held at a potential VJ with respect to side 2 by a battery.
In the case of carriers with a positive charge, E1 ? E2 = 2eVJ for Cooper pairs,
where ?e = ?1:602?10?19 C is the charge of a single electron. The symmetry of
1This particularly intuitive argument comes from one of Feynman?s famous lectures that he
gave to a group of sophomores, less than two years after Josephson?s original paper. A detailed
discussion and several additional references are given in x1.4 of Ref. [53]. See also xII.1 of Ref. [54].
12
the two equations is preserved by deflning the zero of energy to be between E1 and
E2, yielding
i~@?1@t = eVJ?1 +T?2 and i~@?2@t = ?eVJ?2 +T?1: (2.2)
Substituting?1 = p?1ei 1 and?2 = p?2ei 2, andmultiplyingtheflrstequation
through by ??1 and the second by ??2 gives
d?1
dt +i2?1
d 1
dt = ?i
2eVJ
~ ?1 ?i
2T
~
p?
1?2 e?i (2.3)
d?2
dt +i2?2
d 2
dt = i
2eVJ
~ ?2 ?i
2T
~
p?
1?2 ei ; (2.4)
where  ?  1 ? 2 is the phase difierence across the junction. In the presence of a
magnetic fleld, this quantity must be gauge invariant and include a term involving
the vector potential.
Taking the real part of Eqs. (2.3) and (2.4) gives the time dependence of the
pair density as
d?1
dt = ?
2T
~
p?
1?2 sin and
d?2
dt =
2T
~
p?
1?2 sin : (2.5)
The rate _?2 is equal to ?_?1 and is just the current IJ  owing through the junction.
Thus we have
IJ = I0 sin : (2.6)
Here, the constant of proportionality (2Tp?1?2=~) is known as the critical current
I0, which depends on the junction material and fabrication. Incidentally, although
Eq. (2.5) suggests _?1 6= 0, the presence of the battery ensures that neither side of
the junction charges up; Eq. (2.6) gives the value of the current that the battery
supplies.
13
Finally, the imaginary parts of Eqs. (2.3) and (2.4) give
d 1
dt = ?
eVJ
~ ?
T
~
r?
2
?1 cos and
d 2
dt =
eVJ
~ ?
T
~
r?
1
?2 cos (2.7)
d 
dt =
d 1
dt ?
d 2
dt = ?
2eVJ
~ ?
T
~
r?
2
?1
 
1? ?1?
2
?
cos : (2.8)
If both superconductors are the same, then ?1 ? ?2 and neither will change due to
IJ, as mentioned above. Therefore, the magnitude of the voltage across the junction
is
VJ = '02? d dt ; (2.9)
where '0 = h=2e = 2:07?10?15 T?m2 is the superconducting  ux quantum.
If no voltage is applied across the junction, then Eq. (2.9) says that the phase
difierence will not change. What is amazing is that Eq. (2.6), known as the dc
Josephson relation, predicts that a steady current of Cooper pairs can  ow through
the junction. The phase difierence will adjust itself to accommodate currents be-
tween ?I0 and I0. Typically, we deal with bias currents on the order of 10 ?A, so
that the value of  corresponds to the coherent motion of 1013 Cooper pairs per
second.
If there is a constant voltage across the junction, then Eq. (2.9), known as the
ac Josephson relation, predicts that the phase will evolve linearly in time. In this
case, the current through the junction will oscillate rapidly according to Eq. (2.6).
A dc voltage of 1 ?V corresponds to an oscillation frequency of 484 MHz. We will
see that this behavior gives a Josephson junction many interesting high frequency
properties.
Incidentally, some of the mystery of the quantum mechanical phase difierence
14
can be removed by solving for  in Eq. (2.9):
 (t) =  (0)? 2?'
0
Z t
0
dt0VJ (t0): (2.10)
The phase difierence is just the time integral of the voltage that has appeared across
the junction and is thus a perfectly well-deflned physical quantity, once the origin
of time is chosen.
A useful picture of the junction can be developed by taking a time derivative
of Eq. (2.6) and then using Eq. (2.9) to eliminate d =dt. One flnds
VJ = LJ dIJdt ; (2.11)
where
LJ = '02? 1I
0 cos 
= '02? 1pI2
0 ?I2J
: (2.12)
As Eq. (2.11) looks just like the expression for the voltage across an inductor, LJ
is known as the \Josephson inductance." As Eq. (2.12) shows that LJ depends on
the current, this is a non-linear inductor. The combination of this element with
a parallel capacitor forms an anharmonic resonator, whose energy levels we use as
states of a qubit. The energy stored in the magnetic fleld by a solenoid depends on
its physical dimensions, leading to the term \geometrical inductance" to describe
its properties. Here, energy is being stored in the  ow of Cooper pairs (see x6.1 of
Ref. [55]) and thus the origin of the Josephson inductance is quite difierent.
2.2 RCSJ Model
The resistively and capacitively shunted junction (RCSJ) model has been
widely used to describe the behavior of current-biased junctions [55{59]. In this
model, the parallel combination of three elements is driven by a current bias Ib, as
15
Ib CJI0RJ
Figure 2.2: RCSJ model circuit diagram. A Josephson junction with critical current
I0 is driven by a current bias Ib. The resistance RJ is responsible for all dissipation
in the system and the capacitance CJ is often due mainly to the geometry of the
junction.
shown in Fig. 2.2. The cross signifles an ideal junction, i.e. an object that exactly
obeys the two Josephson relations with critical current I0. Usually, the parallel
plate geometry of the junction itself gives rise to the shunting capacitance CJ. The
resistance RJ comes from intrinsic dissipation mechanisms in the junction (such as
quasiparticles), normal metal shunts that have been added across the junction, and
contributions from the biasing network. As the RCSJ model represents our qubit
quite faithfully, I will discuss it in some detail. I will start ofi with a derivation of the
Hamiltonian, using the method outline in Appendix A, and end with a mechanical
analog that gives an intuitive picture of its classical dynamics.
2.2.1 RCSJ Hamiltonian
In the RCSJ model, the bias current splits between the three parallel paths
formed by the junction, capacitor, and resistor. Current conservation gives
Ib = VJR
J
+CJ dVJdt +I0 sin : (2.13)
Substitution of the Josephson voltage relation, Eq. (2.9), yields the equation of
motion for this circuit,
CJ
 '
0
2?
?2 d2 
dt2 = ?
d
d 
?
?'02? (I0 cos +Ib )
?
? 1R
J
 '
0
2?
?2 d 
dt: (2.14)
16
This describes the motion (along coordinate x) of a flctitious particle of mass
mJ in a one dimensional potential U with a damping force ??v, where v is the
velocity. Equation (2.14) can be rewritten in the form
md
2x
dt2 = ?
dU
dx ??v; (2.15)
with the following identiflcations
x ?!  (2.16)
v ?! d dt (2.17)
mJ ?! CJ
 '
0
2?
?2
(2.18)
U ?! ?'02? (I0 cos +Ib ) (2.19)
? ?! 1R
J
 '
0
2?
?2
: (2.20)
The potential U is called the \tilted washboard potential" because of its shape;
as shown in Fig. 2.3, the tilt of this potential is set by the current bias and the
corrugation is determined by the critical current. U gives the potential energy of
the junction as a function of the phase difierence  , which is why the flctitious
particle is often referred to as the \phase ball" or \phase particle." I will frequently
refer to the motion of this particle in describing the dynamics of junction devices.
Equation (2.14) was multiplied through by the  ux quantum to give U the correct
dimensions; however, the mass of the particle has dimensions of action times time
and the force has dimensions of action.
Using the particle analogy, the Lagrangian for the system with no dissipation
(RJ = 1) and with  taken as a generalized coordinate is
L( ; _ ) = T ?U = 12
 '
0
2?
?2
CJ _ 2 + '02? (I0 cos +Ib ): (2.21)
17
U
?
(a)
?p 
?U
(c)(b)
Figure 2.3: The tilted washboard potential. (a) The dynamics of a current bi-
ased junction are analogous to a particle moving in a 1-D potential U, where the
phase difierence  across the junction is identifled with the position of the particle.
Classically, when the junction is in the supercurrent state, the particle undergoes os-
cillations of frequency !p in a well with barrier height ?U. The tilt on the potential
is determined by the bias current, which in this case is 0.4 I0. (b) We perform ex-
periments very close to the critical current, where the well is very shallow compared
to the inter-well energy difierence; here Ib = 0:9925 I0, I0 = 30 ?A, and CJ = 5 pF.
(c) The junction dynamics can be approximated by a single one of these wells.
18
With this Lagrangian, the Euler-Lagrange equation [see Eq. (A.1)] generates the
equation of motion, Eq. (2.14). From the Lagrangian, one can also identify the
conjugate momentum and generalized velocity:
p = mJ _ =
 '
0
2?
?2
CJ _ = ~2eCJVJ = ?~NCP ) _ =
 2?
'0
?2 p
CJ : (2.22)
Thus the momentum is equal to the number of Cooper pairs NCP that have tunneled
through the junction, apart from a factor of ?~to get the right dimensions and sign.
H can be expressed in terms of the number-phase conjugate pair with [NCP; ] = i
(see x2.2 of Ref. [2]).
Finally, the Hamiltonian can be obtained from Eq. (A.3) and expressed in a
number of equivalent forms, including
H = 12
 '
0
2?
?2
CJ _ 2 ? '02? (I0 cos +Ib ) (2.23)
= 12
 2?
'0
?2 p2
CJ ?
'0
2? (I0 cos +Ib ) (2.24)
= p
2
2mJ ?mJ!
2
p0 cos ?
'0
2?Ib (2.25)
= 4EC~2 p2 ?EJ (cos +Ir ) (2.26)
= 4ECN2CP ?EJ (cos +Ir ); (2.27)
where Ir ? Ib=I0 is the reduced current bias. In Eq. (2.25), !p0 ? p2?I0=CJ'0
is the frequency of small oscillations in a well of the washboard at zero bias, as
shown in the next section. In Eq. (2.26), EJ = '0I0=2? is known as the Josephson
coupling energy and is a measure of the maximum energy that can be stored in the
JosephsoninductanceLJ. Italsoisanalogoustothe\springconstant"thatdescribes
the harmonic component of the potential at zero bias. EC = e2=2CJ = ~2=8mJ is
the charging energy, which is the energy required to charge the capacitor with one
19
electron solely from electrostatic considerations. We perform experiments on large
area junctions, where EJ  EC, so that the phase difierence  is a relatively well
deflned quantity compared to NCP, i.e. ? ? 1 and ?NCP  1. Typical parameter
values for our qubits are I0 = 30 ?A and CJ = 5 pF, for which EJ = 9:9?10?21 J =
15 THz?h = 720 K?kB and EC = 2:6?10?27 J = 3:9 MHz?h = 180 ?K?kB.
2.2.2 The Tilted Washboard Potential
In this section, I will discuss some of the basic properties of the tilted wash-
board potential. Although these are classical results, they will provide considerable
intuition about the dynamics of the system.
The potential energy as a function of the phase difierence across the junction
is
U = ?'02? (I0 cos +Ib ) = ?EJ (cos +Ir ): (2.28)
The derivative of the potential disappears when sin = Ir (and thus cos =
?p1?I2r), so that a local minimum and maximum occur at phases
 min = sin?1 Ir and  max = ? ?sin?1 Ir: (2.29)
This gives the barrier height as a function of the current as
?U ? U ( max)?U ( min) = 2EJ
?p
1?I2r ?Ir cos?1 Ir
?
(2.30)
? 4
p2
3 EJ (1?Ir)
3=2 ; (2.31)
where the approximation [60] is valid for Ir . 1.
The second derivative of the potential, evaluated at  min, is k = EJp1?I2r.
Therefore, using the particle mass found in the previous section, the frequency of
20
small oscillations (known as the plasma frequency) is
!p =
r
k
m =
r 2?I
0
CJ'0
?1?I2
r
?1=4 = p8ECEJ
~
?1?I2
r
?1=4 (2.32)
in the absence of dissipation. Notice that !p is just 1=pLJCJ. For a planar junction,
both I0 and CJ are proportional to the junction area, so the plasma frequency
depends only on the critical current density.
Although the capacitor does not appear explicitly in the expression for the
potential, it is playing a critical role in motivating its use. Imagine there were no
capacitor and we just had an ideal current-biased junction. For any value of Ib < I0,
the phase difierence would adjust itself in accordance with Eq. (2.6) and  would
stay at a flxed value.
Adding the capacitor (but leaving out the resistor for the moment) gives rise to
a flnite-mass phase particle. Consider the case of Fig. 2.3(c). Classically, the particle
can sit at the minimum of the potential well at  0=2? = 0:23 with no kinetic energy,
in a situation identical to the one described above. I will refer to this situation as
the classical ground state, for lack of a better term. However if the system is given
some energy (with a little kick of bias current, for example), then the phase particle
will undergo plasma oscillations about the minimum, as sketched in Fig. 2.4.
The current through the junction will then oscillate about the equilibrium
value [Ir = 0:9925 in the case of Fig. 2.3(c)] in phase with  . With a time-
varying phase difierence, a voltage appears across the junction, which drives a
current through the capacitor. From Eq. (2.9), this current is 180? out of phase
with the  oscillations. If the phase oscillates with a (small) amplitude ~ , then both
currents will have an oscillatory amplitude I0~ cosIr, so that the total current is Ib,
as required by Eq. (2.13). In other words, it is the redistribution of current through
the capacitor that allows  to explore the tilted washboard and gives rise to the
21
t
?0
t
IJ
IC
Ib
(a) (b)
Figure 2.4: Plasma oscillations of a capacitively shunted junction. (a) For flxed Ib,  
oscillates about the equilibrium value  0 after the system is kicked. (b) The current
through the junction (solid line) and capacitor (dotted) also oscillate, but sum to
the value of the dc current bias, Ib.
dynamics that will be discussed throughout this thesis. Alternatively, one can view
the oscillation as energy exchange between the junction and charge stored on the
capacitor, or between potential and kinetic energy in the mechanical analog.
2.2.3 Junction IV Curve
The nature of the current vs. voltage characteristic (IV) curve for a current-
biased junction depends strongly on RJ. The strength of the damping is character-
ized by the (dimensionless) Stewart-McCumber hysteresis parameter [56,57]
flC = 2?'
0
I0 R2J CJ = (!p0 RJ CJ)2 = Q(0)2 ; (2.33)
where !p0 is the plasma frequency at zero bias and the quality factor Q(Ib) =
!p RJCJ is identical to that of a traditional parallel RLC oscillator. For our devices,
which have large flC and are therefore \underdamped," the IV curve will resemble
the hysteretic one shown in Fig. 2.5 [53].
The difierent regions of this curve can be understood using the analogy of the
particle and tilted washboard potential. Starting at Ib = VJ = 0, the potential
contains minima for Ib < I0, which can trap the particle (see Fig. 2.3). This is
22
Ib
VJ
I0
Irt
2? / e
Figure 2.5: IV curve of a current-biased junction. Once the current exceeds the
critical current I0 a voltage develops across the junction. In an underdamped junc-
tion the curve is hysteretic, not returning to the supercurrent branch until Irt. The
resistance in the sub-gap has been exaggerated for clarity. The dashed line shows
the normal state resistance.
known as the trapped, supercurrent, or zero-voltage state and corresponds to the
vertical branch of the IV curve at VJ = 0. We always operate our qubits in this
regime, but the rest of the IV curve is useful in determining junction parameters.
Once Ib > I0, all the well minima disappear and the particle is free to roll
down the potential. The result is a continuous evolution of the phase that gives rise
to a voltage, which increases rapidly as the particle picks up kinetic energy. When
the voltage reaches the value of the full superconducting gap 2?=e, the system has
su?cient energy to break Cooper pairs. Pair-breaking produces a large source of
damping that rapidly brings the system to a steady state voltage. In underdamped
junctions, the jump to this so-called flnite voltage (or running) state is extremely
fast (as suggested by the dotted horizontal line in Fig. 2.5), a feature which we
will exploit in doing qubit state measurements. Classically, switching out of the
zero-voltage state can occur for Ib < I0 if the particle is thermally excited over the
barrier [61].
As the bias increases after the junction switches, the additional current is
23
mainlyduetoquasiparticlesuperconductor-insulator-superconductortunneling, flrst
observed by Giaever [62], and asymptotically approaches the value set by the normal
state junction resistance Rn. For an ideal BCS superconductor, the product of I0
and Rn obeys [63]
I0Rn = ??(T)2e tanh
 ?(T)
2kBT
?
; (2.34)
which depends on the energy gap ? and temperature T of the junction, but not
its size. In the normal state, Ib = 4I0=? when VJ = 2?=e, which relates I0 to the
height of the quasiparticle branch of the IV curve at T = 0.
The well minima return once the bias current is lowered beneath I0. However
if the particle is in the running state, its kinetic energy prevents it from immediately
\retrapping" into the zero-voltage state, leading to hysteretic behavior. As the bias
is lowered, the particle will slow down and the voltage will decrease. Although we
always operate our qubits in the supercurrent state, the sub-gap resistance Rsg will
govern how energy is dissipated when a small ac voltage appears across the junction
during plasma oscillations [64{66]. Thus we are particularly interested in this section
of the curve. In an unshunted junction, the sub-gap resistance can be quite a bit
larger than Rn, as transport may solely be due to quasiparticles thermally excited
across the gap. At the retrapping current [55]
Irt ? 4I0=?Q; (2.35)
the damping is su?ciently large that the junction jumps back to the zero-voltage
state.
24
U
?
0
1
3
2
?0
?1
?2
?3
??12
??01
Figure 2.6: Quantum states of the tilted washboard. The RCSJ Hamiltonian gives
rise to metastable quantum states jni, which can tunnel to the continuum of voltage
states with escape rates ?n. The unequal energy level spacings (!01 > !12) means
that speciflc transitions can be driven. The potential parameters are the same as in
Fig. 2.3.
2.3 Quantum Mechanical Properties of the Josephson Junction
The previous discussion was based on a classical picture of the current-biased
junction. However, this system is suitable for studying macroscopic quantum phe-
nomena, which partly motivated the original theoretical work on the device [67].
As sketched in Fig. 2.6, a well of the tilted washboard potential will contain a cer-
tain number of reasonably well-localized quantum energy levels jni (of energy En).
The exact number depends on Ib, the junction parameters, and what is meant by
well-localized. The ground j0i and flrst excited state j1i serve as the states of our
qubit [36], although higher states are always present and can be both beneflcial and
harmful.
There are two key properties of the potential that determine the nature of
these states. First, the nonlinearity of the junction gives rise to an anharmonic well.
The energy spacing between jni and jmi, ?Enm =~!nm, decreases higher up in the
well. This allows the possibility of selectively driving a transition between speciflc
levels. In practice, we can do this by applying a microwave current drive on the
25
bias line with a frequency that is resonant with the desired transition. A 0 ! 2
transition is drawn in the flgure. The in uence of such a drive on the other states is
a major theme of this thesis. Shown also in the flgure is one of the decay processes
(in this case, 2 ! 1) that are a result of the junction being able to dissipate energy
to its environment.
The second important property is that the potential falls ofi towards minus
inflnity for increasing  , creating the barrier ?U. As a consequence, there are
no true bound states of the system (see Chapter 3 of Ref. [2]). While scattering
states exist for a continuous spectrum of energies, the density of allowed states
will peak at so-called resonances, corresponding to a high probability of being in
the well. Normalizable wavefunctions, which I will label jni, can be formed by
taking superpositions of the states near these resonance energies. The states jni are,
however, metastable and will eventually leak out of the well, as all the eigenstates
extend to  = +1.
Classically, the phase particle can be thermally driven over the barrier. Quan-
tum mechanically, it can tunnel through a flnite barrier (i.e. for Ib < I0) even at zero
temperature. I will deflne ?n as the tunneling escape rate of the n-th level; the rate
can be thought of as the inverse of the lifetime of the metastable state. Experiments
distinguishing tunneling from thermal escape were flrst performed 25 years ago [23]
and observation of tunneling from difierent states followed soon after [24]. Both of
these processes leave the junction in the voltage state. In quantum computation,
tunneling destroys the information stored by the qubit. While we would want to
avoid such an event during a gate operation, studying the tunneling behavior is the
easiest way we have of measuring the state of our devices at the moment. As sug-
gested in Fig. 2.6, the states higher in the well have very high tunneling rates, with
these short lifetimes corresponding to broad energy levels. In fact, there are reso-
nances above the barrier (see Fig. 3.26 of Ref. [2]), although they may be thought of
26
as smeared into a continuum. Deep in the well, the resonances can be rather sharp
(long-lived) and thus I will refer to these as eigenstates jni with discrete energies
En.
To analyze and simulate the junction dynamics, we need to know the energy
level spacings ~!nm and tunneling escape rates ?n as a function of Ib. In addition,
if we add a microwave signal to the current bias, the inter-level transition rate will
depend on hnjIb ^ jmi, as seen in the Hamiltonian of x2.2.1. Therefore, we also
need the matrix elements of the phase operator,  n;m ? hnj^ jmi. Although the
Hamiltonian looks simple enough, it is the lack of discrete stationary states that
makes the problem di?cult. Below, I will only summarize some of the techniques
that have been used in the group, as details have been presented by their authors
elsewhere.
2.3.1 Harmonic Oscillator Approximation
The lowest-order approach is to ignore the barrier and assume the curvature
at the bottom of the well deflnes a harmonic potential. This results in equally
spaced energy levels En =~!p (n+1=2), where n is a non-negative integer and the
bias-dependent plasma frequency is given by Eq. (2.32). For the potential shown in
Fig. 2.6 (typical operating parameters), !p=2? = 7:5 GHz, or 360 mK in terms of
temperature. Therefore, our experiments must be performed at lower temperatures
to prevent thermal excitation of the system from one level to the next. I flnd
it remarkable that this quantum mechanical result, simple enough to have been
derived essentially from flrst principles in the previous pages, can be verifled in the
laboratory with a rather crudely made macroscopic object.
27
This harmonic model also provides the matrix elements  n;n?1 to lowest order:
hn?1j^ jni = hnj^ jn?1i ?
 ~
2m!p
?1=2p
n (2.36)
=
 2e2
~!pCJ
?1=2p
n =
 2E
C
EJ
?1=4?
1?I2r??1=8pn;
where all other elements are zero. In the true anharmonic potential, these other
elements (including the diagonal ones) are non-zero and can be quite signiflcant
when the well is shallow.
For the full washboard potential, it is useful to measure the barrier height in
terms of the harmonic level spacing. This gives an estimate of the number of levels
in the well as
Ns ? ?U~!
p
= 1p2
 E
J
EC
?1=2h?
1?I2r?1=4 ?Ir?1?I2r??1=4 cos?1 Ir
i
: (2.37)
At zero bias, Ns ? pEJ=EC which is greater than 1000 for our qubits. On the
other hand, we typically perform experiments at a value of Ib where Ns ? 3.
2.3.2 Cubic Approximation
Fairly accurate results can be obtained by approximating a well of the wash-
board with a cubic polynomial (see x2.2 of Ref. [3] and x2.4 of Ref. [1]). As with the
tilted washboard, a cubic well supports resonances rather than true bound states
and is anharmonic. It can therefore supply all of the quantities we are interested
in. Furthermore, a cubic does a fair job of representing the potential even for more
general forms of the dc Josephson relation (see Appendix A of [68]). While it has
some deflciencies (such as failing at high Ib), this approach has the tremendous
advantage of providing analytical solutions which help build intuition over a wide
range of device parameters, aiding in the design of qubits.
28
The strength of the cubic component of the potential is governed by the pa-
rameter ? = 1=p54Ns. The cubic contribution can be treated as a perturbation
to the harmonic oscillator (see x4.3.1 of Ref. [2]). This yields approximate eigen-
states inside the well, which can be used to evaluate the needed matrix elements.
To second order in ?, some of the matrix elements of ^x for the lowest states are
x0;1 = 1p2
 
1+ 114 ?2
?
x1;2 = 1+ 112 ?2 x2;3 =
r3
2
 
1+ 334 ?2
?
(2.38)
x0;0 = 32 ? x0;2 = ? ?p2 x1;1 = 92 ? x2;2 = 152 ?; (2.39)
where ^x = pmJ!p=~?^ ?sin?1 Ir?. As the states are orthogonal, the matrix ele-
ments of ^ are proportional to those of ^x. The elements given in Eq. (2.39) vanish
in the harmonic limit, but grow more rapidly with ? than the \transition" matrix
elements of Eq. (2.38).
In order to capture the behavior of the wavefunctions outside of the well and
calculate ?n, the WKB method can be used (see x4.4 of Ref. [2] and Appendix A
of Ref. [1]). The boundary conditions that allow for out-going waves result in a
discrete spectrum of complex energies. For an eigenstate with a complex eigenvalue
v = fi + ifl, the time dependence of the probability density is flfle?ivt=~flfl2 = e2flt=~.
We want the tunneling escape rate to be a measure of the decay of this probability,
which motivates the deflnitions
En ? Re(h?njHj?ni) and ?n ??2~Im(h?njHj?ni); (2.40)
for eigenstates j?ni. The tunneling rate in the absence of damping can be found
from the probability current density  owing through the barrier as [2]
?n =
p2? [60(7:2N
s)]
n+1=2
n!
!p
2? exp(?7:2Ns): (2.41)
29
Notice that Ib, I0, and CJ only enter implicity through !p and Ns. This sort of
universal scaling makes the expression quite useful. The expression also shows that
the tunneling increases dramatically for states higher in the well, as ?n+1=?n =
432Ns=(n+1).
Including damping in the problem suppresses the tunneling rate. To flrst order
in 1=Q, the ground state escape rate for the cubic potential is [22]
?0 =
p
120?(7:2Ns) !p2? exp
?
?7:2Ns
 
1+ 0:87Q
??
: (2.42)
2.3.3 Full Tilted Washboard
Going beyond the cubic approximation, there are several ways to treat the
resonances of the full washboard potential. What has been used most often in the
group has been to solve Schr?odinger?s equation with appropriate boundary con-
ditions at the turning points of the well. In the method of complex scaling (see
x3.3.2 of Ref. [2]), a transformation is made to complex coordinates:  !  ei and
p ! pe?i . While the usual commutation relation is still satisfled, the Hamilto-
nian becomes non-Hermitian, which allows for complex eigenvalues. Eigenstates are
found as a superposition of harmonic oscillator states, with  adjusted for stability.
Equivalent results are obtained by solving Schr?odinger?s equation on a grid
with simple boundary conditions corresponding to decay to the left of the well and
free oscillation to the right; see x2.4 and x3.3.2 of Ref. [1] and Appendix B for
the MATLAB code used generate the solutions. Starting from a trial energy and
wavefunction (such as a harmonic oscillator state), inverse iteration can be used to
numerically relax to a solution (see, for example, x11.7 of Ref. [69]). Equation (2.40)
provides the interpretation of the resulting complex eigenvalue. In practice, we run
this sort of simulation for a single set of typical parameters and express the escape
30
rates and energy levels as
?n = (7:2Ns)n+1=2 !p2? exp[?7:2Ns +fn? (Ns)] (2.43)
!n;n+1 = !p fn! (Ns): (2.44)
The idea here is to retain the bias-dependent scaling of the cubic potential solutions,
while the functions fn? and fn! contain corrections. These functions, when expressed
as a function of Ns, appear to be universal. As they do not follow any simple
functional form, they can be parameterized as a series of cubic splines (in 1=Ns).
In principle the wavefunctions found with this method could be used to calculate
the matrix elements, but I found it simpler to use the cubic approximation results,
which are su?cient for my purposes.
I will now show a selection of results for a junction with I0 = 30 ?A and
CJ = 5 pF for values of the reduced current bias Ir of interest. Figure 2.7 shows
the discrete number of levels within the barrier ?U (solid line), as determined from
Eq. (2.44) (and the value of E0 from the simulation). There are resonances above
the barrier, but they are so short-lived, they probably do not afiect the dynamics
signiflcantly. This number is well approximated by the continuous function Ns
(dashed), given in Eq. (2.37).
The solid line in Fig. 2.8 is the classical plasma frequency !p. The anharmonic-
ity of the well leads to smaller transition frequencies !01 (dashed), !12 (dash-dot),
and !23 (dotted) between adjacent levels. An upturn in frequency occurs as a level
leaves the well. The levels are so wide at these points that we have not experimen-
tally observed such features.
Tunneling escape rates ?0 (solid), ?1 (dashed), ?2 (dash-dot), and ?3 (dotted)
areshowninFig.2.9(a). Alloftheescaperates increaseapproximatelyexponentially
with the current bias, as even the simplest theories predict. However, the escape
31
s48s46s57s56 s48s46s57s57 s49s46s48s48
s48
s52
s56
s32
s32
s35
s32
s76
s101
s118
s101
s108
s115
s73
s114
Figure 2.7: Number of levels of the tilted washboard potential. The number of
levels (solid) within the barrier ?U as a function of the reduced current bias Ir
as calculated from the quantum simulation of a junction with I0 = 30 ?A and
CJ = 5 pF is discrete. It is reasonably well approximated by the number of harmonic
levels in the well Ns (dashed), given by Eq. (2.37).
s48s46s57s56s53 s48s46s57s57s48 s48s46s57s57s53 s49s46s48s48s48
s52
s53
s54
s55
s56
s57
s32
s32
s110
s109
s32
s47
s32
s50
s32
s40
s71
s72
s122
s41
s73
s114
Figure 2.8: Energy level transitions of the tilted washboard potential. The solid
line shows the classical plasma frequency !p of a junction with I0 = 30 ?A and
CJ = 5 pF. Equation (2.44) gives the transition frequencies !01 (dashed), !12
(dash-dot), and !23 (dotted).
32
(a)
(b)
Figure 2.9: Tunneling rates of the tilted washboard potential. (a) Equation (2.43)
gives the tunneling escape rates ?0 (solid), ?1 (dashed), ?2 (dash-dot), and ?3
(dotted) for a junction with I0 = 30 ?A and CJ = 5 pF. (b) The ratios ?1=?0
(dashed), ?2=?1 (dash-dot), and ?3=?2 (dotted) are strong functions of the reduced
current bias Ir.
33
Figure 2.10: Ground state escape rate. ?0 is plotted as a function of the reduced
current bias for a junction with I0 = 30 ?A and CJ = 5 pF. The solution of the full
potential (solid) agrees with the cubic approximation (dashed) at all but the highest
escape rates. With RJ = 50 ?, damping causes a reduction in tunneling (dotted).
rate is roughly bounded by !p (which can be thought of as an attempt frequency2),
although the second method described in this section does not entirely capture this
efiect. Again, because the lifetimes of states are so short at these high biases, it
is di?cult to accurately measure the way the rates roll-ofi. Ratios ?1=?0 (dashed),
?2=?1 (dash-dot), and ?3=?2 (dotted) are plotted in Fig. 2.9(b). The ratios are
quite large when the levels are deep in the well, which in principle makes it possible
to distinguish tunneling from difierent states. This becomes much more di?cult as
levels reach the top of the barrier and the escape rates rapidly saturate.
The solid line of Fig. 2.10 is the same ground state curve as the one shown in
Fig. 2.9(a). The dashed line is the tunneling rate for the cubic approximation, from
Eq. (2.41). The agreement is quite good below 107 1=s. Above this rate, the solution
of the full potential rolls ofi. As the solution of Schr?odinger?s equation (using the
2Simulations suggest that the numerical value of !p is the maximum escape rate [2], but the
units of the escape rate are, for example, 1=s rather than rad=s.
34
(a)
(b)
Figure 2.11: Matrix elements of the tilted washboard potential. (a) The matrix
elements  0;1 (dashed),  1;2 (dash-dot), and  2;3 (dotted) in the cubic approximation
of a junction with I0 = 30 ?A and CJ = 5 pF are plotted as a function of the reduced
current bias Ir, using Eq. (2.38). Each has been multiplied by (2EC=EJ)?1=4. (b)
From Eq. (2.39), the ratios of  0;0 (solid),  0;2 (dashed),  1;1 (dash-dot), and  2;2
(dotted) to  0;1 are flnite for the anharmonic potential.
35
technique described above) breaks down at high bias, the asymptotic value of ?0 at
Ir = 1 was artiflcially imposed by the choice of f0? (Ns) in Eq. (2.43). The dotted
line in Fig. 2.10 is also in the cubic approximation, but with RJ = 50 ?. As Eq.
(2.42) shows, damping suppresses tunneling. A constant shunting resistance leads
to a bias-dependent Q, but a nearly equivalent ?0 is obtained for Q = 12 (which is
the value it takes at Ir = 0:992 for the plotted curve). Our qubits appear to have
shunting resistance in excess of 1 k?. With this value of RJ, damping lowers ?0 by
? 10 % at Ir = 0:99; alternatively, this can be viewed as a bias current shift of 1 nA.
At Ir = 0:994, where many of my experiments were performed, these corrections
are ? 5 % and 0.5 nA. I have ignored the efiects of damping on ?n in analyzing
the data, but it could have had a non-negligible efiect for our most poorly isolated
qubits.
Finally, several matrix elements of the cubic potential are plotted in Fig. 2.11,
calculated with Eqs. (2.38) and (2.39). Figure 2.11(a) shows the transition elements
 0;1 (dashed),  1;2 (dash-dot), and  2;3 (dotted), where each has been multiplied by
(2EC=EJ)?1=4. From Eq. (2.36), these are 1, p2, p3, respectively, in the harmonic
limit. The anharmonic well causes large deviations, particularly at high bias. Figure
2.11(b) shows  0;0 (solid),  0;2 (dashed),  1;1 (dash-dot), and  2;2 (dotted), scaled by
 0;1. For a harmonic oscillator, all of these would be zero. That they are flnite gives
rise to some of the rich non-linear behavior that is described in the next chapter.
2.4 Asymmetric dc SQUID Hamiltonian
A single current-biased Josephson junction has many characteristics that make
it an attractive candidate for use as a qubit. However, it is typically strongly cou-
pled to the environment through its bias leads, which can be a signiflcant source
of dissipation and decoherence. A variety of isolation techniques have been imple-
mented to decrease the efiects of decoherence, as discussed later in this thesis. One
36
Ib
If
CJ1 CJ2I02I01
L1
Mf
Mb
I1 I2
J
?A
RJ1
L2
RJ2
Figure 2.12: dc SQUID circuit diagram. The junction on the left is thought of as the
qubit, while the one on right forms an inductive current divider with the geometrical
inductance L1. The branch currents I1 and I2 can be independently controlled by
simultaneously applying a  ux and current bias.
such approach uses an auxiliary inductance and junction in parallel with the main
qubit junction [39], as shown in Fig. 2.12. The junction on the left is thought of as
the qubit and the one on the right is part of the isolation network.
The motivation for adding these components is to create a broadband inductive
current divider that fllters noise from the current bias leads. For this approach to
work well, L1 must be much greater than L2. For the devices that I measured,
I01 > I02, although this is not a strict requirement. A longer discussion of circuit
parameters can be found in x4.3. The inclusion of the isolation network results in
a dc SQUID that is asymmetric in the loop inductances L1 and L2, as well as the
junction critical currents and capacitances. However, using the method described in
x6.4, the device can be made to behave much like a single current-biased junction. In
the derivation of the Hamiltonian that follows [4,70,71], I will ignore the resistances
RJ1 and RJ2, as their role in dissipation will be discussed separately.
37
The SQUID is controlled by the usual current bias Ib and an additional current
If. Each line generates  ux in the loop through its own mutual inductance, giving
a total of MbIb + MfIf. As the  ux line dominates in almost all situations, I will
use the symbol 'A to represent its contribution. The current bias splits between
the two arms of the SQUID; each branch current (I1 and I2) further splits between
its junction and capacitor as in x2.2.1, so that
Ib = I1 +I2 (2.45)
I1 = CJ1 ddt
 '
0
2?
d 1
dt
?
+I01 sin 1 (2.46)
I2 = CJ2 ddt
 '
0
2?
d 2
dt
?
+I02 sin 2: (2.47)
It is the coupling of the two junctions by the SQUID loop inductance that
makes the dynamics of this device so rich. The naive strength of the coupling is
given by 1=fl, where fl is the modulation parameter
fl = L(I01 +I02)'
0
: (2.48)
Here, L = L1 + L2 is the total loop inductance. An additional constraint on the
system comes from  ux quantization in the loop. This condition can be found from
the general  ux-phase relationship for a loop interrupted by N junctions [54],
X
N
 N = 2?
 
n? 'T'
0
?
; (2.49)
where n is an integer and  N is the phase difierence across the N-th junction. The
path taken through each of the junctions sets the sign of phase difierence to be
consistent with the Josephson current relation. The convention used in Eqs. (2.46)
and (2.47) is that a clockwise path (which corresponds to negative  ux in this case)
38
around the SQUID loop crosses the qubit with + 1 and the isolation junction with
? 2. The total  ux in the loop 'T is the applied  ux plus the  ux induced by
currents  owing in the loop.3 Applying Eq. (2.49) to the asymmetric SQUID yields
 1 ? 2 = 2?
 
n+ 'T'
0
?
= 2?'
0
('A ?L1I1 +L2I2 +MbIb)+2?n: (2.50)
With two junctions, there will be two equations of motion. They come from
eliminating one of the branch currents in favor of the total current and plugging in
Eqs. (2.46) and (2.47):
I1 = CJ1'02? ? 1 +I01 sin 1 (2.51)
= ?'02? 1L
 
 1 ? 2 ?2?n? 2?'
0
('A +MbIb)
?
+ L2IbL
I2 = CJ2'02? ? 2 +I02 sin 2 (2.52)
= '02? 1L
 
 1 ? 2 ?2?n? 2?'
0
('A +MbIb)
?
+ L1IbL :
The terms on the far right are the branch currents you would expect from a simple
inductance divider. They are modifled by a term, equal to (L2I2 ?L1I1)=L, that is
due to  ux quantization in the loop. For a SQUID with inductive symmetry, this
term is just equal to the circulating current, J = (I2 ?I1)=2. Now the force terms
can be solved for, after multiplying through by '0=2? [with mJ1 ? CJ1 ('0=2?)2
and mJ2 ? CJ2 ('0=2?)2]:
mJ1 ? 1 = '02?
 
?I01 sin 1 + L2IbL
?
?
 '
0
2?
?2 1
L
 
 1 ? 2 ?2?n? 2?'
0
('A +MbIb)
?
(2.53)
3The Josephson inductance presented in x2.1 does not contribute to the loop inductance in this
case, as it does not store magnetic energy.
39
mJ2 ? 2 = '02?
 
?I02 sin 2 + L1IbL
?
+
 '
0
2?
?2 1
L
 
 1 ? 2 ?2?n? 2?'
0
('A +MbIb)
?
:
(2.54)
The flrst terms on the right hand sides would be identical to those for isolated
junctions, if it were not for the ratio of inductances multiplying the current bias.
Because of the interaction terms, the Langrangian cannot be expressed as a sum of
single junctions. Nevertheless, the full Langrangian can be written down directly,
by integrating to get the potential:
L = 12mJ1 _ 12 + 12mJ2 _ 22 ?U ( 1; 2;Ib;'A) (2.55)
U ( 1; 2;Ib;'A) = ?'02? (I01 cos 1 +I02 cos 2)? '02? IbL (L2 1 +L1 2) (2.56)
+
 '
0
2?
?2 1
2L
 
 1 ? 2 ?2?n? 2?'
0
('A +MbIb)
?2
:
The two-dimensional potential resembles the tilted washboard along the  1
and  2-axes, but it is also curled up along the  1? 2 direction. The last term in the
potential is equal to ('T ?'A ?MbIb)2 =2L and represents the energy stored in the
loop inductance by the bias currents. The large-scale curvature of the potential is
inversely proportional to this inductance. The n that was introduced in Eq. (2.50)
is discussed in the next section, but setting it equal to 0 is inconsequential for our
purposes.
The Lagrangian yields momenta p1 = mJ1 _ 1 and p2 = mJ2 _ 2, so the Hamilto-
nian takes on a simple form,
H = 12mJ1 _ 12 + 12mJ2 _ 22 +U ( 1; 2;Ib;'A) (2.57)
= p
2
1
2mJ1 +
p22
2mJ2 +U ( 1; 2;Ib;'A): (2.58)
40
2.5 Classical SQUID Behavior
While the circuit diagram for the dc SQUID is not much more complicated
than that of the single junction, the behavior of the device certainly is. In this
section, I will give a brief review of some classical properties of the dc SQUID that
are relevant to its operation as a qubit.
The sensitivity of the SQUID to magnetic  ux leads to complications and is
a drawback for quantum computing. Many of the  ux characteristics are related to
the value of fl. With no loop inductance (fl = 0), the device is identical to a single
junction with a critical current that is modulated by the applied  ux. For very large
fl, the constraint of Eq. (2.50) can be satisfled in a number of ways, and the two
junctions become uncoupled. We have studied devices with fl between 20 and 250.
I will start out by describing a more strongly coupled device. This simplifles the
discussion, but also accentuates low fl properties, which only moderately afiect our
qubits.
Figure 2.13(a) shows the potential energy U of a symmetric SQUID with
fl = 4:8, drawn using Eq. (2.56) with Ib, 'A, and Mb all zero. The axes are  + =
( 1 + 2)=p2 and  ? = ( 1 ? 2)=p2, which make the symmetry very clear. Along
 ? = 0, the potential resembles a simple tilted washboard, even when the device
is biased. Wells separated by 2?p2 in the  + direction are equivalent, as both
junctions have advanced in the same sense by 2?. The wells in the  ? direction are
physically distinct, as each corresponds to a difierent circulating current and trapped
 ux in the loop. The front and back cuts of Fig. 2.13(a) are plotted in Fig. 2.13(b)
as a solid and dashed line. Although the modulation due to the Josephson energy is
present throughout the potential, there are no local minima for large j ?j. Locations
with local minima (stable wells) will lose those minima (becoming unstable) as the
current and  ux bias change.
41
?  / 2pi?2
?  / 2
pi?2
?  / 2pi?2
0
-2
-1 +1
+2
(a)
(b)
Figure 2.13: Potential of a symmetric dc SQUID with respect to symmetry axes.
(a) The potential of a SQUID with fl = 4:8, L1 = L2, and I01 = I02 is plotted at
Ib = 'A = 0 as a function of  + = ( 1 + 2)=p2 and  ? = ( 1 ? 2)=p2. The
potential has been scaled by U0 ? '0 (I01 +I02)=2? and shifted vertically by the
same amount. (b) Cuts at  + = 0 (solid line) and ?p2 (dashed) can be used to
identify the flve distinct  ux states, which are labeled by N'.
42
(b)
(a)
?1 / 2pi
? 2 / 2
pi
0
-2
-1 +1
+2
?1 / 2pi
0a
0c
0e
-1b
-1d
-2a
-2c
-2e
+2a
+2c
+2e+1d
+1b
Figure 2.14: Potential of a symmetric dc SQUID with respect to junction axes. (a)
The potential for the unbiased SQUID with fl = 4:8 in Fig. 2.13 is plotted as a func-
tion of  1 and  2. The small numbers indicate the  ux state N'; indistinguishable
wells of the same  ux state are labeled with letters, e.g. -1b and -1d. (b) In many
cases, all of the  ux states can be identifled on a single cut along  2 = 0.
43
A contour plot of the same potential is shown in Fig. 2.14(a), plotted as a
function of  1 and  2, with a single line cut in Fig. 2.14(b). Adjacent wells along
one of these directions correspond to the phase of one junction advancing by 2?,
which gives rise to a change of a  ux quantum in the loop.
The location of local minima are marked with crosses on Fig. 2.14(a), with
values listed in Table 2.1. These were calculated numerically using Newton-Raphson
iteration; see, for example, x9.6 of Ref. [69]. As with the single junction, the system
can sit (in the \classical ground state") at the bottom of one these wells with no
kinetic energy. In this case, there are no displacement currents through the junction
capacitances and Ib = I01 sin 1 +I02 sin 2. For an unbiased symmetric device, this
means that the location of the minima satisfy  1 = ? 2. Thus, the two cuts along
 ? are guaranteed to run through all of the minima of the potential. Each of the
letters used to label the wells corresponds to a difierent value of  1 +  2. With no
inductance, the minima lie at intervals of 2? along  1 and  2. With flnite inductance,
the curling of the potential shifts the minima toward the  + axis, so that they are
not located along any constant value of  1 or  2. A single cut of the sort shown in
Fig. 2.14(b) can therefore miss shallow wells corresponding to high  ux states.
The seventh column of Table 2.1 shows the total  ux 'T = LJ due to the
circulating current J = (I2 ?I1)=2 for each of the  ux states. For flnite fl, adjacent
(distinct) states generate  uxes that difier by ?' < '0. The number of these  ux
units N' = 'T=?' is used to label the wells in Figs. 2.13 and 2.14.4 Notice that ?'
decreases slightly as the total generated  ux increases. The sign convention given
in Fig. 2.12 sets states with positive N' to have I1 < 0. Incidentally, the number of
allowed  ux states is set by the maximum circulating current. For large fl, ?' ? '0
and the number of states at zero bias is 2(L1 +L2)I02='0 + 1, provided I01 > I02
as in the qubits I studied in this thesis.
4In Ref. [72], this quantity is known as n; there, N' is used to denote the total number of
allowed  ux states.
44
Table 2.1: Flux state properties of a low fl dc SQUID. The second and third columns
are the locations of the minima of the potential in Figs. 2.13 and 2.14, with the flrst
column giving the labels used there. The (purely) circulating current generates a
non-quantized total  ux 'T, though the  uxoid is always zero as the sum of the
flnal two columns shows.
N'  1=2?  2=2? ( 1 + 2) I1=I01 I2=I02 'T='0 ( 2 ? 1)=2?
-2c -0.871 0.871 0 0.726 -0.726 -1.742 1.742
-1d 0.060 0.940 2? 0.367 -0.367 -0.880 0.880
0c 0 0 0 0 0 0 0
+1d 0.940 0.060 2? -0.367 0.367 0.880 -0.880
+2c 0.871 -0.871 0 -0.726 0.726 1.742 -1.742
+2a -0.129 -1.871 ?4? -0.726 0.726 1.742 -1.742
+2e 1.871 0.129 4? -0.726 0.726 1.742 -1.742
For each  ux state, the sum of the flnal two columns of Table 2.1 is zero,
consistent with the  uxoid quantization condition [54], which can be simply taken
as an inverted version of the  ux-phase relationship of Eq. (2.45). Here, a counter-
clockwise path is chosen so that the corresponding  ux is positive, yielding  2 ? 1.
Although the  ux in the loop changes for the difierent states, all of them give a
 uxoid equal to the constant n in Eq. (2.56). I should point out that several of the
above statements are true only for 'A = 0. In general, if 'A = N''0, then a well
labeled N' is at a global minimum. All of the arguments may be repeated with
respect to this new minimum.
This example gives me an opportunity to discuss two interpretations of n. So
far, Ihaveallowed 1 and 2 tobe unbounded. Knowledge ofthe  uxoid quantization
up to multiples of 2? in Eq. (2.50) can be incorporated into the quadratic term of
the potential as shifts in the two coordinates. The remaining terms are periodic in
 1 and  2, so these shifts have no physical consequences. When the system jumps
to a new  ux state, voltages appear across the junction, which results in the phases
45
evolving. However, the  uxoid remains constant. Although n is not needed at all,
difierent values could correspond to difierent histories of these voltages.
Alternatively,  1 and  2 can be restricted to [??;?], with n providing com-
pensation. In that case, Fig. 2.13(a) would look quite difierent (and confusing!) as
blocks of the potential would be shifted back to the origin, not unlike the reduced
zone scheme used in band structure diagrams. Switching to difierent  ux states is
then encoded in n (which is always the value of the  uxoid). In this case, N' = n.
For  ux states in the  + direction, n is constant and the states truly are indistin-
guishable, i.e. there is no way to label the path that the system took to get to a
state. Although this picture does provide a clear deflnition of N', I will otherwise
use the \extended zone scheme" (with n = 0) which I flnd easier to visualize.
As an example, consider the flrst state listed in the Table 2.1. For this state,
 1=2? = ?0:871,  2=2? = +0:871, and n = 0. The same value of the potential
[given in Eq. (2.56)] can be obtained with  1=2? = +0:129,  2=2? = ?0:129, and
n = ?2. As noted in the previous paragraph, with the phases restricted, the entries
for the flnal three  ux states in the table would be identical.
So far, I have only discussed the situation at zero bias. As the  ux bias is
increased, the potential \rolls" in the  ? direction, resulting in a shift of 2?=p2 for
every additional  ux quantum of applied  ux. Although the shape of the potential
is the same at these  ux quantum intervals, the N' labels of the wells will change.
As the potential rolls, once stable wells become unstable and are replaced by newly
stable wells. If the phase particle is in a well that loses its minimum, it will flnd a
new well in a random fashion. We exploit this process in initializing the  ux state,
as described in x6.5. In practice, we cannot easily check if there is any background
 ux biasing the loop (from trapped vortices, for example), so 'A is measured with
respect to If = Ib = 0.
When the current bias increases, the potential tilts in the  + direction, much as
46
the 1-D tilted washboard does. At some point, the well barrier will disappear. The
conditions for this to happen are described in the next section. This way of making
a well unstable is somewhat difierent than just changing the  ux. As before, the
system can simply switch to a difierent  ux state. However, if the tilt is su?ciently
large, the SQUID will switch to the voltage state. The important difierence between
the SQUID and the single junction is that the SQUID potential gives the particle
two directions to escape in. As described in x6.4, we adjust the biases so that escape
always happens in the  1 direction when operating the device as a qubit. Although
the particle begins to move in this direction, it quickly gains enough energy so that
both phases evolve (corresponding to both junctions being in the voltage state).
For quantum computation, we are interested in the quantum properties of the
dc SQUID. As with the single junction, there are metastable resonances that we will
use as the states of a qubit. Similar techniques as those described in x2.3.3 can be
applied to the 2-D potential [73]. A full description of the difierent solutions could
occupy an entire chapter and is beyond the scope of this thesis. However, in the very
speciflc way in which we bias the SQUID, it behaves much like a single junction.
This is because the two junctions are generally held well out of resonance with each
other and the qubit junction is always made to switch to the voltage state before
the isolation junction [39,72]. Therefore, I will apply the single junction results to
the SQUID. While there are important difierences between the two [73], the single-
junction solution will be su?cient to describe nearly all of the results presented in
the experimental chapters that follow.
2.6 Current-Flux Characteristics
Unlike the current-biased junction, a dc SQUID will not necessarily switch
to the voltage state at a single value of Ib. There are two reasons for this. As
described in the previous section, difierent  ux states correspond to difierent values
47
of circulating currents in the loop. In addition, the applied  ux serves as a bias for
the SQUID, independent of the current bias. Therefore, the IV curve of a single
junction is replaced by current- ux characteristics, Ib vs. 'A, for the SQUID.
For each value of 'A, the critical current can be found by allowing  1 and  2
to adjust themselves to maximize Ib [given in Eq. (2.45)]. The solution can be found
by using the method of Lagrange multipliers, with Eq. (2.50) as the constraint on
the phases [74]. This yields the following condition, which will be satisfled at an
extremum of Ib:
cos 2 = ?
 I
02
I01 cos 1 +
2?
'0 (L1 +L2)I02
??1
: (2.59)
Notice that this is independent of Mb.
This equation yields three qualitatively difierent types of characteristics. How-
ever, in high fl devices of the sort that we use where
(L1 +L2) I01I02I
01 +I02
? '02?; (2.60)
the relationship between the phases will resemble the one shown in Fig. 2.15(a),
which is for the symmetric device of the previous section. This pattern will repeat
every 2? in both phases. On both the solid and dashed curves Ib is maximized (for
some implicit value of 'A), as the second derivatives of Eq. (2.45) conflrm. However,
for the points on (and within) the dashed loop, the potential of Eq. (2.56) is at a
maximum. They are not physically stable solutions and only the loop centered
about (0, 0) will be considered below.
In order to convert this loop to the characteristic of Fig. 2.15(b), Eq. (2.45)
can be used to calculate Ib and Eq. (2.50) can be inverted to give the necessary
48
(a) (b)
a
d c
b
e
a
d
c
b
e
Figure 2.15: Current- ux characteristics of a symmetric SQUID. (a) The total cur-
rent bias Ib has extrema along physical (solid) and unphysical (dotted) branches
for the device introduced in Fig. 2.13. (b) The resulting characteristic shows the
closed area where the system can remain in N' = 0. The small letters show the
correspondence between the phases in (a) and currents in (b).
applied  ux (in the ground state):
Ib = I01 sin 1 +I02 sin 2 (2.61)
'A = '02? ( 1 ? 2)+(L1 ?Mb)I01 sin 1 ?(L2 +Mb)I02 sin 2: (2.62)
Equation (2.59) is always satisfled for  1 =  2 = ??=2, so the characteristic will
have a maximum at jIbj = I01+I02 and critical points at ?(I01 ?I02). In fact, these
are the only four points on the characteristic where either of the junctions is biased
at its critical current. A particular  ux state (N' = 0 for this flgure) is stable if the
system stays within the loop. If any of the boundaries are crossed, then the phase
particle is forced to flnd another well. For su?ciently large values of Ib, the system
will not be able to re-trap and will go to the running state.
Figure 2.16 shows the characteristics for a SQUID with I02=I01 = 0:4, L1 =
49
5:8'0=I01, L2 = 0:2'0=I01, and thus fl = 8:4. While the critical current and
inductive asymmetries are similar to our qubit devices, I have chosen a relatively
low fl for clarity. Figure 2.16(a) shows the physical phase loop, repeated at multiples
of 2?. The corresponding Ib vs. 'A loops are shown in Fig. 2.16(b), for positive Ib.
In both panels, only a selection of an inflnite family of loops is shown; as before, a
line of loops in Fig. 2.16(a) map to the same loop in Fig. 2.16(b) as indicated by
the small numbers. The \top" of the loops are indicated with solid lines; the dashed
lines are obtained by letting  1 !? 1 and  2 !? 2, or Ib !?Ib and 'A !?'A.
Notice that the point Ib = 'A = 0 is enclosed by flve loops, which is the number
of possible  ux states and is coincidentally the same number as for the symmetric
fl = 4:8 device. There is, however, a small region at flnite 'A where six states are
allowed.
Although Fig. 2.16(a) resembles Fig. 2.14(a), the nature of the information
they contain is quite difierent. The former shows the relationship between  1 and
 2 for all possible  ux biases that maximize Ib. The small numbers are used to label
the wells and (as mentioned earlier) are the number of applied  ux quanta needed
to bring the well to the global minimum; these labels extend to inflnity in both
directions. The latter shows only the stable wells under one particular set of bias
conditions. Although they extend to inflnity in the  + direction, the  ? direction
distinguishes the allowed  ux states. N' not only serves as a label, but also indicates
the  ux generated by circulating currents in the loop.
In a high fl asymmetric device, it is sensible to identify the qubit and isolation
junction branches. Comparing Fig. 2.16(a) and (b), each of the steeper solid lines
correspond to  1 ? ?=2(mod 2?) and therefore represent the qubit switching to the
voltage state; they extend from Ib = I01 + I02 to I01 ? I02. In doing experiments,
it is these branches that we will want to cross in order to measure the properties
of the qubit junction. On the other hand, the shallow solid lines correspond to
50
-3
(a)
(b) -2 -1 0 +1 +2 +3
+2c+1b0a +3d
-2c 0e-1d-3b
+1d0c-1b +2e-2a
-2e-3d-4c
+2a +3b +4c
Figure 2.16: Current- ux characteristics of an asymmetric dc SQUID. (a) The crit-
ical points for a SQUID with I02=I01 = 0:4, L1 = 5:8'0=I01, L2 = 0:2'0=I01, and
fl = 8:4 repeat at intervals of 2?. The numbers that label each loop specify the  ux
state N'; loops centered about a common value of  ? are labeled with the same
letter. Panel (b) shows seven of an inflnite series of characteristics. The numbers are
N' and match the labels on panel (a). Experimentally, the bias trajectories that we
use cross the solid lines; the dashed lines may be found by inverting the coordinates
and complete the loop of each  ux state. The inset shows an expanded view of the
intersection of two branches; the box has a width of 0.007 and height of 0.0006.
51
(a)
(b)
Figure 2.17: Slopes of the current- ux characteristics of an asymmetric dc SQUID.
The inverse of the slope of the characteristics shown in Fig. 2.16 for the (a) isolation
and (b) qubit branch (in units of '0=I01) have a curvature due to the Josephson
inductance of the qubit and isolation junctions, respectively. The horizontal axis
shows the full extent of each branch for N' = 0. The dot-dashed lines show the
contribution of the geometrical inductances.
52
 2 ? ?=2(mod 2?) and switching of the isolation junction. This branch extends
from Ib = I01 +I02 to I02 ?I01, although we do not experimentally measure its full
extent, as discussed in x6.3.
The slope of the characteristic can be understood as follows [4]. Consider
a point on the steep (qubit) branch. If 'A increases slightly, then the circulating
current J (shown in Fig. 2.12) becomes increasingly negative in response. Therefore,
a smaller current bias is needed to push the qubit past its critical current, resulting
in a negative slope.
An expression for the inverse of this slope can be found by taking a derivative
of Eq. (2.62) and substituting in the Josephson inductance of Eq. (2.12):
d'A
dIb =
'0
2?
 L
1 +LJ1 ?Mb
LJ1
?d 
1
dIb ?
'0
2?
 L
2 +LJ2 +Mb
LJ2
?d 
2
dIb : (2.63)
One of the phase derivatives may be eliminated using Eq. (2.61) to yield
d'A
dIb = ?(L2 +LJ2 +Mb)+
'0
2?
 L
1 +LJ1 +L2 +LJ2
LJ1
?d 
1
dIb : (2.64)
Along the qubit branch, the second term on the right can be neglected,5 so the
inverse of the slope is ?(L2 +LJ2 +Mb) to a very good approximation. Although
the same result is obtained if d 1=dIb = 0, this limit is not strictly valid, as seen
by the slight curvature of the qubit branch in Fig. 2.16(a). For low fl devices, both
terms of Eq. (2.63) make comparable contributions, but the approximate expression
for the slope is nevertheless still valid. Analogous arguments give the inverse of the
slope of the isolation branch as L1 +LJ1 ?Mb.
Figure 2.17(a) and (b) show the inverse of the slope of each of the branches of
the example device, in units of '0=I01. While these curves were calculated from the
numerical solutions of Eq. (2.59), they are indistinguishable on this scale from the
5This can be shown numerically for a large range of device parameters.
53
sum of the geometrical and Josephson inductance for each arm (with the former?s
contribution indicated by a dot-dashed line). The qubit branch has a much more
noticeable curvature because I02 and L2 are small. It is steeper than the isolation
branch because I have assumed L1  L2, as with the devices that I measured.
2.7 Capacitively-Coupled Junction Qubits
A nice feature of phase qubits is that they can be coupled together with a
simple capacitor. In this section, I will derive the Hamiltonian for a system of
two coupled junctions, which will describe the spectroscopy experiments of x8.6.
Detailed derivations and further discussion can be found in Refs. [41,75{77], x8.1 of
Ref. [3], Chapter 8 of Ref. [1], and Chapters 6 and 7 of Ref. [2].
Figure 2.18 shows two current-biased junctions, whose elements are labeled
with a superscript A or B, coupled together with a capacitor CC. The junctions
have critical currents IA0 and IB0 , which need not be the same, but I will assume
that their capacitances are equal (CJ = CAJ = CBJ ). The junctions are biased
with independent currents, IAb and IBb . We found, somewhat accidentally, that the
stray inductances LAC and LBC, together with CC, form an LC harmonic oscillator
mode [78]. The inductances can be ignored, as I will do at flrst, if the resonant
frequency of the mode is above the region of interest. I have also dropped the
junction shunting resistances normally included in the RCSJ model.
Each bias current splits between its junction and junction capacitance, as well
as the coupling capacitor, giving rise to the equations
IAb = CJ ddt
 '
0
2?
d A
dt
?
+IA0 sin A +IC (2.65)
IBb = CJ ddt
 '
0
2?
d B
dt
?
+IB0 sin B ?IC; (2.66)
54
CCLCA
CJA I0AIbA IbBCJBI0B
LCB
IC
Figure 2.18: Circuit diagram for two LC-coupled junctions. Two current-biased
junctions, similar to one shown in Fig. 2.2, are coupled together with a capacitance
CC, through which current IC  ows. Stray inductances LAC and LBC can have a
signiflcant impact.
where the current through the coupling capacitor is
IC = CCdVCdt = CC'02? ddt
 d A
dt ?
d B
dt
?
: (2.67)
Thus IC travels from junction A to B and depends on the rate of change of the
voltage VC across CC. As the coupling current can be expressed in terms of the two
junction phase difierences, no additional degrees of freedom have to be introduced.
With rearrangements similar to those for the single junction, the equations of motion
for the system become
mJ? A = '02? ??IA0 sin A +IAb ??mC ?? A ? ? B? (2.68)
mJ? B = '02? ??IB0 sin B +IBb ?+mC ?? A ? ? B?; (2.69)
where the efiective masses are mJ ? CJ ('0=2?)2 and mC ? CC ('0=2?)2.
In each of the equations of motion, the flrst term on the right is generated
by the Lagrangian for a single junction, given by Eq. (2.21). The simplest way to
55
generate the second term is with the coupling Lagrangian
LC = 12mC ?_ A ? _ B?2 : (2.70)
The conjugate momenta, given by Eq. (A.8), are
pA = (mJ +mC)?_ A ??0_ B? and pB = (mJ +mC)?_ B ??0_ A?; (2.71)
where the dimensionless coupling constant is
?0 = CCC
J +CC
: (2.72)
The full Hamiltonian has the simple form H = HA + HB + HC according to Eq.
(A.10), where each term is expressed in terms of the generalized coordinates and
velocities. The single junction Hamiltonians HA and HB are given by Eq. (2.23)
and the coupling contribution, obtained from Eq. (A.11), is
HC = 12mC ?_ A ? _ B?2 : (2.73)
To put the Hamiltonian in canonical form, the generalized velocities may be
eliminated in favor of pA and pB by inverting Eq. (2.71). The system Hamiltonian,
expressed in terms of  A and  B and their conjugate momenta, simplifles to
H = (p
A)2
2m +
(pB)2
2m +
?0
mp
ApB ? '0
2?
?IA
0 cos 
A +IA
b  
A?? '0
2?
?IB
0 cos 
B +IB
b  
B?;
(2.74)
where the efiective system mass is deflned to be
m = mJ
 
1+ mCm
J +mC
?
=
 '
0
2?
?2
CJ (1+?0): (2.75)
56
The flrst three terms in H represent a kinetic energy contribution, which is the
total energy stored by the three capacitors. The mass of the phase particle increases
due to the coupling and leads to a downward shift of all of the energy levels. The
flnal two terms give the two-dimensional potential, which resembles an egg crate,
in that it is a tilted washboard in each of the junction axes. In the case of the dc
SQUID, discussed in x2.4, the coupling of junctions modifled the potential, while
here the coupling modifled the kinetic energy. In particular, CC has introduced a
momentum coupling term.
To represent the states of this two-qubit system, I will use the notation jABi
to indicate a direct product of the uncoupled states of junctions A and B. The
ground state of the uncoupled system is j00i. If the two junctions have the same
plasma frequency !p, then j00i has energy ~!p in the harmonic limit. With no
coupling, both j01i and j10i would have energy 2~!p. The coupling capacitor lifts
this degeneracy, leaving the maximally entangled Bell states (j01i?j10i)=p2 as
the flrst two excited states, with energies (2??0=2)~!p. The momentum coupling
term in the Hamiltonian gives the symmetric state (+) the higher of the two energies.
Having discussed the simpler circuit, I will now summarize the results when the
coupling inductors in Fig. 2.18 are included [78]. The LC resonator that is created
contributes an additional degree of freedom,  C = 2?LCIC='0, which corresponds
to the current IC that  ows through the total coupling inductance LC = LAC + LBC.
The renormalized angular frequency of the harmonic LC mode that is created is
!C =
?
LC
 C
CCJ
CJ +2CC
???1=2
: (2.76)
If both junctions are brought into resonance with this frequency, then the three
57
degenerate levels are split by an amount proportional to the coupling coe?cient
?p
2 =
r C
C
2CJ +4CC: (2.77)
If instead, both junctions are in resonance with each other at a frequency !p well
below !C, then the coupling is dominated by the capacitor and can be described by
the frequency-dependent coe?cient
? (!p) = ?
2
1??2 ?!2p=!2C =
?0
1?(1+?0)!2p=!2C: (2.78)
As !p=!C decreases, the coupling weakens, eventually reducing to the purely capac-
itive expression given in Eq. (2.72).
58
Chapter 3
Dynamics of Quantum Systems
In this chapter, I will describe the time evolution of a simple quantum system.
The nature of this system is motivated by the solutions of the junction Hamiltonian
given in x2.3, but most of the discussion is rather general. In fact, much of the
machinery and terminology is borrowed from nuclear magnetic resonance [79] and
atomic physics [5].
The system consists of several states jni of energy En, which have a lifetime
that depends on energy relaxation (on a time scale T1) and tunneling (given by the
rates ?n) which takes the system outside of jni. In addition, phase information is
lost on a time scale T2. We are interested in following the coherent dynamics due
to induced transitions and interaction with the environment.
I will start out with an ideal two-level system, for which exact solutions can be
found, and then move on to a more realistic three-level system. Once interactions
with the environment are included, the density matrix approach must be used. I will
give some background on this subject, outline the analytical solutions, and describe
how we simulate experiments numerically.
3.1 Bloch Sphere
Often in this chapter, I will limit the discussion to a single isolated two-level
system with states j0i and j1i. This is exactly what we want for a qubit, although
our real system has higher levels that cannot be ignored. A generic wavefunction
will be a superposition of the two states,
j?i = aj0i+bj1i = cos  2 j0i+ei` sin  2 j1i: (3.1)
59
0
1
z
x
?
?0 1+i
?2
y
i0 1
0 1+
?2
?2
Figure 3.1: Bloch sphere representation of a two-level system. A vector on the unit
sphere can be used to represent a normalized superposition of j0i and j1i, where the
relative phase between the two basis states is complex.
In the middle expression, the complex numbers a and b must satisfy the normaliza-
tion condition pa2 +b2 = 1. This condition is enforced in the last expression by the
real angles 0 <  < ? and `. The advantage of this last form is that the state of the
system may be graphically represented as a vector on the so-called Bloch sphere,
as shown in Fig. 3.1 (see, for example, x1.2 of Ref. [80]). The polar angle gives the
relative weight of the two states and the azimuthal gives the relative phase. The
state j0i points along the +z-axis; j1i is along the ?z-axis. A 90? rotation in  
of either of these basis states results in an equal superposition of j0i and j1i. The
location on the equator gives the relative phase: the states (j0i?j1i)=p2 point
along ?x and (j0i?ij1i)=p2 point along ?y.
Unfortunately, there is no simple graphical analog for a system with more than
two states.
60
3.2 Two-Level Rabi Oscillations
Transitions in a two-level system can be induced by a time-dependent poten-
tial. A perturbation with a well-deflned frequency will result in a phase coherent
manipulation of the system, corresponding to rotating the Bloch vector along a sin-
gle trajectory [5,81]. Let the system evolve under the Hamiltonian H = H0 +Hi.
H0 is purely diagonal and accounts for the energies E0 and E1 of the orthonormal
ground and excited states j0i and j1i, with
H0j0i = E0j0i and H0j1i = E1j1i: (3.2)
Transitions are due to a potential Hi = A^ycos(!rft), which oscillates at angular
frequency !rf. I will assume that the operator ^y is purely ofi-diagonal and Hermitian,
where the matrix elements of A^y are
h0jA^yj1i = h1jA^yj0i? ? A01: (3.3)
The state of the system at any time is given by
j?i = a0(t)e?iE0t=~j0i+a1(t)e?iE1t=~j1i: (3.4)
The time dependence of the weighting coe?cients a0 and a1 can be found by taking
the inner product of h0j and h1j with Schr?odinger?s equation jHij?i = i~ _j?i. Some
simpliflcation yields
da0
dt = ?i
A01
2~
h
e+i(!rf?!01)t +e?i(!rf+!01)t
i
a1 (3.5)
da1
dt = ?i
A?01
2~
h
e?i(!rf?!01)t +e+i(!rf+!01)t
i
a0; (3.6)
61
where~!01 = E1?E0 is the energy difierence between the two states. If the system
is driven near resonance, then the flrst term in brackets for each equation will vary
slowlywhilethesecondwilloscillaterapidly. Inwhatiscommonlycalledtherotating
wave approximation, the slow (flrst) terms are kept and the rapid (second) terms
are set to zero [82]. Notice that in Eq. (3.5), the e+i!rft contribution from cos(!rft)
is kept, while e?i!rft is kept in Eq. (3.6).
Taking an additional time derivative and plugging in _a0 and _a1 uncouples the
two amplitudes, resulting in
d2a0
dt2 ?i(!rf ?!01)
da0
dt +
jA01j2
4~2 a0 = 0 (3.7)
d2a1
dt2 +i(!rf ?!01)
da1
dt +
jA01j2
4~2 a1 = 0: (3.8)
Onresonance(!rf = !01), theequationsyield ?a0+(?201=4)a0 = ?a1+(?201=4)a1 =
0, where ?01 ?jA01j=~is known as the bare Rabi  opping frequency (between states
j0i and j1i). Assuming that the system starts out in the ground state at t = 0, the
solution is a1(t) = sin(?01t=2). Thus the occupation probability of the excited state
is
ja1(t)j2 = sin2
 ?
01 t
2
?
: (3.9)
If the perturbation is left on for a time t = ?=?01, then the system will make a
transition from the ground state to the flrst excited state with 100% probability.
This is referred to as a ?-pulse and can be thought of as a NOT gate for quantum
computation. If ^A is left on longer, the system will fully return to the ground state
at t = 2?=?01, repeating the cycle indeflnitely in a phenomenon known as a Rabi
oscillation.
With an incoherent drive (which does not correspond to a speciflc path on the
Bloch sphere), the analogs of the stimulated emission and absorption rates would
62
balance each other, resulting in an equal superposition ja1j2 = 1=2 for all time (if
spontaneous emission can be neglected).
With an ofi-resonant drive, Eqs. (3.7) and (3.8) yield amplitudes
a0(t) = ei(!rf?!01)t=2
?
?i(!rf ?!01)?
01
sin
 ?
01t
2
?
+cos
 ?
01t
2
??
(3.10)
a1(t) = ?ie?i(!rf?!01)t=2 ?01?
01
sin
 ?
01t
2
?
; (3.11)
where the efiective Rabi frequency is
?01 =
q
?201 +(!rf ?!01)2: (3.12)
The occupation probability of the excited state becomes [5]
ja1(t)j2 =
 ?
01
?01
?2
sin2
 ?
01 t
2
?
: (3.13)
Thus with detuning, the oscillation frequency increases (as a function of j!rf ?!01j)
and the amplitude of the oscillation decreases.
3.3 Three-Level Rabi Oscillations
As I will discuss in Chapter 9, under a strong microwave drive, we have seen
clear evidence in our qubits for signiflcant population in the second excited state
j2i. A strong drive has two efiects. For one, additional transitions become relevant
to the system dynamics. These are the single photon 1 ! 2 and two-photon 0 ! 2
transitions. In addition, the third level perturbs the 0 ! 1 transition. This is
particularly important at high power, where ?01 is close to !01?!12. This regime is
of interest for quantum computation, because fast gates require high power. Thus
understanding the dynamics of three-level systems is important. The rotating wave
63
approximation is also useful in describing this system; see Ref. [83] and references
therein.
Following the derivation outlined in Appendix C, the Hamiltonian of a three-
level system under a time-dependent perturbation A^ycos!rft in the rotating wave
approximation can be written in matrix form as
H =~
0
BB
BB
@
0 ?01=2 ?02=2
?01=2 !01 ?!rf ?12=2
?02=2 ?12=2 !02 ?2!rf
1
CC
CC
A
; (3.14)
where
?01 = Ay0;1~
?
J0
 A(y
0;0 ?y1;1)
~!rf
?
+J2
 A(y
0;0 ?y1;1)
~!rf
??
; (3.15)
?02 = Ay0;2~
?
J1
 A(y
0;0 ?y2;2)
~!rf
?
+J3
 A(y
0;0 ?y2;2)
~!rf
??
; (3.16)
?12 = Ay1;2~
?
J0
 A(y
1;1 ?y2;2)
~!rf
?
+J2
 A(y
1;1 ?y2;2)
~!rf
??
: (3.17)
Here, the energy level spacing between statesjniandjmiis~!nm, Jn is the nth order
Bessel function of the flrst kind, and yn;m = hnj^yjmi. Despite the notation I have
used, in Eq. (3.15) through Eq. (3.17), the terms outside of the square brackets are
the bare Rabi frequencies; the Bessel function terms make a frequency-dependent
correction.
For the speciflc case of the current-biased junction, the unperturbed Hamil-
tonian is given by Eq. (2.23). Assuming that the total bias current is the sum of
a dc current Ib and a high frequency current ?I?w cos!rft, the amplitude of the
perturbation is given by
A^y = '02?I?w^ ? I?wI
0
EJ
s
8EC
~!p ^x; (3.18)
64
where several matrix elements of ^x in the cubic approximation are given in Eqs.
(2.38)and(2.39). Equation(3.16)showsthatadirecttwo-photontransitionbetween
j0i and j2i, whose strength is given by ?02, is only possible if the diagonal matrix
elements (which I set to zero in the previous section) are flnite. In this case, the
system need not have a state j1i. While all the matrix elements are non-zero for the
current-biased junction, j2i can also become populated with ofi-resonant 0 ! 1 and
1 ! 2 transitions. It turns out that this mechanism outweighs the direct two-photon
process for the power regime we performed experiments in. As the two mechanisms
occur at the same frequency !rf ? (!01 +!12)=2, I will refer to both as two-photon
transitions.
In x3.2, the perturbation caused transitions between flxed energy levels. Here,
in the case of the current-biased junction, the perturbation adds to the current bias.
Thus, the junction?s energy levels, which depend on the total bias, will oscillate with
time. This efiect is accounted for by the non-zero diagonal matrix elements of ^ .
For the persistent current qubit, the variation of the energy levels due to a strong
microwave drive leads to dramatic efiects [84].
As usual, the eigenvalues ei of the Hamiltonian determine the evolution of the
system. For example, in Fig. 3.2(a), the three eigenvalues are plotted for a current-
biased junction with I0 = 30 ?A and CJ = 5 pF, at Ib = 29:8 ?A as a function of
frequency !rf. I have done this for a microwave current of amplitude 5 nA (solid
lines) and 15 nA (dashed lines). It appears that there are three avoided level crossing
at !rf=2? = 6:65, 6.08, and 5.51 GHz. These are the values of !01=2?, !02=4?, and
!12=2? at Ib = 29:8 ?A. If a system is in a superposition of eigenstates near an
avoided crossing, it will undergo oscillations between the states with a frequency
proportional to the magnitude of the splitting. Thus, in this picture, driven Rabi
oscillations appear as Larmor oscillations of non-stationary states.
Consider the simple example of a two-level system. The upper left 2?2 block
65
(a)
(b)
Figure 3.2: Three-level rotating wave approximation. Properties of a current-biased
junction with I0 = 30 ?A and CJ = 5 pF at Ib = 29:8 ?A are plotted for an applied
microwave current of frequency !rf and magnitude I?w = 5 nA (solid) and 15 nA
(dashed). (a) The eigenvalues ei of the rotating wave Hamiltonian show avoided
crossing at microwave frequencies equal to !01=2? = 6:65 GHz, !02=4? = 6:08 GHz,
and !12=2? = 5:51 GHz. (b) The difierence of eigenvalues gives the efiective Rabi
frequency for the three transitions and show that the oscillation frequency increases
with detuning.
66
of Eq. (3.14) gives the Hamiltonian in this case, which has eigenvalues
e1;2
~ =
!01 ?!rf ?
q
?201 +(!01 ?!rf)2
2 : (3.19)
The oscillation frequency between the two states is ?01 = (e1 ?e2)=~, which is equal
to
q
?201 +(!01 ?!rf)2. This is just the expression for the efiective Rabi frequency
in Eq. (3.12).
For the three-level example, the difierence of the eigenvalues is plotted in
Fig. 3.2(b). There are three minima (for each value of I?w), corresponding to on-
resonance single photon 0 ! 1, two-photon 0 ! 2, and single photon 1 ! 2
Rabi oscillations, from right to left. The oscillation frequency increases away from
resonance, with a stronger detuning dependence for the two-photon process.
Comparing the solid and dashed lines, the oscillation frequency of all of the
transition frequencies increases with microwave power, as expected. However a
smaller efiect appears at high power: the location of the minimum frequency un-
dergoes a shift. For I?w = 15 nA, the slowest 0 ! 1 oscillation occurs at !rf=2? =
6:70 GHz, which is 50 MHz greater than !01=2?. This is due to the presence of the
third level and is analogous to the ac Stark shift in atomic physics. The other two
transitions in Fig. 3.2(b) move to lower values of !rf at high power.
I will refer to the minimum oscillation frequency between states jni and jmi
as ?R;nm. The distinction must be made because, for example, at high power in a
multi-level system, ?R;01 will difier from ?01 (which depends on the matrix elements
of the operator connecting the states). And seen above, this new resonance condition
need not occur at !rf = !nm. Ofi resonance, the efiective Rabi frequency will be
denoted as ?R;nm. In the simple two-level example given in x3.2, ?R;01 = ?01 and
?R;01 = ?01 at all drive powers.
In x9.2, I will compare experimental data to the numerical solution of Eq.
67
(3.14), where the matrix elements are given by Eqs. (2.38) and (2.39) and the energy
levels come from Eq. (2.44). However, analytical solutions of the Hamiltonian can
be obtained by flnding the roots of its cubic characteristic equation. From the point
of view of treating the presence of j2i perturbatively, the strength of the coupling of
j2i depends on !01?!12. When this quantity is small compared to ?01, the following
approximations hold [83]. In addition, the energy levels and matrix elements have
been evaluated in the cubic approximation of the tilted washboard potential.
For flxed !rf, the minimum Rabi frequency does not occur at !01 = !rf, but
at !01 = !rf ??!R;01, where
?!R;01 ? ?
2
01
2(!01 ?!12): (3.20)
Conversely, for flxed !01, the resonance moves to !rf > !01, as seen in Fig. 3.2(b).
The minimum oscillation frequency on resonance is
?R;01 ? ?01
 
1? ?
2
01
4(!01 ?!12)2
?
: (3.21)
Thus, the third level suppresses the frequency.
Finally, the two-photon 0 ! 2 transition has a minimum oscillation frequency
?R;02 ?
p2?2
01
!01 ?!12: (3.22)
While the Rabi frequency for a single photon process increases as the square root of
the microwave power, it increases linearly for a two-photon process. The oscillation
frequency increases with detuning as
?R;02 =
q
?2R;02 +(!02 ?2!rf)2; (3.23)
68
which is a general result that holds for two-photon processes.
3.4 Dissipation
Inter-level transitions can occur even when we do not apply a microwave signal
to a junction. This is because the junction is able to exchange energy with the heat
bath that it is in thermal equilibrium with (see x3.2.1 of Ref. [1] and x2.5 of [3]).
In the RCSJ model introduced in x2.2, the strength of the coupling between the
junction and its environment is characterized by an efiective resistance RJ. The
situation is analogous to spontaneous and stimulated emission in atomic systems
[5,81], with the broadband Johnson-Nyquist current noise of RJ replacing the energy
density of Planck blackbody radiation as the source of the transitions.
For large RJ, the thermal noise current It can be treated as a perturba-
tion of the junction Hamiltonian of Eq. (2.23), characterized by the term Ht =
?('0=2?)It ^ . The stimulated emission and absorption rate between jii and jji
(proportional to the Einstein B coe?cient of atomic physics) can be found with
several techniques (see, for example, Appendix B of Ref. [1]1) as
Wstij = Wstji = ~!ij2R
Je2
jhij^ jjij2
exp(~!ij=kBT)?1 ; (3.24)
where !ij is the angular frequency spacing between the two levels.
In addition to these equivalent rates that are strongly temperature dependent,
there is a spontaneous emission rate that persists to T = 0 (analogous to the Einstein
A coe?cient). From the principle of detailed balance, the total emission must exceed
the absorption so as to result in a Boltzmann distribution between any two levels
(in the absence of additional transition mechanisms). From this, one flnds that the
1In Ref. [1], Wt is used to represent the stimulated rate. I have chosen to reserve that symbol
for the total thermal rates given in Eqs. (3.26) and (3.27).
69
spontaneous emission rate is
?ij = ~!ij2R
Je2
jhij^ jjij2 ? 2R
JCJ
!ij
!p jhij^xjjij
2 ; (3.25)
where several matrix elements of ^x in the cubic approximation are given in Eqs.
(2.38) and (2.39). The total thermal emission and absorption rates are
Wtij = Wstij +?ij = ?ij1?exp(?~!
ij=kBT)
(3.26)
and
Wtji = Wtij exp(?~!ij=kBT) = Wstji = ?ijexp(~!
ij=kBT)?1
; (3.27)
where j > i. While there will be transitions between each pair of states, the sponta-
neous emission rate between the ground and flrst excited states serves as a standard
characterization of the system. Its inverse
T1 ? 1?
10
(3.28)
is known as the relaxation or dissipation time.
For our junction devices, an estimate for the dissipation can be obtained in the
harmonic approximation of the 1-D tilted washboard potential. From Eq. (2.36),
?n;n?1 = n=RJCJ and T1 = RJCJ, as one would expect classically.
3.5 Tunneling
A current-biased junction will eventually tunnel from a metastable supercur-
rent state to the flnite voltage state, where the occupation probability ?i of an energy
level jii decays with a tunneling rate ?i. While it is di?cult to determine which
level the junction was in when it tunneled, it is straight forward to measure the
70
total escape rate out of all of the metastable states
? = ? 1?
tot
d?tot
dt ; (3.29)
where ?tot = Pi ?i and the ?i do not include the flnite voltage state that results
after tunneling. If we think of ?i as being deflned for an ensemble of junctions, then
? should only depend on those elements of the ensemble that have yet to escape.
Therefore, the total escape rate is the weighted sum
? =
X
i
?i
?tot?i =
X
i
Pi?i; (3.30)
where
Pi = ?i?
tot
(3.31)
is the normalized probability of being in state jii. While the total population ?tot
will decay to zero with time due to tunneling, Pi Pi (t) = 1 for all times.
3.6 The Density Matrix
If we are describing an isolated system interacting with a single mode of radia-
tion, then Schr?odinger?s equation is su?cient to model the dynamics. However, the
state space of our junction qubits is much larger; in practice, the junction interacts
strongly with its environment. For example, the energy dissipation described in
x3.4 involves losing energy to a huge number of quantum states that constitute a
thermal bath at the temperature of the dilution refrigerator mixing chamber. In-
teraction with these states and those associated with all the bias circuitry leads
to dissipation and decoherence. In addition, we depend on tunneling discussed in
x3.5 for state readout. The flnal voltage state lies outside our simple qubit space
and may be di?cult to describe quantum mechanically [85]. If we are unable to
71
obtain the Hamiltonian that describes this complicated system, then we must aban-
don Schr?odinger?s equation and the hope of describing the system with a single
wavefunction.
Although we may not understand every process at a microscopic level, we do
know how to describe them in a phenomenological way. For example, dissipation is
governed by T1 and tunneling is characterized by the rates ?n. These time constants
describe the dynamics for an ensemble of systems, rather the evolution of a single
particle. This is particularly applicable to the type of data presented later in this
thesis, where we repeat a particular experiment many times to extract statistical
information. The standard approach to analyzing this situation is to use the density
matrix formalism, which allows us to focus on the quantum evolution of only the
subsetoftheuniversethatweareinterestedin, whiletakingintoaccountinteractions
with external degrees of freedom.
I will begin by listing a few basic properties of the density matrix [5,86{88].
The density matrix (or more accurately, density operator) is deflned as
? =
X
i
wijiihij; (3.32)
where wi gives the probability of an element of the ensemble to be prepared in the
state jii; wi is non-negative and Pwi = 1.
In general, this description of the system is quite difierent than the wavefunc-
tion ? = pw0j0i + pw1j1i + :::, which can only be written down if the phase
relationship between the basis vectors is well-deflned.
The density operator ? is Hermitian and can be diagonalized with real eigen-
values. If every element of an ensemble is prepared identically, then wi is equal to
1 for one value of i and zero for the rest. Then ?2 = ? and Tr(?2) = 1, which is the
deflnition of a pure state. Here, Tr(O) = PhijOjii is the trace of an operator O.
72
If more than one of the wi is non-zero (when ? is diagonalized), then Tr(?2) < 1
and the system is said to be in a mixed state or mixed ensemble. The expectation
value of an operator O is given by
hOi = Tr(?O): (3.33)
In order to follow the evolution of an ensemble, the time dependence of the
density matrix is needed. The weights in Eq. (3.32) will be constant, so only the
state vectors jii will vary. The time derivative of ? is
@?
@t =
X
i
wi
? @
@t jii
?
hij+jii
 @
@t hij
??
(3.34)
= ?i~[H;?]: (3.35)
Here, Schr?odinger?s equation and its adjoint were substituted to simplify the flrst
line. This is known as the Liouville-von Neumann equation, as von Neumann used it
to describe quantum evolution, although it resembles Liouville?s theorem for classical
phase space density.2
3.6.1 The Reduced Density Matrix
I will now return to the case of describing our qubit along with the rest of the
universe. Let jii and jji be orthonormal sets that span the space of the qubit and
the rest of the universe, respectively. Assuming that the universe is in a pure state,
the wavefunction and density matrix describing it are [5,86]
j?i =
X
i;j
cij jiijji: (3.36)
2This looks like the Heisenberg equation (that gives the evolution of an operator), but difiers
by more than just a sign; we have been working in the Schr?odinger picture, where it is the state
vectors that carry the time dependence.
73
and
? = j?ih?j =
X
i;j
X
i0;j0
cijc?i0j0 jiijjihj0jhi0j: (3.37)
Imagine we want the expectation value of an operator Oi that only acts on
the qubit (system i). Equation (3.33) gives
hOii = Tr(?Oi) =
X
i
hij
?X
j
hjj?jji
!
Oijii: (3.38)
The term in parentheses is the partial trace of ? over system j and is known as the
reduced density matrix for system i,
 i ? Trj (?) =
X
i;i0;j
cijc?i0j jiihi0j: (3.39)
With  i in hand, the expectation value
hOii = Tri ( iOi) (3.40)
can be found with no direct reference to system j.
If the cij in Eq. (3.36) represents a product wavefunction between the two
systems, then  i will still represent a pure state. However, if the two systems are
entangled (certainly the case for our experiments), then even a pure ? will result in
a mixed  i. In fact, interactions with the environment (that we cannot accurately
describe microscopically) will be included as terms in the Hamiltonian that quite
clearly create mixed states. This leads to non-unitary evolution, for example, as the
junction tunnels to the flnite voltage state, which is outside the desired state space.
From now on, when I refer to the density matrix ? of a junction device, I really
mean the reduced density matrix  i, having traced over the rest of the universe.
74
3.7 Optical Bloch Equations
In this section, I will revisit the two-level system of x3.2, where Schr?odinger?s
equation gave the time dependence of a particular wavefunction. Now the Liouville-
von Neumann equation gives the evolution of an ensemble of systems described by
a density matrix [1,3{5,88]. This will reproduce the prior results, but also allows
the inclusion of the non-unitary transformations that motivated the density matrix
approach. Although the two-level model is too simple to describe our devices accu-
rately (particulary for high power microwave drives), it does qualitatively reproduce
many of the phenomena that we are interested in. Expressing Eq. (3.35) in a matrix
representation in the basis of the eigenstates of H0 [deflned by Eq. (3.2)] yields
0
B@ _?00 _?01
_?10 _?11
1
CA = ?i
~
2
64
0
B@ E0 0
0 E1
1
CA+
0
B@ 0 ~?01
~?01 0
1
CAcos(!
rft);
0
B@ ?00 ?01
?10 ?11
1
CA
3
75;
(3.41)
where I have assumed ?01 is real. The diagonal elements of the density matrix,
?00 and ?11, represent the populations of the ground and excited states, while the
so-called ofi-diagonal coherence terms, ?01 and ?10, describe correlations between
them. In x3.5, I referred to the diagonal element ?ii as ?i.
Thematrixequationcanbesplitintofourequationsthatdescribetheevolution
of each of these elements:
_?00 = ?i?01 (?10 ??01)cos(!rft)+ ?11T
1
(3.42)
_?01 = ?i?01 (?11 ??00)cos(!rft)+i!01?01 ? ?01T
2
(3.43)
_?10 = +i?01 (?11 ??00)cos(!rft)?i!01?10 ? ?10T
2
(3.44)
_?11 = +i?01 (?10 ??01)cos(!rft)? ?11T
1
: (3.45)
On the right hand sides, the terms involving the microwave drive frequency !rf
75
and energy level splitting !01 reproduce the unitary evolution of the Schr?odinger
equation (as those terms came from a valid Hamiltonian).
In addition, I have included the energy dissipation (T1) and coherence (T2)
times in a phenomenological way. T1 only afiects the state occupancy, increasing
the probability that the junction will decay to the ground state. Thermal excita-
tions could have also been included, but for now, I will stay in the limit of zero
temperature.
For a single qubit in a superposition of j0i and j1i, the relative phase between
the basis states becomes ill-deflned on a time scale T2, which only afiects the ofi-
diagonal terms of the density matrix ?. The decoherence rate is often expressed as
the sum
1
T2 =
1
2T1 +
1
T`: (3.46)
Here, phase information is lost as the system relaxes to the ground state; in the way
that T2 enters the Bloch equations, the factor of 2 is needed to ensure consistency
(see x2.6.1 of Ref. [3] and Refs. [5,88]). T` is known as the dephasing time and
characterizes processes that do not change the energy of the qubit. Simply deflned,
T` is the mean time for a system in the pure quantum state aj0i + bj1i (which
has a well-deflned phase) to evolve through interactions with the environment to
the mixture specifled by the density matrix jaj2j0ih0j + jbj2j1ih1j (which has the
same occupation probabilities without any phase information) [21]. In addition,
decoherence in a real (multi-state) system may involve a change of the amplitudes
a and b and leakage into states outside of the qubit space.
Once again, the system will respond at the drive frequency, so it is convenient
to move to a frame rotating with this frequency. The density matrix becomes
? =
0
B@ e?00 e?01e+i!rft
e?10e?i!rft e?11
1
CA; (3.47)
76
which is Hermitian. To proceed, it is also convenient to express Eqs. (3.42) to (3.45)
in matrix form, converting the density matrix to a vector. After some simpliflcation,
the resulting matrix equation becomes
0
BB
BB
BB
B@
_e?00
_e?01
_e?10
_e?11
1
CC
CC
CC
CA
=
0
BB
BB
BB
B@
0 i?te+i!rft ?i?te?i!rft 1T1
i?te?i!rft ?i? ? 1T2 0 ?i?te?i!rft
?i?te+i!rft 0 i? ? 1T2 i?te+i!rft
0 ?i?te+i!rft i?te?i!rft ? 1T1
1
CC
CC
CC
CA
0
BB
BB
BB
B@
e?00
e?01
e?10
e?11
1
CC
CC
CC
CA
;
(3.48)
where ?t ? ?01 cos(!rft) and ? ? !rf ?!01 is the detuning of the microwave drive
from resonance. In the matrix, all of the time dependence is contained in terms of
the form
cos(!rft)e?i!rft =
 e+i!
rft +e?i!rft
2
?
e?i!rft = 12 + 1+e
?i2!rft
2 ?
1
2; (3.49)
where the rotating wave approximation is made by neglecting the term with fre-
quency 2!rf, as it will time average to 0.
With these simpliflcations, the equations of motion become [4,5,88]
_e?00 = +i?01
2 (e?01 ?e?10)+
e?11
T1 (3.50)
_e?01 = ?i?01
2 (e?11 ?e?00)?
e?01
T2 ?i?e?01 (3.51)
_e?10 = +i?01
2 (e?11 ?e?00)?
e?10
T2 +i?e?10 (3.52)
_e?11 = ?i?01
2 (e?01 ?e?10)?
e?11
T1 ; (3.53)
which are known as the optical Bloch equations. Although there are four elements
of the density matrix specifled here, there can only be three independent equations,
77
as ? is Hermitian. The change of variables
u = 12 (e?01 + e?10) (3.54)
v = 12i (e?01 ?e?10) (3.55)
w = 12 (e?11 ?e?00) (3.56)
puts Eqs. (3.50) to (3.53) in a simple form. The time derivatives of Eqs. (3.54) to
(3.56) give the equations of motion in the new coordinates,
_u = ?v ? uT
2
(3.57)
_v = ??u??01w? vT
2
(3.58)
_w = ?01v ? 1T
1
 
w + 12
?
: (3.59)
These are equivalent to the classical Bloch equations that govern the motion of a
spin 1/2 system in a magnetic fleld (NMR). In Eq. (3.59), I have assumed that
e?00 + e?11 = 1 for all time. While dissipation and decoherence are non-unitary
processes, they are trace (i.e. population) preserving, as seen in Eqs. (3.50) to (3.53).
Several interesting phenomena can be found from the Bloch equations; see, for
example, x4.4 of Ref. [1] and Ref. [89]. In steady state (all time derivatives of Eqs.
(3.50) to (3.53) or Eqs. (3.57) to (3.59) set to zero), the excited state population is
e?eq11 = ?
2
01T1T2=2
1+(!rf ?!01)2 T22 +?201T1T2: (3.60)
Equation (3.18) shows that the Rabi frequency ?01 is proportional to the microwave
current I?w; thus, ?01 increases as the square root of the applied microwave power.
Using Eq. (3.60), e?eq11 is a Lorentzian resonance centered at !01 for a flxed power.
78
The full width at half maximum of the peak is
?! = 2T
2
q
1+?201T1T2: (3.61)
This shows that the resonance width will increase with ?01, an efiect known as
power broadening.
The solid lines in Fig. 3.3(a)-(c) show the excited state population as a function
of detuning, calculated with Eq. (3.60), for increasing microwave power. For low
powers, the width of the resonance is 2=T2, while the height of the peak increases
quadratically with Rabi frequency (linearly with power). The dashed curves of Fig.
3.3(a)-(c) show the resonances for a toy model, where these two trends hold at all
powers.
The on-resonance excited state population is plotted in Fig. 3.3(d). The solid
line is for the two-level system; it saturates at 0.5. The dashed is for the toy model;
in this case, the excited state population can be greater than 1 at high power, which
is unphysical.
At large detuning, the solid and dashed curves in Fig. 3.3(a)-(c) are nearly
identical. However, close to the resonance, the excited state population is limited to
0.5. This restriction pushes the full width out, as indicted by the horizontal arrows.
In Fig. 3.3(b), notice that both curves have nearly the same width at the excited
state population of 0.125. However, the toy system does not reach half maximum
until 0.25, where its width is still 2=T2. Thus, power broadening can be viewed as a
consequence of two-level saturation [5].
The time-dependent solution of the Bloch equations gives the form of a Rabi
oscillation. In the absence of dissipation and decoherence, the expressions for the
amplitude and frequency are the same as in Eqs. (3.13) and (3.12). With T1 and T2,
the oscillations decay in amplitude, eventually reaching steady state. On resonance,
79
(a) (b)
(c) (d)
? 11eq~ ? 11eq~
? 11eq~ ? 11eq~
Figure 3.3: Power broadening in a two-level system. The excited state population
e?eq11 is plotted against the angular frequency detuning !rf ?!01 (in units of 1=T2) for
?201T1T2 equal to (a) 0.01, (b) 1, and (c) 4. The solid lines are for a two-level system,
while the dashed are for a toy system that does not saturate; the horizontal arrows
indicate the full width ?! for each of the resonances. Panel (d) shows the excited
state population on resonance for the two-level (solid) and toy (dashed) systems as
a function of ?201 (in units of 1=T1T2).
80
the excited state population is
e?11 (t) = e?eq11 ?e?eq11 e?t=T0
"
cos??01t?+ sin
??
01t
?
T0?01
#
; (3.62)
where the equilibrium level e?eq11 is given by Eq. (3.60) (evaluated at !rf = !01), the
time constant of the decay envelope is
1
T0 =
1
2T1 +
1
2T2; (3.63)
and the efiective on-resonance Rabi frequency with dissipation is
?01 =
s
?201 ?
 1
2T1 ?
1
2T2
?2
: (3.64)
Tunneling can also be included in the optical Bloch equations in a simple way.
We have chosen to add a term ?(?i +?j)=2 to the right hand sides of _?ij in Eqs.
(3.42) to (3.45). For the diagonal element ?ii, this corresponds to a decay rate of
?i, which (at least for ?0) we can observe directly in experiments. In addition,
tunneling leads to a loss of phase coherence.
The Bloch equations can be solved in a similar way, resulting in a resonance
full width of [4]
?! = 2T0
2
q
1+?201 T01 T02; (3.65)
where the efiective relaxation and coherence times are deflned by 1=T01 ? 1=T1 +?1
and 1=T02 ? 1=T2 +(?0 +?1)=2.
In the limit of low power (?01 ?pT01T02) and dissipation-limited decoherence
(T2 = 2T1), Eq. (3.65) reduces to ?! = 1=T1 + ?0 + ?1. Thus, the full width is
the sum of the rates of all of the transitions that depopulate either of the levels. In
other words, the resonance width (in terms of angular frequency) is the inverse of
81
the lifetime, as one might expect. This can be generalized for the resonance between
levels jni and jmi, which has a full width
?! =
X
j
Wtjn +
X
j
Wtjm +?n +?m +2 I@!nm@I
b
; (3.66)
where Wtnm are the thermal rates given in x3.4 and ?n is the escape rate from jni;
see x3.2 and x7.2 of Ref. [3] and Refs. [38,90{92]. The flnal term in the sum gives the
approximate contribution from low frequency current noise with an rms value  I. If
there is a slow  uctuation in Ib, then the junction will be biased at a difierent value
on each trial, which leads to a smearing of all of the escape rate features (calculated
for an ensemble of systems). The assumption in Eq. (3.66) is that the noise enters as
a simple sum with the other broadening mechanisms. A more careful treatment that
takes the frequency dependence of the current noise into account involves modeling
the system with the stochastic Bloch equations [93].
The width of a resonance is often characterized by the spectroscopic coherence
time,
T?2 ? 2?!: (3.67)
As T?2 is bounded below by 2T1, it is a simple way to characterize the impact of
dephasing, tunneling, and inhomogeneous broadening.
These analytical solutions will be useful when attempting to extract T1 and
T2 from experimental data.
3.8 Multi-Level Density Matrix
In the regime that we usually operate in, our junction qubits must be described
by a model with at least three levels. In this case, analytic solutions of the Liouville-
von Neumann equation with dissipation and decoherence can be di?cult to obtain.
82
However, it fairly straight forward to write the equation in the form
d?
dt = P?; (3.68)
where, for a system with N levels, ? is a vector of N2 elements and P is an N2?N2
matrix that describes its evolution. ? can propagated forward a small time ?t as
?(t+?t) ? ?(t)+ d?(t)dt ?t (3.69)
? (I +P?t)? (3.70)
? eP?t?; (3.71)
which can be iterated numerically to flnd ?(t). Typically, Eq. (3.71) is stable while
Eq. (3.70) is not, so an e?cient algorithm for matrix exponentiation is required. To
perform simulations, I used the MATLAB function expm, which uses Pad?e approxi-
mation with scaling and squaring (see Ref. [94], method 3). The structure of P for
the simulations I performed is given in xD.1.
Incidentally, this integration naturally gives rise to the two-photon transitions
described in x3.3. This can be seen by taking the time derivative of Eq. (3.35),
which gives
@2?
@t2 = ?
i
~
?@H
@t ;?
?
? 1~2 [H;[H;?]]; (3.72)
which contains terms quadratic in H. Of course, the Hamiltonian itself must allow
these transitions, but no special provisions need to be made for them to appear in
the numerical solution.
83
Chapter 4
Qubit Design and Fabrication
This chapter contains basic information about the qubits I measured for this
thesis. I will begin with an overview of the fabrication process that was used to
make the chips, so it will be clear how various structures were made when I discuss
speciflc devices. I studied two types of qubit isolation, one using a simple LC fllter
and another using an inductive current divider that showed more complex behavior.
I will summarize the physics of each technique and give details about how they were
actually implemented. I will also give a brief outline of the experimental set-up
while each device was studied. A complete review of the equipment used will be
presented in the next chapter.
4.1 Hypres Fabrication Process
The three devices that I studied were fabricated by Hypres, Inc., in Elmsford,
New York [95]. Their multi-layer process made it easy to design a variety of struc-
tures. The process has a Nb/AlOx/Nb trilayer (with a critical current density of
either 30 or 100 A=cm2) and three superconducting wiring layers with two additional
metal layers. Our devices were made on oxidized silicon substrates.
Figure 4.1 shows a schematic of a chip cross-section and the circuit it forms.
Mask layers M0,1 M1, M2, and M3 are niobium. They are separated by sputtered
SiO2 insulation layers and can be connected by vias I0, I1B, and I2. The vias
have to be larger than a minimum size, but small enough that the wet etch process
that forms the opening stops where it was intended to. Mask layer I1A specifles
1Structures located on layer M0 actually specify where the ground plane is etched away. I will,
however, refer to M0 as the ground plane for convenience.
84
M0
M2
I0
I2
R2
I1B
M3
M1
R3
I1A
M3
I1A
R3
M2 / M1
R3
Figure 4.1: Hypres fabrication process. The LC-isolated circuit diagram at the top
could be implemented with a chip whose cross-section is shown (although the third
dimension is not shown). Gray and hatched areas represent metal; each layer is
labeled by its mask name. Open areas are SiO2 insulation layers labeled by the via
that can pass through them. The anodization layer that is currently part of the
Hypres process was not used when the devices I have studied were made.
where not to remove the counter electrode of the trilayer, thus leaving a junction
behind. R2 is molybdenum (used for resistive shunts in other applications) and R3
is a thick Ti/PdAu layer for contacts. While the thicknesses in the flgure are to
scale, the widths are highly compressed, which exaggerates the layout of the vias.
Nonetheless, it is important to keep in mind thickness variations as layers are added.
For example, R2 should not be deposited on top of any edges, as it is thin enough
to be become disconnected.
Figure 4.1 also shows how we implement some common structures. Some
of the devices used the lowest ground plane (M0), which provided some shield-
ing and reduced the number of wire bonds needed. Most capacitor plates were
85
formed by M1/M2 (0:208 fF=?m2), although M2/M3 (0:080 fF=?m2) and M1/M3
(0:058 fF=?m2) are also useable. Most wiring is located on M1 (which starts out as
the junction base electrode) and M2. Although we did have some lines deflned on
M3 (which has the highest critical current density), it is inadvisable to put impor-
tant structures on this layer, as it comes so late in the fabrication process.2 Wire
bonding pads were made on R3.
We used R2 or R3 to form quasiparticle traps [96]. When a niobium junction
switches to the voltage state, it is able to break Cooper pairs. The resulting quasi-
particles will be at an energy at least ?Nb above the Fermi surface. The idea of the
traps is to place a material in contact with a junction electrode that has available
density states at energies less than ?Nb. Once the quasiparticles \drop" into this
area, they do not have enough energy to come back to the junction, where they are a
source of dissipation. The hope is that by using molybdenum (which superconducts
at about 1 K and has ?Mo ? ?Nb=10), the quasiparticles not only get trapped, but
also recombine quickly. We did not performed separate tests to determine how well
the traps worked in practice.
4.2 LC-Isolated Phase Qubit
A major concern for quantum computing with superconducting devices is the
di?culty of excluding all noise in the microwave frequency range from the bias lines.
Even a small amount of noise power at the plasma frequency !p of the junction
could alter its quantum state. Figure 4.2(a) shows one technique we used to reduce
the bias noise. In the schematic, a junction with critical current I0, capacitance
CJ, and intrinsic resistance RJ is biased by current source Ib, which has a noise
component characterized by resistance Rbn. The LC circuit formed by inductor
Li and capacitor Ci protects the junction at high frequencies using non-dissipative
2We received many useful tips such as this from Anna Kidiyarova-Shevchenko.
86
CJ I0
Li
RJIb CiRbn
(a)
(b)
Xeff I0CJReffIeff
Figure 4.2: LC isolation of a current-biased junction. (a) A junction with critical
current I0, capacitance CJ, and intrinsic resistance RJ is biased by current source
Ib. The fllter formed by Li and Ci isolates the junction from the dissipative element
Rbn at high frequencies. (b) The circuit can be viewed in an equivalent way, where
the junction is driven by an efiective current source Ieff and shunted by Reff and
Xeff, which are purely resistive and reactive. I have chosen to represent Xeff with
an inductor, but its value can be of either sign.
elements. A complete discussion of this approach with more realistic models can be
found in Ref. [97], Chapter 7 of Ref. [3], and x5.2 of Ref. [1].
One point of view is to treat the LC circuit as a current divider, as a high
frequency signal from the current source will get shunted to ground before reaching
the junction, which is thought of as the load. For this to work, the cut-ofi frequency
of the fllter 1=pLiCi must be set well below !p.
An alternate way to characterize the isolation of a junction circuit is shown in
Fig. 4.2(b). Here the biasing circuitry and junction resistance have been replaced
by the parallel combination of an efiective shunting resistance Reff (!) and shunting
reactance Xeff (!). In addition, Ib has been replaced by an efiective source Ieff. These
efiective quantities are, in general, functions of frequency !. In this picture, the
junction is treated as a source that drives Reff and Xeff. The ability of the junction
87
to dissipate energy is due to the resistance and quantifled by the relaxation time
of the system (introduced in x3.4), which is roughly T1 = ReffCJ. The expression
suggests that a large CJ is desirable, which is one reason that our qubit junctions
have relatively large 100 ?m2 areas. Recently, concerns about the quality of oxide
barriers have brought this approach into question [98]. In particular, if Reff is
dominated by intrinsic dissipation in the junction caused by dielectric loss, then
Reff ? RJ will increase with the junction area; in this case T1 = ReffCJ would be
independent of the area.
IfacomplexadmittanceY (!)appearsacrossthejunction, thenReff = 1=Re(Y)
and Xeff = ?1=Im(Y).3 If Xeff > 0, then the efiective reactance is most simply
thought of as an inductor with a value of Leff = Xeff=!. This element appears in
parallel with the Josephson inductance of the junction. Leff has a small efiect on the
junction resonance if Li is chosen to be large (see x6.2 of Ref. [3]). If Xeff < 0, then
it can be thought of a capacitor with Ceff = ?1=!Xeff. In this case, the relaxation
time becomes T1 = Reff (CJ +Ceff).
Without the LC network, 1=Reff = 1=Rbn + 1=RJ. Determining Reff in a real
device is not straightforward. We expect Rbn ? 50 ? at high frequencies, due to
coaxial lines and the output impedance of the function generator that is responsible
for Ib. It could however be quite difierent, as the network of fllters close to the
junction is somewhat complicated. Nonetheless, assuming Rbn = 50 ? and taking
CJ = 5 pF, T1 is predicted to be 250 ps in the absence of any isolation. The relevant
time scale for quantum computation is the inverse of the plasma frequency, which
is roughly 200 ps for our usual operating conditions. Thus, some form of flltering is
required to make the qubit useable.
The function of the LC isolation network is to boost the impedance. The
3If the total shunting impedance is Z, then according to Fig. 4.2(b), Re(Z) =
X2effReff=
?
X2eff +R2eff
?
6= Reff. Re(Z) would be the correct quantity to consider if Reff and
Xeff appeared in series in the dissipation model.
88
shunting admittance across I0 and CJ in Fig. 4.2(a) is
Y (!) =
?
i!Li + 11=R
bn +i!Ci
??1
+ 1R
J
: (4.1)
Neglecting RJ, which is probably not a good approximation, the efiective parallel
resistance and reactance are
Reff (!) = !
2L2
i
Rbn +Rbn
?1?!2L
iCi
?2 (4.2)
Xeff (!) = !
2L2
i +R
2
bn (1?!
2LiCi)2
!Li ?!R2bnCi (1?!2LiCi): (4.3)
Figure 4.3 shows Reff and Ceff for the LC isolation circuit used with device LC2
(described in the next section). With Li = 8:2 nH and Ci = 84 pF, the resonance
frequency (in the absence of any other resistance) is 190 MHz. In Fig. 4.3(a),
the efiective resistances for Rbn = 50 ? (solid) and 15 ? (dashed) are plotted as a
function of frequency. Typically, our qubits are biased such that !p=2? ? 5 GHz. In
this frequency range, Reff > 1 M?, which should result in T1 > 5 ?s for CJ = 5 pF,
againassumingRJ !1. AsFig.4.3(b) shows, theimpedancetransformation factor
Reff=Rbn is dependent on Rbn only near the LC resonance. This quantity may also
be regarded as the factor by which noise power on the bias line is attenuated. From
this point of view (where Ib is the source), the loading of the fllter network by the
junction itself is ignored.
Below the fllter resonance, Xeff looks capacitive. These values are plotted
against the left axis of Fig. 4.3(c). For the larger value of Rbn, Ceff ? Ci at low
frequency. The value is greatly reduced as Rbn decreases; when Rbn < pLi=Ci ?
10 ?, the reactance does not look capacitive at any frequency. Above resonance, the
efiective inductance is plotted (right axis). In both cases Leff ? Li at high frequency.
Figure 4.3 implies that the LC fllter should adequately protect the junction
89
(a)
(b)
(c)
Figure 4.3: Efiective impedances of an LC isolation network. The theoretical prop-
erties of a fllter with Li = 8:2 nH and Ci = 84 pF are plotted as a function of
frequency, for Rbn = 50 ? (solid) and 15 ? (dashed). (a) The efiective resistance
Reff is greater than 1 M? for !=2? > 2 GHz. (b) The normalized resistance Reff=Rbn
is the same for both cases, except for frequencies near the LC resonance. (c) For
frequencies less (greater) than the fllter resonance, the capacitance Ceff (inductance
Leff) is plotted with respect to the left (right) axis. For all plots, the intrinsic
resistance of the qubit junction was ignored.
90
at its plasma frequency of a few GHz. However, serious problems occur at lower
frequencies. In particular, the network has no efiect on Reff at dc, allowing any low
frequency noise to reach the junction without attenuation. This frequency range
must be addressed with another flltering scheme. Also, at the LC resonance, the
isolation fails dramatically and Reff . Rbn. It is possible that current noise in this
frequency range causes inhomogeneous broadening in qubits using LC isolation (see
x6.1 of Ref. [3] and x5.2.3 of Ref. [1]).
4.2.1 Device LC2
Figure 4.4 shows a photograph of two capacitively coupled LC-isolated phase
qubits, a device which I will refer to as LC2.4 The chip was made using design
\ajblc13" and was taken from Hypres Mask 297, Lot 43112, Wafer KL556, which
was fabricated with their 100 A=cm2 process. The two qubits, denoted LC2A and
LC2B, are (nearly) mirror images of each other. I will use a superscript A and B to
label the various quantities for each qubit, as in Fig. 2.18. For example, the qubits
were controlled with two independent current biases IAb and IBb .
Table 4.1 lists important parameters for each qubit of LC2. The top plate
of the isolation capacitor Ci was formed by molybdenum layer R2 to increase the
capacitance per unit area (and not because we wanted a quasiparticle trap). Taking
advantage of the multi-layer process, the isolation inductor is formed by a 122 ?m?
122 ?m square spiral of 7.25 turns. The lines are 2 ?m wide and separated by 2 ?m.5
I estimated the inductance Li as 8.2 nH using Ref. [100]. The spiral is expected to
self-resonate at a relatively high frequency. A large fraction of the capacitance that
shunts Li comes from the return path from inside the spiral; see x6.5 of Ref. [3] and
4The same device is called LCJJ-Nb2 in Ref. [1] and hypres2 in Ref. [3]. Results from this
device are given in Refs. [41,77,78,99].
5I believe that the current minimum allowed spacing on layer M1 of 2:5 ?m was established
after this device was fabricated.
91
(a)
(b)
(c)
Ci
Li
CC
I0 , CJ
100 ?m
50 ?m
10 ?m
Figure 4.4: Photographs of device LC2. (a) Two LC-isolated junctions are coupled
together with two capacitors. The series combination of the two capacitors give the
coupling capacitance CC. The ground plane has been removed from the entire chip.
Remnants of wire bonds are seen on the four pads at the bottom. Close-up views of
(b) the right qubit and (c) its junction show more details. The scale bars are based
on the CAD drawings.
92
Table 4.1: Parameters of device LC2. The two coupled qubits of this device have
nominally identical properties. The isolation of the qubit is provided by Ci and Li.
The qubit junction has capacitance and critical current CJ and I0. Coupling of the
qubits is achieved by LC and CC (the flnal row refers to each of the capacitors).
The third column specifles the layers used in fabrication; commas separate layers
that are in electrical contact (or vias between metal layers), while a slash indicates
a capacitance. Values in the second and fourth columns are based on drawings; the
flfth shows extracted values from experimental results at 20 mK, as described in the
text.
Element Size (?m??m) Layers Design Value Expt. Value
Bond Pad 280?280 R3, M3
Ci 450?450 M2, I1B, R2 / M1 84 pF
Li 122?122 M1 8.2 nH
CJ 10?10 M2, I1B, I1A / M1 4.15 pF 4.85 pF
I0 10?10 I1A / M1 97 ?A 130 ?A
LC 780?90 M1 and M2 2.6 nH 1.7 nH
2CC 60?62 M2 / M1 770 fF 660 fF
x5.2.2 of Ref. [1]. I estimate an overlap area of 170 ?m2, which gives a capacitance
of 35 fF and a self-resonance frequency near 10 GHz. I did not have a simple way
of experimentally measuring either Ci or Li.
The qubit junction is 10 ?m ? 10 ?m, which is the size used for all of the
qubits I studied. By including the \missing area" estimated by Hypres, the design
value of the critical current is 97 ?A at 4.2 K. In contrast, the measured difiraction
patterns of Fig. 6.3 show that both IA0 and IB0 are 130 ?A at 20 mK in the absence
of a suppression fleld.
The design value of the junction capacitance CJ is 4:15 pF; most of this comes
from the junction area itself, but the overlap area of layers M1 and M2 does make a
small contribution. The coupling of the qubits is due to two capacitors, each of which
is designed to be 770 fF. The total coupling capacitance CC is half of this value,
93
because the two capacitors are in series. As the qubits are nearly 1 mm apart, the
inductance due to the lines attaching the capacitors is signiflcant. The inductance of
a 2 ?m trace, located well above the grounded sample box, is roughly 1:5 pH=?m;6
the coupling lines form a 780 ?m?90 ?m loop, resulting in LC ? 2:6 nH.
From Eq. (2.76), the frequency of the LC mode created by the coupling el-
ements is !C=2? = 5:5 GHz. Using Eqs. (2.77) and (2.78), we expect that the
coupling strength varies from ?=p2 = 0:20 at the triple degeneracy to ?0 = 0:085
when the plasma frequency of the junctions is well below !C. Experimental val-
ues for CJ, CC, and LC were extracted by fltting spectra when the qubits were
brought in resonance with each other [78]. Using these values, !C=2? = 7:2 GHz,
?=p2 = 0:17, and ?0 = 0:064.
We studied the same device during three runs of the refrigerator, designated 40
(January to November 2003), 41 (November to December 2003), and 42 (December
2003 to April 2004).
During Run 40, a single line went from the junction to the top of the refriger-
ator, which was used to supply Ib and measure the junction voltage. This two-wire
measurement made detecting the running state a bit challenging, as the voltage
drop across the lossy manganin line was signiflcant [as seen in the IV curve of Fig.
6.2(a)]. Detection was performed by high pass flltering the voltage and looking for
a step. Starting with Run 41, a separate voltage line was added, as described in the
next chapter. Although not essential, we high pass flltered the signal on this new
line for detection.
In addition, beginning with Run 41, two changes were made to greatly improve
the timing resolution of the escape rate measurement (see x6.2). The capacitance of
the discrete LC fllter at the mixing chamber was reduced to decrease the charging
time of the voltage line. Also, the voltage ampliflers and their power supplies were
6I used the calculator available at http://www.emclab.umr.edu/pcbtlc/microstrip.html to esti-
mate this value.
94
upgraded to the ones described in x5.4.
When I refer to a measurement on junction LC2A, the IBb lead was generally
shorted to the refrigerator common after its bias resistor (at the top of the refrig-
erator). With no current through LC2B, its plasma frequency was usually greater
than 20 GHz (depending on its critical current), well out of resonance with LC2A.7
With this arrangement, it is possible that LC2B could escape to the voltage state
at some time and sit in the sub-gap region (see x4.1.2 of Ref. [3]). We did a few
tests where LC2B was reset on every trial and never saw an efiect.
4.3 dc SQUID Phase Qubit
Viewed as a current divider, the LC circuit described in the previous section
is a simple way to isolate a junction with non-dissipative elements. The perfor-
mance could be improved signiflcantly by replacing the capacitor with a small su-
perconducting inductor, which would remove the ofiending resonance and provide
broadband isolation down to zero frequency. However, our usual measurement tech-
nique involves monitoring the bias line to determine when the junction tunnels to
the voltage state. During a switch, most of this voltage would appear across Li; at
dc, the bias line would be shorted out by the new inductor. Nonetheless, the re-
sulting structure resembles an rf SQUID, which has been used in a variety of qubit
designs [33].
In the circuit shown in Fig. 4.5, the qubit junction (JJ1) has critical current I01,
capacitance CJ1, and resistance RJ1. Compared to the LC isolation technique, here
Ci has been replaced with an auxiliary Josephson junction (JJ2) with parameters
I02, CJ2, and RJ2, as flrst proposed in Ref. [39]. When this isolation junction is
in the zero-voltage state, its Josephson inductance LJ2 (see x2.1) acts with L1 to
7The presence of the coupling capacitor still had a signiflcant impact, particulary with the
spectrum of excitations; see Ref. [78] and x8.1.
95
I1 I2
J
?A I
b
If
CJ1 CJ2LJ2I01
L1
Mf
Mb
RJ1
L2
RJ2
Rbn
Rfn
C?w
I?w
Figure 4.5: dc SQUID phase qubit. The junction on the left (JJ1) with critical
current I01, capacitance CJ1, and resistance RJ1 is thought of as a phase qubit. The
junction on the right (JJ2) with parameters I02, CJ2, and RJ2 forms an isolation
network with inductances L1 and L2. To emphasize this role, I have replaced the
ideal junction with its Josephson inductance LJ2. The qubit can be biased with
current Ib and  ux 'A = IfMf (and a smaller contribution IbMb). The total currents
on each of the SQUID arms are I1 and I2; the circulating current is J. The isolation
protects the device from noise characterized by resistances Rbn and Rfn. Energy
level transitions are driven with the microwave current I?w, which couples to the
qubit through C?w.
form a frequency-independent current divider. I replaced the usual junction symbol
by an inductor to suggest this function. When the qubit junction tunnels to the
flnite-voltage state, the flnite impedance on its arm causes current to  ow through
the isolation junction, making it switch as well if we choose I02 < I01. Therefore,
after the qubit junction switches the current bias line will be held at the gap voltage
of the superconductor,8 which can be measured as usual.
There are a few complications to this approach. When the current bias Ib
increases, most of the current will get shunted to ground (just as we want for any
8In Ref. [39], a 50 ? shunting resistor forced the junction to stay in the sub-gap region, to
minimize quasiparticle generation.
96
current noise), causing the isolation junction to switch flrst. However, if all of the
connections are superconducting, then the resulting structure is just an asymmetric
dc SQUID, which was described in x2.4. It can be controlled with an independent
 ux bias 'A = MfIf, where the current If generates a  ux through mutual in-
ductance Mf. If both Ib and If are ramped in the proper proportion (see x6.4),
then only the qubit is biased, resulting in a device that behaves in many ways as
a single isolated current-biased junction. Details of this approach can be found in
Refs. [39,72,101,102] and x5.4 of Ref. [1].
The efiect of the dissipative element Rbn can be found with the model in Fig.
4.2(b). In fact, the circuits shown in Figs. 4.2(a) and 4.5 have nearly the same
structure [1]. In the SQUID circuit, Li is replaced by L1. The other branch of
the divider consists of L2 and the parallel combination of LJ2 and CJ2. The total
impedance is purely reactive and may be thought of as a capacitance (although it
could be negative). This leads to the substitution
1
Ci ! ?!
2
 
L2 + LJ21?!2L
J2CJ2
?
??!2 (L2 +LJ2); (4.4)
where the approximation holds for ! well below the plasma frequency of the isolation
junction, 1=pLJ2CJ2. In Eq. (4.4), I have ignored RJ2 and the mutual inductance
Mb from the bias line to the SQUID, for simplicity. In the low frequency limit, the
efiective shunting resistance is given by Eq. (4.2) as
Reff = !
2L2
1
Rbn +Rbn
 L
1 +L2 +LJ2
L2 +LJ2
?2
? rRbn; (4.5)
where I have ignored RJ1 and r = [(L1 +L2 +LJ2)=(L2 +LJ2)]2 is known as the
isolation factor [101]. The flrst term in the middle expression is typically small
enough to ignore.
From Eq. (4.5), we see that to provide good isolation, L1=(L2 +LJ2) should be
97
large. This will lead to a large modulation parameter fl [deflned in Eq. (2.48)] and
many possible  ux states, as described in x2.5. This complication can be addressed
with the  ux shaking procedure to be discussed in x6.5. The distinct beneflt of a
large inductance is that the coupling between the junctions becomes small [72]. For
this reason, we can think of one SQUID junction as the qubit and the other as simply
providing isolation. To further weaken the coupling, we bias the SQUID with Ib and
If in such a way that the current I2 on the isolation branch is small. In this case, LJ2
is held near its minimum value of '0=2?I02. Thus, a large I02 is also desirable, but to
use our standard measurement technique, we typically require I02 < I01. However,
switching may not always occur even with this choice of parameters. In the devices
I studied, I01 > 2I02, so that the qubit branch of the current- ux characteristic (see
x2.6) extends over a relatively large range of Ib. On the other hand, for large I02, it
can happen that when the  ux state corresponding to zero trapped  ux (N' = 0)
becomes unstable, the current bias will not be large enough to force the entire system
to the flnite voltage state. In this case, our simple detection scheme would not work
and an alternate method would need to be used [4].
Equation (4.5) shows that it is possible to change the level of isolation from
the bias leads provided by JJ2 by varying the Josephson inductance LJ2 [101,102].
This can be accomplished by adjusting the isolation junction current I2. The ability
to vary the isolation turns out to be quite useful in investigating the source of
decoherence in these devices and placing bounds on Rbn. I will show some results
of experiments on varying r in x7.2.2.
Figure 4.6 shows the real and imaginary components of the efiective parallel
shunting impedances of a dc SQUID phase qubit. For these plots, I used Eqs.
(4.2), (4.3), and (4.4), with Rbn = 50 ? (solid lines) and 15 ? (dashed). The
chosen parameters (L1 = 3:52 nH, L2 = 25 pH, CJ2 = 2:09 pF, I02 = 4:4 ?A
with I2 = 0 so that LJ2 = 75 pH) describe SQUID DS1 under suppression fleld
98
(a)
(b)
(c)
Figure 4.6: Isolation of a dc SQUID phase qubit from its bias line. The properties of
a fllter created by inductors L1 = 3:52 nH and L2 = 25 pH and an unbiased junction
(with I02 = 4:4 ?A, CJ2 = 2:09 pF) are plotted as a function of frequency, for Rbn =
50 ? (solid) and 15 ? (dashed). (a) The efiective resistance Reff is uniformly large
below typical values of the qubit plasma frequency. (b) The normalized resistance
Reff=Rbn is the same for both cases, except near LC resonance frequencies. (c) The
efiective shunting reactance is always inductive; Leff is dominated by L1 (dotted
line). For all plots, the intrinsic resistance of the qubit junction was ignored.
99
#2 (see Table 6.1), which are the conditions under which most of the data on this
device were taken. Reff is nearly constant below 10 GHz, which protects the qubit
at its plasma frequency and attenuates low frequency noise. For Rbn = 50 ? and
15 ?, Reff = 65 k? and 20 k?, leading to predictions for T1 of 270 ns and 80 ns,
respectively, for CJ1 = 4:17 pF. These values of T1 are quite sensitive to L2, which
is di?cult to measure accurately. While relaxation times of this order would make a
wide variety of experiments possible (on the assumption that the decoherence time
is not signiflcantly shorter), considerably longer times will ultimately be required
for quantum computation.
Examination of Fig. 4.6(a) shows that Reff has strong resonant features, just
as with the LC isolation network. However, these can be designed to occur at very
high frequencies. In this case, there is a dip at the plasma frequency of the isolation
junction 1=2?pLJ2CJ2 = 13 GHz and a spike at 1=2?pL0CJ2 = 22 GHz, where L0
is the parallel combination of LJ2 and L2. The dip in the isolation could cause a
problem, if this frequency is resonant with a higher order transition, such as 0 ! 2.
The value of Reff at its minimum and the slight upturn at high frequencies is due
to the flrst term in Eq. (4.2).
The efiective inductance Leff, shown in Fig. 4.6(c), is mostly due to L1 (drawn
as a dotted line). Below its plasma frequency, the isolation junction makes an
inductance contribution, while it looks capacitive at higher frequencies. Ironically,
the isolation is most efiective at these higher frequencies, where the isolation network
resembles an LC fllter.
The qubit can also dissipate energy through the resistance Rfn, associated
with the  ux line [1,39]. The shunting impedance due to Rfn can be found using
the equivalent circuit shown in Fig. 4.7, where the junction has been replaced by a
voltage source V1 (and the efiective impedance Rbn on the current bias line is taken
to be inflnite). The circulating current J in the SQUID loop induces a current Jf
100
J
V1
L1
Mf 1  
CJ2
L2
Rfn
LJ2
Jf
Lf
Mf 2  
Figure 4.7: Flux line coupling to SQUID. The shunting impedance from the  ux
noise resistance Rfn can be found by considering the circulating current J due a
voltage source V1 that replaces the qubit junction. J induces a current Jf that
 ows through Rfn and the  ux line inductance Lf. Mf1 and Mf2 are the mutual
inductances between Lf and the SQUID geometrical inductances L1 and L2.
in the loop formed by the  ux line inductance Lf and Rfn. Kirchhofi?s voltage law
at frequency ! for the two loops gives (see, for example, x11.5 of Ref. [103])
V1 =
 
L1dJdt +Mf1dJfdt
?
+
 
L2dJdt +Mf2dJfdt
?
+ i!LJ21?!2L
J2CJ2
J (4.6)
0 =
 
Lf dJfdt +Mf1dJdt +Mf2dJdt
?
+JfRfn; (4.7)
where Mf1 (Mf2) is the mutual inductance between Lf and L1 (L2). The Josephson
inductance of the isolation junction does not store magnetic energy and only enters
as a simple impedance assuming all signals are small. Assuming that the currents
have a time dependence ei!t and that Mf = Mf1 + Mf2, the junction shunting
admittance is [1]
Y (!) = JV
1
=
?
i!Ltot + !
2M2
f
Rfn +i!Lf
??1
; (4.8)
101
where
Ltot = L1 +L2 + LJ21?!2L
J2CJ2
: (4.9)
From Eq. (4.8), the efiective parallel shunting resistance and reactance due to
the  ux line are
RMeff = 1Re(Y) = Rfn
 L
tot
Mf
?2
+ !
2?M2
f ?LtotLf
?2
M2fRfn ? Rfn
 L
tot
Mf
?2
(4.10)
XMeff = ? 1Im(Y) = !L
2
tot R
2
fn +!
3?M2
f ?LtotLf
?2
Ltot R2fn ?!2Lf ?M2f ?LtotLf? ? !Ltot; (4.11)
where the approximations hold for perfect coupling between the inductors, when
Mf = pLtotLf.
For good isolation, then, Mf should be chosen to be much smaller than Ltot.
However, if it is too small, then the required value of If could be so large as to cause
heating at the mixing chamber of the dilution refrigerator. With the simultaneous
bias trajectory shown in Fig. 6.7(d) and discussed in x6.4, the maximum  ux current
we need to apply is roughly (L1=Mf)(I01 +I02).
The efiective impedances for SQUID DS1, calculated with Eq. (4.9) and the
approximations in Eqs. (4.10) and (4.11), are plotted in Fig. 4.8. I have assumed
the inductors are perfectly coupled for simplicity, but this is clearly untrue. From
experimental measurements, Ltot is larger than 3.5 nH. If this were perfectly coupled
to Lf with a resulting mutual inductance Mf = 51 pH (also from measurements),
then Lf would be less than 1 pH. Given the length of the  ux line (shown in Fig.
4.9), this prediction is far too small. Nonetheless, the approximation in Eq. (4.10)
is a lower bound to the full expression, so the simpliflcation is a useful one.
Near !=2? = 5 GHz, the efiective shunting resistance due to the  ux line is
RMeff = 250 k? and 75 k? for Rfn = 50 ? and 15 ?, respectively. With a qubit
junction capacitance of CJ1 = 4:17 pF, T1 is roughly 1000 and 310 ns for the two
102
(a)
(b)
(c)
Figure 4.8: Isolation of a dc SQUID phase qubit from it  ux line. The properties
of the coupling between a SQUID (with L1 = 3:52 nH, L2 = 25 pH, I02 = 4:4 ?A,
I2 = 0, CJ2 = 2:09 pF) and noise resistances Rfn = 50 ? (solid) and 15 ? (dashed)
by Mf = 51 pH are plotted as a function of frequency. (a) The efiective resistance
RMeff is constant, except near the plasma frequency of the isolation junction. (b) The
normalized resistance RMeff=Rfn is the same for both cases and is exactly zero at a
single frequency. (c) The efiective shunting reactance is dominated by L1. For all
plots, the intrinsic resistance of the qubit junction was ignored.
103
cases, if the  ux line is the only source of dissipation. As discussed earlier, the
efiective shunting resistance due to the current bias line is Reff = 65 k? and 20 k?
for Rbn = 50 ? and 15 ?. The total shunting resistance is given roughly by the
parallel combination of Reff and RMeff, which is dominated by the current bias line
in this case (assuming comparable Rbn and Rfn). For both the current and  ux
lines, the isolation fails at the plasma frequency of the isolation junction. This is
potentially more serious for the  ux line, because (in the simplifled model I have
used) RMeff goes to zero near this frequency.
4.3.1 Device DS1
A single dc SQUID phase qubit, which I will refer to as device DS1,9 is shown
in Fig. 4.9. This device is located on chip design \ajblc13" and was fabricated at
Hypres during the same run as device LC2, although it was on a physically difierent
chip.
The two control currents, Ib and If, were returned on the ground plane M0,
which is the light area in the middle of the photograph of Fig. 4.9(a). The bonding
pad connected to M0 was then connected to the sample box, which was electrically
connected to the main body of the refrigerator.
The applied currents If and Ib were used to  ux-bias the SQUID loop through
the mutual inductances Mf and Mb. I estimated these values by assuming that
the lines were one dimensional and neglecting the efiect of the ground plane. The
mutual inductance M is given by the Neumann formula (see, for example, x11.4 of
Ref. [103]) as
M = ?04?
I I dl
1 ?dl2
r ; (4.12)
where a line integral is taken around each of the structures being considered and r
9The same device is called LJJJ-Nb in Ref. [1]. Results from this device are given in Refs.
[72,101,102].
104
(a)
(b)
I01 , CJ1I02 , CJ2
C?w
L1
If Ib I?w GND
N/C
100 ?m
30 ?m
Figure4.9: PhotographsofSQUIDDS1. (a)ThedcSQUIDphasequbitiscontrolled
by three current lines, each of which is returned on a common ground plane, which
appears as a light area. Four bonding pads are labeled with the signal line that
they were wired to. (b) A close-up view shows the qubit junction (10 ?m?10 ?m,
with critical current I01), the isolation junction (7 ?m?7 ?m, I02), the 6-turn loop
forming L1, the  ux line which carries If, and the microwave coupling capacitor C?w.
These photographs show a device nominally identical to the one actually studied.
105
Table 4.2: Device parameters of SQUID DS1. The capacitance and critical current
of the isolation junction are CJ2 and I02, while those for the qubit are CJ1 and I01.
The geometrical inductances of the SQUID arms are L2 and L1. Microwaves are
coupled to the qubit through C?w. The experimentally measured quantities come
from Table 6.1 and microwave spectroscopy.
Element Size (?m??m) Layers Design Value Expt. Value
Bond Pad 280?280 R3, M3
L2 40 ?m M1 and M2 < 40 pF < 30 pF
CJ2 7?7 M2, I1B, I1A / M1 2.09 pF
I02 7?7 I1A / M1 46 ?A 51.7 ?A
L1 84?84 M1 3.3 nH 3.5 nH
CJ1 10?10 M2, I1B, I1A / M1 4.17 pF 4.43 pF
I01 10?10 I1A / M1 97 ?A 107.9 ?A
C?w 4?4 M3 / M1 0.9 fF
is the distance between the two difierential elements. In practice, I evaluated the
integral numerically with the MATLAB function dblquad along paths that started
and ended at bonding pads or that traced around the SQUID loop.
The return path of If through the ground plane has a signiflcant impact on
the value of Mf. Under the assumption that current  ows from the via of the  ux
line [shown in the lower left corner of Fig. 4.9(b)] in a straight line to the via of
the ground connection [shown in the lower right of Fig. 4.9(a)], Eq. (4.12) returns
Mf = 53:8 pH. Less than half of this comes from the intended path on layer M1.
We measured a value of 51 pH from current- ux characteristics (see Table 6.1), so
the simple ground path appears to be a good approximation. With reference to the
photograph, a positive If (one that  ows from bonding pad ?If? to ?GND?) generates
a  ux out of the plane of the page at L1. The SQUID responds with a circulating
current that spirals outwards and adds to the current through the qubit. This is just
what a positive applied  ux 'A does, based on the sign conventions shown in the
106
circuit diagrams of Figs. 2.12 and 4.5. That Mf is positive conveys that a positive
If generates a positive 'A.
The current bias Ib also applies a small amount of  ux to the SQUID. The
current  ows on a line on layer M1 that leads to the SQUID and returns on the
ground plane. By assuming a direct via to via return path, Mb is about 16 pH,
although the chosen boundary between the bias line and the SQUID loop (which
are electrically continuous) afiects this value.
Table 4.2 lists key parameters for this device. The qubit junction (10 ?m ?
10 ?m) is the same size as in device LC2. The isolation junction was chosen to
be roughly half its area. However, we almost always operated this device in a
suppression fleld Bk (and the two junctions present a difierent cross-sectional area
to the fleld), so this ratio was not maintained for the bulk of the measurements I
will discuss. The experimental values for the critical currents are from Table 6.1.
I estimated the qubit junction capacitance CJ2 from microwave spectroscopy. It is
di?cult to do the same for the isolation junction, because it has a lower quality
factor Q.
We designed DS1 to have a large inductance L1 on the qubit arm of the
dc SQUID. This was accomplished with a 6-turn spiral with 2 ?m line width and
spacing. Using Ref. [100], the prediction is L1 = 3:3 nH, in good agreement with the
value of 3.5 nH obtained experimentally by fltting the current- ux characteristics.
Actually, only L1?Mb can be measured with this technique, but L1 dominates this
difierence. In this device, the crossover capacitance is much smaller than it is for the
spirals of LC2. With an area of 20 ?m2, the shunting capacitance is roughly 4 pF
for a self-resonance above 40 GHz. In this case, the resonance may be determined
by other factors, such as the total length of the coil.
The inductance L2 is due to the stray inductance on the isolation arm and is
di?cult to predict and measure. Given that the Ib feed and return lines are separated
107
by 40 ?m, L2 is likely to be of order 40 pH. From Table 6.1, the experimental value
of L2 + Mb varies from -5.2 to 32.6 pH for difierent values of Bk. In this case, Mb
could be making a signiflcant contribution, particularly if the value is negative. In
many of the devices studied in the group at present, Ib and its return are located
near each other on parallel paths to minimize both L2 and Mb.
In order to resonantly excite the junction, we applied a microwave current I?w
to the qubit junction through a small capacitor C?w [see the circuit diagram of Fig.
4.5 and the lower right of the photograph of Fig. 4.9(b)], designed to be about 1 fF.
We were not careful in providing an impedance matched and shielded path for the
microwaves, so it is very likely that the capacitor was not solely responsible for the
coupling strength of I?w to the junction.
We studied device DS1 during Run 43 of the refrigerator (April 2004 to De-
cember 2004).10 As the  ux line was required to carry up to 15 mA, we installed a
new superconducting line for If. In addition, the previous copper sample box was
replaced with an aluminum one, to provide magnetic shielding to the device. More
details on these topics are given in the next chapter.
4.3.2 Device DS2
The flnal device I will discuss, DS2, consists of two dc SQUID phase qubits
coupled by a capacitor.11 It is located on chip design \umdqc04," taken from Hypres
Mask 308, Wafer KL700, which was fabricated with their 30 A=cm2 process. Pho-
tographs of a nominally identical device made during the same run are shown in
Fig. 4.10. As with LC2, I will label properties of the individual SQUIDs (DS2A and
DS2B) with a superscript A or B.
As comparison of Figs. 4.9 and 4.10 shows, each SQUID in DS2 is geometrically
10Two resonantly isolated devices (see x9.2 of Ref. [3] and x5.3 of Ref. [1]) were wired for study
at the same time.
11Results from this device are given in Refs. [83,104]
108
(a)
(b)
L1B
IbA
IbB
GND
IfB
I?wB
I?wA
IfA
CC C?w
CJ2BI02B ,
CJ1BI01B ,
200 ?m
30 ?m
Figure 4.10: Photographs of coupled SQUIDs DS2. (a) Two dc SQUIDs are coupled
by a single capacitor and share a common ground plane. (b) Each SQUID is similar
to DS1, with the exception of a quasiparticle trap located between the junctions.
These photographs are of a device nominally identical to the one actually studied.
109
Table 4.3: Parameters of device DS2. Each of the two coupled qubits has the same
design values, as listed below. The size of all of structure are the same as DS1,
however the junction properties are difierent because of the lower 30 A=cm2 process
used in the fabrication. For the critical currents I01 and I02, the two experimental
values listed are for DS2A and DS2B. Coupling of the devices is achieved by the
single capacitor CC and stray inductance LC.
Element Size (?m??m) Layers Design Value Expt. Value
Bond Pad 280?280 R3, M3
L2 40 ?m M1 and M2 < 40 pF < 30 pF
CJ2 7?7 M2, I1B, I1A / M1 1.92 pF
I02 7?7 I1A / M1 13.8 ?A 3, 6 ?A
L1 84?84 M1 3.3 nH 3.4 nH
CJ1 10?10 M2, I1B, I1A / M1 3.82 pF 4.4 pF
I01 10?10 I1A / M1 29.1 ?A 24, 20 ?A
C?w 4?4 M3 / M1 0.9 fF
LC 280 ?m M0, M1, and M2 ? 200 pH
CC 30?30 M2 / M1 190 fF 130 fF
QP Trap 56?20 R3, M3, I2, M2
identical to device DS1 (although DS2B is its mirror image). However, because of
the lower critical current density, I01 and I02 are much smaller. The capacitances of
the junctions are also slightly smaller in DS2. Because the plasma frequency of the
junctions, when biased so that tunneling to the voltage state was measurable, was
roughly 6 GHz, we never applied a suppression fleld when studying this device. For
the experimental values of the critical currents, the two values listed are for DS2A
and DS2B, based on IV curves and current- ux characteristics. It is odd that they
difier from the design values by such a large amount, particularly in the case of the
isolation junctions. In addition, these values changed several times over the course
of data taking, by as much as 2 ?A in the case of DS2B. Some of these events were
operator-induced, but the critical currents often appeared to change slowly over
110
time or after helium transfers. One possibility is that there was a vortex trapped
in the fllms near the junction. However, as all the critical currents were low, this is
unlikely.
The base electrode of each of the qubits is coupled by a 190 fF capacitor CC.
The counter electrodes are connected by the ground plane. As the lines leading
to capacitor are directly above the ground plane, their self-inductance is likely to
be small. The coupling inductance LC is probably dominated by the ground plane
connection. The grounding vias for the two SQUIDs are separated by 280 ?m. As
the current path between SQUIDs could be quite wide, the total LC is probably of
order 200-300 pH. The frequency of the LC coupling mode is !C=2? = 27 GHz,
using Eq. (2.76) with the design values. This is higher than the zero-bias plasma
frequency of the qubits. If the qubit could be brought into resonance with this mode,
Eq. (2.77) gives a coupling strength of ?=p2 = 0:15. If (as usual) the qubits are in
resonance well below !C, the coupling is reduced to ?0 = 0:047, calculated with Eq.
(2.78). The experimental values of CJ and CC come from a flt to the spectrum of
the coupled system, shown in x8.6.
The ground plane was extended to run underneath the junctions, in the hope of
providing some extra isolation. A quasiparticle trap was also added, by connecting
the wide (low inductance) lead that forms half of the SQUID loop to an island on
layer R3 (Ti/PdAu). I also have not considered the capacitance of the wiring lines
to the ground plane. Hopefully, any resonant modes created are at su?ciently high
frequency to ignore.
The  ux biasing of this device was somewhat complicated, due to the large
cross mutual inductance between the SQUIDs. The total  ux12 in each SQUID due
to the two current (IAb and IBb ) and two  ux (IAf and IBf ) bias lines can be expressed
12With the convention I have chosen, the applied  ux 'A does not contain a contribution from
a SQUID?s own bias line. The total  ux 'T is due to all four control currents and the circulating
current.
111
in terms of the matrix equation 'T = MI or
0
B@ 'AT
'BT
1
CA =
0
B@ MAAf MAAb MBAf MBAb
MABf MABb MBBf MBBb
1
CA
0
BB
BB
BB
B@
IAf
IAb
IBf
IBb
1
CC
CC
CC
CA
: (4.13)
For example, MBAf is the mutual inductance between  ux line B and SQUID A. Us-
ing Eq. (4.12) (with the assumptions about return currents discussed in the previous
section), I estimated the mutual inductance matrix as
M =
0
B@ +90:7 +28:3 ?0:18 +2:7
?62:5 ?45:7 +13:4 ?27:5
1
CApH: (4.14)
The large asymmetry between DS2A and DS2B is due to the ground path which
runs very close to the loop of B. Unfortunately flflMABf flfl > flflMBBf flfl, so biasing DS2A
without signiflcantly afiecting DS2B was di?cult. One approach to the problem is
the subject of x5.2.2. Finally, there is a mutual inductance between the two SQUID
loops, which is predicted to be 3:0 pH. This can be an issue if the  ux state of one
device (randomly) changes, which then afiects biasing of the other.
We measured the mutual inductance matrix using data from the current- ux
characteristics as described at the end of x6.3 and found
M =
0
B@ +71:92 ? +0:44 ?1:10
?52:82 ?54:94 +17:73 ?
1
CApH: (4.15)
The quantitative disagreement between experiment and theory is not terribly sur-
prising. For one, the mutual inductance of a bias line to its own SQUID cannot be
measured with the experimental method we used. Secondly, the presumed return
112
current path may be an oversimpliflcation, as it does run very close to the SQUIDs.
In addition, the presence of the ground plane and its proximity to the SQUIDs alters
the  ux coupling in a way that is not taken into account by Eq. (4.12).
It is worth noting that if more than a few of these devices are to be coupled
together, the issue of cross mutual inductance needs to be seriously addressed. This
not only applies to If and Ib, but also to the microwave current I?w. In both LC2
and DS2, when we intended to excite one device with a microwave drive, the other
also responded (but to a smaller extent). Some of this was undoubtedly due to
the connection provided by the coupling capacitors, but the paths leading from the
wire bonds were not properly shielded. While providing individual current returns
should provide a great improvement to these problems, even small levels of cross
talk will still be rather di?cult to deal with using the present design. A possible
solution is to encase each qubit in an on-chip superconducting shield.13
We studied device DS2 during Run 44 (March 2005 to June 2006) of the
refrigerator. Before this run, the wiring for DS1 was duplicated to accommodate
the additional SQUID. Two new discrete LC fllters and four new copper powder
fllters (which are highly attenuating) were mounted in new housings. We kept the
same sample box, but anchored the copper plate on which the chip was mounted
directly to the mixing chamber with a copper wire.
As with LC2, almost all of the experiments on DS2 that I will discuss involved
justoneofthequbits. Often, wegroundedallofthelinesoftheotheratthetopofthe
refrigerator. However, we noticed that this seemed to afiect the biasing somewhat,
presumably due to changes in the mutual inductance matrix from difierent ground
paths. Therefore, we usually kept all of the lines connected, but held the unused
bias and  ux current lines at zero voltage (at the function generators) and placed a
large resistor to ground on the voltage readout line.
13A device of this sort designed by Anna Kidiyarova-Shevchenko will be tested in the group
soon.
113
4.4 Summary
A simple current-biased junction would make a poor qubit because it could
dissipate energy through the efiective shunting resistance Reff due to its bias leads.
Therefore, an isolation scheme must be used to boost this resistance, so quantum
superpositions can be maintained. The quality of the isolation can be characterized
by the relaxation time T1 = ReffCJ, where CJ is the junction capacitance.
An LC fllter, with a resonance near 200 MHz, will protect a qubit at its plasma
frequency !p=2? ? 5 GHz. For device LC2, consisting of two coupled LC-isolated
phase qubits, the prediction for T1 is roughly 5 ?s. However, this time may not be
realized because the fllter fails at its resonance frequency and below.
In the dc SQUID phase qubit, one junction serves as the qubit, while the other
forms an arm of a inductive current divider. While this does provide broadband
isolation, our implementation does not boost Reff to the levels of the LC isolation.
For the single qubit DS1 and the coupled device DS2, the prediction for T1 is 250
ns due to the stepped up lead impedance.
Both T1 predictions were made by assuming that the bare bias line resistance
was 50 ?. However, it is di?cult to determine this value accurately, particularly
over the wide frequency range that could be important. Also signiflcant is that
the dissipation could be dominated by intrinsic processes in the junction, which the
isolation techniques cannot prevent.
114
Chapter 5
Instrumentation and Experimental Apparatus
This chapter contains a description of the electronics and other equipment I
used in the experiment. In terms of electronics, I will mainly focus on the measure-
ment of the tunneling escape rate of a junction to the voltage state, where the vast
majority of the data in the rest of the thesis comes from. Other experiments only
called for minor changes. I will also discuss some of the special precautions that
had be taken when assembling the wiring and flltering, as the devices operated at
20 mK on a dilution refrigerator. Throughout this chapter, I will refer to various
runs of the refrigerator when difierent components were used. The devices studied
during these runs are described in x4.2.1, x4.3.1, and x4.3.2.
Figure 5.1 shows a block diagram overview of the electronics used in the es-
cape rate experiments. The device (a dc SQUID in this case) was mounted on the
mixing chamber of a dilution refrigerator, located inside an rf shielded room, which
attenuates electromagnetic flelds above a few kHz.1 The device was biased by two
currents, Ib and If, created by dropping the voltage from arbitrary waveform gen-
erators (AWGs) across resistors Rb and Rf. Each of the bias lines was flltered at
the shielded room wall by a pi fllter and passed through a unity gain bufier to avoid
ground loops. Additional protection was provided by an LC fllter and a copper
powder fllter at the mixing chamber of the refrigerator (on both bias lines) and
the on-chip isolation techniques described in Chapter 4. Resonant transitions were
driven with a microwave current I?w. If pulses were required, the microwave source
was gated by a pulse generator.
The Ib and If AWGs were triggered by a square wave from an additional func-
tion generator (labeled ?Master Clock? in Fig. 5.1). The trigger line was optoisolated,
1Testing of the shielded room was performed by Musaddiq Awan.
115
Sync
Tim
er
Sta
rt
Opt
o-
Iso
lat
or
I?w
So
urce
Sto
p
Ib A
WG
Tri
g
Out
pi Filte
r
LPF
Sw
itc
h
LC Filte
r
Po
wderFilte
r
Rb
If A
WG
Tri
g
Out
pi Filte
r
LPF
Sw
itc
h
LC Filte
r
Po
wderFilte
r
Rf
Ma
ste
r
Clo
ck
Opt
o-
Iso
lat
or
Di
lution 
Re
frig
era
tor
JFE
T
OP
A
Sh
ielded
 Ro
om
C?
w
AND
Ga
te
Out
Tri
g
Co
mput
er
Data
L
L L
L
L
Pu
lse
Ge
ner
ato
r
L
L
B
B
B
B
De
tec
tor 
/
Tra
nsmitt
er
B
B
Opt
o-
Iso
lat
or
B
Bu
ffe
r
B Bu
ffe
r
B
DA
C
Optic
al
Re
ceiv
er
B
Chip
Figure
5.1:
Ov
erview
of
exp
erimen
tal
set-up.
The
blo
ck
diagram
sho
ws
the
typical
arrangemen
tof
electronics
for
measuring
the
escap
erate
of
adevice
to
the
voltage
state.
Tw
osync
hronized
arbitrary
wa
veform
generators
(A
W
Gs)
biased
aSQUID
through
aseries
of
fllters,
including
asingle-p
ole
low
pass
fllter
(LPF).
By
monitoring
the
amplifled
junction
voltage
and
sending
an
optical
pulse
ov
er
aflb
er
(dashed
line),
the
time
(with
resp
ect
to
the
sync
of
the
AW
G)
that
the
device
switc
hed
was
recorded.
A
micro
wa
ve
pulse
could
be
sen
tto
the
junction
to,
for
example,
induce
Rabi
oscillations.
Activ
ecomp
onen
ts
are
lab
eled
with
an
?L?
or
?B?
to
indicate
whether
they
were
line
or
battery
po
wered.
When
measuring
single
LC
-isolated
junctions,
the
 ux
bias
circuitry
was
remo
ved.
When
measuring
coupled
qubits,
the
biasing
circuitry
(and
detection
circuitry
,at
times)
was
duplicated.
116
Table 5.1: Commercial electronics used in the escape rate measurement. The table
lists the most common instruments I used. \SRS" is Stanford Research Systems.
Function Instrument
If, Ib Agilent 33120A Arbitrary Waveform Generator, Opt. 001
Master Clock Dynatech Nevada Exact 628 Function Generator
Timer SRS SR620 Universal Time Interval Counter, Opt. 01
Pulse Gen. SRS DG535 Digital Delay Generator, Opt. 01
I?w Hewlett-Packard 83731B Synthesized Signal Generator
Hewlett-Packard 83732B, Opt. 1E1, 1E2, 1E5, 1E8, 1E9, 800
GPIB National Instruments PCI-GPIB
DAC / ADC National Instruments PCI-6110 Data Acquisition Card
& BNC-2110 BNC Connector Block
in an attempt to keep the AWG as noise-free as possible. When studying device
DS2, the bias circuitry was duplicated for the second SQUID. In this case, all four
AWGs were synchronized. When studying the LC-isolated devices, the  ux bias
circuitry was absent, but the rest of the set-up was the same.
All commercial line-powered instruments (indicated with an ?L? in Fig. 5.1)
were located outside of the shielded room and (with the exception of the microwave
source and computer) were powered through an isolation transformer that broke the
ground connection from the wall. A list of the equipment in given in Table 5.1. The
active circuits built in-house2 were powered by sealed lead acid batteries (indicated
with a ?B?), which had to be recharged after roughly 50 hours of operation.
Ourbasicexperimentinvolvedmeasuringthetimeatwhichajunctionswitched
to the voltage state, with respect to the start of a particular trial. The various
signals that are involved in the measurement for a single cycle are sketched in Fig.
2The majority of the electronics that I used were designed and assembled by Huizhong Xu and
Andrew Berkley, although several other people contributed as well.
117
t
t
t
t
t
t
t
t
(a)
(h)
(g)
(f)
(e)
(d)
(c)
(b)
AWG
Sync
Ib
(RT)
Vb 
(RT)
Detector
Out
Receiver
Out
Ib
(MXC)
I ?w
Video Out
(RT)
I ?w
(MXC)
0
tb
tc
td
te
tf
tg
th
Figure 5.2: Timing diagram for escape rate measurement. The cartoons show the
time dependence of several signals at points where they are measurable at room
temperature (RT) and at the device, near the refrigerator?s mixing chamber (MXC).
(a) The reference time t = 0 is determined by the timer, when the AWG sync exceeds
a threshold. The combination of a (b), (c) bias current and (d), (e) microwave
current pulse force the junction to switch to the voltage state. When the (f) junction
voltage exceeds a set value, the (g) detector sends a pulse that is relayed to the (h)
receiver, whose output triggers the timer to stop. See x5.5 for a full discussion.
118
5.2. Some of the signals are shown both at room temperature (RT) where they
can be measured and at the device (MXC). Delays and distortion between the two
points need to be considered when performing current calibrations.
Anintervalcounter(?Timer?inFig.5.1)wastriggered bytheisolated syncfrom
the bias AWG. Ib began to increase at this time, deflned to be t = 0. Typically, the
ramp was linear, reaching the junction critical current in about 1 ms. The voltage
on the bias line Vb was continuously monitored by a two-stage amplifler. At some
point, the junction would switch to the voltage state. Because of the relatively large
capacitance on the bias line, Vb increased somewhat slowly after the device switched,
which was a major factor in determining the timing resolution of the experiment.
When Vb surpassed a preset trigger threshold (roughly 200 ?V), a light pulse was
sent over a flber optic cable to a receiver located outside of the shielded room. The
output of the receiver then stopped the timer.
The timer sent the single time interval to a computer, which was equipped
with a National Instruments PCI-GPIB card. A program written with National
Instruments LabVIEW 6.1 was used to collect, display, and record data from a
large number of trials (anywhere from several thousand to several million). The
master clock signal (not the computer) controlled the repetition frequency, which
was on the order of 200 Hz. The method used to convert a histogram of switching
times to escape rate is discussed in x6.2.
If the experiment required a microwave drive, we also wanted data taken with-
out microwaves, so that the escape rate enhancement could be calculated. The
computer coordinated this by sending an isolated TTL signal to the gate of the
microwave source. Generally, 10000 cycles with microwaves were followed by the
same number without. If we wanted a microwave pulse, then the output of a logical
AND between the computer?s signal and a pulse generator (which outputs a \dc
pulse") was sent to the gate of the microwave source. The computer was also used
119
to automatically update the waveforms of the AWGs, change the frequency and
power of the microwaves, and control the parameters of the pulse generator. The
GPIB connections that allowed this communication are not shown in Fig. 5.1.
5.1 Cryogenics
We used an Oxford Instruments model 200 dilution refrigerator, shown in Fig.
5.3, to cool the devices to as low as 20 mK. The refrigerator has several distinct
vacuum sub-systems. The dilution unit, shownin Fig. 5.3(a), is enclosed in a vacuum
can (which attaches to the 4 K  ange) and the entire refrigerator is immersed in a
dewar fllled with liquid 4He. The dewar is surrounded by a cylinder of mu metal,
to help shield magnetic flelds. The vacuum in the can (initially created with a
difiusion pump, but greatly improved by contact to the 4He bath) thermally isolates
the various stages of the refrigerator during operation.
Cooling of these stages is provided by the closed circulation of a 3He/4He
mixture [105]. When the mixture flrst enters the refrigerator, it is cooled by contact
with the 1 K pot, which is an isolated volume of 4He that fllls from the bath and is
pumped by a dedicated rotary pump (Leybold S40B or Alcatel 2063A). The mixture
liquifles at the pot and cools as it passes through two stages of heat exchangers
before reaching the mixing chamber. Below 870 mK, the mixture undergoes a phase
transition and separates into 3He-rich and 3He-dilute phases. The dilute phase
extends from the mixing chamber to the still, where it cools incoming mixture and
is pumped by the combination of a roots blower (Leybold RUVAC WSU 501) and a
sealed rotary pump. For the rotary pump, we used either a Leybold S65B (specially
retrofltted for Oxford) or an Alcatel 2063H. The evaporative cooling of 3He atoms
crossing the phase boundary to keep the dilute phase in equilibrium is what provides
the cooling power.
The 3He gas that is pumped out of the still passes through nitrogen and helium
120
1 K Pot
Still
Cold
Plate
Mixing
Chamber
(a)
Mixing
Chamber
(b)
Sample
Box
Copper
Powder
Filter
LC
Filter
5 cm 3 cm
4 K
Flange
Patch
Box
Figure 5.3: Refrigerator photographs. (a) The labeled stages reach progressively
colder temperatures, ending with the mixing chamber at 20 mK. (b) The discrete
LC fllters, copper powder fllters, and sample box are attached directly to the mixing
chamber. These pictures were taken before run 44, when device DS2 was studied.
121
traps before returning to the refrigerator, allowing the system to operate indeflnitely,
as long as the main 4He bath is replenished every 60 hours. We have been able to stay
at base temperature for months at a time, interrupted only by having to clean the
traps or service vacuum pumps. There is a small leak into the vacuum can (due to
ports for the microwave lines at room temperature and other unidentifled sources),
so we placed a small amount of absorbing charcoal at the bottom of the can. We
have run the refrigerator for well over a year without saturating the charcoal.
As seen in Fig. 5.3(b), the sample box, copper powder fllters, and discrete LC
fllters were each housed in separate metal boxes, to provide some modularity. They
were mated with SMA connectors and mounted to the base plate of the mixing
chamber with additional copper plates (not seen in the photograph) for mechanical
support and thermal contact. The sample box was mounted closest to the mixing
chamber, to provide it with the strongest thermal link. This meant that the current
and  ux lines had to be carefully bent to make connections at the bottom of the
LC fllters.
We used the system?s original vacuum can and thermal shield (bolted to the
still plate) that flt inside the small bore of a high fleld magnet, even though that
magnet was removed from the system. For this reason, the fllters had to flt within
a 2 inch diameter and the lines had to be carefully arranged so as not to cause any
thermal shorting.
A superconducting NbTi magnet was mounted on the outside of the vacuum
can. The magnet produces 11.13 mT/A, up to a maximum of 50 mT, along the
central axis of the refrigerator. It was centered about the sample box, which is
described next. The devices were mounted vertically, so that the fleld was in the
plane of the junctions. This magnet was only used for devices LC2 and DS1, when
we generally applied a fleld to suppress the critical currents of the junctions. Once
adjusted, the magnet was persisted, although the current lines were permanently
122
1 cm
If Ib
I?w
Figure 5.4: Sample box photograph. The dc SQUID phase qubits were mounted
in an aluminum box to provide a superconducting magnetic shield at milliKelvin
temperatures. Wire bonds connected the center conductor of each SMA connector
to the Hypres chip. The chip was mounted to a copper slab to provide thermal
anchoring. This picture was taken after run 43, when DS1 (whose spiral inductor is
barely visible in the lower left of the chip) was studied. The bent wire on the right
side is an antenna used to couple microwaves to two devices that were wired at the
same time.
attached. We never saw any evidence for fleld drift, but did not check carefully.
A picture of the sample box used for DS1 and DS2 is shown in Fig. 5.4. It
is an aluminum box, which shields devices from external magnetic flelds once it
becomes superconducting. Visible in the photograph is an indium O-ring used to
complete the shielding when the cover is screwed on. With the refrigerator at 20 mK,
we saw no efiect on the switching histograms while changing the fleld of the NbTi
magnet. This was somewhat of a surprise, as there are holes in the box where the
SMA connectors are mounted, and the switching experiment is extremely sensitive
to junction critical currents. The refrigerator had to be heated above 500 mK for
the fleld to penetrate, even though the critical temperature of aluminum is 1.1 K.
Upon cooling back down, the critical currents often changed, suggesting that the
123
fleld lines in the box moved once the box became superconducting.
A thin layer of GE varnish on the bottom of the sample chip was used to
attach it to a small copper plate. In addition, a small amount of silver paint was
applied around the edges of the chip to provide a stronger mechanical and thermal
connection. The copper plate was screwed into the aluminum box with two brass
screws (see Fig. 5.4). For run 44, this plate was thermally anchored to the mixing
chamber with a copper wire that passed through a small hole in the box. Al/Si wire
bonds connected the SMA center conductors to pads on the chip. The pad(s) for
the current return was bonded to the copper plate. A concern for this arrangement
is that the presence of the normal metal copper plate under the SQUID could have
coupled in magnetic noise, adversely afiecting the qubit.
The sample box used for LC2 was similar in design, but made from copper.
It only had two SMA connectors, which was su?cient, because microwaves were
coupled to the bias lines through the copper powder fllters. As the box was made
of copper, it provided less magnetic shielding. This was not a serious issue since the
current-biased junctions of LC2 were not SQUIDs.
5.1.1 Thermometry
Standard resistance thermometers were located on every stage of the dilution
refrigerator. Their values were measured using a Picowatt AVS-47 resistance bridge.
The data taking computer could communicate using GPIB with an AVS47-IB, which
in turn was optically connected to the bridge.3 This ensured that no digital noise
from the computer reached the thermometry lines, which could potentially cause
heating in addition to noisy signals.
Originally, a calibrated RuO2 resistor (denoted R6) from Scientiflc Instruments
3We used a LabVIEW program written by Sam Reed to display and log thermometry data.
This was particularly useful while initially cooling the refrigerator down.
124
(a)
(b)
Figure 5.5: Mixing chamber thermometer calibration. The temperature dependence
oftheresistanceofthemixingchamberRuO2 thermometer, asobtainedfromacross-
calibration with a commercially calibrated resistor, is plotted on a (a) log and (b)
linear scale. Each symbol represents data taken on a difierent day and potentially
difierent excitation voltage. The spread in the points is indicative of the uncertainty
of the flt (solid line) given in Eq. (5.1).
125
was mounted along side an uncalibrated resistor (R7) of the same type on the mixing
chamber of the refrigerator. After R6 was removed for repair, our main thermometer
became R7. To determine the temperature, I used a cross-calibration with R6, the
results of which are shown in Fig. 5.5. The functional form of the flt (provided by
the manufacturer) is
lnT = a+ blnR + clnRR ; (5.1)
with a = -7.98, b = 53.2, c = 354, where R is the resistance in ohms and T is the
temperature in kelvin.
The spread in the data points is due to incomplete thermalization of the ther-
mometers and varying bridge excitations. However, the majority of the experiments
we performed required being well below the energy level spacing of the junction,
so knowledge of the exact temperature was not critical. A few experiments did re-
quire elevated temperature to create excited state population, so the thermometer
calibration was a source of error.
5.2 Current and Flux Bias
The current bias (and the  ux bias for the dc SQUIDs) was generated by
dropping a voltage across a room temperature resistor, Rb (and Rf); see Fig. 5.1.
The voltage came from an Agilent 33120A Arbitrary Waveform Generator (AWG),
whose digital-to-analog converter can produce waveforms of 16000 points with a
12-bit vertical resolution.4 For historical reasons having to do with the design of the
voltage detector, I usually used a negative voltage waveform. With the exception
of Fig. 5.13, which shows an example of waveforms, I will invert the actual sign
of the current and  ux bias, which seems more natural. In order to maximize the
4The instrument has a serious \phase creep" bug that causes arbitrary waveforms to drift with
respect to the sync over the course of hours. It can flxed by sending the command \DIAG:POKE
16,0,1".
126
signal-to-noise ratio, our goal was to set the voltage of the waveform to about 5 V.
Thus, for a typical critical current of 30 ?A, Rb was chosen to be about 200 k?.
Because the required current for the  ux line was much higher, Rf ? 1 k?.
The bias voltage generally entered the shielded room through a pi-section fllter
(Spectrum Control 9001-100-1010), with a roll-ofi frequency near 20 kHz. This fllter
attenuated high frequency noise that might have been present on the biasing lines
and also smoothed out steps in the AWG output. If a higher bandwidth was required
(for example, for high frequency  ux shaking; see x6.5), the signal entered directly
into the shield room, although we could have used a fllter with a higher roll-ofi.
For device DS2, the steps in the waveform did cause a problem, which is addressed
below.
A battery-powered bufier amplifler was used inside the room, so that the bias
voltage was referenced to the refrigerator (and not the shielded room) to eliminate
ground loops.5 For most of my experiments, an AMP03 difierential amplifler with
unity gain served as the bufier; see x6.3.1 of Ref. [1] and x4.1.2 of Ref. [3] for circuit
diagrams. A single-pole low-pass RC fllter was used to remove high frequency
components from the bufier output. The roll-ofi frequency for this was anywhere
from 20 to 100 kHz, depending on the situation.
Just before entering the refrigerator, each line was connected to a switch. With
the switch closed, the line was connected to the refrigerator wiring. With it open,
both sides saw the refrigerator ground. The switches were opened whenever pow-
ering up or reconflguring the room temperature electronics to protect the junction.
I forgot to do this several times, with no adverse efiects; either I was lucky or the
large-area niobium junctions are quite robust.
5The e?cacy of this can be checked by measuring switching histograms with a very fast repe-
tition rate of an LC-isolated junction and looking for a 60 Hz signal in the time series. However,
due to multiple  ux states, this is di?cult to do with a high fl SQUID.
127
5.2.1 Refrigerator Wiring
The current bias lines entered the refrigerator on coated manganin wires, cho-
sen to minimize the heat load (see Fig. 5.6). The wiring was done in twisted pairs,
but we only used one member of each pair, as there were no dedicated returns for
each line either \above" or \below" this point. Instead, all currents were returned
on the refrigerator itself. The wires were inserted into Te on tubing and then a
Cu/Ni tube, which was placed in a wiring port that ran to the vacuum can; i.e. all
of bias lines for the devices were in vacuum below the room temperature plate of
the refrigerator and were never in direct contact with the helium bath space.
Upon entering the vacuum can, each wire was patched to a length of Ther-
mocoax 1 Nc Ac 05, which was thermally anchored at each stage (1 K pot, still,
cold plate) until reaching the mixing chamber of the refrigerator. This coaxial cable
has a stainless steel outer conductor, MgO dielectric, and NiCr center conductor.6
It was designed to be used as a heating element, but has gained popularity in low
temperature physics in situations where lines that have high attenuation at mi-
crowave frequencies are required [106]. At room temperature, the combination of
the manganin and Thermocoax had a dc resistance of 90 ?.
As we were most concerned with high frequency performance of the Thermo-
coax, we characterized a section of cable with an Agilent 8722D network analyzer.
Figure 5.7(a) shows the transmission coe?cient jS21j, which gives the attenuation of
a signal on the bias line. Above 5 GHz, the transmission is below the noise  oor of
the analyzer. Thus, Thermocoax provides a simple way to prevent high frequency
noise at the top of the refrigerator from reaching the junction.
The network analyzer can also measure the impedance of an element. If a load
impedance ZL terminates a transmission line with characteristic impedance Z0, the
6Over time, several sections of our cable developed shorts on the order of 10 M?. It is possible
that the dielectric absorbed water; this problem can be avoided by sealing the ends of the cable
when putting connectors on it and storing it.
128
Mixing Chamber
1 K Pot
Still
Cold Plate
Ch
ip
Sa
mple B
ox
Po
wder F
ilter
LC
 Fil
ter
Room Temperature Plate
4 K Flange
Patch Box
Ib
VJIfI?w
Th
erm
oco
ax
Nb C
oax
Th
erm
oco
ax
Lak
eS
hor
e
Lak
eS
hor
e
SS
 Coax
Ma
nganin
Figure 5.6: Schematic of refrigerator wiring. Lines for the current bias Ib, junction
voltage VJ,  ux bias If, and microwave current I?w begin at room temperature at
SMA connectors. As they proceed down the refrigerator, they are mechanically
clamped to each refrigerator stage, before reaching the SQUID at the mixing cham-
ber. All lines are either commercial coaxial cable or a wire inside of a metal tube.
The  ux line was not in place for LC2; all needed lines were duplicated for the
coupled devices, LC2 and DS2.
129
(a)
(d)
(b)
(c)
Figure 5.7: High frequency properties of Thermocoax. (a) The measured transmis-
sion coe?cient jS21j reaches the noise  oor of the network analyzer above 5 GHz for
an 80 cm length of cable at room temperature. As the analyzer measures voltages,
jS21j expressed in dB is given by 20log10
q
[Re(S21)]2 +[Im(S21)]2. The efiective
parallel (b) resistance Reff, (c) inductance Leff, and (d) capacitance Ceff were calcu-
lated from S11, using Eqs. (5.3) and (5.4).
130
complex re ection coe?cient is [107]
S11 = ZL ?Z0Z
L +Z0
: (5.2)
Here, Z0 represents the measurement lines of the analyzer and is equal to 50 ?. By
inverting this equation, the resistive and reactive parts of the load are given by
RL = 1?[Re(S11)]
2 ?[Im(S
11)]
2
[1?Re(S11)]2 +[Im(S11)]2 (5.3)
XL = 2 Im(S11)[1?Re(S
11)]
2 +[Im(S
11)]
2; (5.4)
where ZL = RL+iXL. For a two-port device (like a section of cable), the value of the
one-port impedance ZL will depend on how the other port is terminated, equaling
Zsc if it is short-circuited and Zoc if it is left open. The characteristic impedance
of the device is pZscZoc [107]. In the case of the Thermocoax, the attenuation was
high enough that the termination had little efiect on S11.
The discussion in x4.2 of the impedance that shunts the junction and Fig.
4.2(b) apply here as well. When the analyzer measures S11, port 2 is terminated
with 50 ?, so using S11 to calculate ZL is perhaps a good approximation to the
arrangement that the junction sees. The efiective parallel resistance and reactance
are Reff = 1=Re(Y) and Xeff = ?1=Im(Y), where Y = 1=ZL. Reff never dips
below 20 ?, as shown in Fig. 5.7(b), and will contribute to the shunting resistance
that determines the relaxation time T1 of the junction. The efiective inductance and
capacitance are shown in Fig. 5.7(c) and (d). This particular section of Thermocoax
has a few resonances and looks inductive near typical junction plasma frequencies.
We avoided using highly resistive manganin and Thermocoax for the  ux bias
lines of the SQUIDs, in order to avoid excess heating from the large currents required
on these lines. For the  ux lines, a flrst section of LakeShore CC-SR-10 coax went
131
from room temperature to a patch box on the still plate, as sketched in Fig. 5.6.
The conductors are made of steel, which is a poor thermal conductor and provided
some attenuation to high frequency noise. Inside of the copper patch box, each line
entered a small cavity. This design allows for the insertion of a fllter. However, we
instead just used the box to heat sink each line to the refrigerator?s still, which has
a large cooling power. This was done by attaching a coated wire to a wall of the
cavity with GE varnish.
The rest of the  ux bias path was formed using homemade coax.7 A niobium
wire served as the center conductor, which was threaded through Te on tubing.
This combination was inserted into stainless steel tubing that formed the outer
conductor. It was di?cult to precisely control the geometry with this arrangement,
but this line only carried relatively slow signals, so impedance mismatches were not
a major concern.
To provide additional protection against high frequency noise, each current
and  ux bias line passed through an LC fllter and a copper powder fllter at the
mixing chamber of the refrigerator; see x4.2.2 of Ref. [3] and x6.2.2 of Ref. [1] for
additional details. The LC fllter, shown in Fig. 5.8, is made of discrete components
and is designed to have a cut-ofi frequency near 10 MHz. It was originally designed
for use with the LC-isolated qubits and protects the junction over the region where
the on-chip isolation fails (see Fig. 4.3). We did, however, use a fllter of this sort on
each of the current and  ux bias lines for the dc SQUID phase qubits, as shown in
Fig. 5.6.
Network analyzer measurements of an LC fllter are shown in Fig. 5.9. The
isolation, as measured by jS21j is not particularly good above 500 MHz. Difierent
fllters, made with nominally identical components, had characteristics that were
quite difierent. Of more concern is that Reff drops below 10 ? at several points,
7The coax was designed and assembled by Roberto Ramos.
132
2.2?
33p 68p
2.2?1?1? 0.1? 0.1?(a)
(b)
Figure 5.8: Discrete LC bias fllter. (a) Each T-fllter was composed of six inductors
and two capacitors, as shown in the circuit diagram. (b) The photograph shows one
of the fllters that was used during runs 43 and 44. The components were assembled
on a piece of copper foil, which was mounted on a copper housing. This arrangement
provides good thermal anchoring for the bias lines.
even though it surpasses 1 k? at others. Needless to say, measuring S11 of the entire
bias line (at operating temperature, rather than in sections at room temperature)
would give the most accurate picture of the environment that the qubit and its
on-chip isolation see, as each stage of flltering may load others in non-trivial ways.
At very high frequencies, stray capacitance and inductance lead to a failure of
the discrete components. To make up for this, we followed the LC fllters with copper
powder fllters (see Refs. [108{110] and x6.2.2 of Ref. [1]). These work by passing a
signal wire near small grains of metallic powder. At high enough frequencies, the
skin depth becomes comparable to the grain size and the large surface area of the
powder leads to strong damping. We used two types of fllters. The flrst, which I will
refer to as the \short fllter," was used during runs 40 to 43. For run 44, when DS2
was studied, we were concerned that the flltering of the bias lines was insu?cient
and switched to the \long fllter."
133
(a)
(d)
(b)
(c)
Figure 5.9: High frequency properties of an LC fllter. (a) The transmission coe?-
cient jS21j, measured by a network analyzer, is small below 1 GHz, but increases as
the discrete components fail. The efiective parallel (b) resistance Reff, (c) inductance
Leff, and (d) capacitance Ceff were calculated from S11.
134
1 cm
Figure 5.10: Photograph of a long copper powder fllter. Niobium wire was wrapped
around a core, with a 4.3 mm diameter, formed by Stycast 2850FT and copper
powder. The turns reverse direction half-way down the fllter to try to minimize
pick-up from stray magnetic flelds. This piece was inserted into a copper housing
and potted with epoxy. Another SMA connector was attached to flnish the fllter.
There are roughly 300 turns, for a total wire length of about 400 cm.
The short powder fllter resembled the one described in Ref. [108]. The signal
wire passed through a cavity that was fllled with a mixture of Stycast 2850FT epoxy
and copper powder. For the LC-isolated junctions, microwaves were coupled to the
bias line by inserting an antenna into the cavity.
The long powder fllter was based on the design presented in Ref. [110].8 First,
a 50-50 mixture (by mass) of Stycast and 200 mesh copper powder was cast into a
cylindrical core with a diameter of 4.3 mm. The signal wire (3 mil niobium wire,
with a copper cladding) was wound around the core, with the direction of the turns
reversed at the midpoint, as shown in the photograph of Fig. 5.10. The assembly
was then placed in a hole in a copper housing and fllled with more epoxy to provide
mechanical stability and thermal anchoring.
Once again, we used the 8722D network analyzer to characterize the fllters
at room temperature. Results are shown for a short (solid circles) and long (open)
fllter in Fig. 5.11. It is unclear how the performance of the long fllter changes when
it becomes superconducting. The extra length of wire increases the attenuation
of the long fllter over the short one, particularly at low frequencies. Nonetheless,
neither fllter performs as well as the Thermocoax at 10 GHz. The efiective parallel
8The procedure was reflned with extensive testing, including varying the powder size and den-
sity, in our lab by Patrick Detzner, Tristan Guha-Gilford, and Tara McCarron.
135
(a)
(d)
(b)
(c)
Figure 5.11: High frequency properties of copper powder fllters. A network analyzer
was used to measure a short (solid circles) and long (open) fllter at room temper-
ature. (a) The transmission coe?cient jS21j is smaller for the long fllter over the
entire frequency range. The efiective parallel (b) resistance Reff, (c) inductance Leff,
and (d) capacitance Ceff were calculated from S11.
136
impedances are plotted in Fig. 5.11(b)-(d). As the sample box was directly attached
to the copper powder fllters, it is possible that Reff for this fllter set the bare noise
resistance Rbn in Fig. 4.2(a). Unfortunately, Reff varies from an acceptable 400 ? to
less than 20 ? at several frequencies, which could result in a high level of dissipation.
At a typical plasma frequency of 5 GHz, the flltering on the bias lines appears
to provide a total of at least 200 dB of attenuation. I tested this by using a bias tee
at the top of the refrigerator to couple a microwave current onto the current bias
line during run 44.9 Even with a source power of 1 ?W at 5 GHz, no change was
observed in the switching histograms. A signal of this strength on the dedicated
microwave line would have produced a Rabi oscillation in excess of 500 MHz. To
test the e?cacy of the shielded room, we performed several experiments with its
door open. This never had an observable impact on Rabi oscillations, which are the
most sensitive measurement that we performed. The high level of attenuation on
the bias lines may be su?cient at the present time. (The shielded room did provide
acoustic isolation, which lead to greater stability for the refrigerator.) It is puzzling,
then, why we saw clear evidence for high frequency noise at frequencies above !p
on the bias lines, as discussed in x7.2. This suggest that the noise was not coming
from room temperature, but from a source within the refrigerator that still needs
to be investigated.
The behavior of the fllters at lower frequencies is also of interest. Figure 5.12
shows the voltage-biased transfer function of the long copper powder fllter (dashed),
LC fllter (dotted), and both fllters in series (solid), measured with the circuits drawn
as insets. The usual AWG was used to provide an oscillating voltage Vs cos(!t),
where !=2? was limited to 15 MHz. A Tektronix TDS 3054B oscilloscope measured
the input voltage Vi cos(!t+ i), output voltage Vo cos(!t+ o), and phase shift
9I recommend saving this test for the end of a run. When I lowered the drive frequency
to roughly 1 GHz, the critical currents of the junctions suddenly changed during one of events
obliquely referred to in x4.3.2.
137
(b)
(a)
(c)
(d)
DUTVs VoVi 10
DUTVs VoVi
Figure 5.12: Low frequency properties of the bias fllters. The measured (a) mag-
nitude and (b) phase of the transfer function H = Vo=Vi for the long powder fllter
(dashed), LC fllter (dotted), and combination of the two (solid) is plotted as a func-
tion of frequency. (c), (d) The attenuation of the fllters is visible when a load of
10 ? is placed on their outputs.
138
? =  o? i. I used this simple conflguration to measure several of the components
in the experiment. To accurately determine their behavior, however, the input and
output impedances would have to be set the values actually seen in operation.
The magnitude of the transfer function H (dBm) = 20log10 (Vo=Vi) is plotted
for (a) an open circuit and (c) with a 10 ? load to simulate the junction. The atten-
uation below 1 MHz in the latter case is nearly the same for both fllters, suggesting
that this was due to the measurement set-up. A current-biased measurement might
have given more accurate results. The most prominent features in all cases are the
sharp resonances near 10 MHz. In the case of the powder fllter, I believe these are
due to the inductance and stray capacitance between the large number of closely
spaced windings in the fllter. Further testing is needed to determine the positive
and negative efiects of the various forms of flltering.
5.2.2 Biasing of DS2
Three additional factors afiected the biasing of DS2, which had two coupled
SQUIDs. The flrst had nothing to do with the device itself, but the flltering used
when it was studied. The long copper powder fllters installed before run 44 intro-
duced an LC resonance that could be excited by the steps in the discretized output
of the AWG for the  ux current. The resulting switching histograms [calculated with
the technique discussed in x6.2] had numerous spikes, rendering them useless. To
overcome this, we low-pass flltered IAf and IBf (where the superscript distinguishes
the two SQUIDs, DS2A and DS2B) with the pi fllters at the shielded room wall
and, at times, with additional RC fllters.
Although the current bias lines were flltered with the same sort of copper
powder fllters, current steps on these lines did not appear to have easily detectable
adverse efiects. Nonetheless, as we almost always wanted Ib / If (for reasons
discussed in x6.4), it was convenient to fllter Ib the same way as If.
139
The second problem was that this device showed very strong heating efiects
when a junction was left in the voltage state. This could be remedied with a slow
repetition rate (50 Hz or so), which made data taking quite inconvenient. Alterna-
tively, the rate could be increased to over 200 Hz if the junction was forced to retrap
within a few microseconds of tunneling. While this could have been accomplished
with additional circuitry [111], we chose to include the quick shut-ofi in the Ib wave-
form, so that the device saw identical biasing on every cycle. For example, when
DS2B was left in the voltage state for about 300 ?s (corresponding to a dissipation
of 22 pJ) with a repetition rate of 230 Hz at 20 mK, the escape rate was nearly
identical to what was measured at 95 mK when the junction was forced to quickly
retrap. I do not know why this efiect in SQUID DS1 was far less pronounced.
It was impossible to force a quick retrap when the bandwidth of the current
bias lines was reduced by the pi fllter. Therefore, we used pi fllters on the  ux lines10
and left the bias lines unflltered at room temperature. As the current and  ux lines
then had difierent bandwidths, the waveforms for the AWGs had to be modifled to
give Ib / If after the fllters.
Finally, the cross mutual inductances between the two devices were large
enough to cause problems. Ideally, when performing an experiment with coupled
qubits, we would have liked to determine the biasing of each device independently.
The current return line and proximity of the devices of DS2, as discussed in x4.3.2,
made this impossible.
To address the second problem, I flrst measured the complex transfer function
H (!) of each of the four bias lines between the AWG and the bias resistor R.11
Imagine that the goal was to have If = mIb. I flrst chose a voltage time series
Wb (t) that the AWG for the current bias would output, which was generally some
10Because of the reduced bandwidth, If decayed to zero relatively slowly at the end of each
cycle. However, as it is only Ib that controls retrapping, this had no efiect on heating.
11In making these measurements, it was quite important to consider the phase shift across the
output impedance of the AWG, as well as the fllters and bias resistor.
140
sort of ramp. A prediction for the current that this generates comes from taking a
fast Fourier transform, applying the transfer function, and transforming back:
Ib (t) = FFT
?1 (Hb (!)FFT(Wb (t)))
Rb : (5.5)
I used LabVIEW?s Complex FFT.vi and Inverse Complex FFT.vi to do this calcu-
lation, but kept only the positive frequency components to force the answer to be
real-valued. The voltage time series for the  ux bias AWG was then set to
Wf (t) = FFT?1
 FFT(I
f (t)Rf)
Hf (!)
?
; (5.6)
where If (t) = mIb (t) in this case.
As the bandwidth of the current bias line was higher than than of the  ux
bias, I tried to \round" Wb (t) to remove high frequency components. I did not use
any sophisticated algorithm to do this, so if all of the Fourier components were kept
in Eq. (5.6) (the Nyquist frequency of the time series was generally about 2 MHz),
the resulting Wf (t) oscillated quite a bit. Therefore, I usually cut the transforms
ofi above 100 kHz. This procedure worked reasonably well (in that a high repetition
rate yielded nearly smooth histograms with none of the signatures of heating) and
was used for much of the data that will be presented on DS2.
Biasing both of the devices at the same time proved to be more of a challenge.
Again, I flrst selected voltage waveforms for the current biases and calculated IAb (t)
and IBb (t) using Eq. (5.5). If the goal had been to bias either of the devices indi-
vidually, then the necessary  ux currents would have been IAf0 (t) = mAIAb (t) and
IBf0 (t) = mBIBb (t). In order to produce the corresponding  uxes in both devices
simultaneously, Eq. (4.13) yields
IAf (t) = M
BA
f
?MAB
b I
A
b ?M
BB
f I
B
f0
??MBB
f
?MBA
b I
B
b ?M
AA
f I
A
f0
?
MAAf MBBf ?MABf MBAf (5.7)
141
IBf (t) = ?M
AA
f
?MAB
b I
A
b ?M
BB
f I
B
f0
?+MAB
f
?MBA
b I
B
b ?M
AA
f I
A
f0
?
MAAf MBBf ?MABf MBAf ; (5.8)
where each of the currents is a time series. As the same current biases are used for
both the independent and coupled situations, MAAb and MBBb do not enter these
expressions. The voltage waveforms for both of the  ux AWGs were then found by
using Eq. (5.6).
The coupled spectra ofx8.6 were produced with  ux waveforms generated with
this procedure (see Fig. 5.13). In this experiment, device DS2A was flrst quickly
ramped and then slowly ramped starting at 500 ?s. DS2B was linearly ramped so
that the qubits were degenerate at 780 ?s. After the bias was reset to allow the
junctions to retrap,  ux shaking (see x6.5) was used to simultaneously initialize the
 ux state of each device.
Figure 5.13(a) shows the waveforms output by the  ux AWGs. The inverse
Fourier transforms used to calculate them yielded many high frequency components,
particularly in producing the nearly  at section for DS2A. When I measured the
currents at the bias resistors [see Fig. 5.13(b)], these components had been flltered
out and the waveforms matched those for the current bias. That the  ux shaking
oscillations do not begin with a constant amplitude shows that the procedure has
 aws. The  ux applied to each SQUID predicted by Eq. (4.13) (using measured
values of IAb , IBb , IAf , and IBf ) is plotted in Fig. 5.13(c). Notice that because IAf has
a large efiect on DS2B, a bi-linear IBf must be used to get a linear 'BT . Also, because
MBBf was rather small, IBf was somewhat large to be sending to the mixing chamber
of a dilution refrigerator. The superconducting coax used for this line worked quite
well; although the sample thermometer did register a small increase, there were
minimal signs of heating in the data. Incidentally, the little jump in 'BT near 780 ?s
is due to both current biases being shut-ofi, as this is the time that the junctions
tunneled to the voltage state.
142
(a)
(c)
(b)
Simultaneous
Biasing
Reset
Flux Shaking
Figure 5.13: Avoided crossing biasing of device DS2. (a) The waveforms used by
the AWGs for the  ux bias of DS2A (solid) and DS2B (dashed) have many high
frequency components. (b) These components are flltered out at the shielded room
wall and are not seen in IAf and IBf , as measured at the top of the refrigerator.
(c) The predicted  ux biasing, which takes into account cross mutual inductances,
roughly stabilizes DS2A, while DS2B is linearly ramped.
143
The idea behind this process was to be able to set the biasing of each device
independently and then combine them so that both SQUIDs would switch to the
voltage state at the same time. In practice, they would miss by as much as 100 ?s
for the sort of waveforms shown in Fig. 5.13. Because of the shallow slope used for
DS2A, a small error in the  ux level would lead to a large shift in the switching
time. Therefore, a fair bit of trial and error was used to arrive at waveforms that
produced a good degeneracy. A large source of error is that I was only able to
measure the transfer functions of the lines and the currents at the top of the re-
frigerator. The hope was that the cold flltering did not greatly afiect the relatively
low frequency waveforms. The origin of the signiflcant error in the coupled biasing
remains somewhat of a mystery.
5.3 Microwaves
The microwave current I?w was generated by Hewlett-Packard 83731B and
83732B synthesized sources. The 1 Hz resolution and 1:5?10?9 fractional drift per
day of the 83732B were far better than we required. The one feature that we did
take advantage of was a TTL gate that could produce a pulse of under 10 ns, for
frequencies above 1 GHz. While this gate was convenient to use, more sophisticated
pulse shaping will ultimately be required for quantum operations [112,113].
The pulses that gated the microwaves were created by a Stanford Research
Systems DG535. The instrument has the unique combination of a long total time
range of 1000 s with a remarkable 5 ps precision. The rise time of the pulse is
a rather slow 3 ns, but this did not cause any noticeable efiects. As described in
the beginning of this chapter, we used a logical AND between the pulse and a dc
level from the computer (from the digital output of a National Instruments PCI-
6110 data acquisition card) to interleave data with and without microwaves. A
CD74HCT132E chip was used to do this and a faster chip might have improved the
144
timing resolution.
Inside the refrigerator, we used coaxial cable for the microwave lines. Initially,
there were multiple segments of coax with fllters inserted along the way, to pro-
tect the junction from noise and to thermalize the center conductor. However this
introduced impedance mismatches along the path, which gave the lines a strong
frequency dependence. Starting with run 41, a single length of UT-34-SS-SS coax
went from the room temperature plate to the mixing chamber, as shown in Fig. 5.6.
The stainless steel outer and center conductors ofiered some attenuation. The outer
jacket of the cable was clamped to the refrigerator at multiple stages, allowing some
cooling of the center conductor through the Te on dielectric. At room temperature,
each line had a dc resistance of 50 ?.
5.4 Voltage Detection
The success of the escape rate measurement hinged on the ability to precisely
detect when a junction switched to the voltage state. During run 40, the voltage
of the junctions was measured on the same line that provided the current bias. In
this two-wire conflguration, there was a signiflcant contribution to the total voltage
from the drop along the bias line, due its relatively resistive manganin section.
Nonetheless, it was still possible to detect the switch by looking for the fast edge
that it produced. In addition, the Johnson noise due to the manganin section was
large enough to limit the timing resolution of the experiment.
Starting with run 41, a dedicated line was used to measure the voltage across
the device (see Fig. 5.6). The upper section of this line was LakeShore CC-SR-10
coax, which went to the patch box on the still plate, just as with the  ux bias lines.
Its stainless steel center conductor was less resistive than the manganin wire, but
still not too thermally conductive. From the box, roughly 1 m of Thermocoax was
used to go to the mixing chamber, where the current bias line was tapped. At room
145
temperature, the dc resistance of the LakeShore and Thermocoax sections were 8
and 60 ?, respectively. The resistance on the current bias line from the voltage
tap to the junction was under 1 ? when the refrigerator was cold. At the mixing
chamber, noise on both the current bias and voltage lines was attenuated by a length
of Thermocoax, a discrete LC fllter, a powder fllter, and on-chip isolation.
As the gap 2?=e in Hypres-deposited niobium is only 2.8 mV, it was necessary
to amplify the voltage across the device at room temperature before it could be used
to trigger the timer to stop. At difierent times, we used a combination of commercial
(usually a Stanford Research Systems SR560) and homemade ampliflers. For the
data that I will present, the most common choice was two stages of homemade
circuits. Although they had flxed gain and bandwidth, their voltage noise was lower
than any commercially available instrument we could obtain. See x4.1.4 of Ref. [3]
and x6.3 of Ref. [1] for detailed descriptions and circuit diagrams.
A common-source JFET inverting amplifler was used as the flrst stage. The
voltage noise of the amplifler was reduced to less than 0:3 nV=pHz by using sixteen
2SK117 transistors wired in parallel. The transfer function, measured with the same
procedure that produced Fig. 5.12(a), is shown in Fig. 5.14(a). The bandwidth is
about 5 MHz.
The flrst stage?s gain of 40 is not quite su?cient to produce a voltage that
is easy to monitor. Therefore, we followed the JFET amplifler with a second stage
that used an AD829 op-amp inverting amplifler. The transfer function of this stage
alone is shown in Fig. 5.14(b). The gain is about 50 above 10 kHz and rolls ofi
above 1 MHz, due to the open-loop gain of the op-amp. The input of the amplifler
has a 10 kHz high pass fllter, created with an additional AD829, that serves two
purposes. For one, it removes the roughly 5 V ofiset created by the dc biasing of the
JFET amplifler. In addition, it minimizes the contribution to the voltage measured
at the top of the refrigerator due to the drop along the bias lines.
146
(a)
(c)
(b)
Figure 5.14: Gain and phase shift of the homemade voltage ampliflers. Measure-
ments are shown for the (a) flrst stage amplifler, (b) the second stage amplifler, and
(c) the two in series, where in each plot the left axis corresponds to the gain jHj
(solid line) and the right to the phase shift ? (dashed line).
147
The gain of both ampliflers in series is shown in Fig. 5.14(c). The maximum
value of 1700 amplifled the junction voltage to nearly 5 V, which was convenient to
work with. To further improve the amplifler performance, both stages used battery-
operated low noise power supplies that were regulated with active feedback.
The ampliflers produced a large voltage step when the junction switched to
the running state. The flnal step was to get this signal from the ampliflers at the top
of the refrigerator to the timer located outside of the shielded room. If we had done
this by sending the amplifled junction voltage directly to the timer, we would have
had to fllter the signal at the shielded room wall to prevent noise and grounding
problems. Unlike the biasing lines, which carry relatively low frequency signals, we
wanted a very high bandwidth on the voltage detection line. Therefore, we converted
the output of the ampliflers to a digital signal by sending their output to a Schmitt
trigger (i.e. a comparator with hysteresis), made from a high speed CLC420 op-
amp. When the trigger?s input decreased below an adjustable negative value, it
produced a pulse that drove an LED, which in turn was coupled to an optical flber.
The flber left the shielded room through a narrow waveguide and coupled to an
optical receiver. The signal was flnally converted to a TTL-compatible pulse with
another CLC420.
5.5 Timing
The escape rate measurement required currents and voltages from a variety
of instruments (see Fig. 5.1) that were physically separated by non-negligible dis-
tances. Precisely determining the temporal relationship between the difierent signals
is somewhat involved. Figure 5.2 shows what the signals qualitatively look like for
one cycle of the experiment where a microwave pulse excites the junction, forcing
it to escape to the voltage state. The proflles are not drawn to scale and I have
changed some of their polarities for clarity.
148
A cycle started when the bias AWG (Agilent 33120A) received a positive TTL
edge from the master clock, a Dynatech Nevada Exact 628 function generator. To
protect the AWG from noise, the clock was flrst sent to a 6N137 inverting optoiso-
lator. The master clock?s square wave output had a period that varied noticeably
at times. In addition, the AWG had a 25 ns jitter on its trigger input. The origin
of time, however, was set by the sync output of the AWG (not the master clock), so
any delays or jitter in triggering the AWG are unimportant. The trigger jitter does
limit the synchronization of multiple AWGs.
The sync output signal from the AWG was sent to the start input of a Stanford
Research Systems SR620 universal time interval counter (?Timer? in Fig. 5.1). This
instrument can measure the time between two voltage edges with a resolution of 4
ps. Again, to protect the AWG, its sync was optoisolated. This both delayed the
edge (not important) and increased its risetime from 2 ns to about 40 ns, depending
on what the output was connected to. As indicated by a dashed line in Fig. 5.2(a),
the start trigger of the timer was set to the voltage where the sync output was rising
most rapidly, in order to minimize the efiect of the slow risetime. In addition, we
usually ac coupled this input, to minimize trigger waveforms from slow  uctuations
due to the instrument being line powered.
It was critical that the output of the AWG was synchronized with its sync;
I measured the jitter to be better than 100 ps (by using the timer in a simple
experiment). The bias current was (usually) flltered at the shielded room wall,
bufiered, and flltered again. This lead to a distortion of the waveform. For a
particular value of the current, this can be thought of as a time delay tb for the
current bias just before it entered the refrigerator, as shown in Fig. 5.2(b). The
signal was further distorted by the wiring and fllters in the refrigerator; as these
were designed to cut ofi much higher frequencies, the efiective delay tc at the device
was not expected to be too difierent from tb.
149
For Rabi oscillation experiments, I also needed to switch on the microwaves at
a specifled time. The process started when the pulse generator was triggered by the
AWG sync (see Fig. 5.1). In conjunction with a dc level from the computer and a
logical AND gate, the generator created a dc pulse that gated the microwave source.
The source outputted a microwave pulse as well as a TTL ?video out? signal that
was held high for the duration of the pulse. The video out, drawn in Fig. 5.2(d),
was a 5 V signal we could easily measure. There was a delay and jitter on the pulse
generator?s trigger, its output, the inputs and outputs of the AND gate, the gate
of the microwave source, and its video out. These can all be taken into account by
measuring the video out and deflning the pulse to start at td. The jitter on td, due
to the full chain of instruments, is less than 200 ps (and much of this could be due
to the timer, which was used to measure the value).
There is a delay (< 100 ns, according to the manufacturer?s speciflcations)
and jitter between the microwave source?s video out and its rf output, propagation
delay along roughly 5 m of room temperature SMA cable, delay and distortion on
the stainless coax in the refrigerator, and an impedance mismatch at the sample.
All of this leads to the pulse appearing at the junction at a time te that is delayed
with respect to the original trigger signal from the AWG sync.
I will make the assumption that Ib and I?w are su?ciently high to force the
junction to switch to the running state immediately at te. I will also ignore the very
short time that it takes for an unshunted junction to reach the gap voltage (of order
CJ (2?=e)=I0 ? 1 ns). In order for our ampliflers to detect the switch, however, the
discrete LC fllter and about 2 m of cable in the refrigerator had to be charged to
the gap voltage. This lead to a relatively slow rise in the voltage Vb at the top of
refrigerator, as indicated by Fig. 5.2(f).
Figure 5.15(a) shows measurements of Vb as a function of time when junction
LC2A switched to the voltage state, taken with a Tektronix TDS 1002 oscilloscope
150
during run 41, at 20 mK. For this plot, I averaged the data to remove bit noise in the
traces. With the oscilloscope (set to a 1 M? input impedance) connected directly to
the voltage line (dashed line), the signal reaches a maximum in 300 ns. The initial
slope of the voltage and the noise on the line set the resolution for determining
the switching time. The scope averaged 128 traces and the sharp feature at the
maximum voltage is an artifact of the triggering. The 1.5 MHz ringing that follows
is due to the excitation of a resonance, although I was not able to identify the
components that were producing it.
With an SR560 amplifying the voltage [solid line in Fig. 5.15(a)], the charging
time is longer. In making this plot, I divided the trace by the gain of the amplifler
(100), so the slope of the output signal in V/s is actually quite a bit higher than the
direct measurement. However, the bandwidth and input capacitance of the amplifler
do slow the output signal.
The open circles in Fig. 5.15(a) show the (ac coupled) output of the flrst stage
FET amplifler, again scaled by its gain. With its 1 MHz bandwidth, this amplifler
further slows the signal. While the charging time is perhaps a bit shorter than for
the SR560, the maximum scaled slope is deflnitely smaller. The high frequency
ringing has also been attenuated. Finally, the solid circles show the scaled output
of the two-stage homemade amplifler. The charging time has been increased even
further and the signal decays due to the high pass fllter on the second stage.
The voltage noise at the amplifler output and the switching slope set the
resolution for determining the switching time. By measuring the voltage line directly
with an oscilloscope, the noise was under 150 ?Vrms12 and the maximum slope was
about 20 mV=?s, corresponding to a timing resolution of less than 7.5 ns. The
output of the SR560 with a gain of 100 (and the input hooked to the voltage line)
12The dashed line in Fig. 5.15(a) was taken with averaging on the oscilloscope, which is why it
looks very quiet. I made some of the voltage noise measurements well after the curves in that flgure
were taken. As the instrumentation was nearly identical, the values should be fairly accurate.
151
(a)
(c)
(b)
Figure 5.15: Charging of the voltage line. (a) For junction LC2A during run 41,
the voltage on the bias line was measured with an oscilloscope directly (dashed
line), an SR560 amplifler (solid line), the JFET amplifler (solid circles), and the two
homemade ampliflers connected in series (open circles). (b) The initial slope of the
voltage, as measured with the scope directly (solid) and an SR560 amplifler (open),
varies with the junction critical current I0 (indicated by the gray line, with the right
axis). (c) The voltage versus time for SQUIDs DS1 (solid) and DS2B (open) shows
difierent ringing frequencies.
152
had a noise of 900 ?Vrms. Even though there was more noise on the SR560, the
slope of its output during a switch was 900 mV=?s, so the resolution was a much
improved 1 ns. The output of the flrst stage amplifler had a noise of 170 ?Vrms;
with a switching slope of 300 mV=?s, the resolution was about 600 ps. The gain
of the flrst stage amplifler was smaller than that of the SR560, but the low noise
of the homemade amplifler lead to a better timing resolution. Finally, with both
homemade stages hooked to the voltage line, the output had a signal of 720 ?Vrms.
The switching slope was 10 V=?s, for a resolution of under 100 ps, which was the
best of the four scenarios shown in Fig. 5.15(a). This value could be improved by
increasing the bandwidth of the homemade ampliflers or by using cold ampliflers
inside the refrigerator.
The (maximum) slope of the signal on the voltage line is set by the combination
of three factors (see x6.2.3 of Ref. [1]). The fastest rise time possible is determined
by the inductance and capacitance on the voltage line that must be charged to the
gap voltage. However, this time might not be realized if the bias current which is
available to charge the line is not su?ciently high. Finally, as Fig. 5.15(a) showed,
the bandwidth of the detection ampliflers may also limit the slope.
We attempted to observe the difierent efiects by varying the critical current I0
of junction LC2A with an in-plane suppression fleld Bk, which results in a difiraction
pattern (see x6.1). The traces in Fig. 5.15(a) were taken in the absence of a fleld,
where I0 = 130 ?A. The wide gray curve in Fig. 5.15(b) shows the critical current
variation as a function of Bk, with the right axis. Plotted with the left axis is the
maximum slope of the junction voltage, as measured directly with the scope (solid
circles) and with an SR560 amplifler (open circles).
The slope has a clear dependence on the critical current, but it is not a linear
one over the full range. At low values of I0, the charging is limited by the critical
current and the capacitance it has to charge. Here, the slope is linear in I0 and both
153
measurements of the slope agree. As the critical current increases, the bandwidth
required to measure the slope also increases. At some point the slope saturates, as
the measuring instruments are no longer fast enough to follow the charging. As the
SR560 has a lower bandwidth than the scope, the SR560 clips at lower value of I0
(about 10 mV=?s) than the scope (about 20 mV=?s).
It is possible that the saturation seen with the scope is not due to its band-
width, but to the components inside the refrigerator that must be charged. When
these data were taken (Run 41), the component value for the discrete LC fllter at
the mixing chamber were 1.1 ?H-100 pH-1.1 ?H; c.f. Fig. 5.8(a). Assuming the fllter
is responsible for most of the inductance and that the total capacitance of the fllter
and coaxial lines is about 0.5 nF, the analysis presented in x6.2.3 of Ref. [1] predicts
a voltage slope of roughly 25 mV=?s. This agrees well with the solid circles in Fig.
5.15(b), so it is plausible that scope was not the limiting factor in this case.
To sum up, for low critical currents, it is important to keep the capacitance on
the voltage line low, while still providing su?cient flltering for the junction. At high
critical currents, fast ampliflers are needed to take advantage of the faster charging
times.
Figure 5.15(c) shows the junction switching voltage versus time from SQUIDs
DS1 (solid circles) and DS2B (open). The output of the two-stage amplifler was
recorded with a TDS 3054B oscilloscope and the traces have not been scaled by the
amplifler gain. The charging time is comparable to that seen for junction LC2A,
although it is shorter for DS2B, because of its smaller critical current. The frequency
of the ringing is difierent for the two devices, perhaps due to the difierent powder
fllters used while measuring each.
When the amplifled junction voltage surpassed a set threshold at time tf, the
Schmitt trigger outputted a pulse (with some jitter and a small delay) at time tg,
as sketched in Fig. 5.2. It was then converted to an optical pulse, traveled through
154
about 4 meters of flber optic cable, reconverted to an electrical signal, and flnally
input to the receiver?s comparator to create a fast edge. The output from the
receiver, sketched in Fig. 5.2(h), was sent to the stop trigger on the timer. When
the voltage crossed an adjustable threshold, the timer stopped (with some jitter).
The output from the timer was read by the computer, which ended the cycle.
I measured some of the delays and estimated the others. The current calibra-
tion technique, described in the next section, measures the bias current at the top
of the refrigerator. While this accounts for the delay ta, tb can only be inferred by
looking at the switching data. The delay on the detector side can be measured with
a microwave pulse. We usually applied a short 10 ns ofi-resonant microwave pulse,
adjusting its time to coincide with a large number of counts in the switching his-
togram. The start of pulse at td (and its width) can be measured at the microwave
source?s video out using an additional SR620 timer. With a su?cient number of
repetitions, the detected time th can be read from the switching histogram. The
detector delay th ?te, plus the small propagation delay te ?td, was usually about
500 ns.
Figure 5.15 reveals a potential weakness of our detection scheme. We just used
a single threshold value to determine the switching time, treating Vb (t) as if it had
a sharp edge. The time resolution could be substantially improved by using the full
time-dependent waveform provided by an oscilloscope. What made this impractical
was the slow repetition rate (under 10 Hz) needed in order to digitize and record
each trace. With both a Tektronix TDS 3054B and TDS 7254B, a limiting factor
was the time needed to arm the scope trigger on each cycle. Nevertheless, even with
a small amount of data, the full trace information can be valuable for diagnostic
purposes. On a few occasions, we noticed that the voltage proflle was not consistent
from cycle to cycle. By recording each trace, it was possible to determine if speciflc
features in the calculated escape rate were due to switches of a certain form.
155
Short of digitizing the entire waveform, some improvement in the resolution
can be obtained by using two Schmitt triggers (and two receivers and SR620 timers)
set at difierent threshold values. Assuming a roughly linear voltage, a line flt through
the two points can be extrapolated to flnd the switching time. As shown in Fig.
5.15(b), the slope of Vb depends on the value of Ib at the time of the switch. By
calculating the slope based on the two points, we could see the variation of slope
across a single histogram peak that was 150 nA wide, suggesting that the relative
time resolution was quite good. Even with this technique, we saw no change (from
using using a single detector) in the escape rate during Rabi oscillations. While there
are several places where we could improve the time resolution in the experiment,
the overall resolution appeared to be shorter than 1 ns (based on being able to see
sharp features in the measured escape rate on this time scale) and did not appear
to limit any of the experiments I will present.
The issues of timing could be avoided by using a scheme that more closely
resembles the usual notions of measurement (as discussed in x6.6) together with
a readout that could be performed after an arbitrarily long delay [114]. We are
currently pursuing both objectives in the group.
5.6 Current Calibration
When performing experiments such as Rabi oscillations, we only needed the
switching time information. In measuring a spectrum, though, knowledge of the
current that the junction switched at was also important.
We most often used a simple measurement to calibrate Ib and If as a function
of time. Even if this method sufiered from systematic errors, the energy levels
and escape rates could still be parameterized with slightly modifled values for the
critical current and capacitance of the junction. For this calibration technique,
we simply recorded the voltage drop across the room-temperature bias resistances
156
Rb and Rf as a function of time; we used the analog-to-digital converter (ADC)
on the computer?s PCI-6110 data acquisition board, which has a 12-bit vertical
resolution and a full range set to ?5 V. Simultaneously, the optoisolated sync from
the bias AWG was digitized to provide the origin of time. Sampling the voltages
at a rate of 4 MS/s generally provided an adequate proflle for the waveforms. For
these measurements, the junction grounding switches at top of the refrigerator were
closed (junctions connected), to try to measure the currents just as they would be
during data taking. The resistance of the wires inside of the refrigerator (above the
point where they are thermally clamped to a stage) changed by a few Ohms with
the level of the helium bath. Therefore, the accuracy of the current calibration was
limited by these  uctuations. For this reason, we saw no reason to use anything
more stable than ordinary 1% metal foil resistors for Rb and Rf.
The voltage across the resistors was amplifled with a Stanford Research Sys-
tems SR560 in difierential mode, without flltering. As the output line was attached
to the shielded room wall, the presence of the amplifler perhaps provided some pro-
tection to the devices (although breaking the ground connection between the input
and output of the amplifler would have been preferable). The input to the SR560
is limited to 1 V, so for the bias resistance Rb we used the series combination of a
large resistor and roughly 1 k?, across which the voltage was measured. With a
careful choice of the Rf resistors, the SR560 could be kept on the same gain setting
(roughly 20) during the calibration of Ib and If. There was no concern when this
was not possible, because we experimentally found that the dc gain of the amplifler
varied by less than 0.5% from nominal values.
This electronics conflguration was also used to measure IV curves, such as
the ones shown in x6.1. Typically, the voltage was measured directly with another
SR560 (with a gain of 500 and no fllters), whose output was sent to the computer?s
ADC. The usual homemade ampliflers could not be used, because the flrst stage
157
produced a large voltage ofiset and the second stage had an ac coupled input.
Returning to the current calibration, the various instruments in the experiment
lead to signiflcant timing delays (deflned in Fig. 5.2), as described in the previous
section. When the data collection computer registered a count at time th, the switch
actually occurred at an earlier time te. This has to be taken into account when the
current calibration is used to convert the computer?s switching times to Ib and If.
This was particularly important for high speed ramps.
We also used an alternative approach to create a reflned current calibration,
which took advantage of the junction being an extremely sensitive detector. The
assumption that we commonly make is that the 0 ! 1 resonance occurs at a flxed
value of Ib and is independent of the bias ramp rate (and temperature and the
trapped  ux in a SQUID, in other circumstances). Thus the slope of a linear Ib ramp
can be found by varying its dc ofiset and tracking the location of the resonance at
a particular microwave frequency. In doing this, none of the delays in Fig. 5.2 have
to be considered, as all measurements are made with respect to the time axis of the
computer. To flnish the calibration, the \ofiset" of the bias ramp needs to be found.
This can be done by stabilizing the bias at a large value where the escape rate is
measurable. For a steady bias current, the current through Rb should be very close
to the actual current  owing through the device. The value of Ib at other escape
rates can then be found with the slope previously measured.
As an example, Figs. 7.1 and 7.12 show escape rate data of junction LC2B
taken with ramp rates of 0.07 and 0.93 A/s, respectively. While the switching times
difiered by an order of magnitude, the escape rates plotted as a function of Ib should
coincide. However, usingthesimplecalibrationmethod, thecurvesdifieredbynearly
1 ?A. With the reflned escape rate calibration just described, the disagreement was
only 15 nA. The results for the measurement of the slope were fairly good, but
determining the value of Ib with the stabilized ramp turned out to be quite tricky.
158
The escape rate method mostly improved the calibration for the faster ramp rate,
which is more sensitive to the frequency response of the lines and electronics delays.
Aside from Fig. 7.12, all of the data that I will present were taken at relatively slow
rates, so I ordinarily just used the simple current calibration.
159
Chapter 6
Device Characterization and Measurement
Techniques
This chapter is the flrst devoted to experimental results. I will begin by de-
scribing the current-voltage (IV) characteristic curves of the devices and discuss
the various parameters that can be extracted from them. Almost all of the data in
the chapters that follow come from measurements of the rate at which a junction
tunnels to the flnite voltage state. Therefore, I will describe the technique in some
detail, although it is well established in the fleld. The escape rate, which was flrst
introduced in Chapter 2, is useful because it is a measure of the population in the
excited states of the qubit. By measuring the escape rate while varying system pa-
rameters (such as the bias current, temperature, or microwave power), a remarkable
amount of information about the device can be extracted.
The IV curves and tunneling measurement technique are covered in the flrst
two sections of the chapter and apply to both the LC-isolated junctions and dc
SQUID phase qubits described in Chapter 4. The three sections that follow are
speciflc to the dc SQUIDs. In particular, I will describe how the current- ux (Ib
vs. 'A) characteristics can be found from escape rate measurements. From these,
we can determine many of the device parameters. The Ib vs. 'A curves are also
essential in determining how the SQUID should be biased so that it behaves much
like a single junction. Finally, I will describe the technique used to initialize the
SQUID to a particular  ux state.
The last section contains a general discussion of quantum state readout meth-
ods for current-biased junctions. While most of the data that will follow come
from escape rates, we have also used two pulsed techniques that provide valuable
160
information and conform to more traditional ideas of quantum measurement.
6.1 IV Curves
As described in x2.2.3, even a simple IV curve contains useful information
about a junction. As an example, Fig. 6.1(a) shows data from device LC2A, taken
during Run 42 (see x4.2.1 for a description of this device and the runs during which
it was studied). The IV curve for LC2B was quite similar. The measured voltage Vb
is across the junction and a small series resistance (? 0:1 ?) that is on the line. For
this measurement, the current bias Ib is sinusoidally swept with a frequency of 20
Hz (see x5.6 for a description of the instrumentation used for these measurements).
The junction stays in the supercurrent state up to the critical current I0 ? 124 ?A,
when it switches to the voltage state where Vb = 2?=e ? 2:8 mV. The IV curve
then traces out the quasiparticle branch, not retrapping until a very low current
bias. The current rise, here up to 210 ?A, and its hysteresis are visible in all of
our devices and is independent of the current sweep frequency. Whenever there is a
sharp jump in the voltage, the \overshoot" is due to ringing on the voltage line.
Based on the slope of the IV curve in Fig. 6.1(a) at high voltage, the normal
state resistance Rn is roughly 14 ?. From this value and Eq. (2.34), the expected low
temperature critical current is 160 ?A (much higher than the design value of 97 ?A
given in Table 4.1). That the measured value is 20% smaller could be an indication
of the junction quality. However,it could be a result of the size of the junction.
The assumption I have made so far has been that the phase difierence  is constant
across the face of the junction. This breaks down for large junctions, where the fleld
generated by the supercurrents gives rise to a current screening analogous to the
Meissner efiect in bulk superconductors [115]. The length scale for this screening is
161
(a)
(b)
Figure 6.1: IV curves of device LC2. (a) The IV curve of junction LC2A, taken
with a 20 Hz sinusoidal current sweep at 20 mK, shows the quasiparticle branch
from which the normal state junction resistance can be calculated. (b) The sub-
gap region, shown for junction LC2B with a 3.5 mHz excitation, is important in
preliminary evaluations of the device for use as a qubit.
162
given by the Josephson penetration depth
?J =
s
~
2e?0J0 (2?+d); (6.1)
where J0 is the critical current density of the junction, the junction barrier has
an efiective thickness of 2? + d to an applied fleld, ? is the usual superconducting
penetration depth, and d is the thickness of the oxide (generally thin enough to
ignore). The idea is that the supercurrent will be conflned to a distance ?J from the
edge of the junction. For small junctions, the flelds, currents, and phase difierence
will be uniform over the area of the junction. On the other hand, if the junction
is larger than 2?J in either dimension, the current will be signiflcantly screened
from the center of the junction and the maximum critical current will be smaller
than expected from the critical current density and area. For device LC2, Hypres
quotes a nominal J0 of 100 A=cm2 and ? ? 100 nm for their niobium fllms, yielding
?J ? 36 ?m. Thus, for a 10 ?m?10 ?m junction, the size should not be an issue,
leaving the smaller than expected I0 as a possible concern.
Figure 6.1(b) shows the sub-gap region of the IV curve for LC2B, taken during
Run 40, when the voltage was measured on the current bias line. The contribution
from a series resistance of 140:7 ? has been subtracted in calculating the junction
voltage VJ. A careful measurement of the sub-gap characteristic can be performed
by voltage-biasing the junction with its critical current suppressed [64]. However,
the only change we made in taking this data was to use Wavetek model 20 analog
function generator to avoid a discretized current bias.
Accurately identifying Ib = 0 is di?cult given the ofisets of the measuring
instruments, but it appears that Ib ? 1 nA when VJ = 1 mV. Assuming that the
curve is roughly linear for small voltages, the sub-gap resistance Rsg should be well
above 100 k?. This is comparable to values reported for similar junctions used for
163
quantum computation [64,66,96,116].
Although the junction appears to retrap at VJ = 0:5 mV, I can only say
that the retrapping current is smaller than about 1 nA. Using Eq. (2.35), the
quality factor Q is greater than 1:5 ? 104. Both from the design of the device
and spectroscopy measurements, we know the junction capacitance is CJ ? 5 pF.
Equation (2.33) gives RJ > 100 k? (obtained from the retrapping current and the
RCSJ model), consistent with the sub-gap resistance (measured directly). It will
be interesting to keep this value in mind as we use other techniques to measure the
dissipation of our qubits. However, we should expect RJ to depend on frequency,
and the mV scale at which we have obtained RJ corresponds to roughly 500 GHz,
well above the operating frequency of the qubit.
By applying a fleld in the plane of a junction (denoted Bk), its critical current
can be suppressed [117]. This efiect occurs because of interference which is analogous
to single-slit difiraction in optics. The critical current follows the Fraunhofer pattern
[118],
I0?'k? = I0 (0)
flfl
flfl
fl
sin??'k='0?
?'k='0
flfl
flfl
fl; (6.2)
where 'k is the applied  ux in the junction barrier due to Bk and I have assumed
the junction has a uniform critical current density.
Figure 6.2 shows IV curves at three difierent flelds, taken on LC2B during
Run 40. Figure 6.2(a) shows the voltage on the bias line at zero fleld, while in
Fig. 6.2(b) the contribution from the series resistance has been subtracted ofi. The
combination of this correction, bit noise from the function generator supplying Ib,
and the discrete values from the ADC measuring Vb lead to the noticeable scatter in
the data. With no applied fleld, the junction switches to the full gap value, just as in
Fig. 6.1. Panels (c) and (d) show curves for two difierent values of Bk. As expected,
the critical current (indicated by an arrow) decreases. However, the flrst jump in
164
(a)
(d)(c)
(b)
Figure 6.2: IV curves of junction LC2B as a function of fleld. The IV curves were
taken by applying a 1 Hz sinusoidal current drive at 25 mK, while measuring the
voltage on the bias line. An arrow indicates the critical current. (a) The slope of
the zero-voltage branch is due to a series resistance of 142 ? on the bias line. In all
other panels, its contribution has been subtracted away. (b) At Bk = 0, there is a
single switch to the full gap voltage. By increasing Bk to (c) 1.32 mT and (d) 1.41
mT, the critical current decreases, as the system moves down the main difiraction
peak. For low I0, the junction goes to the gap in two steps. Although the currents
for the two jumps depend on the fleld, the voltage of the flrst switch is 800 mV in
both cases. The location of this feature is also evident on the quasiparticle branch.
165
junction voltage is only to 800 mV, with the jump to the gap voltage occurring at
a higher current. The origin of these sub-gap features is unknown, although similar
things have been seen before in our group (see x2.4 of Ref. [3]).
By taking IV curves over a range of Bk, the critical currents can be assembled
into a difiraction pattern. Figure 6.3(a) shows two such patterns taken on Device
LC2B. The open points show an asymmetric curve that never exhibits full sup-
pression. At the same time, the device was not switching in a reproducible way to
the voltage state while we were measuring the escape rate (described in the next
section). This is typical behavior when trapped  ux is present in the junctions.
After cycling the sample to 20 K, the solid data points were taken and the switching
became much cleaner. Suppression of I0 to less than 1% of its maximum value is
indicative of a junction with a uniform barrier. From these results, one can see that
the difiraction pattern is a valuable diagnostic tool for evaluating the fabrication
quality and condition of a Josephson junction.
Because the maximum critical current shown here corresponds to a large
plasma frequency, we generally operated device LC2 with a suppression fleld. In
general, we tried to leave the device at one of the difiraction peak maxima, where
it was least sensitive to fleld  uctuations. However, this was not always possible as
the switching characteristics were not necessarily stable at these points.
The dashed line on panel Fig. 6.3(b) is a flt of Eq. (6.2) to the main peak.
This peak is reproduced nicely, but the flt underestimates the height and period of
the higher order peaks. Although there is uncertainty in the value of the fleld Bk
(because the sample is not exactly on the axis of the solenoid, for example), it is
almost certainly ofi by a simple multiplicative constant, so the discrepancy in the
widths of the difierent peaks is puzzling. As the junction is small compared to ?J,
it is unlikely that this is due to non-uniform fleld penetration. However, the base
electrode of the junctions is comparable in thickness to ?, so the efiective barrier
166
(a)
(b)
(c)
Figure 6.3: Difiraction patterns of device LC2. (a) The current at which a voltage
flrst appeared is plotted as a function of the in-plane suppression fleld Bk. For junc-
tion LC2B, an initially odd pattern (open circles) was restored (closed) by thermally
cycling the device above Tc. (b) A flt to the simple theory (dashed line) shows qual-
itative difierences with the data (circles). (c) For low I0, there were two switches,
where the flrst one (solid circles) marked the departure from the supercurrent state
and the second (open) ended at the full gap voltage. This occurred in both devices,
but data from junction LC2A is shown. The data were taken at roughly 30 mK.
167
thickness could depend on the value of Bk. In addition, Eq. (6.2) applies to an
exactly rectangular junction; the rounded corners of our junctions could cause some
deviations [53]. Nevertheless, the flt gives 'k = ('0=1:45 mT)Bk or a cross-sectional
area for the junction of 1:4 ?m2. For a 10 ?m wide junction, this gives an efiective
barrier thickness of 140 nm. As we expect the thickness to be 2? ? 200 nm, this
value is not unreasonable.
For LC2A, the corresponding patterns before and after thermal cycling were
nearly identical to the one shown in Fig. 6.3(b), even though the measurements
were separated by flve months. This curve matches the one for LC2B (after it was
cycled), suggesting that both junctions on the coupled device are nearly identical.
In Fig. 6.3(c), I0 is indicated with solid circles, while the secondary switches [of the
sort shown in Fig. 6.2(c) and (d)] and are shown with open circles. Although the
sub-gap state occurred at roughly a single voltage, the current at which the system
jumped to the gap voltage varied with Bk.
For the dc SQUID phase qubits, IV curves are more di?cult to interpret
because the two junctions have difierent critical currents. Data taken on device
DS2 are shown in Fig. 6.4, with If = 0. Because of the difierent allowed  ux states,
there is a wide range of Ib at which the device switched to the voltage state. The
full range of these \critical currents" is indicated by the gray rectangles. Although
the maximum value (expected to be the sum of the critical currents of the two
junctions, I01+I02) is similar in the two devices, the minima (which can be no small
than I01 ?I02) are quite difierent, which is an odd combination. One possibility is
that the devices were exposed to difierent levels of noise either during switching or
retrapping. After thermally cycling the devices, the critical current ranges changed
slightly, but were still unequal, suggesting that these nominally identical devices
difiered in a signiflcant way. Despite the difierence in the critical current modulation,
both of the SQUIDs have essentially the same quasiparticle branch, with a normal
168
(a) (b)
Figure 6.4: IV curves of device DS2. The quasiparticle branches for SQUIDs (a)
DS2A and (b) DS2B are nearly identical, for curves taken with a 1 Hz sinusoidal
current drive at 20 mK. However, the ranges of critical currents are quite difierent,
as indicated by the gray rectangles. If many IV curves were plotted on top of each
other, the gray rectangles would contain several well deflned lines, corresponding to
switching from the difierent allowed  ux states.
state resistance Rn ? 48 ?.
Because the SQUIDs were mounted in an aluminum box, it was di?cult to
controllably vary the magnetic fleld and measure a difiraction pattern. In addition,
although Bk was nominally in the plane of the SQUID loop, it undoubtedly was
biasing the SQUID as well as suppressing the junction critical currents. In ret-
rospect, it might have been useful to add single junctions to the Hypres chips so
that the fabrication quality of a particular foundry run could be evaluated with the
measurements described in this section.
169
6.2 Escape Rate Measurement
To measure the escape rate, our standard measurement sequence proceeds in
the following way, which is nearly identical to the method developed more than 30
years ago [60]. Each repetition starts with the junction unbiased, in the zero-voltage
state. At time deflned to be t = 0, Ib is increased. Usually, this is done in a linear
fashion, although some of my experiments required a slightly more complicated
waveform. For the SQUIDs, the  ux-bias current If is ramped simultaneously with
the current bias Ib, as described in x6.4. At some point during this process, the
device escapes to the flnite voltage state and stays there. The time (with respect to
t = 0) at which this switch occurs is recorded to a precision better than 1 ns. The
bias current is then reset to a slightly negative value to ensure retrapping to the
supercurrent state. The reset occurs at a predetermined time and is independent of
the switching time.
This process is then repeated anywhere from a few thousand to a few million
times, depending on the desired precision in the escape rate. Depending on the sit-
uation, we varied the repetition frequency from 1 Hz to 1 kHz, typically using about
200 Hz. For rates above 1 kHz, heating efiects from the junction being in the voltage
state were evident. Each repetition may be regarded as a phase particle evolving in
the tilted washboard potential. It is convenient to think of N serial experiments as
an ensemble of N identically prepared phase particles. For this reason, I will use the
terms \state occupation probability" and \population" interchangeably. Tunneling
causes the population in the supercurrent state to decrease exponentially with time.
The escape rate of the device is thus comparable to the inverse of the mean life-
time of a radioactive sample, where switching to the voltage state is analogous to
decay. The distinction is that in the case of the Josephson junction, the decay time
is dependent on Ib and therefore time t, if the current through the junction is being
170
ramped. This time-dependent total escape rate ? is deflned through the relationship
dN(t)
dt = ??(t)N(t); (6.3)
which integrates to Z
t2
t1
?(t)dt = ln
?N(t
1)
N(t2)
?
; (6.4)
where N(t) is the number of elements of the initial ensemble of N(0) trials that
remain in the zero-voltage state at time t.
To calculate ? numerically, a histogram h of the N(0) switching times is flrst
made with bin size ?t, where h(ti) is the number of counts in the bin centered at
time ti. An example of such a plot with ?t = 2 ns is shown in Fig. 6.5(a), in the
absence (open circles) and presence (solid) of an applied continuous microwave drive.
The histograms show a small number of counts at short switching times because the
escape rate is small at these bias currents. The number of counts increases as the
escape rate increases exponentially with time. Eventually, the number of counts
decreases again, as all members of the ensemble have already tunneled out. A
microwave signal can induce transitions from the ground state j0i to the flrst excited
state j1i. The open circles show that in this case a well-deflned bump appears at
a deflnite current in the histogram, when the microwaves are resonant with the
junction.
Under the assumption that ? is constant during ?t, the second expression
of Eq. (6.4) can be discretized to yield (see Ref. [60], x4.4 of Ref. [3], and x2.5 of
Ref. [1])
?(ti) = 1?t ln
? N (t
i)
N (ti+1)
?
= 1?t ln
" P
j?i h(tj)P
j?i+1 h(tj)
#
: (6.5)
Notice that ?(t) is independent of the distribution of counts before time t. In the
limit of small time bins, when h(ti) ? N (ti+1), the logarithm reduces to the ratio of
171
counts in a particular bin to the total number left in the zero-voltage state. If h(ti)
and N (ti+1) are regarded as uncorrelated variables governed by Poisson statistics,
the uncertainty in the escape rate is
 ? (ti) = 1?t
s?
1
N (ti)
?2
h(ti)+
? 1
N (ti+1) ?
1
N (ti)
?2
N (ti+1): (6.6)
Equation (6.5) was used to generate the escape rate curves shown in Fig. 6.5(b)
from the histogram. Here, the x-axis has been converted into bias current using the
escape rate calibration method described in x5.6; the full axis corresponds to the
same time interval shown in Fig. 6.5(a). In Fig. 6.5(a), there are more counts at
43.18 ?s in the absence of microwaves than with microwaves. This is because some
fraction of the ensemble has already escaped to the voltage state in the region of
the microwave resonance, while the total number of counts in both experiments is
the same. However, in Fig. 6.5(b), both escape rates at the corresponding current
33:48 ?A are the same, which indicates that the microwaves are far detuned from
resonance at this point. In this way, ? is a particularly convenient way of comparing
data sets taken under difierent conditions or with difierent numbers of total counts.
It is also useful to deflne the escape rate enhancement as
??
? ?
???bg
?bg ; (6.7)
where ?bg and ? are the escape rates without and with applied microwaves. Figure
6.5(c) shows ??=? for the data in Fig. 6.5(b). The microwave resonance at Ib =
33:43 ?A is nearly Lorentzian. The scatter at low bias currents is due to poor
counting statistics from the low escape rates there. In Chapter 8, I will show how
a spectrum of transitions can be measured by mapping out the resonance as the
microwave frequency is varied. The application of the escape rate measurement to
state readout is discussed in x6.6.
172
(a)
(b)
(c)
Figure 6.5: Switching experiment. (a) The histograms h of the time when the
junction switched to the voltage state during a linear current ramp with (open
circles) and without (solid circles) microwave activation have the same number of
totalcounts. (b)Fromthesedata, theescaperate?ofthedevicecanbecalculatedas
afunctionofcurrentbias. (c)Theescaperateenhancement??=?duetomicrowaves
shows the resonance at this one frequency. This data set was taken with a microwave
frequency of 5.9 GHz at 20 mK on junction LC2B.
173
6.3 Current-Flux Characteristics
The SQUID Ib vs. 'A curves discussed in x2.6 are useful in identifying  ux
states and bias trajectories. In this section, I will show that there are several ways to
measure these characteristics and that each provides information about the device.
Figure 6.6 shows an Ib vs. 'A characteristic for SQUID DS1. Switching his-
tograms were taken by sweeping Ib at 280 flxed values of If. Rather than the single
peak shown in Fig. 6.5(a), SQUID histograms contain several peaks corresponding
to difierent  ux states. In Fig. 6.6, each of the histograms forms a vertical line of
the grayscale map. Because of the inductive asymmetry, most of the current bias
gets shunted to the isolation junction, so the shallow branches corresponding to it
switching to the voltage state are clearly visible. Although there is a very rich struc-
ture in these lines, I will focus on the  ux states and the extraction of parameters
that determine the e?cacy of the isolation.
The data look similar to Fig. 2.16(b), but no counts occur for Ib < 30:7 ?A,
whereas the isolation junction branch should continue to Ib = 0. This happens
because the potential must be tilted at least a certain amount before the phase
particle will switch to a continuously running voltage state when its well becomes
unstable (in the sense described in x2.5).
The situation is sketched in Fig. 6.7(a), which shows simulated characteristics
of the fl = 8:4 SQUID introduced in Fig. 2.16. The solid vertical line indicates a
path when the current bias is swept, with the  ux bias held constant. A particular
 ux state becomes unstable if the path begins inside of a characteristic loop and
crosses a critical line. A switch to the voltage state will occur (thus forming a
histogram peak) if the  ux state is occupied and Ib is above the horizontal gray line.
This outcome is indicated by open circles. If the isolation branch is crossed below
the gray line (indicated by a square), the well does become unstable, but the system
174
Figure 6.6: Ib vs. 'A characteristic of SQUID DS1. Histograms, measured with flxed
 uxat20mK,arestackednexttoeachothertoformthemap, whereblackrepresents
alargenumberofcounts. Theshallowlinesrepresenttheisolationjunctionswitching
from difierent  ux states. The steeper qubit junction branches are not deflned as
clearly. The solid lines are drawn for I01 = 34:23 ?A, I02 = 4:28 ?A, L1 = 3:535 nH,
L2 = 10 pH, Mb = 0, for N' = ?60;?59;?58 (top three curves) and N' =
?48;?47;?46 (bottom three curves) with an ofiset of ?0:34 '0.
175
(a)
(b)
(c)
(d)
Figure 6.7: Bias trajectories for a dc SQUID. The heavy black lines indicate paths
where (a) Ib is swept, (b) If is swept negatively, (c) If is swept positively, and (d) Ib
and If are swept simultaneously. When a critical line is crossed, the system either
must flnd a new  ux state (squares) or switch to the voltage state (circles). The
isolation junction will switch to the voltage state if Ib is above the gray bar, with
the possible exception of biasing very close to the gray critical bar, as in (b).
176
quickly retraps in one of the stable  ux wells. The minimum value of Ib required to
produce switching depends on the damping in the device, but I have not performed
modeling of the retrapping process. In the example shown in the Fig. 6.7(a), the
histogram will contain flve peaks, with one corresponding to the qubit switching to
the voltage state flrst. Notice that the highest four  ux states that switch were not
stable at the beginning of the ramp at Ib = 0. They only have the possibility of
becoming occupied during the retrapping indicated by the two squares.
The value of measuring the characteristic by sweeping Ib is that the maximum
switching current is very close to I01+I02. I say close, because a junction will tunnel
through its potential barrier for Ib < I0; there is no direct way to measure I0. In
addition, the slope of the isolation branch gives the value of L1 and the periodicity
of the branches gives Mf. Calibration of If is straightforward in this case.
Similar information can be obtained with the bias path shown in Fig. 6.7(b).
In this case, Ib is ramped up and stabilized at some level, after which If is ramped
negatively. The full characteristic is mapped out by varying the Ib level. However,
during the If ramp, only the isolation junction can switch, as the qubit branches are
crossed in the unstable-to-stable direction. Typically, once Ib crosses the gray line,
no retrapping can occur and a maximum of flve histogram peaks (in this case) will be
seen, even though more isolation branches are crossed. If Ib is stabilized close to the
gray line, though, either switching or retrapping can occur, depending on the level of
noise and dissipation in the system. In Fig. 6.7(b), I have indicated the uncertainty
of the outcome with a square and a circle for the isolation branches crossed near
the gray bar. For example, the  ux state that switches at 'A = ?1:25 '0 can only
become occupied if a retrapping event occurred earlier on the bias path. The result
of many sweeps is a histogram with more than flve peaks.
Figure 6.8(a) shows a characteristic for DS1, measured by sweeping the  ux
negatively. To the right of the dashed line, Ib is being stabilized, so the x-axis should
177
be thought of as time in this region. Once the  ux ramp begins, only the isolation
junction switches, with the total number of branches decreasing as Ib increases.
Switching disappears completely below Ib = 31 ?A and many peaks are seen for
31 ?A . Ib . 32 ?A, which is the bias range where both switching and retrapping
can occur.
To obtain information about the qubit junction, the bias path shown in Fig.
6.7(c) can be used. First Ib is stabilized and then If is ramped positively. During
the flrst part of the ramp, crossing of the isolation branches leads to retrapping and
switching to the voltage state. Once Ib crosses the gray line, the total number of
histogram peaks is flxed. During the If ramp, the qubit is guaranteed to switch. If
Ib is stabilized at a low value, it is possible to observe all of the stable  ux states
switch on their qubit branches with repeated sweeps. This is only true if there is
nothing analogous to the gray line for the qubit; i.e. the SQUID always goes to
voltage state when a qubit branch is crossed.
An Ib vs. 'A characteristic, obtained experimentally by sweeping If positively,
is shown in Fig. 6.8(b). Again, the dashed line separates the Ib and If ramps. In
these data, the steep qubit branches are well deflned and their slopes and periodicity
give L2 and Mf. The lack of counts in the lower left corner may be due to retrapping.
The complex structure at the bottom of each branch, which is barely visible in the
flgure, may be related to the part of the characteristic drawn with dashed lines in
Fig. 6.7.
As with junction LC2, we generally operated SQUID DS1 with a suppression
fleld Bk to reduce the plasma frequency of the qubit. The data of Figs. 6.6 and
6.8 were taken at what I will refer to as fleld #2. While the magnet was set to
Bk = ?2:9 mT with the aluminum sample box at elevated temperature, the critical
current deflnitely changed as the box became superconducting (see x5.1). Unless
otherwise noted, alldata inthe thesis on DS1 weretakenat this fleld value. However,
178
(a)
(b)
Ib
If
time
Figure 6.8: Swept- ux Ib vs. 'A characteristic of SQUID DS1. (a) By flrst stabilizing
Ib and then ramping If negatively, only the isolation branch is crossed. The diagram
shows a cartoon of the bias waveforms; the dashed line marks the division between
the bias and  ux ramps. (b) Similarly, by sweeping If positively, only the qubit
branch is seen.
179
we did look at the three types of current- ux characteristics with no suppression
fleld and two other values of Bk. For flelds #1 and #3, Bk was set to 2.5 and 0.5
mT, respectively. Information gleaned from linear flts to the junction branches of
the characteristics is summarized in Table 6.1. For the characteristics where Ib is
swept, the values refer to the isolation branch.
The periodicity in If gives Mf for all trajectories. Mf should be independent
of the fleld. The average of the twelve measurements is 50:90?0:08 pH, where the
variation is due to uncertainty in the If current calibration.
As mentioned, the maximum possible switching current is I01+I02. If the entire
qubit branch is visible (as I am assuming), then it ends at I01 ?I02. Therefore, the
range of Ib for which switches were observed is indicated in rows labeled \Max Ib"
and \Min Ib." For each fleld, the overall maximum and minimum were used to
calculate I01 and I02 and the zero-bias Josephson inductances L0J1 and L0J2. The
uncertainty in Ib is under 1%.
As discussed at the end of x2.6, the inverse of the slope of the isolation branch
is Ltot1 ? L1 + L0J1 ? Mb and the negative of slope of the qubit branch is Ltot2 ?
L2 + L0J2 + Mb. I have assumed that the dependence of the Josephson inductances
on I1 and I2 is only important at the ends of the branches. Due to the dependence
of L0J on the junction critical current, Ltot1 and Ltot2 are functions of Bk, although
Ltot1 is only weakly so.
One way to flnd the slope is to measure the spacing of the branches along
Ib, assuming a periodicity of '0 and that the branches are nearly straight lines;
values obtained from this method are denoted Ltot (?Ib). Alternatively, the slope
can be measured directly and Mf can be used to convert the dimensionless ratio
to an inductance; the result of this method is denoted Ltot (slope). The statistical
uncertainty in Ltot from fltting all of the branches of a particular set is generally
less than 10 pH, though it is quite a bit smaller for a few. Some of the variation
180
Table 6.1: Current- ux characteristic properties of SQUID DS1. The properties of
the primary branch for three bias trajectories are listed for four suppression magnetic
flelds. Unless otherwise noted, data in this thesis were taken at fleld #2. Entries
marked with an asterisk are anomalous; see text.
Zero Field Field #1 Field #2 Field #3
Ib swept
Mf (pH) 51.30 50.30 50.61 50.72
Ltot1 (?Ib) (nH) +3.506 +3.517 +3.530 +3.501
Ltot1 (slope) (nH) +3.506 +3.517 +3.531 +3.504
Max Ib (?A) 159.6 48.0 38.7 136.3
Min Ib (?A) 70.0 22.4 30.7 51.0
If swept negatively
Mf (pH) 50.89 51.15 51.07 51.01
Ltot (?Ib) (nH) -1.150? +3.509 +3.530 -0.0254?
Ltot (slope) (nH) -1.176? +3.522 +3.535 -0.0256?
Max Ib (?A) 154.2 47.6 38.4 135.8
Min Ib (?A) 69.4 22.9 31.1 52.0
Jumps? (?A) 106.7? 31.4, 39.0? 82.2?
If swept positively
Mf (pH) 51.04 50.87 50.95 50.92
Ltot2 (?Ib) (pH) -37.4 -42.5 -70.9 -35.3
Ltot2 (slope) (pH) -39.6 -42.2 -69.5 -37.3
Max Ib (?A) 156.9 47.8 38.4 135.4
Min Ib (?A) 56.1 23.1 29.9 38.4
# peaks 177 43 15 167
Extracted parameters
I01 (?A) 107.9 35.6 34.3 87.4
I02 (?A) 51.7 12.5 4.4 49.0
L0J1 (pH) 3.0 9.2 9.6 3.8
L0J2 (pH) 6.4 26.4 75.2 6.7
L1 ?Mb (nH) 3.503 3.508 3.520 3.499
L2 +Mb (pH) 32.6 15.6 -5.2 30.3
# states 177 43 15 167
181
for the swept- ux characteristics is due to Ib not quite stabilizing when the If ramp
started (largely due to the flnite bandwidth of the lines).
With the total inductances and critical currents (and thus L0J) in hand, L1?Mb
and L2 +Mb may be calculated. These values should be independent of the critical
currents and thus the suppression fleld. For the qubit side, the average value is
L1 ?Mb = 3:508?0:004 nH. On the isolation side, the average value is L2 +Mb =
18:3?8:7 pH. The large uncertainty in L2 + Mb is disappointing, as this value (in
conjunction with LJ2 and L1) determines the level of isolation; see 4.3. That the
value appears to scale with the critical currents suggests that the discussion in x2.6
needs a small correction or that there is a systematic error in the measurement of
the characteristics. On the experimental side, calibration of If is the most likely
culprit, although no similar pattern is seen in Mf. Incidentally, using this method,
it is not possible to extract a value for Mb (aside from its sign), but that is not an
important issue because Mb only appears as L1 ?Mb or L2 +Mb.
For the trajectory where If is swept positively, the total number of peaks is an
indication of the number of allowed states (and thus fl), so this value is listed in the
table as \# peaks." The theoretical number is calculated by counting the number
of stable wells of the potential for the extracted device parameters at zero bias, and
is listed as \# states." Although the two numbers agree, I essentially enforced this
by assuming that we could see the entire qubit branch.
Finally, there was an odd feature in some of the characteristics of DS1 where
If was swept negatively. For flelds #1 and #2, the isolation branch was seen as
expected. For fleld #1, the branches were not completely straight, with two small
jumps at the Ib values listed in the table. It is possible that they were artifacts
of the measurement. For zero fleld and fleld #3, not only did the slope of the
observed branch not match the isolation branch, the sign of the slope was wrong.
In addition, there were fairly noticeable jumps near I01. In the case of fleld #3, the
182
unphysical solution mentioned in x2.6 happens to have a slope of -23.3 pH and a
discontinuity at I01. It is unclear how crossing this branch could result in a switch
to the voltage state. The ? 1:2 nH slope of the zero fleld data remains a complete
mystery. However, the ?sweep  ux +? characteristics always appeared as expected,
so there is no reason to believe that the high fl devices cannot be used as qubits.
In Fig. 6.6, I have drawn solid lines for six branches using the parameters
extracted from the characteristics for fleld #2. However, a few small adjustments
had to made to the device parameters (given in the flgure caption), as this data set
was not the one used to calculate the values listed in Table 6.1. With the modifled
parameters, there is good agreement for the isolation branch and there is a high
density of qubit switches that fall under the characteristic. However, there are quite
a few qubit switches above the critical line, which should never happen, so the flt
(or our understanding of the switching dynamics) is not perfect.
We performed similar measurements on DS2 to extract parameters for both
SQUIDs. The isolation branch had an inverse slope of 3.35 and 3.39 nH for DS2A
and DS2B, respectively, so the spiral inductors are nearly identical. As discussed in
x4.3.2 and x5.2.2, the cross mutual inductances were signiflcant for these devices. A
matrix equation relating the four control currents (IAb , IBb , IAf , and IBf ) to the  ux
in each SQUID loop is given by Eq. (4.13).
For example, one of the matrix elements is MBAf , which is the mutual induc-
tance between  ux line B and SQUID A. We measured switching histograms of
device A by ramping IAb at flxed values of IBf (and IAf = 0). These were stacked to-
gether to form a Ib vs. 'A characteristic, whose periodicity gave MBAf = ?0:18 pH.
Similar characteristics were constructed to measure flve other elements, the results
of which are given in Eq. (4.15). With this method, the mutual inductance of a
bias line to its own SQUID (MAAb and MBBb ) cannot be measured. Although these
elements can be large, the  ux generated by the small current biases is not very
183
large. More importantly, these elements are not needed in the procedure I used to
bias both devices simultaneously, as shown in Eqs. (5.7) and (5.8).
6.4 Simultaneous Biasing
It greatly simplifles the operation of a dc SQUID phase qubit if it can be
treated as a simple current-biased junction, leaving the auxiliary junction to provide
broadband isolation, well out of resonance. However, if we were to just ramp Ib as in
x6.2, then almost all of the current would go through the isolation junction, causing
it to switch to the voltage state flrst. Instead, in order to measure the escape rate
of the qubit junction, we simultaneously ramp both Ib and 'A in an attempt keep
 2 flxed at some initial value. Usually, we choose the initial Ib and If to be zero, so
I1 ? Ib and I2 ? 0.
As we are interested in the classical ground state of the SQUID (no displace-
ment currents), Eqs. (2.61) and (2.63) can be used to flnd the relationship between
small changes in currentbias (?Ib)and  ux bias (?If) needed to ensure d 2=dIb = 0.
The corresponding change in the applied  ux is
?'A = Mf?If = (L1 +LJ1 ?Mb)?Ib: (6.8)
The implication of this result is that keeping  2 flxed corresponds to following a
trajectory parallel to the isolation branch of the current- ux characteristic.
What makes this di?cult to do experimentally is that the Josephson induc-
tance LJ1 is a function of Ib. Although LJ1 only changes appreciably when I1 is very
close to I01, this is exactly the region where we want to measure the escape rate.
In practice, we simply ramp both Ib and If linearly with a ratio experimentally
determined from the middle of the shallow branch of the current- ux characteristic.
This corresponds to Eq. (6.8) evaluated roughly at the zero-bias value of the qubit?s
184
Josephson inductance L0J1. At flrst, all of the bias current will go through the
qubit as desired, but as the ramp proceeds and LJ1 increases, a small amount of Ib
will get shunted through the isolation junction. The trajectory on the current- ux
characteristic is shown in Fig. 6.7(d). As none of the isolation branches are crossed,
there are never any retrapping events and each  ux state is guaranteed to switch to
the voltage state when occupied.
No matter what the bias trajectory is, we would like to be able to calculate
the current through each of the junctions. The current division for an incremen-
tal increase in the biases will depend on the instantaneous value of the Josephson
inductances. An integral can be taken along the bias path to follow the branch
currents [4]. Alternatively, as the flnal state is independent of the path taken, the
SQUID potential at the Ib and 'A of interest will provide the same answer. The
location of the local minimum for the well of the  ux state N' gives the classical
ground state values of  1 and  2. Thus, the branch currents can be found without
explicitly invoking the Josephson inductance, as in Table 2.1 (which was computed
for zero bias).
Calculations for a typical device are shown in Fig. 6.9, under the simple si-
multaneous ramp [i.e. Eq. (6.8) evaluated at L0J1]. The simulation parameters are
I01 = 30 ?A, I02 = 5 ?A, L1 = 3:5 nH, L2 = 50 pH, Mb = 0, and the qubit capaci-
tance is CJ = 5 pF. Figure 6.9(a) shows the current through the isolation junction.
Ideally, this would always be zero, but it reaches about 50 nA or 0.1% of Ib by the
end of the ramp. As this is a very small fraction of I02, the isolation junction should
hardly be afiected.
In Fig. 6.9(b), the calculated ground state escape rate ?0 of the qubit junction
is plotted with a solid line. I have assumed that this value depends only on  1 and
not  2, so that Eq. (2.43) may be used. The dashed line shows what ?0 would be
for a perfect simultaneous ramp where I1 = Ib. The two lines would coincide if they
185
(a)
(b)
(c)
Figure 6.9: Simultaneous biasing of a dc SQUID phase qubit. Simulations are
shown for a SQUID with I01 = 30 ?A, I02 = 5 ?A, L1 = 3:5 nH, L2 = 50 pH, and
Mb = 0, biased with a simple simultaneous ramp. (a) The current I2 through the
isolation junction starts at zero, but increases as the Josephson inductance of the
qubit increases. (b) This results in a shift of the ground state escape rate (solid line)
with respect to what it would be if I1 = Ib (dashed) (calculated for CJ = 5 pF).
Values for a SQUID under the simple ramp are reproduced by a single junction
with efiective parameters (circles). (c) A shift also occurs between the two biasing
conditions for the level spacing between the ground and flrst excited states, !01.
186
were plotted as a function of I1 instead of Ib. Figure 6.9(c) shows similar curves for
the lowest energy level spacing !01, calculated with Eq. (2.44).
It is generally di?cult to calculate I1 accurately from experimental data, be-
cause precise values of all the inductances and the background  ux must be known.
For the simple simultaneous ramping, the current shunted through the isolation
junction results not only in a shift in the current axis, but also a small change in the
slope of ?0 and !01 with respect to Ib. Therefore, it might seem that plots made as
function of Ib (which is relatively easy to measure) could not be compared to single
junction theoretical results. However, ?0 for the SQUID under the simple ramp
is nearly identical to ?0 for a single current-biased junction with I0 = 30:048 ?A
(instead of 30 ?A) and CJ = 4:662 pF (instead of 5 pF), which is plotted with
solid circles in Fig. 6.9(b). For !01, agreement is found for I0 = 30:044 ?A and
CJ = 5:045 pF [see solid circles in Fig. 6.9(c)]. These efiective junction parameters
also do a good job of describing the escape rates and energy levels of the higher
states jni and difierent  ux states.
To summarize, if we use a simple simultaneous biasing designed to make the
qubit junction switch, I1 < Ib during the ramp. Nonetheless, we expect the qubit
properties to resemble those of a single junction when plotted as a function of Ib.
However, thecriticalcurrentandcapacitanceofthemodeljunctionwilldifierslightly
from the actual qubit values and no single model will describe both the escape rates
and energy levels. The discrepancies, however, are fairly small in our devices, which
makes the independent junction approximation useful.
In practice, I often flt the experimental base temperature escape rate as a
function of Ib to extract I?0 and C?J and independently flt the 0 ! 1 transition
frequencies to extract I!0 and C!J. These parameters could then be used to predict
?n and !nm for higher levels over a range of Ib.
A potentially serious  aw in the previous discussion is that I used classical
187
arguments (based on properties of the 2-D potential) to determine the branch cur-
rents I1 and I2. I then assumed that I1 determined the energy levels and escape
rates. Quantum mechanically, the expectation value h 1i of the phase difierence
across the qubit junction determines I1. Because the potential wells are anharmonic
and the metastable states jni have a signiflcant weight outside of the well, h 1i is
closer to the potential barrier than the well minimum. Thus, the redistribution of Ib
will difier for classical and quantum treatments. In fact, full quantum simulations
show that the magnitude of the branch currents depends on the state of the qubit
junction [73]. Therefore, even if the single junction model with certain values of
I0 and CJ describe ?0 for the SQUID?s qubit junction, they may not describe ?1
correctly. Since the difierence in hI1i for j0i and j1i is predicted to be a few nA for
our devices [73], I have ignored this efiect. Nonetheless, it is small source of error
for the simulations in the chapters that follow.
6.5 Flux Shaking
As I discussed in x6.3, our SQUIDs can retrap in difierent  ux states. When
performing the simultaneous ramp, each of these states sits in a slightly difierent
potential and will switch to the voltage state at a difierent current (due to the
circulating current from the trapped  ux). Of course, I01 and I02 are independent of
the  ux state, but I will refer to the  ux states as having difierent critical currents
for simplicity. For the phase qubit, satisfying the DiVincenzo criterion concerning
qubit initialization [20,21] usually refers to occupation of the ground state j0i of a
tilted washboard well. However, for our SQUIDs, there are multiple wells that are
distinguishable and each has its own j0i. In this case, the \classical" initialization
to one particular well is also required.
To set the SQUID in a speciflc  ux state, we use a procedure similar to one
developed for a symmetric low fl dc SQUID [119]. In those experiments, a \shaker"
188
circuit was used to oscillate the current bias with an amplitude slightly less than the
maximum critical current. If the device was in the maximum critical current state,
it would remain there throughout the oscillations. However, if it was not, then it
would switch to the voltage state and eventually retrap when the bias was reset.
This switching and retrapping would continue until the device happened to retrap
in the highest critical current state.
For reasons discussed below, we chose to oscillate the  ux bias to perform the
initialization [72], using the process sketched in Fig. 6.10. The applied  ux has the
form 'A(t) = ?'A + e'A sin(!At+`)=2, where ?'A is a static  ux ofiset and e'A
is the peak-to-peak amplitude of a sine wave of angular frequency !A. The value of
` is chosen so that 'A(0) = 0; the oscillations run for a total time T, where !AT=2?
is an integer. To explain the process, I will use the picture of the SQUID developed
in x2.5. Consider the simple case of a symmetric dc SQUID with fl = 4:8 at Ib = 0.
The left column of Fig. 6.10 shows cuts in the  1 direction for difierent values of the
 ux. Notice that in each case, the potential can support flve  ux states, although
this range is centered about a value dependent on 'A.
In the sequence shown, the  ux oscillates about an ofiset of ?'A = 1 '0, with
a peak-to-peak amplitude of e'A = 4 '0 (although three snapshots at integral values
of 'A='0 are not drawn). With this choice, the well marked by a vertical arrow is
stable at all times and is therefore the one that the shaking will tend to occupy.
The solid circles show an example of what the phase particle might do during
an oscillation. In this example, the device starts out in the N' = ?2 state after
retrapping from the voltage state. When 'A(t) = '0, this  ux state becomes
unstable and the particle must settle in one of the states that are then allowed
(N' = ?1 to 3). I will assume that the retrapping is random and that in principle
the particle could end in any allowed well. Say it goes to 0. At 'A(t) = 2 '0,
this state is still stable and nothing happens. At 3 '0, the well is unstable and
189
U
+1 ?0
0
+2 ?0
+3 ?0
?1 ?0
0
?A
?1 
Figure 6.10: Schematic of the  ux shaking procedure. The left column shows a cut
through the potential of a symmetric SQUID with fl = 4:8 as the applied  ux is
oscillated to force occupancy of the N' = 1 state (vertical arrows). The solid circles
represent what the  ux state might do for any one trial. The right column shows the
probability of occupying the difierent  ux states for an ensemble of measurements,
starting from the simple (flctitious) distribution at the top. This distribution also
governs how the occupation probability of a well that becomes unstable is redis-
tributed, as the shaking proceeds. The solid part of each bar represent population
carried over from the previous step; the open part is due to redistribution.
190
the particle must switch again. If, as in the flgure, the device settles in N' = 1,
then it will stay there for the remainder of the oscillation (and any shaking that
follows). If it retraps in any other well, it would be forced out within one cycle and
the whole process would repeat. Each time the system switches  ux states, it enters
the running state for a brief amount of time (on the order T1 ? 50 ns). On the
other hand, if we had used current shaking [119], the system would remain in the
voltage state longer and cause heating and possibly decoherence due to quasiparticle
generation or other efiects.
The right column of Fig. 6.10 shows the probability p'(N') for occupying  ux
stateN', fromasimplemodeloftheshakingdynamics[4]. Theflrsthistogramshows
an example of what p' could be upon retrapping from the flnite voltage state. These
initial vales are given by the discrete probability distribution ?0'. In the example, I
have taken ?0'(N') = 0:125;0:2;0:35;0:2;0:125 for N' = ?2;?1;0;1;2, respectively
(with all other values equal to zero). As the N' = 0 well is at the lowest energy, it
has the highest probability of being occupied. The desired N' = 1 state starts out
at p'(1) = 0:2.
For simplicity, I will assume that wells become unstable when the applied  ux
is equal to an integral number of  ux quanta. At 'A(t) = '0, N' = ?1 through 2
remain stable. Thus p' for these states are carried over from 'A = 0, represented
by the solid bars in the second histogram. However, N' = ?2 has become unstable
and any system in that state must settle in one of the newly allowed states. The
redistribution is governed by the discrete probability distribution ?'(N' ?'A='0).
A key assumption in the model is that ?' = ?0'. Thus at 'A(t) = '0, the proba-
bilities p'(?2)?'(N' ?1) are added to p'(N'), which are indicated by open bars
in the second histogram. Here, p'(?2) = 0:125 is the occupation probability of
N' = ?2, before it became unstable.
This process continues, with a difierent set of flve  ux states having a non-zero
191
occupation probability every time 'A increases by a  ux quantum. For example,
at 'A(t) = 3 '0 the N' = 2 state inherits a probability of 0.225 from the previous
step. In that previous step, p'(0) = 0:402, 20% of which now goes to N' = 2,
bringing its total to p'(2) = 0:306.
For the one oscillation shown in the flgure, p'(1) increases from 0.2 to 0.488.
Notice that the \lower" states end with flnite probability, while the \higher" are
completely empty. This situation would be reversed with a  ux oscillation of the
opposite polarity. In either case, the occupation probability of the desired well
increases quite dramatically with further shaking. A difierent well could have been
selected simply by changing the ofiset ?'A of the oscillations, whereas when shaking
the current bias, the system can be initialized to only the highest critical current
state.
Before continuing, I should point out that Fig. 6.10 is slightly misleading. A
cut along the  1-axis was chosen in an attempt to show all of the  ux states using
a single graph. However, as indicated in x2.5, this cut could miss  ux states that
have shallow wells. More importantly, the flgure suggests than when the system
jumps to a new  ux state, it is only  1 (or  2, if the perpendicular cut is chosen)
that changes value. In reality, when a  ux well becomes unstable, it leaves the
phase particle with su?cient energy to explore a large region of the 2-D phase space
(which, incidentally, is one reason why we assume the redistribution probabilities ?'
are close to the retrapping ones ?0'). However, as the measurement is only sensitive
to the flnal  ux state and not the chaotic trajectory that was taken to get there,
the simplifled cartoon conveys the essence of the technique.
Our SQUIDs are highly asymmetric (in critical currents and inductances) and
can have many  ux states. Nonetheless, the  ux shaking procedure still works. As
we do not know the device parameters exactly, the peak-to-peak amplitude e'A and
ofiset ?'A of the oscillations need to be found experimentally.
192
Figure 6.11 shows an example of data for SQUID DS1, measured at 20 mK.
We measured switching histograms with the usual simultaneous current and  ux
bias ramps. However, before starting the ramps, ten sinusoidal  ux oscillations
were applied to the device at Ib = 0. Cartoons of the waveforms used for If are
shown in the flgure. The left and right side panels of Fig. 6.11(a) show the resulting
histograms for e'A = 0 and 14:4 '0. Each vertical line of the central panel of Fig.
6.11(a) is a histogram of this sort, where dark colors represent a large number of
counts. As the simultaneous ramp is designed not to change the initial  ux state
of the SQUID, the number of counts in a histogram peak is a good measure of the
occupation of the corresponding  ux state.
In Fig. 6.11(a), e'A (indicated by the vertical arrow on the sample waveform)
was varied in order to flnd the optimal value. In addition, ?'A was set to e'A=2 in
each case, so that the  ux was always positive. I have labeled the x-axis in both
current and  ux, where MfeIf = e'A (and Mf = 51:2pH). The histogram with no
 ux oscillations is centered about Ib = 33:5 ?A, as shown in the left panel. As
the oscillation amplitude and ofiset increase, the critical current of the maximally
occupied state increases. In fact, every time e'A increases by a  ux quantum, the
range of occupation shifts up by one state. This trend continues until the highest
critical current state is occupied with a probability of 0.473 at e'A = 14:4 '0. The
histogram at this oscillation amplitude, for which only one allowed state is always
stable, is shown in the right panel of Fig. 6.11(a). This state had zero counts in the
initial distribution. Once e'A > 15 '0, no single potential well is stable throughout
a full cycle of an oscillation, so the  ux shaking does not isolate any state.
To flnd the best ofiset for selecting a single state, we next varied ?'A with
flxed e'A = 14:5 '0 [see Fig. 6.11(b)]. In this flgure, Mf?If = ?'A. As the
ofiset increases, the critical current of the selected state increases, as expected. The
SQUID could be initialized to any of the sixteen observed  ux states, if the ofiset
193
(a)
(b)
0 ?0 14.4 ?0
N?0
8
0
-7
~
~
Figure 6.11: Experimental determination of shaking amplitude and ofiset. The
grayscale maps show switching distributions. (a) Ten oscillations at 44 kHz that are
entirely positive (inset) occupy the  ux state of SQUID DS1 with the highest critical
current with increasing probability as the oscillation amplitude (eIf or e'A) increases.
(b) By changing the ofiset (?If or ?'A) of the oscillations (with e'A = 14:5 '0),
the SQUID can be initialized in any of the sixteen visible  ux states.
194
was chosen carefully. At ?'A = 7:2 '0, the flfteenth and sixteenth states are
nearly equally occupied, as are the fourteenth and flfteenth at 6:2 '0. Only near
?'A = 6:7 '0 is the shaking efiective in isolating the flfteenth state. Naively, one
would expect the best performance to occur at an integral number of  ux quanta.
There is an apparent 0:3 '0 shift on the ?'A-axis. This could be due to the fact
that the  ux waveform resets slightly negatively to match the current waveform.
Also, it could simply be due to a calibration error or to an additional source of  ux
near the device, such as trapped vortices. This ofiset tended to vary with time and
particularly after helium transfers, so the parameters for the  ux shaking often had
to be flne tuned.
An unbiased SQUID made from conventional s-wave superconductors is always
able to support an odd number of  ux states, so this device ought to have at least
seventeen states. If we identify the  ux state that is isolated with an ofiset of
?'A = ?0:3 '0 as N' = 0, the highest critical state is N' = 8, consistent with
this number. In addition, the N' = 0 state would have a critical current consistent
with the value of I01 listed in Table 6.1, as required. It is possible that N' = ?8
is \invisible" because the system does not escape to the voltage state when that
state?s critical current is exceeded during the simultaneous ramp. It is, however,
curious that the initialization worked for a peak-to-peak amplitude of 14:5 '0. For
seventeen states, the minimum of e'A = 15 '0 would occur only for a fortuitous
value of fl.1 The inconsistency between the experimentally measured value and the
prediction based on the number of states is perhaps due to a calibration error with
the former. That the amplitude is so close to the critical value explains why the
ofiset had to be adjusted so carefully to pick out a single state. I suspect that if
we had used a slightly larger e'A in Fig. 6.11(b), each  ux state would have been
1This is true when the \end"  ux state is being isolated and there is no  ux ofiset. For any
other state, it is possible that a peak-to-peak amplitude of 14 '0 would isolate one of seventeen
states.
195
isolated for a wider range of ?'A.
The data shown in Fig. 6.11 for DS1 were taken under suppression fleld #2.
The values listed in Table 6.1 might not be exactly right, as they imply flfteen
 ux states at zero bias. The discrepancy could be due to an error in the current
calibration, misidentiflcation of I01?I02 on Fig. 6.8(b), or a drift in critical currents
during the flve months separating measurement of the current- ux characteristics
and  ux shaking. As I02 only needs to be larger by 200 nA to result in seventeen
states, any of the possibilities is plausible.
An alternate explanation is that there was some background  ux biasing the
SQUID when If = 0, which is also entirely likely. In this case, the SQUID could
support an even number of states, as mentioned in the discussion of Fig. 2.16. As
an example, consider the simultaneous bias path shown in Fig. 6.7(d). It crosses the
qubit branch of flve  ux states, which is the maximum number of peaks that will be
seen in a histogram, independent of  ux shaking. Assume instead that the \slope"
of the trajectory is the same, but that it starts at 'A = ?0:5 '0. In this case the
path crosses six branches, all of which can be seen in a switching histogram, as long
as the  ux shaking is performed in the presence of the overall  ux ofiset; i.e. this
ofiset is independent of the applied ofiset ?'A.
Figure 6.12(a) shows the expected number of  ux states for the parameters
listed in Table 6.1 as a function of the applied  ux. Figure 6.12(b) contains the
same information for a SQUID with a slightly larger I02. In both cases, the number
of states is odd when 'A is near an integral number of  ux quanta and even when it
is near a half-integral number of  ux quanta. Thus if the simultaneous ramp used
to measure the histograms actually began at flnite 'A (which we know it does), it
would be possible to see sixteen peaks and the experimental value of the  ux shaking
amplitude in Fig. 6.11 would make sense. In fact, for a SQUID with the parameters
of Fig. 6.12(b), the bias trajectory would have to be carefully set in order to see an
196
(a) (b)
Figure 6.12: Flux dependence of the number of allowed states. The number of  ux
states for a dc SQUID is plotted as a function of the applied  ux at Ib = 0. The
device parameters are I01 = 34:3 ?A, L1 = 3:520 nH, L2 = ?5:2 pH, and Mb = 0,
with I02 equal to (a) 4:4 ?A and (b) 4:6 ?A.
odd number of states; a small ofiset during the simultaneous biasing would lead to
an even number of histogram peaks. This explanation is not inconsistent with the
flfteen peaks seen in Fig. 6.8(b), as the bias path used to generate that data set was
quite difierent.
We can test the model of  ux shaking sketched in Fig. 6.10, by assuming that
the initial multi-peak histogram obtained without any shaking gives the redistribu-
tion probabilities ?'. I will assume that there are seventeen states. In Fig. 6.13(a-c),
oscillations with e'A = 14:8 '0 and ?'A = ?0:4 '0 were used to initialize the sys-
tem to N' = 0; in Fig. 6.13(d-f), N' = 8 was occupied with the same amplitude
and ?'A = 7:7 '0. The bars give the occupation probability of each of the states
after one, flve, and ten oscillations and the dots indicate the levels predicted by the
redistribution model. The agreement is reasonable in all cases with discrepancies
likely due to the simpliflcation of the redistribution process.
In the model, I assumed that  ux states only become unstable when 'A is
equal to an integral number of  ux quanta. At this point, the potential is the same
197
(a) (b) (c)
(d) (e) (f)
(g)
Figure 6.13: Flux shaking with a variable number of oscillations. The occupation
probability p' of each of the  ux states for SQUID DS1 is plotted for shaking that
occupies N' = 0 after (a) one, (b) flve, and (c) ten oscillations. (d-f) Comparable
results are plotted for occupancy of N' = 8. The solid circles show the result
of the simple redistribution model. (g) The occupation probability of N' = ?7
(open bars), N' = 0 (gray), and N' = 8 (hatched) are nearly the same after
N = 10 oscillations that occupy those states, even though their initial values are
quite difierent.
198
as at 'A = 0 (apart from a shift in the coordinates), which motivated the assumption
?' = ?0'. However, as seen in Fig. 6.12, a  ux state will generally become unstable
at some fraction of a  ux quantum during the oscillations. For an increasing  ux,
this happens at 'A = 0:751 '0 and 0:092 '0 in Fig. 6.12(a) and (b). When the
system is forced to flnd a new well, the potential can be quite difierent than at zero
bias. Thus, it is not surprising that the model does not reproduce the data perfectly.
Figure 6.13(g) shows the occupation probability of N' = ?7, 0, and 8 after
N oscillations whose ofisets isolate those states; for example, for N' = ?7, ?'A =
?7:3 '0. Although the high  ux states start ofi with probabilities under 10?4, all
three states reach comparable levels within just a few oscillations. It is also clear
that  ux shaking has its largest impact in the flrst few oscillations. Again, the
agreement between the data and model (circles) is good, although the model seems
to overestimate the occupancy.
We also studied  ux shaking in device DS1 with suppression fleld #3, where
there were many more  ux states. Figure 6.14(a) shows the initial experimental
probability distribution after retrapping. Note in particular that p'(0) = 0:03. After
just flve oscillations with no ofiset [see Fig. 6.14(b)], the distribution changes quite
dramatically; p'(0) increases to 0.09. As Fig. 6.14(c) shows, after 45 oscillations
p'(0) reaches 0.5. The small dots indicate the results of the simple model, which
are in reasonable agreement with the data. One issue is that ?0' was not determined
with a great deal of precision. The insets to Fig. 6.14 shows the same data (with
experimental values shown with solid lines) plotted on a log scale.
For the data in Fig. 6.14, the  ux oscillations had a frequency of 19.2 kHz
with a peak-to-peak amplitude of 166:4 '0. This implies anywhere from 167 to 169
 ux states. Interestingly, we were able to see 167 histogram peaks by varying ?'A.
That the oscillation amplitude is consistent with the number of peaks seen with the
current- ux characteristic obtained by sweeping the  ux positively (for suppression
199
(a)
(e)
(d)
(c)
(b)
p ?
p ?
p ?
p ?
p ?
Figure 6.14: Flux shaking with a large number of states. The bars indicate the
occupation probability of each of 167 allowed  ux states of SQUID DS1, measured
at 20 mK. The dots show the results of the simple model, although some have
been omitted for clarity. The insets have the same information on a log scale, with
experimental data indicated with solid lines. (a) Only a quarter of the states are
occupied with no shaking. The system can be efiectively initialized to the N' = 0
state with (b) 5 or (c) 45 oscillations or to the N' = 82 state with (d) 5 or (e) 45
oscillations.
200
flelds #2 and #3) is what lead me to assert that the entire qubit branch could be
seen with that bias trajectory.
A  ux ofiset ?'A = 83:7 '0 was used for the data in Fig. 6.14(c) and (d),
which isolated N' = 82. The implication is that the calibration must have had
signiflcant error, as the ofiset is about a  ux quantum larger than expected. After 45
oscillations, p'(82) = 0:48, although it was not seen at all in the initial distribution.
Thus even large fl devices can be initialized efiectively and are viable qubits, insofar
as initialization is concerned. A similar technique has been applied to high fl rf
SQUIDs to provide an on-chip precision  ux bias [120].
Although  ux shaking appears to work well with single devices, we were con-
cerned that there could be problems with the coupled qubits that are ultimately
needed. When one device is forced to switch  ux states, it brie y enters the voltage
state before retrapping. While we were unable to measure any voltages with the
detection electronics, this process still could afiect neighboring qubits. For example,
a current pulse could reach a neighbor through a coupling capacitor, causing it to
switch out of the desired well.
We did some preliminary testing on DS2. First, with DS2B grounded, 45
oscillations (with no ofiset) were applied to DS2A at 17 kHz, initializing it to N' = 0
with a probability greater than p'(0) = 0:999. Next, DS2A was grounded and 45
 ux oscillations were applied to DS2B, thereby initializing it with a probability
of p'(0) = 0:986. Finally, oscillating currents were applied to both  ux bias lines
simultaneously. It was di?cult to determine the occupation probabilities of both
devices using the simultaneous biasing, because the switching of one device to the
voltage state could trigger the other, even if both were not initialized to N' = 0.
While it was clear that the probability of DS2A fell by a small amount, the joint
initialization probability was greater than 0.98. From this we conclude that there is
no reason to believe that simultaneous shaking of multiple SQUIDs will not work,
201
but that further tests are needed to optimize the process.
An unexpected beneflt of the difierent  ux states is that they correspond to
difierent amounts of current through the isolation junction and thus difierent levels
of isolation from the bias leads (see x4.3). In the rest of the thesis, unless otherwise
noted, the SQUID was initialized with at most one trapped  ux quantum, where
the isolation from the current bias leads works best.
6.6 State Readout
If junctions or SQUIDs are to be used in performing quantum gate operations,
a reliable method of determining the state occupation needs to be developed. Even
with a more modest goal of understanding the dynamics of these devices, this knowl-
edge is useful. We have used three schemes, sketched in Fig. 6.15, each of which
has advantages and disadvantages. Unfortunately, what they have in common is
that the qubit junction ends in the voltage state on each measurement cycle. The
resulting large electric fleld across the junction may lead to charge motion efiects
in the barrier in addition to heating. In terms of quantum computation, these are
then destructive measurements and are quite difierent from continuous state mea-
surements, where the system is projected to an eigenstate of interest. As the voltage
state is outside of the qubit basis, error correction cannot be implemented on the
qubit. These techniques could be used with auxiliary measurement qubits, which
are needed for error correction.
In principle, it should be easy to \calibrate" any of these techniques. By driv-
ing the 0 ! 1 transition with su?cient power, it should be possible to saturate the
j1i population at very close to 0.5, which is the value a measurement should return.
Unfortunately, with the short coherence times of our devices, it was di?cult to pro-
duce a precisely known saturation without also exciting a non-negligible occupation
of j2i. As a result, evaluating the performance of the measurements was a bit more
202
Control Readout
?t
0
1
3
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
?01
?01
?01
?12
(a)
(c)
(b)
Figure 6.15: Schemes for state readout. The left column shows the potential, energy
levels, and tunneling rates during state manipulation (microwave activation at !01 in
this case). The right shows the conditions for readout. (a) For the direct tunneling
measurement, the bias current is set so ?0 and ?1 are both measurable. (b) With
a microwave pulse readout, the anharmonic nature of the potential is exploited by
exciting only the 1 ! 2 transition to perform readout. (c) Lower bias currents can
be used with the bias pulse readout, where the bias is held high for a short time
?t, long enough for j1i (but not j0i) to escape. While the potentials have been
drawn for realistic values of Ib, the anharmonic level spacing has been exaggerated
for clarity.
203
involved.
6.6.1 Direct Tunneling
Unless otherwise noted, the method I used for state readout was to measure
the escape rate to the voltage state. The total escape rate, which is experimentally
accessible, is the sum of the individual state populations weighted by the corre-
sponding escape rates. As the ratio of escape rates between adjacent energy levels
is generally quite large, this is a particularly sensitive way of detecting excited state
population.
For example, assuming that only the two lowest states are occupied, Eq. (3.30)
gives the total escape rate as ? = P0?0+P1?1, where Pi is the normalized probability
of being in state jii [see Eq. (3.31)]. Using P0 + P1 = 1, we can write the excited
state population due to a microwave current as
P1 = ???0?
1 ??0
? ?0?
1
??
? ; (6.9)
where the approximation holds for ?1  ?0 and under the assumption that the
escape rate ?bg measured without microwaves is equal to ?0. Thus the escape
rate enhancement ??=? deflned in Eq. (6.7) is proportional to P1, and P1 can be
extracted once ?0=?1 is estimated from theory. A major source of error is that even
a small population in j2i can have a dramatic efiect on ?, due to the size of ?2.
At times, the sensitivity to upper levels makes the technique quite useful. For
example, consider a junction at flxed bias in dynamic equilibrium. In this case, the
normalized probabilities Pi will be constant. Thus, ? will also be constant even as
the occupation probabilities ?i decay to zero due to tunneling. However, if inter-
level transitions or tunneling move the system away from equilibrium, then Pi and ?
will change and the total escape rate will give an accurate picture of the dynamical
204
processes of the junction.
The power of this technique lies in its simplicity and its well-understood char-
acteristics. It only requires that a particular experiment be repeated many times
to build up a histogram of switching time with good statistics. No extra timing or
calibration is required. In some sense, it is a passive measurement, where no poten-
tially disruptive action is taken on the junction, other than the switching itself. As
one histogram will give information over a range of bias currents (if a current ramp
is being used), it is easy to collect a fair amount of data quickly.
The technique also has several drawbacks, particularly with respect to quan-
tum computation. For one, we do not initiate the measurement. The junction can
escape before, during, or after the manipulation that is being performed. Also, tun-
neling can shorten the lifetime of an excited state in a very direct way and thus itself
is a source of decoherence. These decohering efiects are strongest at high escape
rates, where it is easiest to take data.
Perhaps the most serious issue for the work presented here is that it can be
di?cult to extract the populations from the total escape rate, because the individ-
ual rates are not necessarily known beforehand. While ?0 can often be measured
precisely, the higher escape rates are easier to predict than to measure. Work is still
ongoing in the group to obtain accurate experimental data of ?1 and ?2, using the
technique described in x6.6.3 [104].
Lastly, it is only easy to measure escape rates between 102 and 108 1/s, which
restricts the range of bias currents that can be studied. For slower rates, each
repetition will take a long time. To reach faster rates, the bias current has to be
swept quickly so that the junction will not escape prematurely. As ? increases
exponentially, the accessible escape rate range corresponds to a bias current range
of less than one percent of the critical current for the types of devices we study.
205
6.6.2 Microwave Pulse
Someoftheshortcomings oftheprevioustechniquecanbeaddressedbyusinga
microwave pulse to perform state readout [39,44]. This approach takes advantages of
the anharmonic potential, which leads to unequal energy level spacings. The general
idea is that the junction is biased where both ?0 and ?1 are so low that the system
is unlikely to tunnel during the course of a particular experiment. Manipulations
can then be performed between j0i and j1i without fear of tunneling. When it
comes times to perform a measurement, a microwave pulse of frequency !12 < !01
is applied. If the junction is in j1i at this time, it will be excited into j2i. It will
then quickly tunnel to the voltage state, if ?2 is large enough. If the junction is in
j0i at the time of the pulse, the pulse ideally has no efiect and the junction stays in
the zero-voltage state.
Knowledge of individual escape rates is not needed and measurements can be
performed at even lower bias currents by using a pulse of frequency !13. In principle,
by adjusting the length and power of the pulse, this technique can be a \single-shot"
measurement. For each trial, one only has to know if the junction tunneled to the
voltage to know whether it was in j0i or j1i. In addition, the measurement is
performed at the same current bias as the manipulation, which is an advantage that
the technique discussed in the next section does not have.
In practice, we found that the excitation process was too ine?cient to realize
a single-shot measurement. At low microwave power, the excitation rate may not be
su?ciently faster than the relaxation rate from j2i to promote all of the population
in j1i. The slow rates may also lead to a long measurement time, making it di?cult
to resolve fast dynamics (for example, during a Rabi oscillation). At high power,
the resonances broaden and the system will tunnel even if it is in j0i. Also, it can
be somewhat di?cult to determine the resonance condition accurately, especially
for energy levels deep in the potential well. For these reasons, this technique has a
206
limited \measurement fldelity" associated with it that the direct escape rate does
not sufier from. The initial results were not very promising, so I never quantitatively
determined the e?ciency of this technique at optimal conditions or tried a 1 ! 3
transition. A measurement of the relaxation time determined with this technique is
described in x8.7.
6.6.3 dc Bias Pulse
The central idea of the microwave pulse measurement, as previously discussed,
is to selectively force the population in j1i to tunnel to the voltage state on demand
by promoting it to a state with large escape rate. In the dc bias pulse technique,
this is accomplished instead by freezing the populations, but quickly increasing
all of the escape rates by applying a quick pulse of current (or  ux) bias. If the
height and width of this pulse are chosen carefully, the system will only escape
if it started in j1i. This process is somewhat reminiscent of the standard way of
manipulating charge qubits and has been applied to charge-phase [31],  ux [121],
and phase qubits [43,114] successfully.
As with the microwave pulse, the dc bias pulse can be a single-shot mea-
surement. It is, however, easier to implement, faster, more e?cient, and can give
information about the population in j2i in a natural way.
Care has to be taken that the pulse itself does not excite population into higher
states. In practice, a line normally used for microwaves was used to supply the pulse.
The small capacitive coupling from this line to the junction and the limitations of
the pulsing electronics probably kept the Fourier components of the pulse well below
the plasma frequency of the junction.
Another issue with this technique is that it involves sweeping the junction
through a potentially large range of current. If there are, for example, spurious
resonances that couple to the junction (see x8.5), the system will decohere while
207
passing through. This efiect has been claimed to cause a reduction in measurement
fldelity [98]. However, for our devices the pulses are very fast (roughly 2 ns) and the
splittings are very small (less than 5 MHz), so that this efiect should be small [104].
6.7 Summary
I began this chapter by showing IV curves of the junctions. These provide
a fairly straight-forward method of determining the junction quality, including the
barrier uniformity and leakage. It was di?cult to measure the sub-gap resistance,
but it appears to be greater than 100 k? at dc. This sets the minimum intrinsic
dissipation of any of our qubits.
The escape rate measurement which I described in x6.2 is our primary method
of determining the behavior of our junctions. Our basic experiment involves mea-
suring the time at which a junction switches to the flnite voltage, with respect to,
for example, the start of a current bias ramp. By repeating the experiment many
times, we can calculate the rate of tunneling ? from the ensemble statistics. ? is
very sensitive to excited state population and is one way to readout the state of a
qubit. However, the pulsing techniques in x6.6 are more powerful, although they
still need to be optimized.
We would like to operate our SQUIDs so that one junction serves as a qubit
and the other provides isolation, where the qubit acts as a single current-biased
junction. This can be accomplished by initializing the  ux state to a particular
value (x6.5) and applying a current and  ux bias in such a way that adds no cur-
rent through the isolation junction (x6.4). The bias trajectory to follow can be
determined experimentally by measuring the current- ux characteristics (x6.3).
208
Chapter 7
Tunneling Escape Rate Measurements
We can learn a great deal of information about phase qubits by studying
tunneling from the zero-voltage state. All of the measurements described in this
chapter involve simply measuring the rate of tunneling (which is a very sensitive
gauge of excited state occupation probability) as a function of the current bias,
as described in x6.2. There are three main goals for these experiments. First of
all, the more experimental results we can explain, the more confldence we have
that our model of the device is accurate. Secondly, we can use the tunneling data
to determine some key device parameters, such as critical currents. Finally, for
quantum computation, measurements of the dissipation time T1 are very important,
as T1 sets an upper bound on the time during which quantum gate operations can be
performed. Estimates for the dissipation can be found by considering the interplay
of the unknown decay rate with other processes that we understand. For example,
changing the temperature of the junction results in thermal transitions governed by
the Boltzmann factor and the bias ramp rate has a dynamical efiect that is fairly
easy to characterize.
Figure 7.1 shows the total escape rate ? of junction LC2B measured at six
temperatures T (as determined by the thermometer at the refrigerator?s mixing
chamber). At 25 mK, ? is roughly exponential in the current bias Ib, as one would
expect for tunneling from the ground state alone (see x2.3.3). As the temperature
increases, higher states become thermally populated. These states have higher tun-
neling rates, leading to an enhanced ?. Because the escape rate measurement works
over a particular range of ?, it was possible to obtain data over a difierent range
of Ib for each temperature. The scatter at both ends of each curve is due to poor
counting statistics. At high Ib, several of the curves appear to approach each other.
209
Figure 7.1: Temperature-dependent escape rates of junction LC2B. For a given cur-
rent bias Ib, the total measured escape rate ? increases quite dramatically with the
temperature of the mixing chamber as higher energy levels are thermally occupied.
The data was taken at a ramp rate of 0.07 A/s with the suppression fleld Bk set to
2.7 mT, by Huizhong Xu.
The goal of much of this chapter is to describe the various features of data sets
of this kind. I will flrst show that a classical model of the junction reproduces the
high temperature data. However, to describe the lowest temperatures, the quantum
results of x2.3 must be employed. With the two extremes covered, I will then
summarize the master equation formalism that can be used to describe experiments
in which the dynamics occur on a time scale much longer than the coherence time T2.
Finally, I will describe how both slow and fast bias ramps can be used to measure
the dissipation time T1.
210
7.1 Thermal Activation and Macroscopic Quantum Tunneling
One approach to understanding the temperature series shown in Fig. 7.1 is to
treat the junction classically, as was flrst done over 30 years ago [60]. Deviations
from classical behavior formed some of the flrst evidence for macroscopic quantum
tunneling in these systems [23,122,123]. This approach can provide a conflrmation
of the description of the circuit and a simple method for determining the dissipation
in the system [111,124].
The classical analysis begins with the assumption that the Josephson relations
and the properties of the tilted washboard given inx2.2.2 are valid. The assumptions
will be justifled if the resulting model describes the data faithfully. The classical
rate at which thermal activation allows the phase particle to escape the potential
well is [61,125]
? = at !p2? exp
 
? ?Uk
BTesc
?
; (7.1)
where !p is the plasma frequency [given by Eq. (2.32)], ?U is the barrier height of
the well [given by Eq. (2.30)], and at is a classical thermal prefactor that param-
eterizes damping. Tesc can be thought of as an efiective escape temperature that
characterizes the lifetime of the supercurrent state. In a classical system, Tesc will be
equal to the refrigerator temperature T and will be independent of Ib.1 In describing
experimental data with this model, neither may be true.
Tesc can be calculated for each value of ?, by inverting Eq. (7.1) [68,123]. The
di?cultyofthismethodisthatthejunctioncriticalcurrentI0 mustbeindependently
measured in order to evaluate !p and ?U. Alternatively, Eq. (7.1) may be rewritten,
1Alternatively, Tesc can be deflned by Eq. (7.1) with at = 1 [123]. In this case, Tesc 6= T in the
presence of damping and it could be a function of Ib.
211
using the approximation for the barrier in Eq. (2.31), as
h
ln
?at !p
2??
?i2=3
=
?
4p2
3
EJ
kBTesc
!2=3
I0 ?Ib
I0 : (7.2)
If Tesc = Tesc (T) is taken to be a constant with respect to Ib, then the expression on
the right side is linear in Ib and the equation can be written as !? = c1Ib+c0, where
!? ? [ln(at !p =2??)]2=3. Using an initial guess for I0, the data can be flt to a line,2
with the fltting parameters giving the critical current and escape temperature as
I0 = ?c0c
1
and Tesc = ?4
p2
3
'0
2?
1
kB
1
c1pc0 : (7.3)
This extracted value of I0 can then be used to recalculate !? and another flt per-
formed. This process converges in just a few iterations to self-consistent values of
I0 and Tesc [111,124]. A value for the junction capacitance CJ is still needed to
calculate !p, but I have found that Tesc generally changes negligibly when CJ is
varied by as much as 50%.
7.1.1 LC-Isolated Phase Qubits
I performed the Tesc fltting procedure on the data shown in Fig. 7.1. !? and
the best flt line (whose parameters were used to calculate !?) are plotted in Fig.
7.2(a). The open points in Fig. 7.3(a) and (b) show the extracted values of Tesc and
I0 from the flts, as a function of temperature (for more values than shown in Fig.
7.1). Based on the quality of the flts, the uncertainties in Tesc and I0 are roughly 3%
and 4%, respectively. In doing the analysis for Fig. 7.1(a) and the open circles in Fig.
2I used the Levenberg-Marquardt method for these flts (with uncertainties determined from
counting statistics of the histograms); see x15.5 of Ref. [69]. The algorithm generates a covariance
matrix; the uncertainties in the fltting parameters are given by the square root of the main diagonal
of this matrix. Unless otherwise noted, this is always the algorithm I used for nonlinear fltting or,
as in this case, for linear fltting with unequal weighting.
212
(a)
(b)
Figure 7.2: Thermal activation in junction LC2B. Each of the curves of Fig. 7.1 is
roughly linear when plotted as !?, as predicted by Eq. (7.2). The solid lines show
the end results of an iterative fltting procedure, with the reduced ?2 shown in the
insets (for more temperatures than shown in the main flgure). The quality of the
flts at high temperature is worse (a) assuming no damping (at = 0) than for (b)
RJ = 2 k?. The reverse is true at low temperature.
213
7.3(a) and (b), I have assumed that at = 1. In addition, I have taken CJ = 4:2 pF,
consistent with the design value and the spectroscopic measurements discussed in
the following chapter.
I will flrst discuss the results for T > 50 mK (with at = 1). Figure 7.1(a)
shows that !? is roughly linear in Ib, suggesting that the junction is displaying
classical behavior. However, the data do show some curvature and other features,
as re ected in the large reduced chi-square (or ?2 per degree freedom) ?2?, shown
in the inset to Fig. 7.2(a), particularly at low temperature. In addition, Eq. (7.2)
predicts that the linear flts will coincide at the point (!? = 0, Ib = I0), but they do
not. The lowest temperature curves appear to disagree the most. In fact, the best
flt value of I0 [open circles in Fig. 7.3(b)] varies quite strongly with T. Although
Tesc does increase in proportion to T, it does not agree as well as one might expect
if the theory were complete.
Some of the deflciencies of the classical theory can be addressed by including
the efiects of damping. In the limit of low damping, the thermal prefactor is [61]
at = 4hp
1+(QkBT=1:8?U)+1
i2 ; (7.4)
where Q = !p RJCJ is the bias-dependent quality factor of the junction, determined
by the efiective shunting resistance RJ of the RCSJ model (see x2.2). Notice that
at = 1 if Q = 0, and this is what I used for the flts discussed in the previous
paragraphs.
I repeated the iterative fltting procedure with RJ = 2 k?. A selection of the
flts are shown in Fig. 7.2(b) and self-consistent values of Tesc and I0 are plotted
as solid circles in Fig. 7.3(a) and (b). Figure 7.3(c) shows the calculated values of
at (Ib) for the lowest (solid) and highest (dashed) temperatures.
Although the flts at low temperature are somewhat worse with the inclusion
214
(a)
(b) (c)
Figure7.3: EfiectiveescapetemperatureforjunctionLC2B. Thebestflt(a)efiective
escape temperature Tesc and (b) critical critical current I0 are plotted as a function
of the refrigerator temperature T, with CJ = 4:2 pF. Several of the escape rate
curves that were analyzed to generate these points are plotted in Fig. 7.1. The open
points correspond to at = 1, while the solid are for RJ = 2 k?. An arrow marks the
prediction for the zero-temperature escape temperature T0esc from Eq. (7.5). (c) The
calculated values of at for RJ = 2 k? at T = 25 mK (solid) and 320 mK (dashed)
are strongly bias-dependent and show that dissipation has a signiflcant efiect on
activation.
215
of damping, ?2? for the four highest values of T are nearly equal to 1 [see Fig.
7.2(b)]. For these values, the extrapolation of the flts do nearly coincide at !? = 0:5.
Furthermore, the solid circles of Fig. 7.3(a) show that the values of Tesc now lie very
close to T, which is how I selected the value of RJ to use. Finally, the variation in
the extracted values of I0 has been reduced to about ?0:1%. All of this suggests
that the classical theory with damping is accurately describing the switching of the
junction to the voltage state at high temperatures.
Nevertheless, there are several sources of error in this experiment that need to
be considered. First of all, there are the usual concerns with the current calibration.
As all of the data were taken with the same nominal bias ramp, this is unlikely to
be a serious problem and likely just results in an overall shift of I0. However, we
often saw shifts in the switching histograms as a function of T that were clearly
unrelated to junction dynamics. Although I was not able to flnd their origin, it is
possible that these shifts were due to the current supplied to the mixing chamber
heater or a temperature-dependent resistance on the current bias line. Fortunately,
even if I shifted Ib by 2% in doing the analysis, the extracted value of I0 would shift
by a comparable amount, but Tesc changed negligibly. As the experimental Ib shifts
were generally less than 20 nA, they should not afiect the analysis signiflcantly. The
other problem is the calibration of the mixing chamber thermometer, as discussed
in x5.1.1. It is quite possible that the values of T could be incorrect by 10% or more
at high temperatures. In addition, the temperature of the sample could be difierent
than that of the thermometer. Hopefully, this error is somewhat smaller than the
corrections introduced by damping. Finally, the junction capacitance is required in
the analysis. Fortunately, as I noted earlier, Tesc has a particularly weak dependence
on CJ. Herein lies the power of this technique. With a simple experiment and an
analysis method that makes only a few assumptions and is relatively insensitive
to experimental parameters, an estimate for the shunting resistance (and thus the
216
relaxation time T1 = RJCJ) can be easily found; in this case, T1 ? 8 ns.
Turning now to the low temperature data, Tesc saturates at T ? 50 mK (both
with and without the inclusion of damping), meaning that even when the thermo-
dynamic temperature was 20 mK, the junction behaved as a classical system would
at 50 mK. If the junction were truly classical, then Tesc would continue to decrease,
as required by Eq. (7.1). At T = 0, there would be no thermally activated switching
and the junction would go to the flnite voltage state only when Ib ? I0 (in the
absence of electrical noise). The saturation behavior is consistent with quantum
tunneling; even at absolute zero with the system in the ground state j0i, it will
tunnel out at a rate ?0.
The theoretical ground state escape rate of a junction was described in x2.3.
By comparing Eqs. (7.1) and (2.42), the efiective escape temperature at T = 0
(where at = 0) for the cubic approximation to the washboard potential is
T0esc = ~!pk
B
1
7:2(1+0:87=Q)
"
1? ln
p120?(7:2N
s)
7:2Ns (1+0:87=Q)
#?1
; (7.5)
where Ns is the harmonic approximation to the number of levels in the well, deflned
in Eq. (2.37). This expression provides an interesting way to predict the low tem-
perature value of Tesc in the quantum limit, as the input parameters (I0 and RJ)
can come from fltting high temperature data to a classical theory. The transition
between quantum and classical character occurs at the crossover temperature Tcr.
In the case of low damping, a simple estimate for this temperature is [126,127]
Tcr ? ~!p2?k
B
: (7.6)
For the devices that I have studied, the base temperature of the dilution refrigerator
was always well below Tcr. With regards to quantum computation, the expression
shows that as the critical current density decreases, the temperature required to
217
initialize a junction to its ground state also decreases.
As tunneling has a difierent functional form than thermal activation, T0esc is a
function of Ib. I will choose to evaluate it where ? can be measured precisely. The
\middle" of the base temperature ? curve in Fig. 7.1 is at about 105 1=s, where
Ib ? 33:41 ?A. Near this value of Ib (with I0 = 33:6 ?A and CJ = 4:2 pF),
!p=2? ? 8 GHz and T0esc is roughly 85 mK, with an uncertainty of about 10 mK.
With RJ = 2 k?, dissipation gives a correction of less than 1 mK; this is consistent
with Fig. 2.10, which suggests that damping does not greatly afiect ?0 in our devices.
An arrow marks T0esc on Fig. 7.3(a), as in all of the Tesc plots in this section. Given
the uncertainty in I0 and CJ, there is perhaps reasonable agreement with the open
circles, which were also calculated with at = 1.3 The estimate for the crossover
temperature is 60 mK at the chosen value of Ib, which is roughly the value of T
where saturation begins.
A useful feature of the LC-isolated devices is that a suppression magnetic fleld
Bk could be used to adjust the junction critical current (see x6.1). At Bk = 4:4 mT,
I0 for junction LC2A was reduced to 21:5 ?A. Even at the base temperature of the
refrigerator, the measured escape rate was exponential in Ib only to a certain value.
Above 5 ? 107 1=s, ? rolled ofi noticeably. It is possible that this was an artifact
of the detection scheme. If the voltage that develops across the junction when it
switches to the running state does not have the same time dependence on each trial,
then a smearing will be introduced in the switching histogram.4 The same efiect
can occur if there is a loss of timing resolution from sources external to the junction.
For example, this data set was taken with a high ramp rate of 0.52 A/s, leading
to short switching times. The highest escape rates correspond to a small number
of counts, so they can be easily distorted. In this particular case, the switching
3The solid circles were calculated using a classical theory of dissipation. Thus, the disagreement
of these points with T0esc and the increasing Tesc with decreasing T at low temperature only suggest
that the theory does not apply in this temperature range, as expected.
4Plots of this voltage, as measured at the top of the refrigerator, are shown in Fig. 5.15(a).
218
(a)
(b)
Figure 7.4: Efiective escape temperature for junction LC2A. The best flt value of
the escape temperature Tesc is plotted against the refrigerator temperature T, for
analysis with (solid) and without (open) damping. (a) When !p=2? ? 6:5 GHz
(Bk = 4:4 mT), near agreement is found between the two temperatures for RJ =
2 k?, although the flts were performed for ? < 107 1=s. (b) At the unsuppressed
critical current, where !p=2? ? 13 GHz, a smaller value of 500 ? is needed. The
insets show the self-consistent values of I0 from the iterative fltting procedure.
219
waveforms did not look out of the ordinary, but the distortion of the escape rates
was quite pronounced. As the lower ? data behaved as expected, I kept ? < 107 1=s
in the analysis that follows.
Figure 7.4(a) shows Tesc and I0 obtained for LC2A from the iterative fltting
procedure under these conditions. The open circles were calculated with at = 1 and
show a hint of saturation at low T. I used CJ = 4:8 pF in this case, which was the
value derived from the spectroscopic method described in the next chapter, but Tesc
is relatively insensitive to small changes in the capacitance.5 At high temperatures,
Tesc agrees with T for RJ = 2 k?. With this value of RJ, the flts of !? vs. Ib are of
higher quality than those with at = 1, but ?2? does not improve as dramatically as
it does in Fig. 7.2. Not shown is that !? for the difierent temperatures do coincide
near !? = 0 with the inclusion of damping, as expected. At T = 24 mK, ? = 105 1=s
at Ib = 21:32 ?A. Taking I0 = 21:45 ?A, Eq. (7.5) predicts T0esc ? 65 mK, consistent
with the data (open circles). At this point !p=2? ? 6:5 GHz, corresponding to a
crossover temperature of 50 mK.
The data used to generate Fig. 7.4(b) were taken in the absence of a suppres-
sion fleld, so the critical current was at a maximum. The escape rates again were
not featureless (although not rounded ofi), but restricting the fltting range had little
efiect on Tesc, so I kept all of the data. The solid dots were calculated for CJ = 5 pF
(again, from spectroscopy) and RJ = 500 ?. Although the flts were not perfect, the
classical model clearly does not work for the value of 2 k? used for the previous data
sets. As before, the coincidence of !? provides a good measure for the quality of the
RJ determination (not shown). At Ib = 121:15 ?A, the base temperature escape
rate is 105 1=s. At this point with I0 = 121:54 ?A, T0esc ? 130 mK, !p=2? ? 13 GHz,
and Tcr ? 100 mK.
5The capacitance of the junction should not be a function of Bk, but the coupling between
the junction and the LC mode created by the coupling capacitor depends on !p and leads to a
renormalized capacitance. It is not entirely clear how this afiects the classical theory.
220
It is perhaps telling that the shunting resistance is smaller at the highest
critical current. The two data sets of Fig. 7.4 were taken within two weeks of each
other (Fig. 7.1 was taken seven months earlier on a difierent junction) and there
were no concerns with the device at this time. In fact, the unsuppressed data were
taken three weeks after thermally cycling the refrigerator above 20 K, after which
the fleld had never been turned on. However, the base temperature escape rate
was not exactly exponential, so it is possible that there were problems. Of course,
a possible explanation for the discrepancy is that the classical activation model is
breaking down in some subtle way. Alternatively, it could be that RJ varies with
frequency. It is not clear whether this is due to the design of the isolation network
itself or, for example, resonances in the sample box or electrical components.
7.1.2 dc SQUID Phase Qubits
The same fltting procedure can be applied to the escape of the dc SQUID
phase qubits. With the simultaneous biasing described in x6.4, the qubit junction
flrst escapes to the voltage state, while the isolation junction remains near zero bias.
In this case, the thermal theory for escape from a 1-D potential should approximate
the behavior of the SQUID. I will give further details of the measurement of ? for
SQUIDs in x7.2.2.
Escape rates for DS1 are shown in Fig. 7.5(a) for a range of temperatures.
The current I1 through the qubit was calculated from the calibrations of the current
and  ux biases, Ib and If, and the device parameters in Table 6.1 for fleld #2. I
assumed that there was no  ux ofiset (at If = 0), which could result in a small shift
in I1. At the lowest temperatures, a noticeable bump appears at I1 = 33:89 ?A. The
feature persists to higher temperatures, but gets washed out as higher states become
thermally occupied. There will be a brief discussion of this and other features in
the next section.
221
(a)
(b)
Figure 7.5: Efiective escape temperature for SQUID DS1. (a) As in Fig. 7.1, high
temperatures show an enhancement in the total escape rate ?. However, in the dc
SQUID devices, additional features are seen at the lowest temperatures. The lines
show flts of the thermal activation theory to the T = 170 mK data, with RJ = 0
(solid), 2 k? (dashed), and 8 k? (dotted). (b) Fits of this sort also yield the efiective
escape temperature Tesc and the self-consistent critical current I0, for at = 1 (open
circles), RJ = 2 k? (solid circles), and RJ = 8 k? (triangles).
222
The extracted values of Tesc and I0 for DS1 are shown in Fig. 7.5(b), for at = 1
(open circles) and RJ = 2 k? (solid circles), with CJ = 4:4 pF. The presence of
the bump does call the results of the flts into question. However, when I restricted
the flts to 105 < ? < 107, Tesc only changed by a few percent at all but the highest
temperature. The agreement between T and Tesc with damping is not as good as
it was for the junction LC2. Also, the SQUIDs were expected to provide better
isolation, corresponding to a larger RJ. Increasing RJ further would bring the
higher temperatures into line, but would send the intermediate temperatures below
the T = Tesc line.
Figure 7.5(b) shows that good agreement is found by restricting the flts to
? < 106 and setting RJ = 8 k? (plotted as triangles). The predicted values of ?
at T = 170 mK from Eq. (7.1) are plotted in Fig. 7.5(a) for at = 0 (solid line),
RJ = 2 k? (dashed), and RJ = 8 k? (dotted). The last two match the data well at
low bias, but the RJ = 8 k? curve clearly underestimates ? at high I1. In addition,
the convergence of !? is not nearly as good as it is for RJ = 2 k? (not shown).
Thus I can only restrict the shunting resistance to the range 2 k? < RJ < 8 k?,
which corresponds roughly to 10 ns < T1 < 30 ns.
At T = 20 mK, the escape rate is equal to 105 1=s at I1 = 33:92 ?A. At this
current with I01 = 34:1 ?A, T0esc ? 80 mK, !p=2? ? 8 GHz, and Tcr ? 60 mK.
Aside from the usual concerns about the temperature calibration and detection
errors, there could be several  aws in the theory. As I will discuss in x7.4, even at
high temperatures, the quantum nature of the junction asserts itself quite clearly at
high current bias, which could explain some of the discrepancies there (and perhaps
why this was not as serious a problem for LC2, where the damping was stronger).
It could also be that RJ is a function of temperature [65,66], but the relatively
small range of temperatures measured should correspond to a negligible variation.
However, the T = 170 mK curve covers a current range corresponding to almost 3
223
Figure 7.6: Efiective escape temperature for SQUID DS2B. The best flt value of
the escape temperature Tesc is plotted against the refrigerator temperature T, for
at = 1 (open) and RJ = 1:5 k? (solid). Signiflcant heating was observed when then
junction was left in voltage state for 400 ?s (squares), as compared to when it was
forced to retrap quickly (circles). The inset shows the self-consistent values of I0
from the iterative fltting procedure.
GHz in !p=2?. Therefore, if RJ were a rather strong function of frequency, it could
result in apparent temperature dependence, as each curve covers a difierent range of
I1 (because the escape rate measurement is sensitive to a flxed range of ?). Finally,
although we bias the device so that it will act as a single junction, the potential
that determines its dynamics is two-dimensional. Particularly at high temperature,
the phase particle could explore a wider range of escape trajectories, as previous
experiments have reported both in the classical [119] and quantum [128] regimes.
The open circles in Fig. 7.6 show Tesc and I0 for SQUID DS2B, for CJ = 4:5 pF
and at = 1. !? was calculated with ? as a function of the current bias Ib, rather
than the qubit current I1, because I did not know all of the device parameters well
enough to extract I1. Based on the previous data set, I found this introduces very
224
little error in Tesc. The solid circles in Fig. 7.6 were calculated with RJ = 1500 ?
(corresponding to about 7 ns) and show a smaller, but still prominent disagreement
between Tesc and T. The disagreement is similar to what I saw in DS1 in Fig. 7.5(b).
If these devices are to be used for quantum computation, it is critical to determine
whether the low values of RJ accurately describe the junction?s isolation or if they
are due to  aws in the theory (or its application). Nonetheless, the agreement is
qualitatively correct and many quantitative aspects also agree well. For example, I
will predict the characteristic temperatures at the bias current where ? = 105 1=s.
At Ib = 19:42 ?A, T0esc ? 60 mK, !p=2? ? 7 GHz, and Tcr ? 55 mK, in good
agreement with the open circles of Fig. 7.6.
One concern in claiming that the low temperature saturation of Tesc is due to
quantum tunneling is that the junctions may not be getting as cold as the mixing
chamber thermometer indicates. That the minimum escape temperature of a par-
ticular device varies with critical current, as in Fig. 7.4, is a good indication that
this is not the case. Applying heat in an alternate way serves as another check,
which I will discuss next.
As described in x4.3.2 and x5.2.2, DS2 showed fairly strong heating efiects if
the junctions were left in the flnite voltage state. In Fig. 7.6, I forced the junctions to
retrap within 2 ?s of the end of the histogram peak, for the values plotted with open
and closed circles. For the squares, I allowed the junctions to stay in the voltage
state for 400 ?s, with Ib linearly decreasing to zero 200 ?s after a switch. Tesc
with at = 1 (open squares) and with RJ = 1500 ? (solid squares) look qualitatively
similar to their unheated counterparts and are nearly equal to them for T > 150 mK.
Below 150 mK, the curves diverge as heating becomes evident. In doing the flts of
!? vs. Ib to extract Tesc (not shown), the expected intersection of the curves at high
bias (of the sort shown in Fig. 7.2) is not as good as for the unheated data. This by
itself might not be a serious cause for concern. However, additional problems are
225
apparent. Based only on the heated data, Ib = 19:37 ?A when ? = 105 1=s, where
T0esc should still roughly be 60 mK, for I01 = 19:62 ?A. At the smaller current bias,
the plasma frequency increases by 500 MHz at most, corresponding to Tcr ? 60 mK.
Both of the characteristic temperature predictions are well below the values seen in
the plots of the heated data. The lack of consistency between T0esc and the observed
saturation of Tesc is one piece of evidence that we are not observing macroscopic
quantum tunneling in the heated data set. For this device, the saturation level is
slightly above T0esc, even when the junctions retrap quickly, so more careful tests are
needed in order to check if smaller heating efiects are present. The slow retrapping
data in Fig. 7.6 also suggest that leaving the junction in the running state has the
same efiect as increasing the refrigerator?s temperature.
As I discussed in Chapter 2, the retrapping current Irt also depends on the
junction shunting resistance [see Eq. (2.35)]. Therefore, it is possible to estimate RJ
by measuring a retrapping histogram [129,130]. This becomes easier at high temper-
atureswherethermallyexcitedquasiparticlesdecreaseRJ bythefactorexp(?=kBT),
where 2? is the superconducting energy gap. For niobium, ?=kB ? 16 K, so exper-
iments would have to be done at relatively high temperatures to obtain an easily
measurable retrapping current. It would be interesting to compare efiective resis-
tance values obtained from both switching and retrapping, but I did not obtain
su?ciently precise measurements of the IV curves to allow this to be done in a
meaningful way.
7.2 Low Temperature Escape Rate
In the previous section, I compared the low temperature (saturated) value of
Tesc to the quantum prediction, in order to judge whether the junction was escaping
due to quantum tunneling from the ground state at ?0. Rather than characterize
the full curve with a single number, I will now examine the bias dependence of ?
226
measuredatthebasetemperatureoftherefrigerator. Unfortunately, simplylowering
the thermodynamic temperature of a junction does not guarantee the absence of
excited state population.
7.2.1 LC-Isolated Qubits
Figure 7.7(a) shows the total measured escape rate ? versus bias current Ib of
the LC-isolated junction LC2B at T = 25 mK. The data plotted as open squares
were taken at a bias ramp rate of 0.07 A/s (identical to the data in Fig. 7.1), while
the solid circles were taken at 0.93 A/s. Because the junction is given less time to
escape at lower ?, a faster ramp rate allows a measurement of higher escape rates.
In order to convert switching time to Ib for this plot, we performed a calibration
using the escape rate method described in x5.6, for both the 0.93 and 0.07 A/s rates.
This gives a reliable value for the slope of the ramp, but the \ofiset" is more di?cult
to measure. As a result, the escape rates for the two sweeps were close but did not
lie on top of each other, although we have every reason to believe they should. The
calibration was more accurate for the slower rate, so I added an ofiset of 15 nA to
the 0.93 A/s data so that the two data sets were coincident.
For comparison, the solid line in Fig. 7.7(a) is the theoretical ?0 for a single
junction with I0 = 33:648 ?A and CJ = 4:24 pF, calculated with Eq. (2.43). At
flrst glance, the good agreement over nearly seven orders of magnitude is quite
encouraging and conflrms our description of the device. However, the iterative fltting
procedure for the thermal theory with damping returned a critical current closer to
33:6 ?A [see solid circles in Fig. 7.3(b)]. A 50 nA difierence over such a small range
is somewhat troubling. A bigger concern comes from comparison to the I0 and CJ
parameters found by fltting the spectrum of excitations to theory (see Fig. 8.1(b)
in x8.1). For that measurement, the extracted parameters are I0 = 33:663 ?A and
CJ = 7:32 pF. ?0 for these spectroscopically derived values is plotted as a dashed
227
(a)
(b)
Figure 7.7: Low temperature escape rate of junction LC2B. (a) With I0 ? 33:6 ?A,
the measured escape rate at T = 25 mK agrees for a 0.07 A/s (squares) and 0.93
A/s (circles) bias ramp rate. A flt to the data (solid line) yields I0 = 33:648 ?A
and CJ = 4:24 pF. Spectroscopic measurements (see x8.1) give I0 = 33:663 ?A and
CJ = 7:32 pF, corresponding to a lower ? (dashed). (b) When the critical current is
further suppressed to I0 ? 15:4 ?A, the escape rate follows that of a junction with
I0 = 15:434 ?A and CJ = 3:4 pF (solid), except near a noise induced feature at
15.22 ?A. Spectroscopy gives I0 = 15:420 ?A and CJ = 5:65 pF, parameters which
again underestimate the measured ?.
228
line in Fig. 7.7(a). It is likely that some of the disagreement is due to the device
(LC2A) coupling to its neighbor, qubit LC2B, through the coupling capacitor CC
and inductor LC.
The efiect of the LC mode created by the coupling networkcould be reduced by
lowering the critical current of LC2B. The circles in Fig. 7.7(b) show the measured
values of ? at T = 20 mK when IB0 ? 15:4 ?A.6 The solid line shows the flt to
?0 using I0 = 15:434 ?A and CJ = 3:4 pF, which agrees well with the data. The
dashedline isdrawnforthejunctionparameters obtained from spectroscopy[see Fig.
8.1(a)]. The disagreement between the curve derived from spectroscopy and the data
is not as bad as it is in Fig. 7.7(a), but still signiflcant. While this is suggestive that
the LC resonance is causing the discrepancy, a full quantum simulation would have
to be done to determine if the coupling capacitor is causing deviations from single
junction behavior and if the escape rates and energy levels can both be described
by a junction with the same \renormalized" parameters.
The escape rate in Fig. 7.7(b) has a very prominent feature at Ib = 15:218 ?A
and a weak oscillation at higher currents. These unexpected features are of par-
ticular interest if they are due to a failure of the isolation network at the high
frequencies where they occur. As these data were taken below Tcr, it is unlikely that
the features are due to a thermal noise current at frequency !01 from an object in
equilibrium with the refrigerator and device. There are a number of other suspects.
I cannot discount the possibility that the smaller oscillation is an artifact of the
biasing (as Ib was generated by a digital voltage source) or detection. For example,
if voltage developed across the junction when it switched to the running state in a
particularly pathological way, it could lead to false features in ?. It is noteworthy
that no oscillations are seen in Fig. 7.7(a), although features with a variety of forms
have been seen for other values of I0. It could be that ?0 for the device is not
6These are conditions under which the spectra of Ref. [41] were taken.
229
the simple exponential that one would expect for a single junction. For example,
resonant coupling [131,132] between the junction and the LC coupling mode or the
other junction might lead to such oscillations. If neither of these are the problem,
then the features are indicative of excited state population created by an external
noise source. As ?1 and ?2 are so much larger than ?0 [see Fig. 2.9(b)], only a
tiny population in the quantum states j1i or j2i would be necessary to explain the
features.
7.2.2 dc SQUID Qubits
For the dc SQUID phase qubits, measurement of the escape rate was somewhat
more involved. While this lead to some experimental complications, it also allowed
us to better pin down the source of deviations in these devices. For one, the bias
trajectory had to be selected. By ramping the current bias Ib with  ux bias If = 0,
the isolation junction generally switches flrst [see Fig. 6.7(a)]. Because the device
can re-trap in a number of  ux states, the histogram shown in the inset to Fig.
7.8(a) consists of several peaks. Each corresponds to a difierent qubit current I1
when the isolation junction switches. By drawing the current- ux characteristics
with the device parameters for fleld #2 in Table 6.1, I found that the peaks marked
with symbols correspond to  ux states N' = -47 (circle) and -53 (square). The
escape rate for each state can then be plotted as a function of the isolation junction
current I2, which was calculated using the method described in x6.4. In order for
I2 not to exceed the input value of I02, I had to assume that there was background
 ux of 0:35 '0 when If = 0.
The escape rate of the isolation junction looks quite difierent than the previous
ones in that it appears to be a double exponential, governed by two rather difierent
constants. Although the Ib ramp was linear, it is possible that I2 varied in some
complicated way. However, with the device parameters that I assumed, one expects
230
(a)
(b)
Figure 7.8: Low temperature escape rate of SQUID DS1. (a) The escape rate of
the isolation junction is not exponential in the current I2. Data are shown for two
 ux states, N' = -47 (circles) and -53 (squares), with the full histogram in the
inset. The line is drawn for I0 = 4:4 ?A and CJ = 2 pF. (b) The escape rate of
the qubit junction in the  ux state N' = 0 behaves as expected at high bias, but
shows an excess tunneling rate at lower bias. The line is drawn for I0 = 34:308 ?A
and CJ = 4:43 pF, parameters which come from the spectrum shown in Fig. 8.2(a)
(except for a +8 nA ofiset). Histograms of multiple  ux states are shown as insets.
231
that I2 is quite linear over the range plotted. In addition, if the current division
depended strongly on the Josephson inductance of the qubit, then ? would vary
for the two  ux states. Instead, they are essentially identical aside from a current
shift. If L1 is taken as 3.515 nH instead of 3.520 nH, then the two measured curves
(and those for other  ux states) are indistinguishable on the scale shown. In prin-
ciple, this type of analysis could be used to determine the device parameters with
higher accuracy in high fl devices where the junctions may be regarded as being
independent.
The solid line in Fig. 7.8(a) shows the expected ground state escape rate of a
single junction with I0 = 4:4 ?A (from Table 6.1) and CJ = 2 pF (roughly the design
value for the isolation junction), calculated using Eq. (2.43). As there is no on-chip
isolation for this junction, it is expected that the observed ? is due to excited state
population created by noise on the bias line. The data is somewhat reminiscent of
that shown in Figs. 7.1 and 7.12, where population was excited thermally. As I will
discuss in x7.4, the value of the relaxation time T1 determines the value of ?0 where
depopulation of all of the excited states occurs. For the short T1 expected for this
junction, the collapse to ?0 should occur at a relatively high current; following the
curve to higher ? (using a faster Ib ramp rate) could elucidate the situation.
By simultaneously ramping Ib and If with the proper ratio, we can guarantee
that the qubit junction switches flrst [see x6.4 and Fig. 6.7(d)]. This is the path that
I used for the thermal activation measurements of x7.1.2. The measured escape rate
using this bias trajectory is shown in Fig. 7.8(b), when the device was initialized in
the  ux state N' = 0.7 This state is marked with a circle in the inset, which shows
the multi-peak histogram. As with the isolation junction, I calculated the current
through the qubit I1 using the device parameters in Table 6.1. For reasons discussed
7Flux shaking was not used in acquiring this data set. The unshaken retrapping probability of
N' = 0 was su?ciently high.
232
in x8.2, I assumed a background  ux of ?0:263 '0.8
From a spectrum taken at the same time as ? [see Fig. 8.2(a)], I extracted
the qubit junction parameters I0 = 34:300 ?A and CJ = 4:43 pF. The solid line in
Fig. 7.8(b) is the prediction of ?0 for a junction with this capacitance, but with a
critical current of 34:308 ?A. The small ofiset brings data and theory into very close
agreement at high bias. As discussed in x6.4, a single value of CJ will describe both
the energy levels and escape rates of the qubit, when these quantities are plotted
as a function I1 (and not Ib). The 8 nA ofiset could be needed due to errors in the
calculation of I1 or because the qubit junction of the dc SQUID device does not
quite behave as an independent junction.
Figure 7.8(b) shows that there is an enhancement above the presumed ground
state escape rate that is even larger than it was for LC2B. Some insight into
the origin of these features can be gained by examining their variation with  ux
state. As long as the same simultaneous bias path is used, the biasing of the qubit
should remain almost unchanged. What does vary with  ux state is I2.9 When
jI2j increases, so does the Josephson inductance of the isolation junction; thus, the
isolation factor introduced in x4.3 decreases and the isolation degrades with an
increasing amount of trapped  ux [101].
The escape rates of flfteen  ux states of DS1, obtained by initializing the device
with  ux shaking (see x6.5), are shown in Fig. 7.9(a). I have shifted the curves along
the current axis for clarity; the x-axis is Ib for N' = 0. I took some spectroscopic
measurements at the same time as these background curves. The arrows indicate
the position of the 0 ! 1 resonance at 7.4 GHz, which we expect to occur at the
same value of I1 if the junctions are su?ciently decoupled.
8The data in Fig. 7.8(a), which required a positive  ux ofiset, was taken just a few days before
the data in Fig. 7.8, which required a negative ofiset. This points to an error in identifying N' or
in the determination of device parameters.
9This can also be accomplished by remaining in the same  ux state and adding a dc ofiset to
the applied  ux [101].
233
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8(a)
(b) -83 -82 -81 -64 -48 -32 -16 0 16 32 48 64 81 82 83
Figure 7.9: Flux state dependence of ? for SQUID DS1. The measured escape rate
? varies quite strongly with the  ux state N' (indicated by the small numbers).
Large values of jN'j correspond to poor isolation and lead to additional excited
state population and features in ?. The difierent curves for (a) suppression fleld
#2 and (b) fleld #3 (see Table 6.1) are shifted for clarity; the x-axis is labeled for
N' = 0. The arrows in panel (a) indicate the bias value where !01=2? = 7:4 GHz.
234
In each curve, which I should emphasize was taken at T = 20 mK without
a microwave drive, there is one or more prominent bumps. There are at least two
distinct phenomena seen in the difierent curves. First of all, the large bump that is
seen for N' = 0 grows in height with increasing jN'j. With reference to the arrows,
the location of the bump (in terms of I1 or !01) is flxed. The fact that the size of the
bump decreases as the isolation improves suggests that the bump is due to current
noise on the bias line that is being flltered by the isolation junction. A convincing
argument for this theory can be made by scaling the escape rate enhancement by
the estimated isolation factor. The resulting curves all lie on top of one another and
are a measure of full noise power on the bias line [101]. It is still unclear to me what
determines the proflle of the enhancement, but presumably it is a combination of
the noise spectra and the isolation junction flltering.
Secondly, for N' = ?7 and 8, a new peak appears to the left of the old one.
That scaled down versions of it are not obvious for the other  ux states suggests that
a new noise source has been introduced. As jI2j increases (by changing  ux states),
LJ2 also increases, which degrades the broadband isolation of the qubit. However,
it is important to remember that one arm of the inductive divider is a junction and
the isolation will fail at its plasma frequency. In an aluminum SQUID studied in our
group (device AL1 in Ref. [101]), it was deduced that noise at the plasma frequency
of the isolation junction was exciting the 0 ! 2 transition of the qubit [102]. In the
case of SQUID DS1 under fleld #2, !01=2? for the isolation junction at zero bias is
roughly 12.5 GHz. From spectroscopic measurements, !02=2? for the qubit junction
is equal to 12.5 GHz when I1 ? 34:1 ?A [see Fig. 7.8(b)], where no distinct feature
is seen. Either there was an error in the calculation of the plasma frequency of the
isolation junction or something else is causing the most prominent feature in Fig.
7.9(a). Also, it could be that the new features that appear at high  ux states are
due to flrst excited state population.
235
-12 -11 -10 -7 -3 0 3 7 10 11 12
Figure 7.10: Flux state dependence of ? for SQUID DS2B. The (shifted) escape
rate is plotted for difierent  ux states. The arrows indicate the location of !01=2? =
5:9 GHz.
Figure 7.9(b) shows ? for a selection of  ux states taken for suppression fleld
#3, when the critical currents of both SQUID junctions were larger. There are sev-
eral prominent features, suggesting that the isolation is poor. In this case, where the
Josephson inductance of the isolation junction is small, the isolation factor should
be large. In device LC2A, the isolation appeared to degrade at high frequencies
as well, as shown in Fig. 7.4(b), although it is not clear if there is a connection.
There is, however, no obvious trend in the size of the enhancement as the  ux state
changes, as one would expect if the broadband isolation were failing. In this case, the
zero-bias plasma frequency of the qubit is closer to 40 GHz. It seems unlikely that
the sharp peak seen for N' = 0 is due to thermal noise at this frequency, so more
careful measurements are needed to determine the origin of the various features.
Finally, Fig. 7.10 shows the low temperature escape rate for several  ux states
of SQUID DS2B. Once again, there are features that grow in size but remain flxed
in location and others that move in frequency as jI2j increases. During this run of
236
the refrigerator, we installed new copper powder fllters with a high attenuation on
all of the lines. There are features even at the best isolation, suggesting that the
excitations are not due to high frequency noise from room temperature. By using a
pulse technique to place an upper bound on the total excited state population (see
x6.6.3 and Ref. [104]), it seems that the features could only be due to population in
j2i or perhaps higher states [133]. For this device, where I02 is relatively large, it
could be that that the plasma frequency of the isolation junction corresponds to !02
of the qubit. Interestingly, ? for N' = ?12 is relatively featureless. This is likely
due to the fact that when I2 is su?ciently large, the flltering action of the isolation
junction fails at a frequency that the qubit is not sensitive to [102].
In order to make a high quality qubit, it will be essential to eliminate the source
of these excitations. This may require stronger flltering or a more careful engineering
of the frequency response of the fllters. On the other hand, from spectroscopy and
Rabi oscillation measurements, the relaxation and coherence times are independent
of the level of isolation from the bias line [101], so other noise sources need to be
addressed flrst.
7.3 Master Equations
I have now addressed the low temperature escape rate with the quantum pre-
diction for ?0 and the high temperature (T > Tcr) data with a classical model,
both with a certain amount of success. As seen from the values of ?2? in Fig. 7.2,
the escape rates near the crossover region cannot be described with either simple
approach. One solution is to deflne a tunneling prefactor similar to the one of Eq.
(7.4), but that applies to intermediate temperatures [68]. Instead, I will use a for-
malism that explicitly takes into account the quantum nature of the junction. The
idea is follow the evolution of a system of quantum states jii subject to a number
of transitions.
237
The density matrix approach introduced in Chapter 3 should accurately de-
scribe the dynamics of the escape rate experiments at any temperature. However,
as the time scales being considered are much longer than the coherence time T2,
the ofi-diagonal terms of the density matrix ? are negligible throughout the bias
sweep.10 As a result, the equations simplify and are equivalent to what are known
as the master equations; see Refs. [91] and [92], x3.1 of Ref. [1], and x2.5 and x2.6.2
of Ref. [3]. This generic approach to describing the dynamics of a quantum system
is quite intuitive, in that the population of each level can be found by solving a cou-
pled set of rate equations. When analytical expressions can be derived, the master
equations provide more transparent expressions than the density matrix formalism.
In addition, it is often faster to numerically integrate the master equations than to
time step the density matrix equations, particularly at low Ib where escape rates are
small (so that the time step may be large).
As in Chapter 3, I will use the simplifled notation ?i to denote the diagonal
elements of the density matrix ?ii. From the references given above, the time rate
of change of the occupation probability of jii is
d?i(t)
dt =
X
j6=i
[?Wji ?i(t)+Wij ?j(t)]??i ?i(t); (7.7)
where ?i is the tunneling escape rate from jii (discussed in x2.3.3 for the tilted
washboard potential) and Wij is the total inter-level transition rate from jji to
jii. I will only consider the thermal transitions described in x3.4, but the efiects
of a microwave drive could also be included in Wij (see x3.2.2 of Ref. [1]). In Eq.
(7.7), the flrst and last terms on the right account for depopulation of jii, while
repopulation from other levels is given by the second.
In order to investigate the behavior of the master equations, flrst assume that
10Measurements of T2 for our devices, which will be described in the next two chapters, yield
values under 20 ns.
238
tunneling can be turned ofi and the junction is held at a flxed bias current. Then,
using detailed balance, the thermal rates will balance each other (given enough time)
resulting in a Boltzmann distribution amongst the states. The resulting equilibrium
populations will therefore only depend on the energy level spacing and temperature.
Now suppose that tunneling is turned back on. The probabilities ?i will ex-
ponentially decay to zero, the only possible equilibrium solution. However, at some
point during this decay, the ratio of the probabilities will hit steady state. This is
also true for the normalized probabilities Pi = ?i=?tot (where ?tot = P?j) of Eq.
(3.31) and thus the total escape rate ? = PPi?i of Eq. (3.30). As the tunneling
and thermal rates must balance each other out, the new equilibrium state will not
be a Boltzmann distribution. If the current bias changes slowly enough, the system
can stay in this dynamic equilibrium, the so-called stationary condition, until the
junction switches to the voltage state (see x3.4.1 of Ref. [1]).
One way to flnd the stationary solution of the master equations is to re-feed
the population that escapes back into the ground state (see x2.5 of Ref. [3] and
Ref. [92]) and to set _?i = 0 (where the dot indicates a time derivative). This is
not physical, but provides su?ciently accurate results as long as the excited state
population is small.
Instead I will use the fact that when the normalized probabilities reach steady
state, their time derivatives vanish in the stationary limit, yielding
dPi
dt =
d
dt
 ?
i
?tot
?
= 1?
tot
d?i
dt +
?i
?tot
 
? 1?
tot
d?tot
dt
?
= 1?
tot
d?i
dt +?Pi = 0; (7.8)
where _?i is given by Eq. (7.7). I will discuss the solution for a two-level system
in detail in the next section. For more than two levels, analytical solutions can be
di?cult to obtain. Numerical solutions are easy to flnd, as the system of equations
of Eq. (7.8) can be solved iteratively using the Newton-Raphson method (see, for
239
example, x9.6 of Ref. [69]). While this method only converges if the initial guess is
fairly close to the flnal answer, a Boltzmann distribution is su?cient in most cases.
In practice, starting from low bias, where the system will be very nearly in thermal
equilibrium, and bootstrapping up a current ramp is often quicker.
When considering a large (non-stationary) ramp rate, corresponding to con-
tinuously evolving Pi, I usually just solved Eq. (7.7) for the populations numerically.
As with the Liouville-von Neumann equation in x3.8, Eq. (7.7) is easily cast into
the matrix form
d?d
dt = p?
d: (7.9)
For a system with N levels, ?d is the main diagonal of the full density matrix (and
is therefore a vector of N elements) and p is an N ? N matrix that describes its
evolution. The structure of p is given xD.2. The evolution of the system can then
be found e?ciently with matrix exponentiation, as shown in Eq. (3.71), with a
potentially time-dependent p.
7.4 Determination of T1 With a Slow Bias Sweep
The relaxation time T1 is one of the key indicators of the quality of isolation
of a qubit. Many experiments to measure T1 have been performed on a variety of
qubits. Perhaps the most common is to create an excited state population with a
microwave pulse and then monitor the decay time back to the ground state [39]. I
will show results of this technique inx8.7, but this technique is not without problems.
One concern we had was that the microwave wiring lines must be flltered or weakly
coupled to the qubit, which could give these leads a complicated frequency response.
If a resonance of this wiring is also excited by the pulse, the observation of the decay
could be determined by the lifetime of the resonance and not just the junction. This
is particularly an issue when trying to measure times of several nanoseconds, as in
240
device LC2. Similarly, the junction can be excited by a \dc pulse" in the bias
current [37], but the bandwidth of the bias and detection lines must be high enough
so as not to in uence the results. Care must also be taken that this extra bandwidth
does not introduce additional noise to the qubit.
The dissipation time may also be found by simply ramping the bias current,
when there is a thermally created excited state population. The escape rate can
be studied classically, as in x7.1, or with the quantum master equations, which
predict features dependent on the junction shunting resistance RJ [134]. In general,
flnding the resistance from such data requires detailed modeling and knowledge of
the system [135], especially when the junction temperature is much higher than its
characteristic frequency. The problem is that the escape rate can depend strongly
on parameters aside from RJ.
The situation is greatly simplifled if the junction is held at a slightly elevated
temperature (near the crossover temperature Tcr), where only the two lowest levels
have a non-negligible probability of being occupied [4,99]. This technique comple-
ments the thermal activation analysis nicely, as each approach works in a difierent
temperature range. In addition to a careful choice of temperature, the bias ramp-
ing rate must be high enough to highlight the desired feature in the data, but low
enough to remain in the stationary limit, so as not to afiect the junction dynamics.
These will be the assumptions in the analysis that follows. If these requirements are
met, an estimate for T1 can be found in a straightforward way.
The basic idea is as follows. Consider the two-level system shown in Fig.
7.11(a). The states j0i and j1i are separated by ?E ? ~!01 and tunnel to the
voltage state with rates ?0 and ?1. Thermal emission and absorption occur with
rates W? ? Wt01 and W+ ? Wt10 [deflned in Eqs. (3.26) and (3.27)]. The inclusion
of tunneling will take the system away from a Boltzmann distribution. We can
(presumably) directly measure ?0 and from it predict ?1. By setting the temperature
241
T, we also know the ratio of W+ to W?. The one true unknown is T1 ? 1=W?, which
can be estimated by flnding the new equilibrium that the four rates create. The
current bias Ib provides a simple way to drastically change the escape rates and it
turns out that at a particular value of Ib, the equilibrium values can be found with
little efiort.
The occupation probabilities ?0 and ?1 can be found with the master equations
of Eq. (7.7), of which there are now two:
d?0
dt = ?(W+ +?0)?0 +W??1 (7.10)
d?1
dt = W+?0 ?(W? +?1)?1: (7.11)
The two-level master equations describe the system on the long time scales we are
interested in. If Ib is varied slowly, then there is the additional constraint that the
normalized probabilities Pi are efiectively constant at any instant. From Eqs. (7.8)
and (7.10), the time derivative of P0 is
dP0
dt = ?(W+ +?0)P0 +W?P1 +?P0: (7.12)
As P0 (t) + P1 (t) = 1, we also know that _P1 = ? _P0. The expression takes on a
convenient form if it is expressed in terms of the ratio of probabilities:
 
1+ P1P
0
?2 dP
0
dt = W?
 P
1
P0
?2
+(?W +??) P1P
0
?W+; (7.13)
where ?W ? W??W+ and ?? ? ?1??0 are the difierences in the thermal and escape
rates, both positive quantities. The stationary condition is imposed by setting the
time derivative to zero, resulting in the exact expression for the probability ratio
P1
P0 =
?(?W +??)+
q
(?W +??)2 +4W+W?
2W? : (7.14)
242
U
?
0
1
?0
?1
?EWW
(a)
?0
?1
?
W
W
(b)
Figure 7.11: Transitions of a two-level system. (a) The two lowest levels of the
washboard, j0i and j1i, are separated by energy ?E. ?0 and ?1 are the tunneling
rates from the two levels and the inter-level transitions W+ and W? keep the system
in thermal equilibrium in the absence of tunneling. (b) The escape rates are plotted
as a function of Ib as dashed lines, while the inter-level transitions for RJCJ = 4 ns
and T = 80 mK are shown with dotted lines. The inset shows the bias dependence
of T1. The total escape rate ? (solid) collapses to ?0 under a stationary sweep when
?1 = W?, resulting in a feature that can be used to estimate T1.
243
I will now examine Eq. (7.14) in a few limits, applicable to data taken under
certain conditions, to highlight the simple behavior dictated by this expression (see
x3.4.2 of Ref. [1] ). At low temperatures, W?  W+, so that P0 ? 1 for all bias
currents, while P1 ? W+=(W? +??). This excited state occupation probability is
qualitatively difierent for high and low values of Ib. At low bias currents, where
?? < ?1 ? W? and thus P1 ? W+=W?, the system is essentially in thermal equi-
librium because tunneling is a small perturbation to the thermal rates. Tunneling
is nonetheless important, as it allows us to gauge the state populations through the
total escape rate, which is ? ? ?0 +?1e??E=kBT.
In contrast, at high currents, ?1 > ??  W?  W+, so that P1 ? W+=??
vanishes and ? collapses to ?0 (even though the temperature of the junction has
not changed). This is due to the depopulation of the excited state as a result of
strong tunneling from this state.11 As shown by the solid line in Fig. 7.11(b) [which
was generated with Eqs. (7.14) and (3.30)], the shift between these two limiting
behaviors occurs near the bias current (denote it I?b) where ?1 = W?. For the case
drawn, I?b = 33:44 ?A, but note that I?b is weakly dependent on temperature through
W?. The corresponding total escape rate at the crossing point is ? ? ?0 + W+=2,
with the (generally valid) assumption that ?1  ?0.
W+ varies slowly with bias current (through the energy level spacing), while ?0
increases exponentially. This results in a clear shoulder in ? at a bias current of I?b.
Therefore, this feature (and thus I?b and W+) can be directly identifled on an escape
rate curve, as long as the temperature is chosen so that the junction behaves as a
two-level system. T1 is then given by e??E=kBT=W+, where microwave spectroscopy
can be used to directly measure ?E(I?b). However, the fact that 1=T1 ? W? = ?1
11This has always struck me as counter-intuitive. A high tunneling rate from the excited level
ought to lead to a large ?. However, ? depends on the product ?1P1. The transition rates (in
particular W+) are unable to maintain a signiflcant P1 which leads to the lower ?. Thus it is not
important that it is tunneling that exceeds W?. Any process that removes population from j1i at
a rate faster than W+e?E=kBT will take the system away from a Boltzmann distribution.
244
occurs at I?b allows a rough estimate for T1 to be obtained from the escape rate
curves alone. At very low temperature, the total escape rate is just ?0(Ib), as P1 is
negligible. While ?1=?0 is a function of Ib, this ratio (evaluated at I?b) is on the order
of 500 for a large range of T1 for the tilted washboard potential [see Fig. 2.9(b)];
thus
T1 ? [500 ?0(I?b)]?1 : (7.15)
At low bias currents and su?ciently high temperatures, the levels above the
flrst excited state will have non-negligible occupation. Each of the levels will empty
out in order, leading to a series of shifts in ?. At slow sweep rates, these tend not to
be as pronounced as the single distinct feature for the two-level case. Nonetheless,
such features have been previously reported [134].
7.4.1 LC-Isolated Phase Qubits
Figure 7.12(a) shows the total escape rate ? as a function of bias current Ib
for junction LC2B, at temperatures between 25 and 320 mK. This data was taken
at the same time as Fig. 7.1, except at a higher Ib ramp rate of 0.93 A/s. Below
50 mK, the escape rate is roughly exponential in the bias current, as expected for
tunneling out of the ground state alone. As the temperature increases, thermally
excited population in the higher states gives rise to higher escape rates. As discussed
above, each of these levels will empty out as the sweep progresses, once its tunneling
rate exceeds the dissipation rate for that state. For large Ib only the ground state
retains any population and all the curves collapse onto ?0. The escape rate at which
this collapse occurs increases with temperature, because W+ increases as well.
The ramp rate was chosen to be just high enough to follow this trend. At lower
rates, the junction is more likely to escape at lower bias currents, so we were unable
to follow ? all the way up to the current where collapse occurs. At higher rates,
245
(a)
(b) ?
0
1
3
2
4
5
6
7
Figure 7.12: Master equation simulation of junction LC2B. (a) The symbols show
the measured escape rate ? at several temperatures T, under the same conditions
as in Fig. 7.1, except that the bias ramp rate was 0.93 A/s. T1 can be estimated
from the shoulder feature (marked with large open circles) at 65 and 85 mK. The
solid lines come from a master equation simulation, with T1 = 4 ns, including
anharmonic matrix elements (second fltting technique described in text). (b) The
simulated contribution to ? by each of the eight levels and the total ? are plotted
for Tfit = 275 mK.
246
the sweep goes further into the non-stationary regime and unnecessarily complicates
the dynamics, which we want to avoid for now. Incidentally, the classical thermal
pre-factor in Eq. (7.4) predicts the same qualitative suppression of ? with increasing
Ib (see x2.5.1 of Ref. [3]), as seen in Fig. 7.3(c). However, the dotted line in Fig.
7.5(a) clearly shows that the classical theory neglects the fact that total escape rate
is bounded below by ?0. This is the reason why I chose the low ramp rate data for
the thermal activation analysis in x7.1.1.
Even at this modest ramp rate, the calibration of the bias current as a func-
tion of time was a serious challenge, as discussed at the beginning of x7.2.1. In
addition to the 15 nA ofiset I mentioned there, I added a small ofiset to Ib for each
temperature (for reasons described in the discussion of Fig. 7.3), to produce the
expected convergence at high bias currents. These adjustments, all less than 15
nA, could be needed because of an incomplete knowledge of the calibration (e.g.,
its temperature dependence), or to small drifts in the detection electronics over the
course of the data taking. The small ofisets could also be needed because the actual
critical current of the junction changed with time. In this case, it is not legitimate
to simply shift escape rate curves, but this is unlikely to introduce major errors for
such small changes.
Examination of Fig. 7.12(a) shows that at 65 and 85 mK, ? does not increase
much over its 25 mK values. This suggests that the two-level analysis is applicable
at these temperatures. That the three escape rates are nearly parallel at low bias
currents supports this claim, as does the single well-deflned shoulder feature at
the elevated temperatures. Using Fig. 7.11(b) as a guide, this features begins at
I?b ? 33:435 ?A for both temperatures (indicated by the large open circles). At this
current, ?0 (I?b) ? 5?105 1=s and thus our rough rule Eq. (7.15) gives T1 ? 4 ns.
The master equation can be used to perform a more quantitative analysis at
all temperatures. I did use the fully time dependent form of Eq. (7.7), although
247
in principle, this data was taken at a low enough ramp rate to use the stationary
solution. The escape rate at 25 mK (which should be nearly independent of RJ, T,
and sweep rate) was flt to the single junction model to give I0 = 33:648 ?A and
CJ = 4:24 pF. I assumed that the tunneling rates from the higher levels were given
by a single tilted washboard with these parameters.
Imadeafewsimplifyingassumptionsinaninitialattemptatmodelingthedata
[99]. I assumed that the energy levels were given by the same junction parameters
as the escape rate. In addition, I used the harmonic approximation in calculating
the spontaneous emission rates ?ij [given by Eq. (3.25)], so that thermal transitions
only occurred between adjacent levels. Finally, ?ij was taken to be independent of
Ib, although Wtij were bias-dependent through the Boltzmann factor.
With these assumptions, the results of an eight-level simulation agree well with
the data, when RJ is set at 1 k?, corresponding to T1 = 4:2 ns.12 It is encouraging
that this more thorough analysis produced a T1 consistent with the rough estimate
above. The values of the temperature I used in the simulation are listed as T(1)fit in
Table 7.1. As expected, they are reasonably close to the temperature of the mixing
chamber T, given the uncertainty of our thermometry. In doing the flts, I added
the values ?I(1)b in Table 7.1 to the current axis for the data, so that the escape
rates would converge at high bias. The quality of the flts is indicated by ?2(1)? in
the table; here, the number of degrees of freedom ? varied from 100 at the lowest
temperature to over 300 at the highest.
We had additional information about junction LC2B in the form of spectro-
scopic measurements. From Figs. 7.7(a) and 8.1(b), it seems that the low temper-
ature escape rate and the 0 ! 1 transition frequency are not described by a single
set of parameters for an ideal current-biased junction. This is most likely due to
12I did not use a formal fltting procedure here, because the unknown Ib ofiset was rather im-
portant, but it was not a truly free parameter. Instead I varied parameters by hand in trying to
minimize ?2?. For a chosen bias ofiset, this was fairly easy to do.
248
Table 7.1: Master equation flt parameters for junction LC2B. The escape rate data
shown in Fig. 7.12(a) were flt with an eight-level master equation simulation, for
six values of the refrigerator temperature T. Using a simple model of inter-level
transitions, the flt temperatures T(1)fit were extracted. A more accurate treatment
yielded T(2)fit. The data were shifted by ?Ib to produce the expected collapse to ?0
and the quality of the flts is indicated by ?2?; these two quantities are listed for both
fltting attempts.
T (mK) T(1)fit (mK) ?I(1)b (nA) ?2(1)? T(2)fit (mK) ?I(2)b (nA) ?2(2)?
25 25 0 1.5 25 0 1.5
65 73 -3.5 1.9 59 -3.2 2.3
85 98 -4.5 1.9 80 -4.5 2.0
115 134 -10.5 2.5 109 -10.3 2.3
190 215 -13 6.4 174 -13.5 5.7
320 340 -15 4.2 275 -15.5 4.1
the harmonic mode created by the coupling capacitor to LC2A. Instead of flnding
the energy levels and escape rates of the entire coupled system, I just let the es-
cape rates be described by I?0 = 33:648 ?A and C?J = 4:24 pF, while the energy
levels and matrix elements behaved as a single junction with I!0 = 33:663 ?A and
C!J = 7:32 pF. A nice feature of the master equation approach is that these exper-
imentally measured parameters can be input in a phenomenological (yet accurate)
way, even if the underlying physics is not entirely understood.
I also tried a second approach to modeling the data. I allowed the spontaneous
rate to be bias-dependent, using Eq. (3.25) and the cubic approximation for the
matrix elements  n;m = hnj^ jmi. From that Eq. (3.25), it might seem that T1
would decrease with increasing Ib because  0;1 increases [see Fig. 2.11(a)]. As the
inset to Fig. 7.11(b) (which was drawn with the improved set of parameters) shows,
T1 actually increases slightly due to a concurrent decrease of !01. W? decreases
as well, although the Boltzmann factor overwhelms this at higher T, where W?
249
increases with Ib. Using the cubic approximation to evaluate  n;m also leads to
inter-level thermal transitions between every pair of states. Some of the matrix
elements are given in Eqs. (2.38) and (2.39).
With this improved model of transitions, the best choice for RJCJ (which is
the value of T1 at low Ib) in the eight-level master equation simulation is 4 ns. The
results are plotted as solid lines in Fig. 7.12. In plotting the data in the flgure, I
added the values ?I(2)b in Table 7.1 to the current axis. It is interesting that both T1
and the quality of the flts (indicated by ?2(2)? in the table) do not change appreciably
from the simpler flrst attempt. In fact, the simulated curves are virtually identical
for the two attempts, although I have only plotted the results for the second one.
However the simulation temperatures for the improved model, listed as T(2)fit
in the table, generally show better agreement with T. The highest temperature
is the only exception, but that could be due to an incorrect choice of ?Ib, as the
data does not extend to the ?0 collapse. The change in flt temperatures is largely
due to the more accurate treatment of energy levels, as I?0 and C?J overestimate
the level spacing; because the device did not behave as a single junction, I!0 and
C!J do not completely parameterize !01 over the entire range of interest. The more
accurate values of ?n;n+1 also had a small impact, while the inclusion of more than
adjacent-level relaxation had a negligible efiect.
The solid lines in Fig. 7.12 do a good job of capturing the features in the data,
for a value of RJ that is independent of temperature and frequency. The only major
failing is the collapse to ?0 at the highest temperatures, which appears to be faster
than the simulation predicts. The reason for this is unclear. The simulation does
predict some gentle oscillations in ?, some of which are seen in the data. The origin
of these features is made clear in Fig. 7.12(b), which shows ? for T(2)fit = 275 mK as a
heavy black line. The contributions from each level ?iPi that sum to this value are
shown as a series of gray lines. As a level reaches the top of the potential barrier,
250
its large escape rate leads to depopulation and a decreasing contribution to ? even
as its escape rate continues to increase. This leads to the ripples in ?. I artiflcially
forced ?i to saturate at roughly 5?1010 1=s (because the quantum simulation that
generates them eventually breaks down), but the particular way in which they roll-
ofi is unimportant due to this depopulation. It is interesting that at any given Ib,
only two or three levels make a substantial contribution to ?.
Given the assumptions of the model, T1 is determined with good precision.
For example, if RJCJ is chosen to be 6 ns, then for the data at T = 65 mK, the
minimum ?2? increases to 3.5 at Tfit = 61 mK. Comparison of the flt to the data
would show that the shoulder feature is not reproduced well.
7.4.2 dc SQUID Phase Qubits
In the order to perform the same type of analysis on SQUID DS1, I flrst had
to adjust the calculation of I1. By assuming a background  ux ofiset of 0.26 '0, ?
at 20 mK of Fig. 7.5(a) could be made to overlap the curve of Fig. 7.8(b). With this
agreement, I then took I?0 = 34:308 uA, I!0 = 34:300 ?A, and C?J = C!J = 4:43 pF,
as in the later flgure. I used a microwave peak at 7.0 GHz to align the higher
temperature curves. This resulted in the expected collapse at high Ib, as seen in
Fig. 7.13, without having to explicitly enforce it with an additional ofiset.
Due to the obvious signs of excited state population at low temperature, it
is impossible to apply the rough estimate of Eq. (7.15). Starting at about 100
mK, it seems that thermal excitations overwhelm the efiects of the bias noise, so
I used the eight-level master equations (with matrix elements calculated in the
cubic approximation and a bias-dependent T1) to describe that data. Unfortunately
at these high temperatures, allowing T to vary somewhat results in a fair bit of
uncertainty in T1. The solid lines in Fig. 7.13 are drawn for RJCJ = 14 ns, which
corresponds to RJ ? 3 k?. Taking the value as 10 or 20 ns results in noticeably worse
251
Figure 7.13: Master equation simulation of SQUID DS1. The symbols show the
same data as in Fig. 7.5(a), with the qubit current I1 adjusted to match Fig. 7.8(b).
The ramp rate is 0.1 A/s, slow enough to stay in the stationary limit. The solid
lines come from an eight-level master equation simulation, with T1 = 14 ns.
agreement, but without a well deflned feature to flt to, it is di?cult to pin down
the value with much more precision. At low temperatures, the calculated escape
rate underestimates the data. This was done intentionally, as the enhancement is
not expected to be entirely thermal in this case. For mixing chamber temperatures
of 20, 65, 90, 110, 130, and 170 mK, the simulation revealed flt temperatures of 20,
74, 93, 117, 138, and 179 mK.
To summarize, the slow bias sweep escape rate technique is easy to perform
and provides an estimate of T1 with little data analysis. The main assumption is that
the device is described by a tilted washboard potential. In the case of the energy
levels, this assumption can be checked. However, verifying the tunneling rates of the
excited states is more di?cult, but it appears that this can be accomplished using
252
a pulsed measurement technique [104]. Deviations from single junction behavior,
absent a better model, will cause problems with the application of Eq. (7.15) and
the full master equations.
7.5 Fast Sweep
In the previous section, I asserted that a slow bias sweep would result in
negligible _Pi. This begs the question, slow compared to what? The answer comes
from recognizing that the escape rates ?0 and ?1 increase nearly exponentially with
bias current. If the bias current is ramped linearly in time,13 then the escape will
be exponential in time as well. It is convenient to assume that the escape rates
have the form ?i = ?0iefit, where fi characterizes the speed of the ramp (see x3.4.3
of Ref. [1]).
The escape rate for a typical junction at elevated temperature, as calculated
with the two-level master equations, is plotted for several ramp rates in Fig. 7.14.
The solid black line is drawn for stationary conditions, where the dynamics are
independent of the sweep rate. As discussed in the previous section, ? begins to
collapse towards ?0 at the bias current where ?1 ? W? ? 1=T1.
The escape rate for a ramp rate of 0.1 A/s is shown as a dotted line and
deviates slightly from the stationary case. The sweep rate parameter fi, numerically
calculated as (1=?0)d?0=dt = dln?0=dt, is plotted as a nearly horizontal gray dotted
line segment. The same information is plotted for 1 (dashed) and 10 (dot-dashed)
A/s. For these faster rates, the shoulder feature moves to higher escape rates.
Rather than it beginning when ?1 = W?, it occurs roughly when ?1 = fi. Thus, it
seems that ? deviates from its stationary values roughly when fi > 1=T1.
In Fig. 7.12, Ib was ramped at 0.93 A/s, which corresponds to fi = 7?107 1=s.
13In the previous section, when escape rates were measured in the stationary limit, the \wave-
form" used for Ib was entirely unimportant. At each value of Ib, the system reached dynamic
equilibrium, so the path taken to get there had no efiect.
253
?0
?1
W
W
(b)
(a)
Figure 7.14: The efiect of sweep rate on ?. (a) The solid black line shows the
stationary escape rate for a junction with I0 = 30 ?A, CJ = 5 pF, T1 = 100 ns, at
T = 100 mK. The dynamics are set by ?0, ?1, W+, and W? (drawn as extended gray
lines), and a shoulder is seen at the value of Ib where ?1 ? W?. The shoulder in ?
moves for ramp rates of 0.1 (dotted black line), 1 (dashed), and 10 (dot-dashed) A/s
to the current where ?1 is roughly equal to the sweep rate parameter fi (drawn as
gray line segments for each of the ramp rates). (b) As the ramp rate increases, the
dynamics move further from the stationary limit, as re ected by the time derivative
of normalized ground state occupation probability.
254
Therefore, the stationary condition is met if T1 is less than 15 ns, which turned out
to be the case for the LC2B. In Fig. 7.13, the 0.1 A/s ramp rate corresponds to
fi = 8?106 1=s. Thus any system with T1 < 125 ns will remain stationary.
Although the method for flnding T1 presented in the previous section no longer
applies, the fast sweep provides another technique. Over some \low" range of sweep
rates, the total escape rate will be independent of fi, as the dynamics stay nearly
stationary. At some point, however, ? will start to deviate; the value of fi at this
rate is roughly 1=T1 [4]. This conclusion can be verifled by continuing to increase
the ramp rate and following the shoulder. While we have seen small deviations in
? as a function of the ramp rate, we have never clearly seen this clear shoulder
movement behavior with our devices. Thus it seems that 1=T1 is faster than the
ramp rates that we can experimentally implement, suggesting that T1 is shorter than
50 ns [133].
The condition for steady state can also be examined using the sweep rate
analysis. Fig. 7.14(b) shows _P0 for the three flnite ramp rates in Fig. 7.14(a); it is
identically zero for the stationary case. The values are quite large near the shoulder
feature for the fastest ramp, indicating that the system is far from stationary when
the depopulation of j1i occurs. The excited state population is in efiect frozen, as
not enough time is allowed for the system to relax to the ground state. At even
higher temperatures, a very fast sweep leads to a sequential emptying of levels,
providing dramatic evidence of energy level quantization (see Ref. [134], x3.4.3 of
Ref. [1], and Ref. [104]).
As the ramp rate increases, not only does the shoulder move, but ? at lower
bias also decreases uniformly. This is due to the fact that as Ib increases, !01
decreases. IfthesystemhastimetoadjusttoaBoltzmanndistribution, thentheflrst
excited state becomes progressively more heavily populated as the ramp proceeds,
leading to the stationary limit. For fast ramp rates, W+ does not have su?cient
255
time to populate j1i during the time when the level spacing is small. Thus, even
though the thermodynamic temperature is the same for all cases in Fig. 7.14, the
fast ramp rate cases appear to be at a lower temperature (at Ib = 29:70 ?A).
This discussion motivates another way of looking at the efiect of the various
transitions rates of the system. The occupation probability of the flrst excited state
can be characterized by an efiective temperature,
Teff = ? ~!01k
B ln(P1=P0)
: (7.16)
Teff is plotted in Fig. 7.15 for the simulations of Fig. 7.14(a). In the stationary case
(solid line), Teff = T = 100 mK at low bias. In this region, tunneling has a negligible
efiect, resulting in a Boltzmann distribution. As Ib and ?1 increase, the flrst excited
state state becomes depopulated, resulting in what appears to be a colder junction.
The situation is quite similar to evaporative cooling, where the hottest particles are
selectively removed from a sample, leaving behind a smaller and colder population.
The general behavior of the efiective temperature at the higher ramp rates
is similar, except that the starting point (on the graph) is lower and the cooling
is delayed to higher Ib. The minimum temperature that the junction reaches is
independent of fi.
This raises an intriguing possibility for initializing a qubit [4]. If the tempera-
ture cannot be made low enough, then Ib could be quickly set to a high value. For a
particular trial, if the system is in j1i, it will be more likely to tunnel to the voltage
state, at which point the trial ends. If it does not tunnel, then the bias can be
lowered to the point where the manipulation is to done, with a certain amount of
confldence that the system is in the ground state. Two issues could make this proce-
dure di?cult to implement. First of all, in order obtain a low Teff, ? must be made
quite large. Thus the fraction of the total number of trials that is useable could be
256
Figure 7.15: \Evaporative cooling" of a Josephson junction. For the escape rates
shown in Fig. 7.14(a), depopulation of the flrst excited state by tunneling leads
to a non-Boltzmann distribution. Thus, the efiective temperature decreases quite
quickly with the current bias.
quite low. More importantly, the source of the heating might still be present. Thus
when Ib is lowered, Teff might initially be low, but W+ could quickly repopulate j1i.
However, the technique could be quite useful for diagnostics. For example, in
x7.2.2, I showed several unusual low escape rate curves. A major question is whether
these odd features are due to complex potentials that determine the dynamics of
these systems or whether they are the result of some small, athermal excited state
population. If the latter is true, then a pulse of Ib that takes the system to a
high escape rate and then back to one of the features, will result in a lower ?. As
mentioned, this will be true as long as the pulse occurs on a time scale shorter than
any re-excitation. Variations on this type of measurement suggest that the measured
escape rate features seen in our devices are due to excited state population [133].
257
7.6 Summary
The amount of information that can be gained from the tunneling escape
rate alone is amazing, which undoubtedly is the reason that it has been used to
study macroscopic quantum phenomena for so long. At very low temperatures, the
experimental escape rate can be flt to theoretical predictions that were calculated
with straight-forward techniques for a simple junction Hamiltonian. The utility
of the escape rate comes from its extreme sensitivity to excited state population.
This is what made it possible to extract T1 from slow sweep experiments in x7.4
at moderate temperatures, where the thermal excitations were minimal. At higher
temperatures, ? follows classical activation theory with damping, as discussed in
x7.1.
The experiments in this chapter provide the flrst evidence (in this thesis) for
the quantum behavior of our devices. In addition, I extracted the relaxation time T1
both in the classical and quantum regimes. For the LC-isolated qubits, the classical
thermal theory gave a shunting resistance of RJ ? 2 k? when the junction plasma
frequency was near !p=2? ? 5 GHz (see Fig. 7.3). With a junction capacitance of
CJ ? 4 pF, T1 ? 8 ns. Using a master equation simulation (and a rough rule, derived
for a two-level system), I determined the relaxation time of the same junction to
be about 4 ns14 (see Fig. 7.12). It is possible that the discrepancy is due to the
efiects of the coupling capacitor or to a frequency-dependent RJ, but a factor of two
difierence with the techniques is hardly a cause for concern at this point. In any
case, the value is far below the 5 ?s predicted in x4.2, presumably due to a failure of
the LC fllter or intrinsic junction dissipation. The isolation also appears to degrade
for plasma frequencies above 10 GHz (presumably for difierent reasons) and it may
be important to determine on what frequency scale RJ varies.
14In the simulation, I assumed that the shunting impedance was independent of frequency (and
thus the current bias Ib), leading to a constant value of RJCJ. However, T1 picks up a bias
dependence through the matrix element h0j^ j1i.
258
Using the thermal activation analysis shown in Figs. 7.5 and 7.6 for the SQUID
phase qubits, RJ appeared to be between 2 and 8 k?, corresponding to T1 between
10 and 30 ns. The master equation simulations suggest that T1 ? 14 ns (see Fig.
7.13). It is possible that the spurious features in the intermediate temperature range
[see Fig. 7.5(a)] led to an incorrect value of classical Tesc there. Placing an upper
bound on ? (which I did in order to get 8 k?) could be necessary because the
classical theory does not predict the collapse to ?0 correctly. The longer T1 is, the
sooner the collapse happens, which is why there might have been more problems
with the SQUID data. Nonetheless it does appear that the broadband scheme did
improve the qubit performance, but again, the measured T1 is quite a bit shorter
than the prediction of roughly 200 ns in x4.3.
259
Chapter 8
Spectroscopy and Non-Coherent Dynamics
The chapters begins with a description of how we measured the spectrum of
transitions for the LC-isolated qubits. The dc SQUID phase qubit requires addi-
tional techniques and analysis, as we were interested in just the properties of one
of the two junctions. With the methodology established, I will then describe the
wealth of information that can be obtained from spectra. For example, the width of
a resonance gives important information about the isolation of the qubit. Spectra
also reveal the nature of the coupling of the qubit to other degrees of freedom; I
will show evidence for two-level systems that interact with the qubit and spectra of
two SQUID phase qubits that are capacitively coupled together. The chapter closes
with measurements of the relaxation time T1, obtained with microwave pulses.
8.1 Spectroscopy of LC-Isolated Phase Qubits
To measure the energy level spectrum of an LC-isolated phase qubit, we flnd
the enhancement ??=? in the tunneling escape rate due to a microwave current
I?w, as described in x6.2. The current bias Ib is linearly ramped while a continuous
wave microwave signal of flxed angular frequency !rf is applied to the junction.
Whenever the microwave drive is resonant with a transition between qubit energy
levels, an excited state population will be generated and this leads to a peak in the
escape rate enhancement ??=?, such as that shown in Fig. 6.5(c). By repeating
this measurement at a series of drive frequencies (usually at intervals of 50 or 100
MHz), the transition can be mapped out.
Figure 8.1(a) shows a spectrum of junction LC2B when the critical current
260
was quite suppressed.1 LC2A was held at zero bias throughout the data taking, so
that it was well out of resonance with LC2B. Each horizontal line of the grayscale
map shows the resonant peak at a particular drive frequency, where dark colors
represent a large escape rate enhancement. However, I have normalized each line to
its maximum enhancement. As this data set was taken at the base temperature of
the refrigerator, the excited states of the qubit should not be thermally occupied.
The microwave signal can therefore only make the 0 ! 1 transition (at frequency
!01) visible, provided the power is not too high.
The circles indicate the current bias at which the enhancement reached a
maximum for each setting of !rf. As expected, !01 decreases with increasing Ib, as
the well of the tilted washboard potential becomes more shallow. These points were
flt to a simulation of !01 for a single junction generated with Eq. (2.44); the solid
line corresponds to a junction with critical current I0 = 15:420 ?A and capacitance
CJ = 5:65 pF. Over the frequency range shown, the flt is rather good. Despite
the presence of the measurement electronics and on-chip circuitry, the junction still
retains its simple quantum nature.
The flt capacitance of 5.65 pF is larger than the value of 4.85 pF listed in
Table 4.1. This is due to the presence of the junction coupling network of LC2,
which created an LC mode at 7.2 GHz [78]. While the zero-frequency coupling
constant is ?0 = 0:064 [see Eq. (2.72)], the efiective frequency-dependent constant is
? (4:8 GHz) = 0:12 [see Eq. (2.78)] at the average frequency of the spectrum. This
leads to an efiective capacitance CJ (1+? (!)) = 5:4 pF [see Eq. (2.75)], which is
fairly close to the flt value. Here, I have included the efiect of the LC mode with an
efiective frequency-dependent CC; the number listed in Table 4.1 came from fltting a
spectrum to a more accurate model, where the LC mode was treated as a quantum
degree of freedom [78].
1The escape rate under the same conditions is plotted in Fig. 7.7(b).
261
(a)
(b)
Figure 8.1: Transition spectra of junction LC2B. (a) The escape rate enhancement
due to microwave activation is plotted as a grayscale map, where dark colors rep-
resent larger values. The white curve is a flt of !01 to the maximum enhancement
at each frequency (plotted as circles), with I0 = 15:420 ?A and CJ = 5:65 pF. (b)
At a difierent suppression magnetic fleld, the plasma frequency of the junction is
close to the resonant frequency of the LC mode created by the coupling network
to LC2A, causing a deviation from single junction behavior. The line is drawn for
I0 = 33:663 ?A and CJ = 7:32 pF. Both data sets were taken at 25 mK by Huizhong
Xu and Andrew Berkley.
262
When taking a spectrum, we typically set the microwave power such that the
escape rate enhancement ??=? was between 1 and 5. This is large enough to ensure
a clearly visible peak, but not so large as to cause power broadening. In the case
of Fig. 8.1(a), the nominal power PS of the microwaves at the room temperature
output port of the source was set to -15 dBm for all frequencies. For other data
sets, in order to account for the frequency-dependent attenuation of the microwave
lines, we had to adjust PS for difierent !rf to obtain an enhancement in the desired
range.
As I noted in x6.2, the escape rate is di?cult to measure precisely when it
is small. For example, at 5.15 GHz the resonance was located at Ib = 15:26 ?A,
where the background escape rate was ?bg ? 2:6?104 1=s. By applying a relatively
strong drive, the escape rate with microwaves ??w was boosted to about 5:5?105,
which was easily measured. However, the large uncertainty in ?bg leads to scatter in
??=?. In an attempt to remedy this problem when plotting a spectrum, I typically
used the background data from all of the frequencies to calculate ?bg. This reduces
the noise at low Ib in Fig. 8.1(a) quite dramatically. Experimentally, this also means
that more time can be devoted to taking microwave data at each frequency.
What can make this averaging process di?cult is that the background switch-
ing time essentially always drifts over the 5 to 50 hours during which a spectrum
is taken. This can be due to the biasing and detection electronics warming up, the
temperature of the lab  uctuating, the level of liquid helium in the refrigerator?s
dewar decreasing, as well as actual changes in I0. As a result, simply using all
the data to calculate a single ?bg would lead to smeared out features and incorrect
values. To monitor the drift, we always interleave background and microwave data.
I usually attempt to correct for the drift with the following simple procedure.
An average switching time t is selected. The average of the flrst N background data
points is calculated to be tbg. Then ?t = t ? tbg is added to each of the flrst N
263
points of both the background and microwave data. This is repeated for each block
of N points. The key is to pick N large enough so that the features of the histogram
are not smoothed over, but smaller than the scale of the drift. Typically, there are
no obvious variations in the enhancements for values of N between 5000 and 20000.
If there are large drifts of I0, there is no justiflcation for this procedure, as it is not
known how various features of a histogram will change. Nonetheless, I did perform
the correction for the spectra presented in this chapter [including Fig. 8.1(a)], where
the drifts were relatively small.
Figure 8.1(b) shows a spectrum of LC2B taken at a difierent suppression fleld
where I0 was larger. These are the same conditions under which the T1 measurement
in Fig. 7.12 was performed. The flt of the circles to a single junction spectrum,
drawn as a solid line for I0 = 33:663 ?A and CJ = 7:32 pF using Eq. (2.44), shows
fairly signiflcant disagreement with the data. This is due to the plasma frequency
of the junction approaching the LC coupling mode frequency of 7.2 GHz. In this
case, ? (6 GHz) = 0:24, which gives an efiective capacitance CJ (1+? (!)) = 6:0 pF,
which is still smaller than the flt value. It may that the coupling strength is changing
so quickly that the efiective parameter will not faithfully reproduce the energy levels.
While the data could have been flt to a system with two degrees of freedom (such
as shown in Fig. 2(b) of Ref. [78]), the single junction flt su?ciently characterizes
the device for the T1 measurement simulation. Notice that the flt capacitance for
the spectrum is quite large, while the value that describes the ground state escape
rate ?0 [4.24 pF, from Fig. 7.7(b)] is much closer to the design value of the junction.
This suggests that the coupling to the LC mode does not have a strong efiect on
the escape rates.
The spectroscopy measurements were always performed with a slow bias ramp2
[0.037 and 0.93 A/s for Figs. 8.1(a) and (b)], so it did not in uence the dynamics.
2A discussion of what is meant by a slow ramp in given in x7.5.
264
Therefore, the same results could have been obtained by sweeping !rf at flxed Ib.
We did not sweep !rf because of two challenges. Given the flnite bandwidth of the
biasing lines, it takes a certain amount of time for Ib to stabilize to a dc value. If
Ib is settling to a value where the escape rate is directly measurable, then many
switching counts will occur before !rf can be swept. The measurements described
in x6.6.2 and x6.6.3 could alleviate this problem, because they can be used at lower
bias. In addition, the microwave lines have a frequency-dependent attenuation. By
sweeping Ib we can flnd the resonance location for a particular frequency and power
of microwaves. If !rf were swept, then we would have to work at flxed PS and
features in the enhancement could be due to the energy level structure of the qubit
or resonances of the microwave lines.
8.2 Spectroscopy of dc SQUID Phase Qubits
The basic procedure for measuring and analyzing a spectrum of a dc SQUID
phase qubit is the same as described above. However, the bias trajectory and  ux
state serve as potentially useful degrees of freedom.
The simplest bias path to take while measuring a spectrum is shown in Fig.
6.7(a), where the current bias Ib is swept at flxed  ux bias If. In this case, I never
saw any clear enhancement peaks. This is because with this trajectory, the isolation
junction usually switches to the voltage state. As it is not protected from noise on
the current bias line, the resonances will be very broad; apparently they are too
broad to be seen.
Instead, the qubit junction can be studied with the simultaneous current and
 ux ramping scheme sketched in Fig. 6.7(d). Figure 8.2(a) shows a spectrum of
SQUID DS1 for  ux state N' = 0 with the simultaneous biasing. The 0 ! 1
transition is clearly visible. Although this was taken at the base temperature of the
refrigerator, there is a hint of the 1 ! 2 transition.
265
(a)
(b)
Figure 8.2: Spectra of SQUID DS1. (a) At 20 mK, the 0 ! 1 transition of the
qubit is clearly visible, under simultaneous biasing of  ux state N' = 0. The circles
indicate the peak enhancement and the lines show !01 and !12 for a single junction
with I0 = 34:275 ?A and CJ = 4:48 pF. (b) The spectrum for N' = ?7 shows much
more structure, including the 0 ! 1 (circles), 1 ! 2 (squares), and 2 ! 3 (triangles)
transitions. Lines are drawn for these three frequencies for I0 = 30:304 ?A and
CJ = 4:30 pF.
266
As the bias path is designed to hold the current I2 through the isolation junc-
tion at nearly zero, the current I1 through the qubit is roughly Ib. Therefore, the
device can be modeled as a single junction biased by Ib, as I discussed in x6.4. The
solid lines show the transition frequencies as a function of Ib for I0 = 34:275 ?A
and CJ = 4:48 pF, calculated with Eq. (2.44); the agreement between the data and
calculation show that the single junction model provides a good parameterization
of the SQUID energy levels.
For Fig. 8.2(b), we initialized the device to N' = ?7. The spectrum has quite
a few features that were not present at N' = 0. The transition for !01 is visible,
although it is not nearly as well deflned as in Fig. 8.2(a). In addition, the 1 ! 2 and
2 ! 3 transitions can be seen in the escape rate enhancement (but not clearly on the
grayscale map), even though the refrigerator was at base temperature. As discussed
in x4.3, this is due to noise on the bias leads and the degradation of the isolation by
the Josephson inductance of the isolation junction. I flt the 0 ! 1 transitions peaks
(marked by circles) with the single junction model (biased by Ib), which yielded
I0 = 30:304 ?A and CJ = 4:30 pF. These same values were used to draw !12 and
!23, which agree fairly well with the data, even though Ib < I1 for this  ux state.
The 0 ! 1 data deviate signiflcantly from the flt near 8.5 GHz, where there appears
to be an avoided crossing. The difierence in Ib for N' = 0 and ?7 suggests that
I2 ? ?4 ?A in the latter case. Assuming that I02 = 4:4 ?A (see Table 6.1) and
CJ2 = 2:09 pF (see Table 4.2), !01=2? for the isolation junction is predicted to be
8.1 GHz at this bias current. Thus the two SQUID junctions could be degenerate at
this  ux state, with the coupling between them leading to an energy level splitting.
The values of I0 are quite difierent for the two  ux states, because the flts were
performed with respect to Ib. As the qubit?s critical current is independent of N',
it is the circulating current due to the trapped  ux that makes up the difierence.
Fits with respect to the qubit branch?s current I1 should return the same value of
267
(a) (b)
Figure 8.3: Spectral flt parameters for SQUID DS1. By fltting the measured spec-
trum of DS1 to !01 for a single junction, the efiective (a) critical current I0 and
(b) capacitance CJ of the qubit can be extracted. The value depends on the  ux
state N' that the SQUID was initialized to. The open circles are the result of
fltting to the current bias Ib. For the solid squares, the qubit current I1 was calcu-
lated with I01 = 34:3 ?A, I02 = 4:4 ?A, L1 = 3:520 nH, L2 = ?5:2 pH, Mb = 0,
Mf = 51 pH, and  ux ofiset '0T = ?0:487 '0. More consistent values of I0 are
found for L2 = 25 pH and '0T = ?0:2 '0 (triangles).
I0. I attempted to verify this with the data set shown in Fig. 7.9(a), where DS1
was initialized to sixteen difierent  ux states with  ux shaking. We took microwave
data for only four frequencies (7.2, 7.3, 7.4, 7.5 GHz), so there was more uncertainty
in the flt parameters than for the data of Fig. 8.2.
I flrst used the single junction model to flt !01 vs. Ib. In this case, I0 increases
by roughly 570 nA when N' increases by 1 (results not plotted). The flt values of
the capacitances are shown in Fig. 8.3(b) as open circles. They vary by almost 20%
over all the  ux states.
Rather than just fltting to Ib, I1 can be calculated using the method of flnding
potential minima described in x6.4, as long as If is known throughout the bias
ramp. Table 6.1 gives the following device parameters for fleld #2: I01 = 34:3 ?A,
268
I02 = 4:4 ?A, L1 = 3:520 nH, L2 = ?5:2 pH,3 Mb = 0, Mf = 51 pH. The solid
squares in Fig. 8.3 show the resulting flt values of I0 and CJ vs. N'. In order for the
flt value of I0 to equal the input value of I01 = 34:3 ?A for N' = 0, I had to assume
that there was a background  ux ofiset of '0T = ?0:487 '0 (which was present when
If = Ib = 0). Ideally, I0 and CJ would be independent of N', but there is quite a
bit of variation in I0. Worse still, with these parameters, the bias trajectory does
not cross the qubit branch for N' = 8.
I performed the same analysis on the spectrum of Fig. 8.2(a). As this data
set was taken flve months after the one shown in Fig. 7.9, the device parameters
appeared to be slightly difierent. A flt to the measured spectrum !01 vs. I1 (for
N' = 0) yielded I0 = 34:300 ?A and CJ = 4:43 pF, for a  ux ofiset '0T = ?0:263 '0.
The extracted parameters for other  ux states, measured simultaneously, varied to
a similar extent as in Fig. 8.3. Nonetheless, these are junction parameters I used in
Figs. 7.8(b) and 7.13 as I!0 and C!J to describe the energy levels of the qubit.
Returning to Fig. 8.3, I tried to improve the consistency by choosing the
modifled values L2 = 25 pH and '0T = ?0:2 '0. The results are shown as triangles
in Fig. 8.3. The values of I0 vary over a much smaller range than before. However,
they are centered about 34.6 ?A, even though I01 was set to 34.3 ?A to calculate
I1. I could not lower this value while maintaining solutions for all of the states.
This inconsistency and the systematic variation of the flt parameters suggest that
the device parameters have not been identifled correctly, the biasing model has been
overly simplifled, the independent junction assumption breaks down for large jN'j,
or there were errors in taking the data. The analysis in the rest of this chapter and
the next will not be signiflcantly afiected by these small errors. The main message
of Figs. 8.2(b) and 8.3 is that assuming Ib ? I1 under the simultaneous ramp will
3Only the sum L2 +Mb can be determined from Ib vs. 'A characteristics. Here, I have chosen
to set Mb = 0, which has no efiect on the biasing calculation. However, L2 can be negative as a
result, if Mb is su?ciently negative (a result of the sign conventions).
269
result in an energy level structure that resembles a single current-biased junction,
with a corresponding capacitance close to CJ1. However, when one takes a close
look at the data, some discrepancies with the model become apparent.
As another test, Fig. 8.4 shows a spectrum of DS1 taken with an intentionally
shallow bias trajectory. While the simple simultaneous biasing described in x6.4
corresponds to a ramp ratio of ?If=?Ib = 69, these data were taken at a ratio of
109. In addition, the mixing chamber of the refrigerator was heated to 110 mK. As
j1i was thermally occupied, the spectrum shows both !01 (circles) and !12 (squares).
For the asymmetric fl = 8:4 SQUID shown in Fig. 6.7, the bias path results
in two types of switches. For low Ib, the path crosses the isolation branch along
the dotted lines; this will result in the SQUID retrapping in another  ux state. For
higher Ib, the qubit will switch to the voltage state in the usual way. As there is
retrapping along the way, the full experimentally measured histogram shown in the
inset to Fig. 8.4(a) is quite asymmetric, unlike what would result from a proper
simultaneous ramp. Interestingly, flfteen peaks are seen in this histogram (although
not all are visible on the plot), with the flrst apparently coming from the isolation
junction switching to the voltage state. I made the arbitrary choice to analyze the
 ux state marked with an arrow.
In Fig. 8.4(a), the transition frequencies are plotted as a function of Ib, which
we ordinarily only do for the proper simultaneous ramp (when Ib ? I1). Nonetheless,
the transition frequencies have the usual dependence on the current bias. A flt to
!01 (solid lines) gives I0 = 31:660 ?A and CJ = 3:57 pF, with reasonable agreement.
However !12 for the same parameters underestimates the data by 500 MHz. The
experimental values of !12 are faithfully reproduced for a single junction with I0 =
31:688 ?A and CJ = 3:52 pF (dashed lines), but the corresponding !01 is too large.
To get sensible flts in this case, If and the trapped  ux of this particular state
must be considered. By using the current- ux characteristics with values listed in
270
(a)
(b)
Figure 8.4: Shallow ramp spectrum of SQUID DS1. At 110 mK, both the 0 ! 1
(circles) and 1 ! 2 (squares) transitions are clearly seen in the spectrum. (a) When
the spectrum is viewed as a function of Ib, both transitions cannot be described
by a single junction model. The solid lines are drawn for I0 = 31:660 ?A and
CJ = 3:57 pF, while the dashed are for I0 = 31:688 ?A and CJ = 3:52 pF. The
inset shows the full (unshaken) histogram, with an arrow indicting the peak that
was analyzed. (b) A good flt to the transitions is found by plotting the spectrum
against I1, which takes If and trapped  ux into account. The grayscale map shows
the normalized enhancement.
271
Table 6.1, it appears that this peak is N' = 27. This state can only be occupied
by retrapping along the bias path, so  ux shaking cannot be used for initialization.
The combination of the circulating current due to trapped  ux, Ib, and If were used
to calculate the current through the qubit I1 as plotted in Fig. 8.4(b). Now a flt
to !01 yields I0 = 34:300 ?A and CJ = 4:51 pF, which also describes the 1 ! 2
transition. These flts are of a higher quality than those shown in Fig. 8.4(a). In
the grayscale map, which is the normalized escape rate enhancement, the 2 ! 3
transition is barely visible. The model junction reproduces this branch as well.
The usual simultaneous biasing does not add any current through the isolation
junction. For the range plotted in Fig. 8.4, I2 is roughly ?2:6 ?A and changes
appreciably. In Fig. 6.7(a), a better flt value of I0 could have been obtained by
shifting the Ib-axis; however this would not have changed the slope and the flt value
of CJ would still have been incorrect.
Agreement between the flt value of I01 and the value from the current- ux
characteristics was enforced by adding a constant ofiset  ux of '0T = ?0:364'0
to the bias path. In calculating I1 in Fig. 8.4(b), I used L2 = ?5:2 pH. With
L2 = 30 pH, the flt values become I0 = 34:275 ?A and CJ = 4:50 pF, which
essentially corresponds to a small shift. Thus, the di?culty in determining L2 is not
a serious problem for extracting I01 and CJ1.
In everything that follows for the SQUIDs, I used the simultaneous current
and  ux biasing as shown in Fig. 6.7(d). This example was given just to show that
the picture of the SQUID that we have developed holds together and to suggest a
technique to verify that the qubit junction is being biased as expected.
8.3 Spectroscopic Coherence Time
Aside from locating the !01 resonance and providing a measurement of I0 and
CJ (which in turn can be used to predict tunneling rates), spectroscopy also can be
272
used to measure the spectroscopic coherence time T?2 of the system; see x3.7. This
quantity, which depends on the relaxation time T1, the coherence time T2, and low
frequency noise sources, is a useful indicator of the quality of the isolation of a qubit
from its environment.
The flrst step in calculating T?2 is to flnd the full width at half maximum of a
resonant transition. If the level spacing were held constant while !rf was swept, the
system only had two levels, and there was no tunneling or current noise, we would
expect to see peaks with Lorentzian linewidths if the excited state population was
plotted as function of !rf [see Eq. (3.60)].
Experimentally, we were only able to gauge the resonance width by measuring
the total escape rate due to a microwave current. As shown in Fig. 6.5(c), when the
microwaves are resonant with a qubit transition, a peak is seen in the escape rate
enhancement ??=?. As the enhancement is roughly proportional to the excited
state population (see x6.6.1), the resonance width can be estimated from ??=?. A
more careful analysis shows how the enhancement is related to the true width (see
x3.5.1 of Ref. [1] and x2.6.3 of Ref. [3]).
I usually flt the measured escape rate enhancement as a function of ramp time
to extract the full width at half maximum ?t and the peaks were often slightly
asymmetric (due to, for example, bias-dependent escape rate ratios) and not de-
scribed well by Lorentzians or Gaussians. I often just identifled the width by eye or
flt to another functional form, even if there was no physical justiflcation for doing
so. I found that the asymmetric double sigmoidal,4
y = A
1+exp
?
?x?xc+w1=2w2
?
2
41? 1
1+exp
?
?x?xc?w1=2w3
?
3
5; (8.1)
had enough degrees of freedom to provide a high quality flt. Here, A is related to
4This function and many others are included in Origin?s fltting utility.
273
the height, xc controls the center, and w1, w2, w3 set the width and asymmetry. No
matter the method, there was typically a 10% uncertainty in determining ?t. It
was often larger at low and high currents, where the counting statistics tended to
be poor.
Using the calibration of Ib as a function of time, I then converted ?t to the
full width at half maximum ?Ib in terms of current. Figure 8.5(a) shows ?Ib as
a function of !01 (assuming a resonance peak is centered about !rf = !01) for the
junction LC2B spectra in Fig. 8.1(a) (open circles) and Fig. 8.1(b) (open squares),
and the SQUID DS1 spectrum in Fig. 8.2(a) (solid triangles). Each spectrum hap-
pened to cover a difierent frequency range, which was dictated by the range of the
measurable escape rates. At the low frequency end of each spectrum (i.e. a shallow
potential well) the widths increase rapidly, while at high frequencies they appear to
saturate.
The spectrum gives !01 as function of Ib, which may be used to convert ?Ib to a
width ?! in frequency. The assumption is that if we had performed the experiment
by holding Ib constant and sweeping !rf, we would have measured a resonance width
?!. Results of this conversion, which are independent of the current calibration
when plotted as a function of !01, are shown with symbols in Fig. 8.5(b). The
qualitative frequency dependence is preserved between ?t and ?!, but there is an
interesting efiect with the two data sets for LC2B. Because Fig. 8.1(b) was taken
over frequencies close to the LC coupling mode, jd!01=dIbj is relatively small as !01
approaches the bias independent mode.5 Therefore even though ?Ib for the squares
is larger than for the circles, ?! for the squares is slightly smaller (at the highest
frequencies). It is possible that the circles are limited by current noise, while the
squares saturated due to an intrinsic mechanism. In general, even if ?Ib saturates
5The flt of the spectrum was quite poor for this data set, so I converted ?Ib to ?! by hand,
rather than by using extracted values of I0 and CJ. At the lowest frequencies, I was unable to do
this, which is why there are fewer squares in Fig. 8.5(b) and (c).
274
(a)
(b)
(c)
LC2B
DS1
LC2B
DS1
LC2B DS
1
Figure 8.5: Resonance widths of LC2B and DS1. Peak statistics are shown for the
junction LC2B spectra in Fig. 8.1(a) (open circles) and Fig. 8.1(b) (open squares),
and the SQUID DS1 spectrum in Fig. 8.2(a) (solid triangles). The (a) full width
?Ib at half maximum of the resonance in current was converted to a (b) full width
?! at half maximum in frequency by using the !01 transition frequencies. The lines
show estimates for dissipation (dotted), tunneling (dashed), and the sum of both
and current noise (solid). (c) The spectroscopic coherence time is T?2 = 2=?!.
275
with increasing frequency, ?! will continue to decrease due to the decreasing value
of jd!01=dIbj.
In the case of DS1, the width at 7.5 GHz is about 75 MHz, for a resonance
quality factor of roughly 100. However assuming T1 ? 15 ns, Q ? 700 using the
deflnitionQ = !p RJCJ giveninx2.2.3. Thislattervalueisbasedonlyondissipation.
That the observed Q is much smaller suggests that other broadening mechanisms are
at work. As shown by Eq. (3.66), ?! is due to all of the transitions that depopulate
either j0i or j1i, assuming that pure dephasing and power broadening are negligible.
Atlowtemperatures, whereWt01 ? 1=T1 andWt10 ? 0, ?! ? 1=T1+?1+2 I @!01=@Ib
[38]. Here, the escape rate ?1 of the flrst excited state is much larger than than that
of the ground state and  I is the rms value of the low frequency current noise.
The dotted lines in Fig. 8.5(b) show the contribution to ?! by dissipation.
Based on the measurements in x7.4, I assumed T1 was 4 ns for junction LC2B and
15 ns for SQUID DS1 (and that it had no frequency dependence). The contribution
from tunneling is indicated by the dashed lines. ?1 increases quickly with increasing
Ib (or decreasing !01), which accounts for the increase in ?! at high bias. Finally,
1=T1+?1+2 I@!01=@Ib is plotted with solid lines.6 For LC2B, I chose  I = 3:2 nA;
this value roughly reproduces ?! for the circles and squares, although there are
signiflcant quantitative discrepancies. For DS1, I set  I = 2 nA, suggesting that the
inductive isolation did reduce the low frequency current noise, as it was designed
to do. It is unclear why there are systematic deviations between the data and
prediction in all cases, but they may be due to the simpliflcations in Eq. (3.66)
and di?culty of accurately calculating ?1. In principle, this method can be used to
determine T1, by choosing parameters that give the best flt; see x7.1 of Ref. [3].
6For the circles, I calculated the energy levels with Eq. (2.44) for junction parameters I!0 =
15:420 ?A and C!J = 5:65 pF; escape rates cames from Eq. (2.43) with I?0 = 15:434 ?A and C?J =
3:4 pF. For the squares, the spectrum could not be accurately flt with the single junction model,
so I extracted vales directly from the data; tunneling rates were calculated with I?0 = 33:648 ?A
and C?J = 4:24 pF. For the triangles, I!0 = 34:308 ?A, I?0 = 34:300 ?A, and C!J = C?J = 4:43 pF.
276
Finally, values of the spectroscopic coherence time T?2 = 2=?! [see Eq. (3.67)]
are plotted in Fig. 8.5(c). The SQUID phase qubit (triangles) is better isolated than
the LC-isolated qubit, as seen in the saturation values of ?! and T?2. In both cases,
T?2 is well below estimates for 2T1, suggesting that dephasing and inhomogeneous
broadening are signiflcant.
It could also be that the saturation of ?Ib is not due to  I or T1, but instead
to the measurement technique. For example, if the timing resolution of the escape
rate measurement was su?ciently poor, then no peak could be narrower than a
certain value. Also, we measure the resonance peak by sweeping Ib linearly to I0.
The linewidth is only reproduced faithfully if the time scale of the sweeping is much
slower than the dynamics of the junction.7 Of lesser concern is the accuracy of the
current calibration, as these errors are largely removed by measuring peak widths
in terms of frequency rather than current.
These issues are addressed by Fig. 8.6, which shows the statistics for the
resonance peaks of SQUID DS2B taken at three difierent ramp rates of Ib.8 As seen
in Fig. 8.6(a), ?t does scale with the ramp rate, increasing by a factor of 4 between
0.0258 A/s (circles) and 0.0041 A/s (triangles). However, when these are expressed
in terms of frequency, they are in reasonable agreement [see Fig. 8.6(b)]. I have
not shown the intermediate step of ?Ib, but these values saturate at about 4 nA.
Thus this check shows that there is no obvious connection between the saturation of
Ib and the experimental technique. The scatter above 5.7 GHz is indicative of the
uncertainty of the peak flts. Certain features, such as the large width at 6.4 GHz,
appear to be real.
As in Fig. 8.5(b), the dotted line in Fig. 8.6(b) shows the contribution to ?!
by dissipation; here, I assumed T1 = 15 ns. The dashed line comes from tunneling
7A similar situation is discussed in x7.5
8The goal was to hold the ratio of If and Ib constant for the three rates, but this was an
additional source of error.
277
(a)
(b)
(c)
Figure 8.6: Resonance widths of SQUIDs DS2A and DS2B with difierent ramp
rates. The peak statistics were calculated from spectra of DS2B measured at 20
mK, with Ib ramp rates of 0.0258 A/s (open circles), 0.0066 A/s (solid squares),
and 0.0041 A/s (open triangles). The solid diamonds come from a spectrum of
DS2A, taken with a 0.018 A/s ramp. While the (a) resonance width in terms of the
ramp time depends on the rate, the (b) width in frequency and (c) spectroscopic
coherence time do not. The lines in (b) show estimates for dissipation (dotted),
tunneling (dashed), and the sum of both and current noise (solid) for DS2B.
278
out of the flrst excited state. The solid line is the sum of both of these contributions
plus that of low frequency noise, with  I = 1:5 nA.9
The values of T?2 plotted in Fig. 8.6 for SQUID DS2B are longer than those
for SQUID DS1 in Fig. 8.5(c) (triangles) and the value of  I in describing ?! is
smaller. I did not do enough careful testing to determine whether this is due to
something intrinsic to the devices (e.g. difierent fabrication process, presence of
the quasiparticle trap on DS2B, suppression fleld on DS1) or an artifact of the
technique (e.g. heavier room temperature and cold flltering for DS2B). The latter
seems likely as other spectra of DS1 showed slightly longer values of T?2 than those
of Fig. 8.5(c).
Also shown in Fig. 8.6, with diamonds, are the resonance statistics for a spec-
trum of DS2A. As the qubit critical current was slightly higher for this SQUID, the
measurable frequency range is higher. The values of ?! and T?2 are in rough agree-
ment with DS2B. This is potentially interesting because, as mentioned in x4.3.2,
the mutual inductance of SQUID A to its own  ux line was 71.92 pH, while that of
SQUID B to its own  ux line was 17.73 pH. Thus the coupling of noise on the  ux
line to the qubit should be about 16 times as strong in DS2A as in DS2B, based
on Eq. (4.10). It would seem then that the level of isolation was not limited by the
coupling to the  ux line, provided that the magnitude of the noise source did not
scale with If. Biasing the same device with the two difierent  ux lines would have
been a more sensitive test, but one we did not do.
In taking the spectrum of DS2A, we did not reset Ib quickly after a switching
event (see x5.2.2), which was done for the spectra of DS2B. Therefore, there was
thermally excited population in j1iand the spectrum did show a clear 1 ! 2 branch.
In order to see a large signal, the microwave power had to be increased over usual
levels, which may explain the increasing ?t seen in Fig. 8.6(a) for this device (dia-
9To calculate energy levels, I used I!0 = 17:754 ?A and C!
J = 4:44 pF. To calculate escape
rates, I used I?0 = 17:765 ?A and C!J = 3:70 pF.
279
monds). We did, however, attempt to cancel out the  ux in SQUID B due to cross
mutual inductances, by applying IAf and IBf simultaneously. In another spectrum
that we took when we did not do this, the peak widths were noticeably larger. As
the conditions were not identical (e.g. difierent room temperature fllters), it is not
possible to draw any flrm conclusions about cross-talk between the devices and its
connection to T?2.
A further test on the description of the peak widths is to examine transitions
of the energy levels higher in the potential well. As discussed in x3.4, the relaxation
rate from jni to jn?1i in the harmonic limit is ?n;n?1 = n=RJCJ. Assuming
dissipation-limited decoherence at T = 0, the resonance widths should occur with a
1 : 3 : 5 ratio for the 0 ! 1, 1 ! 2, and 2 ! 3 transitions [see Eq. (3.66)].
Figure 8.7 shows the full width at half maximum of the (a) 0 ! 1, (b) 1 ! 2,
and (c) 2 ! 3 transitions of SQUID DS1. We heated the mixing chamber to 110
mK so that j1i and j2i were su?ciently occupied to make transitions from them
visible. The data for the three transitions were taken with roughly the same range
of microwave frequencies, but I have plotted the points as a function of !01; for
example, !23=2? is 7:4 GHz at the bias current where !01=2? is 8:68 GHz.
At low frequencies, the widths increase substantially, due to escape rate broad-
ening. At high frequencies, the 0 ! 1 and 1 ! 2 widths appear to saturate, but
at values of 80 and 100 MHz, far from the expected 1 : 3 ratio. This suggests that
current noise (or perhaps pure dephasing) had a signiflcant efiect on the resonance
widths. ?!23 was di?cult to extract due to poor counting statistics, but it may be
saturating near 200 MHz. While ?!23=?!12 is consistent with broadening due only
to dissipation, the values require T1 ? 5 ns, which is lower than the measurements
of Chapter 7 would suggest.
I attempted to perform a quantitative analysis of the widths using Eq. (3.66),
as in Fig. 8.5(b). The dotted lines in Fig. 8.7 indicate the contribution to the
280
(a)
(b)
(c)
Figure 8.7: Resonance widths of SQUID DS1 at elevated temperature. The circles
show the full width ?! of the (a) 0 ! 1, (b) 1 ! 2, and (c) 2 ! 3 transitions
measured at 110 mK, as a function of !01. The lines show estimates for dissipation
(dotted), tunneling (dashed), and the sum of both and an rms current noise of 2 nA
(solid).
281
widths from inter-level transitions. The thermal rates are given by Eqs. (3.26) and
(3.27), which I evaluated (for single level transitions of a four-level system) by using
the cubic matrix elements in Eq. (2.38). I assumed that RJCJ had a frequency-
independent value of 15 ns. At 110 mK and the bias current where !01=2? =
7:5 GHz, the inverse of the sum of the thermal rates is 14.6, 4.84, and 3.27 ns, for
the 0 ! 1, 1 ! 2, and 2 ! 3 transitions. The dotted lines are drawn at these
values, assuming no frequency dependence.
The dashed lines are estimates for the contribution by tunneling from the
higher of the two states in the transition; for example, the dashed line in Fig. 8.7(b)
is ?2, calculated using Eq. (2.43). Finally, the solid lines in Fig. 8.7 are the sum of
the contributions from dissipation, tunneling, and current noise with  I = 2 nA.
As in Fig. 8.5(b), ?!01 is well described by the estimate including current
noise [see solid line in Fig. 8.7(a)]. What strengthens the argument for current noise
is the fair agreement between data and prediction for ?!12 in Fig. 8.7(b). The
agreement is not as good for ?!23 in Fig. 8.7(c). In this case, the widths ?Ib in
current were di?cult to extract due to poor counting statistics in the switching
histograms. In addition, the conversion to ?! was inaccurate due to deviations
in !23 from the single junction model; this lead to underestimates of ?! at low
frequencies. Nonetheless, it does seem that estimates for ?3 based on measurements
of ?0 are reasonably accurate. It would be useful to examine data at lower bias to
determine the saturation value of ?!23.
8.4 Multi-Level and Multi-Photon Transitions
Figure 8.8 shows a spectrum of SQUID DS2B, taken at 110 mK. As with Fig.
8.4, the excited states were thermally populated, even in the absence of microwaves.
As with all of the spectra, data at each frequency were taken at a difierent microwave
power and each horizontal line of the grayscale map has been independently normal-
282
(a)
?01?
12?23?34
?01
2
?12
2
?23
2
?34
2
?02
2
?13
2
?24
2
(b)
Figure 8.8: Low frequency spectrum of SQUID DS2B. At 110 mK, the spectrum
shows a number of transitions between the lowest flve states in the potential well. (a)
The grayscale map shows the escape rate enhancement spectrum, while the dashed
lines are drawn for a single junction with I0 = 17:828 ?A and CJ = 4:51 pF. (b)
The full width at half maximum of some of the transitions are plotted as a function
of frequency. The solid line shows the current width corresponding to a frequency
width of 50 MHz for !01=2 and !01.
283
ized. Difierent features of the spectrum could have been emphasized by the selection
of difierent microwave powers.
Centered about 6 GHz are four branches that correspond to the single photon
transitionsthatIdiscussedintheprevioussection. Asthetiltedwashboardpotential
well is anharmonic, !34 < !23 < !12 < !01 at any given value of Ib. I flt the
0 ! 1 transition to the single junction values given by Eq. (2.44), which yielded
I0 = 17:828 ?A and CJ = 4:51 pF. All of the dashed lines are drawn with these
parameters.
Escape rate enhancement is also present at exactly half of these frequencies.
The branches labeled !01=2, !12=2, !23=2, and !34=2 correspond to two-photon
transitions between levels. Multi-photon transitions, allowed because the potential is
anharmonic, have been previously observed in a number of superconducting systems
[42,43,121]. They are easy to identify from their quadratic power dependence.
Similarly, in between the single photon branches are three narrows ones that
appear at frequencies above 6 GHz. These could have been made more prominent at
lower frequencies by increasing the microwave power. As agreement with the dashed
lines suggests, these occur at frequencies !02=2, !13=2, and !24=2. They correspond
to two-photon transitions between states jni and jn+2i.
Figure 8.9 shows the spectrum of transitions at higher frequencies.10 Most
prominent are the branches labeled !02, !13, !24, and !35, which are single photon
versions of the transitions just described. At higher frequencies still, are transitions
that cover the spacing between four levels with a single photon, namely !03, !14,
and !25. The microwave power was not set high enough to see these transitions
clearly, which explains the noise around 18 GHz. The two-photon version of these
transitions (!03=2 and !14=2) are barely visible near 9 GHz. I have interrupted the
dashed line flt near these branches, so as not to obscure the data. At 13.4 GHz, the
10The frequency axis is compressed by over a factor of 3 as compared to Fig. 8.8. Data were
also taken at larger intervals.
284
(b)
?14?25 ?03
?02?24?35 ?13
?03
2
?14
2
(a)
Figure 8.9: High frequency spectrum of SQUID DS2B. At 110 mK, the spectrum
shows a number of transitions between the lowest six states in the potential well. (a)
The grayscale map shows the escape rate enhancement spectrum, while the dashed
lines are drawn for a single junction with I0 = 17:828 ?A and CJ = 4:51 pF. (b)
The full width at half maximum of some of the transitions are plotted as a function
of frequency. The solid line shows the current width corresponding to a frequency
width of 50 MHz for !02 and !03.
285
power was unintentionally set very high. Three additional peaks appear that seem
to correspond to transition frequencies near 2!01, 2!12, and 2!23.
The dashed lines in Figs. 8.8 and 8.9 correspond to twenty transitions (but
depend on only flve transition frequencies !nm) and were all plotted with the same
values of I0 and CJ from a flt to !01. This is encouraging in that it conflrms our
understanding of the Hamiltonian of the device. However, the presence of all of these
transitions does make a liability of this device clear. We would like to control the
population in j0i, j1i, and possibly j2i (if it used as an auxiliary state) for quantum
computation. However, in the phase qubit, higher excited are not far separated from
the desired states and care must be taken to ensure that they do not become highly
populated.
For all of the branches, the dashed-line flts overestimate the observed values at
low frequencies. As seen most clearly with the single-level, single-photon transitions
between 3.5 and 5 GHz, the branches become nearly vertical. This is at least partly
due to the fact that we measured the resonances by sweeping Ib. Thus the escape
rates (and their ratios) vary across the resonance. The escape rate out of the higher
level of a transition becomes very large as it leaves the potential well, resulting in a
distortion of the shape of the resonance of ??=?. This leads to deviations from the
actual energy level spacing.
In Figs. 8.8(b) and 8.9(b), the full width at half maximum ?Ib of several of the
transitions is plotted as a function of frequency. There is a fair bit of scatter in the
points, at least partly due to the small number of counts taken at each frequency.
At the low frequency end of each branch, the widths increase due to escape rate
broadening. As the goal of this data set was to see all available transitions, the
microwave power was set fairly high, which probably lead to some power broadening
as well. This is particularly true at the high frequency end of each branch, where
the background escape rates are small (and the easiest way to get good statistics
286
on a microwave resonance is to use a high power). Finally, the temperature was
high enough to broaden the transition. Nonetheless, several of the transitions (with
both one and two photons) have a minimum width of roughly 4 nA, suggesting that
there was low frequency current noise that limited the \resolution." As expected,
at a given frequency, transitions deeper in the well are sharper and the two-photon
transitions are sharper than single photon ones (e.g., the two-photon 0 ! 2 is
sharper than the single photon 0 ! 1).
The solid lines in Figs. 8.8(b) and 8.9(b) show the value of ?Ib corresponding
to a full width of ?!=2? = 50 MHz. This is plotted for a transition from the ground
state, but all of the transitions in a given frequency range have similar results. A
two-photon 0 ! 1 transition corresponds to small values of d(!01=2)=dIb. Thus,
deep in the well, a 4 nA width corresponds to a sharp resonance of about 25 MHz.
For a 0 ! 3 transition, d!03=dIb is quite a bit larger, so the same 4 nA corresponds
to a frequency width of over 100 MHz at 20 GHz. A detailed comparison of the
widths for difierent transitions could be performed to see if Eq. (3.66) is obeyed.
8.4.1 Power Dependence
So far, most of the spectroscopic data I have shown were taken at relatively low
microwave power, so the efiects of power broadening could be ignored. One reason
to examine the power dependence of the resonance peaks is determine the maximum
power that can be used without afiecting the width. More importantly, the power
dependence can be compared to the expected behavior to verify our understanding
of the system dynamics and extract important parameters. I will delay a direct
comparison to the expected behavior discussed in x3.7 until x9.1.
The inset of Fig. 8.10(a) shows the measured escape rate of SQUID DS1 taken
without microwaves (solid circles) and with a 7.6 GHz microwave current (open
circles). The nominal power of the microwave source was set to PS = ?26 dBm.
287
(a)
(b)
(c)
Figure 8.10: Power dependence of spectral peaks in SQUID DS1. (a) With a mi-
crowave current of frequency 7.6 GHz and power PS = ?26 dBm, four peaks are
seen in the measured escape rate enhancement; the solid line is a flt with a sum of
Lorentzians. The inset shows the escape without (solid circles) and with (open cir-
cles) microwaves. (b) The maximum enhancement for the single photon 0 ! 1 (solid
circles), 1 ! 2 (solid triangles), and two-photon 0 ! 2 (open circles) transitions
increase as a function of power. The lines show power law flts. (c) The resonance
full widths, in terms of the qubit current I1, also has a power dependence.
288
The microwaves excite four transitions: single photon 0 ! 1, two-photon 0 ! 2,
single photon 1 ! 2, and two-photon 1 ! 3, from right to left. Even though the
refrigerator was at base temperature, there was enough population in j1i to make
the 1 ! 2 and 1 ! 3 transitions quite visible at high power.
The escape rate with microwaves is uniformly larger than the background
values, making it seem that perhaps the microwave current is heating the device.
This hypothesis can be checked by examining the escape rate enhancement, which
is plotted with solid circles in the main panel of Fig. 8.10(a). As opposed to the
usual enhancement of 1 to 5 (typical for taking a spectrum), ??=? reaches nearly
100. The solid line is a flt to the sum of four Lorentzians, which does a good job
of reproducing the peaks and also the valleys. This strongly suggests we are mainly
seeing resonant phenomena and heating due to the microwaves is making a negligible
contribution.
We measured the escape rate for a wide range of source powers PS and flt
the enhancements to extract the height and width of the resonance peaks. The
maximum enhancement is plotted in Fig. 8.10(b), for the single photon 0 ! 1 (solid
circles), 1 ! 2 (solid triangles), and two-photon 0 ! 2 (open circles) transitions.
The values increase with power, with the two-photon process having a stronger
dependence.
The straight lines in Fig. 8.10(b) are a flt to a power law, (PS)fi, where PS is
expressed in Watts rather than dBm. The single photon 0 ! 1 transition begins
with a nearly linear dependence, as shown by the dashed line, which is drawn for
fi = 0:946. It eventually slows down to the dotted line, for which fi = 0:336. This
reduction of the slope with power is expected saturation behavior; an incoherent
microwave drive cannot populatej1ito more than 50% occupation probability. From
Eq. (2.43), I estimate that ?0 = 6:3?104 1=s and ?1 = 3:7?107 1=s, so that ??=?
should never exceed 290. Even at the highest power, where the enhancement is
289
100, this suggests dissipation is preventing a complete saturation of the flrst excited
state. The 1 ! 2 transition appears to follows the same power laws, but the lack of
data at low powers (due to small enhancements) makes it di?cult to be sure. The
solid line is drawn for fi = 1:99, showing that the two-photon process does have the
expected quadratic power dependence.
Figure 8.10(c) shows the full width ?I1 at half maximum of the three res-
onance peaks. The single photon processes have nearly constant values below
PS = ?40 dBm, which corresponds to ??=? ? 25 in this case. I will return to
the dependence of the broadening in x9.1. The two-photon transition (open circles)
gets sharper with increasing power. I do not know if this is artifact of the fltting
procedure, as the difierent peaks begin to overlap at high power.
8.5 Spurious Junction Resonances
Recently, much has been made of the importance of materials science for high
quality junction qubits.11 We like to think of our devices as simple \artiflcial atoms"
consisting of a few tunable energy levels. This model works quite well in the variety
of experiments I have described so far, but it should come as no surprise that the
huge number of real atoms that make up the junctions, wiring, and insulation can
have some efiect on performance.
It has been long established that Josephson junctions can show behavior as-
sociated with coupling to a bath of two-level systems [137,138]. The standard
microscopic picture, now more than 20 years old, is that a group of atoms (or
perhaps a single atom) can tunnel between two stable positions inside the tunnel
barrier or nearby insulating layers. This may result in a  uctuation of the critical
current (which in view of the saturation of ?Ib in x8.3 is di?cult to distinguish
11As the material of the junction barrier has received much of the attention, it would be very
interesting to see if the NbN/AlN/NbN junctions used in Refs. [37,136] had intrinsically lower
dissipation.
290
from current noise). Recent experiments have shown that the quantum properties
of individual two-level  uctuators can be studied when they come into resonance
with a qubit [44,121].
No consensus has been reached on the impact of these objects on quantum
computation, but they have been proposed to be a leading source of decoherence
for large junctions [139]. Turning the problem on its head, if these external degrees
of freedom show su?ciently long coherence, they may be treated as qubits [140].
The basic idea is that the operations can be performed by bringing the junction and
two-level system into resonance, at which point they will coherently evolve.
While we did not come to any hard conclusions on the physical nature of these
objects, in this section I will summarize the results that we did obtain. This was
important to do, not only to fully characterize the system, but also because the
existence of microstates could have a strong in uence on other measurements, such
as Rabi oscillations. I focused on trying to identify spurious resonators through
spectroscopy and conflrming that their origins were intrinsic to the junctions.
Figure 8.11(a) shows a spectrum of SQUID DS2B taken at base temperature.
In order to determine the junction parameters, a spectrum with lines taken every
100 MHz is su?cient. This one, though, was taken at 3 MHz intervals and reveals
several gaps, with no obvious periodicity. This suggests that the junction is coupled
to some extra degrees of freedom. I flt ??=? vs. Ib at each frequency to extract the
resonance location, height, and full width. The solid circles (in this and subsequent
flgures in this section), which show the center of the peaks and are ofiset for clarity,
support the idea that some of these features are avoided crossings. However, even
the largest of these, for example the one at 6.17 GHz, show a splitting of less than 5
MHz. This very weak coupling makes it di?cult to get a clear picture of the origin
and impact of these gaps. Given that we generally have linewidths of about 50 MHz,
it is a bit surprising that we see anything at all. Incidentally, while there are several
291
(a) (b) (c)
Figure 8.11: Fine spectrum of SQUID DS2B. (a) A spectrum taken every 3 MHz at
a constant power of PS = ?59 dBm at 20 mK reveals several small gaps. The centers
of the resonances are indicated by the solid points, all of which are horizontally ofiset
for clarity. (b) The maximum enhancement (solid circles, with the bottom axis) and
full width ?Ib (open, top axis) are plotted for each resonance peak, as is the (c)
area under each peak.
292
odd features in the escape rate measured without microwaves (see S7.2.2), there is
no obvious correlation between these features and the spectral splittings.
It is entirely possible that the curious features in the spectrum are due to res-
onances in the microwave lines. We know that transmission varies on large (GHz)
frequency scales; the question is whether there are sharp features. We can check this
possibility in a few ways. Figure 8.11(b) shows the peak escape rate enhancement
??=? and full width ?Ib for each frequency. The whole spectrum was taken at
nominally flxed microwave power, but the width  uctuates quite a bit, never de-
creasing below 4 nA, just as in Fig. 8.6. Nearly without exception, the enhancement
and width are anti-correlated. In theory, when less microwave power reaches the
junction, the enhancement should decrease and the full width should either remain
constant or get narrower, if there is less power broadening. In the data, the reso-
nances get wider near points where ??=? is suppressed, suggesting that the gaps
are not due to dips in the drive power. Incidentally, the scatter in the widths could
explain some of the variation of T?2 seen in Figs. 8.5 and 8.6, where we took no steps
to avoid these features in the spectrum.
An example of what the escape rate enhancement might look like near an
avoided crossing is plotted in Fig. 8.12(a). In generating this plot, I assumed that
the resonance has a flxed width in frequency; thus, the full width ?Ib in terms of
current is proportional to dIb=d!01, which increases near the splitting. For simplicity,
I have also assumed that the area under the Lorentzian resonance (as a function of
Ib) is constant. In Fig. 8.12(b), the peak enhancement (solid line) and full width
(dashed) are plotted on the horizontal axis, with frequency on the vertical axis. In
this model, the signature of an avoided crossing is ??=? going to zero, while ?Ib
diverges. There are some similarities between this prediction and the data shown
in Fig. 8.11(b), but the relatively large frequency intervals and current noise in the
data lead to less pronounced features.
293
(a) (b)
Figure 8.12: Resonance shape near an avoided crossing. (a) The simulated escape
rate enhancement shows a splitting in the spectrum at a certain frequency. (b) At
this frequency, the maximum enhancement ??=? (solid line) reaches a minimum,
while the full width ?Ib (dashed) of the resonance diverges.
In Fig. 8.11(c), the area under each resonance curve is plotted. We initially
measured the resonance as escape rate enhancement versus current bias, but it would
not necessarily be fair to compare the area at difierent frequencies in terms of these
variables. Instead, I converted the independent and dependent axes to frequency
[using the dependence found in Fig. 8.11(a)] and (??=?)(?0=?1) (which, from Eq.
(6.9), is an estimate of P1) before calculating the area. Here, the escape rate ratio
comes from Eq. (2.43) for a 1-D tilted washboard with parameters determined from
Fig. 8.11(a). Finally, before plotting the results in Fig. 8.11(c), I normalized all of
the values to the maximum area, which occurs at !rf=2? = 5:89 GHz. As it turns
out, the increased width of the resonance at the spectral gaps does not compensate
for the reduced amplitude, as the scatter is still clearly visible in Fig. 8.11(c). We
would expect the area to vary if reductions in the transmitted microwave power
294
were causing the gaps, but this could also be the result of a more subtle efiect.
The smaller areas for all frequencies above 6 GHz could be suggestive of increased
attenuation of the microwave drive there. A better test would be to calculate the
area as a function of !rf for each value of Ib; however in this case, the frequency
response of the microwave lines could lead to false features in the extracted areas.
Figure 8.13 shows three spectra of DS2A taken with difierent conflgurations of
the microwave drive. Fig. 8.13(a) was taken with a source connected to microwave
line A (i.e. the one capacitively coupled to the qubit junction of DS2A).
Figure 8.13(b) was taken under the same conditions, except that an extra 2 m
of SMA cables was added to the room temperature lines. That the features remain
in the same place in both spectra suggest that they are not due to resonances in
the drive lines. As expected, there was an overall decrease in the enhancement in
Fig. 8.13(b) due to attenuation in the additional cables (which has been hidden by
rescaling the grayscale axis). However, relatively speaking, the enhancement at low
frequencies in Fig. 8.13(b) seems to be large. This efiect could be due to resonances
in the extra cables, but the sharp features clearly are not.
For Fig. 8.13(c), the same microwave source was used to drive microwave line
B. Due to the weaker coupling to DS2A from this line, the enhancement was smaller,
which accounts for the noisier signal. Despite this difierence, the locations of the
splittings are unchanged.
The previous two measurements most likely rule out the microwave lines (both
at room temperature and inside the refrigerator) as a source of the spectral reso-
nances. Although the features appear to be intrinsic to the devices, it is possible
they are due to resonances in the sample box. Under the same conditions as in
Fig. 8.11, we measured the spectrum at lower frequencies and saw a splitting at
!rf=2? = 3:085 GHz. This appears to a two-photon transition to the splitting seen
at 6.17 GHz in Fig. 8.11(a). Assuming that the box cavity acts as a harmonic res-
295
(a) (b) (c)
Figure 8.13: Three flne spectra of SQUID DS2A. The splittings are qualitatively
the same when driving (a) microwave line A with a certain length of SMA cable,
(b) microwave line A with cable nearly twice as long, and (c) microwave line B. The
gray scale for each plot has been normalized to the enhancement near 6.65 GHz.
The refrigerator was at 20 mK, but the junctions remained in the voltage state long
enough to cause some heating.
296
onator, it would not have a two-photon response. I would like to think that this
rules out the box as a culprit, but it is possible that any non-linear element in bias
path could lead to an excitation of a higher order mode. Also if the flrst excited
state of the junction is excited by a two-photon process, it could then couple to a
harmonic level of the box. In fact, the resonances at 6 GHz could be higher order
modes. More variations of two-photon excitations would have to be performed to
check this.
With some evidence pointing towards micro-resonators, we can attempt to
develop a model of the interaction. Consider a system consisting of a Josephson
junction (JJ) and a two-level system (TLS), whose energy levels are independent of
the junction?s current bias. In the absence of coupling between the two, I will label
the states jJJTLSi, where TLS can either be 0 or 1. The energies E of the four
lowest excited states, measured with respect to the ground state j00i, are plotted
in Fig. 8.14(a). The junction parameters are I0 = 17:736 ?A and CJ = 4:49 pF, and
the resonator?s excited state is at a frequency 6.69 GHz.
Degeneracies of the uncoupled states occur at 13.70 and 6.69 GHz, as shown
by the solid lines in Fig. 8.14(b) and (c). If the junction and resonator are coupled,
these degeneracies are lifted. Let the states of the coupled system be denoted by
the rounded ket jn). The dashed lines in Fig. 8.14(b) and (c) show the four lowest
excited coupled states, with a 10 MHz coupling between j10i and j01i and between
j11i and j20i.
The combination of the solid and dashed lines in Fig. 8.14(d) show all of the
transition frequencies near 7 GHz between the coupled states; here, ~!nm is the
difierence in energy between jn) and jm). The solid lines roughly indicate where the
transition results in an excitation of the junction. With our experimental technique,
which relies on an enhancement of the tunneling escape rate, only these transitions
can be detected. For example, a transition between j10i and j11i could be allowed
297
(a) (b)
(c)
01
11
10
20
1
2
3
4
?01
?02
?13
?23
?14
?24
?23
(d)
?13 0 1
1 2
Figure 8.14: Model of a spurious junction resonator. (a) The energy levels E of
a junction (JJ) and a two-level system (TLS) (as measured from the ground state
j00i) in the absence of coupling are plotted as a function of the junction?s bias Ib.
The labels indicate the state jJJTLSi. (b,c) The energy levels of the uncoupled
system (solid) are degenerate at 13.70 and 6.69 GHz. The energy levels, denoted
jn), in the presence of a 10 MHz coupling show avoided crossings (dashed). (d)
The transition frequencies !nm between jn) and jm) are drawn as solid and dashed
lines; the escape rate of the system is expected to increase along the solid lines. The
diagrams are drawn for I0 = 17:736 ?A and CJ = 4:49 pF, to match Fig. 8.16.
298
(and result in absorption of the applied microwave power), but it would not cause
? to increase.12 Thus, the solid lines correspond to the 0 ! 1 and 1 ! 2 transitions
of the junction.
In this model, the coupling results in avoided crossings for both the 0 ! 1 and
1 ! 2transitionsatIb = 17:504?A. Whatisinterestingisthattheshapeoftheanti-
crossings are qualitatively difierent.13 Two splittings also occur at Ib = 17:466 ?A,
with the one on the 1 ! 2 branch occurring at the same frequency as the 0 ! 1
splitting at Ib = 17:504 ?A. In addition, there is a transition unafiected by the
coupling at 7.0 GHz. While the transitions between j00i and j10i and between
j01i and j11i are degenerate, only the latter of these is afiected by the interaction
between j11i and j20i at Ib = 17:466 ?A. It may be that this avoided crossing is
di?cult to observe experimentally, for weak coupling strengths.
This single resonator results in the four avoided crossings, as well as one for
the 0 ! 2 transition. The state j01i might also couple to j20i at a higher bias
current, causing an avoided crossing of the usual sort on the 1 ! 2 branch (and an
inverted one on the 2 ! 3); unfortunately, we could not take spectra over such a
large range of current bias with the escape rate measurement. Additional features
will appear if the resonator is a harmonic system, with more than one excited state.
Figure 8.15(a) shows a spectrum of SQUID DS1 at 20 mK. Although this
device was fabricated with a higher critical current density process, the density and
appearance of the gaps are roughly the same as for DS2. The power was adjusted
several times during the course of the data taking (indicated by arrows), so the
sharp jumps in contrast should be ignored.
In order to test the model of Fig. 8.14, there must be some population in j1i.
12The resonance areas in Fig. 8.11(c) might vary for a similar reason.
13The energy level model also applies (in a qualitative way) to the coupling of the states of the
two junctions of the SQUID, where the isolation junction replaces the spurious resonator. In Fig.
8.2(b), where the plasma frequency of the isolation junction is fairly low, it is possible that there
is an \inverted" avoided crossing on the 1 ! 2 branch of the qubit near 8 GHz.
299
(a) (b)
(c)
Figure 8.15: Fine spectra of SQUID DS1 at base and elevated temperatures. Spectra
at (a) 20 and (c) 105 mK show gaps at the same frequencies, even on the 1 ! 2
branch of (c). The two data sets were taken three weeks apart, which may explain
the difierent current biases. Panel (b) was taken under the same conditions as (a),
except that the SQUID was initialized with three fewer  ux quanta. For (a) and (b),
the power was changed at the frequencies marked by arrows to keep the enhancement
roughly constant; panel (c) was taken at constant (albeit higher) power.
300
Ideally, this would be done by flrst resonantly driving the 0 ! 1 transition and then
mapping out the 1 ! 2 branch. I found this di?cult to do, especially over a wide
frequency range. There is also the concern that if the flrst excitation is not done
carefully, it will introduce features of its own in the spectrum.
Instead, j1i can be thermally populated, as in Fig. 8.15(c) [which is plotted
on axes with a difierent aspect ratio than Fig. 8.15(a) and (b)]. Although the
same features appear on the 0 ! 1 branches, the avoided crossings are much more
prominent at low temperatures. In addition, for the data at elevated temperature,
the anti-correlation between the height and width of the resonance peaks is not as
clear as in, for example, Fig. 8.11(b). The energy level spacing is less than 400 mK,
so there is bound to be some thermal smearing at 105 mK. In addition, in order to
see any enhancement at higher temperatures, the microwave power had to be set
higher, which could have lead to some broadening. The 1 ! 2 branch is visible,
although the small initial j1i population and the lower current bias (and thus escape
rate) result in poor statistics. Nonetheless, it is clear that gaps appear at the same
frequencies as for the 0 ! 1 branch, as predicted by Fig. 8.14(d). A stronger test
of the model would be to identify a 1 ! 2 splitting without a 0 ! 1 feature at the
same frequency.
The spectrum in Fig. 8.16(c) is not over a wide enough range to see transitions
on !01 and !12 at the same Ib, a shortcoming that the spectrum in Fig. 8.16 does
not have. An avoided 0 ! 1 crossing is visible at 6.69 GHz. Figure 8.14 predicts an
inverted 1 ! 2 anti-crossing at 6.29 GHz, although nothing obvious is visible there.
It is possible that our spectroscopy technique of sweeping the bias is not sensitive
to this type of feature. For the usual type of anti-crossing, it is possible to sweep
\through" the gap, making it easy to see. In the case of the 1 ! 2 anti-crossing, two
closely spaced peaks would have to be visible over a range of frequencies to identify
the feature. In addition, if the system has 4 nA of low frequency noise, it would
301
Figure 8.16: Fine spectrum of SQUID DS2B at elevated temperature. This spec-
trum, showing !01, !12, and (very faintly) !23, at 105 mK covers a wide enough
range of frequencies to be compared to Fig. 8.14.
302
tend to smear out only the inverted anti-crossing.
The splittings may be di?cult to see at high temperature, because they get
thermally smeared together. Similarly, it has been suggested that spurious res-
onators can become saturated at high powers, at which point they no longer are
able to interact with the junction. I attempted to reproduce this phenomenon by
taking flne spectra at difierent microwave powers. Two spectra separated by a fac-
tor of eight in power showed no qualitative difierence in the contrast of the gaps.
Perhaps a larger range of microwave powers needs to be investigated.
An intriguing possibility is that these resonances are an intrinsic property of
the complicated SQUID potential. One possibility is that we are seeing resonant
tunneling between levels of difierent  ux states [131,132]. However, the well barrier
is tiny with respect to the energy difierence between wells, so this seems somewhat
unlikely. Alternatively, we could be seeing the efiects of interaction with the isolation
junction (within the well of a single  ux state).
To check both of these options, we took spectra when SQUID DS1 was ini-
tialized to difierent  ux states, which should dramatically change the potential that
describes the dynamics of the phase particle. Figure 8.15(b) was taken with three
fewer  ux quanta in the SQUID loop, as compared to (a). This results in a smaller
Ib to switch to the voltage state and a larger current through the isolation junction.
It also strengthens the coupling of the SQUID to the bias lines, which might also
be in uencing the spectral gaps. Despite these changes, the features in the spectra
occur at the same frequencies. Unfortunately, I never took a detailed spectrum of
LC2 which could have deflnitively settled this aspect of the problem, as each of its
qubits contained only one junction.
So either the junction does contain resonators and we were unable to detect
the predicted 1 ! 2 feature, the model is incorrect, or the gaps are real spectral
splittings due to one or more harmonic systems external to the device, such as the
303
sample box cavity. In Fig. 8.13, two splittings are seen near 6.75 GHz for DS2A.
However, in Fig. 8.16, no such features are seen for DS2B even though both devices
were on the same chip and measured during the same cool-down, suggesting that
the box is not the source. A convincing test would be to thermally cycle a single
device and check if the splittings move.
While the spectra can serve to characterize these possible resonators, in the
end, we are only concerned with them if they afiect the qubit performance. For
example they have been reported to afiect the amplitude of Rabi oscillations [44] and
coherent oscillations between a qubit and resonator (in the absence of a microwave
drive) have been observed [114]. These dynamical experiments not only suggest the
efiect of the resonators on the qubit, but can also exclude several possible origins of
the features in the spectra.
For our devices, we saw no clear dependence of a Rabi oscillation on the bias
point. There was a slight dependence of the decay behavior (of the sort shown in
x8.7.2), but it was di?cult to draw any quantitative conclusions. It may be that the
defects are very weakly coupled to the junction, leading to small splittings in the
spectrum and minimal in uence on dynamics. This may also explain the high fldelity
seen with a pulsed-bias readout scheme [104]. Further work is needed to determine if
the apparent weak coupling is a result of the niobium fabrication process or related
to our measurement technique. For example, the junction switches to the disruptive
voltage state on each cycle. The resulting large electric fleld could selectively alter
the defects most strongly coupled to the junction, leaving only those corresponding
to small splittings in the spectra.
8.6 Spectroscopy of Coupled Qubits
Coupling qubits together in a controlled way is essential to realizing a super-
conducting quantum computer. Several groups have used spectroscopy to charac-
304
(a)
(b)
(c)
01
1110
20
02
t
IbA
IbB
1
2
3
4
5
Figure 8.17: Coupled junction spectra. (a) The bias currents IAb and IBb of two
junctions are ramped so that they are degenerate at a particular time. (b), (c) The
solid lines show the transition frequencies !0n (with respect to the ground state)
for uncoupled junctions. The labels jABi denote uncoupled states. A coupling
capacitor lifts the degeneracy at four points for the states shown, resulting in avoided
crossings (dashed lines). States of the coupled system are indicated by a rounded
bracket, jn).
terize coupled systems [41,78,141,142]. This simple but powerful technique can aid
in the development of gate operations.
In x2.7, I described how a capacitor can be used to couple two current-biased
junctions together. Apart from this physical coupling, the efiect of the capacitor is
strongest when the junctions are dynamically coupled by having their energy levels
in resonance.
An experiment to characterize the coupling of junctions is described in Fig.
8.17 [41,75,76,78]. I will flrst give a qualitative description, before returning to
explain how the device parameters were chosen to generate the graphs. The time
305
dependence of the two bias currents IAb and IBb are plotted in Fig. 8.17(a), assuming
for the moment that the junctions are identical. The solid lines in Fig. 8.17(b) and
(c) show the transition frequencies !0n of the system near the time when the currents
are equal, for the case of uncoupled junctions.14 The labels specify the state jABi,
which indicates a direct product of the single junction states. For example, the
state j01i represents junction A in the ground state and B in the flrst excited state.
As IBb increases more quickly with time than IAb , j01i decreases more quickly than
j10i. For the flve lowest states shown, there are four times at which two system
states are degenerate.
With the addition of the coupling capacitor, the system transitions become
those plotted with dashed lines in Fig. 8.17(b) and (c). The states of the coupled
system are indicated with rounded brackets; for example, j1) is the flrst excited
state. The degeneracies are lifted, where the magnitude of the splitting depends on
the coupling coe?cient ?0, deflned in Eq. (2.72). The coupled state j1), or the \lower
branch," in Fig. 8.17(b) is a superposition of the uncoupled statesj01iandj10i. For
small t, this state is roughlyj10i, while it becomesj01iat large t. At the degeneracy
point, it is equal to the maximally entangled Bell state (j01i+j10i)=p2.
I performed this type of spectroscopic experiment with the coupled SQUIDs of
device DS2. Due to the cross mutual inductance of the SQUIDs and their bias lines
and some heating efiects, simultaneous biasing was conflgured with the procedure
described in x5.2.2. The biasing circuitry of Fig. 5.1 was duplicated to control both
qubits. However, the escape rate of the coupled system was measured by only
monitoring the voltage across DS2A. With both qubit junctions near their critical
current, when one device switched to the running state, the sudden voltage step
generated a current pulse that  owed through the coupling capacitor. This forced
the other junction to switch to the running state as well, with a delay of less than
14I am using the ambiguous symbol !0n to indicate the transition frequency between the ground
state and an excited state of either uncoupled or coupled system.
306
200 ps (see Chapter 9 of Ref. [1]). In addition, although the system was excited
with a microwave current that coupled only to DS2B, both junctions were excited.
Much of the microwave cross-talk could be due to the coupling capacitor CC, which
is much larger than the microwave coupling capacitor C?w.
Figure 8.18 shows measured spectra of DS2, taken with the same biasing, for
(a) high and (b) low frequencies. For any given frequency, the enhancement on
each branch varied by a large amount, partly because only one microwave line was
used. Therefore, I chose to apply a difierent gray scale to each branch (and as usual,
to each frequency). A flne solid line divides regions colored with difierent scales.
Experimentally, the x-axis of Fig. 8.18 is time; the corresponding values are plotted
in Fig. 8.17(b) and (c). I converted these times to IAb and IBb (which apply to both
plots of Fig. 8.18) using the simple current calibration method of x5.6. The two
qubit junctions had quite difierent critical currents, so difierent currents had to be
applied to bring their energy levels into resonance.
Five branches are clearly visible, with signs of four avoided crossings. There
are places on each branch where the escape rate enhancement was negligible for the
applied power at that frequency (so rescaling the grayscale axis made no difierence).
Some of these may be a result of the measurement technique, as the spectrum
is assembled by sweeping the bias at flxed frequency. For example, at !rf=2? =
6:85 GHz, the bias trajectory flrst crosses j1) of Fig. 8.18(b), causing many elements
of the ensemble to escape to the voltage state. The remaining members must occupy
the upper state before its branch is crossed a few microseconds later. The possibility
of a causal relationship between the enhancement on each branch could be verifled
by turning on the microwave current after crossing the solid line. The gap in j3) of
Fig. 8.18(a) at !rf=2=? = 12:6 GHz cannot be explained in this way, as it is the flrst
branch crossed. A power dependence study at this frequency might be interesting.
The broad regions of low enhancement below the lowest branches of Fig. 8.18(a)
307
(a)
(b)
01
10
1
2
01
10
11
20
02
3
4
5
02
20
11
Figure 8.18: Avoided crossing spectra of DS2. In this experiment, performed at 20
mK, IAb was nearly stabilized, while IBb was linearly ramped; the x-axis labels apply
to both (a) and (b). The four degeneracies in Fig. 8.17 are lifted by the coupling
capacitor, resulting in anti-crossings. The dashed lines are a flt to the spectrum of
two coupled current-biased junctions, where the device flt parameters are difierent
for (a) and (b). The thin solid lines separate regions that are plotted with difierent
gray scales.
308
and (b) are due to transitions from excited states, suggesting that heating from the
junctions being in the voltage state was not entirely eliminated.
To compare the spectra to theory, I flrst tried to determine the values of the
critical current and capacitance for each qubit junction. Ideally, this would be done
with independent measurements, with the junctions held out of resonance [41,78].
With DS2, that was not easy to do, because of the di?culty of the simultaneous
biasing. Errors in the biasing can be (at least partially) compensated for with the
choice of I0 and CJ. The uncoupled energy levels, generated with Eq. (2.44), are
shown as solid lines in Fig. 8.17. As there appeared to be some drift between the
two data sets (which were taken on difierent days), I allowed the parameters to
vary for the low and high frequency ranges.15 For the low frequency data [Figs.
8.17(b) and 8.18(b)], the critical currents are IA01 = 24:332 ?A and IB01 = 17:709 ?A,
while the efiective capacitances are CAJ1 (1+?0) = 4:2 pF and CBJ1 (1+?0) = 4:5 pF.
These parameters for DS2A do not reproduce the !12 transition frequencies in Fig.
8.18(b) (the light band at 6.2 GHz), which points to an error in the biasing or
analysis. For the high frequency data [Figs. 8.17(c) and 8.18(a)], the parameters are
IA01 = 24:367 ?A, IB01 = 17:707 ?A, CAJ1 (1+?0) = 4:8 pF, and CBJ1 (1+?0) = 4:5 pF.
The parameters for DS2B are nearly the same and are consistent with \uncoupled"
spectra, but the parameters of DS2A had to be changed signiflcantly for the two
spectra.16
From the design parameters given in x4.3.2, the renormalized frequency of the
LC mode created by the coupling capacitor and stray inductance is predicted to
be !C=2? = 27 GHz. By biasing the qubit junctions near !p=2? ? 7 GHz, the
coupling will be largely capacitive. With the simultaneous current and  ux bias,
15Because the two junctions can drift independently of one another, it is impossible to apply the
techniques outlined in x8.1 to remove the drift from a coupled switching histogram. This stresses
the importance of stabilizing both the electronics and device conditions when scaling these qubits.
16I0 and CJ are highly correlated. If the same value of CA
J1 was used for both data sets, the
values of IA01 would also agree, but the \slope" of the energy levels would not quite match the data.
309
the qubit junction of each SQUID behaves much like a single junction. Thus the
capacitively-coupled junction Hamiltonian given in Eq. (2.74) (with two degrees
of freedom) should accurately describe the actual system of LC-coupled SQUIDs
(which has flve degrees of freedom). Solutions of the Hamiltonian can be expressed
as a superposition of single junction states jni. I used the MATLAB code in xB.2
to calculate the energy levels for the coupled system.
The dashed lines in Figs. 8.17 and 8.18 show the coupled transition frequen-
cies, calculated with the individual junction parameters given above and a coupling
constant ? (!) = 0:03. There is reasonable agreement with the data, although the
parameters of DS2A are difierent for the two plots. The only major deviations occurs
at high bias for the two highest frequency branches, whose origins are unclear.
With ? = 0:03, CJ is roughly 4.4 pF, which is somewhat larger than the design
value of 3.82 pF, given in x4.3.2. The value of ? is even smaller than the design
value of 0.047, suggesting that the LC coupling mode has a frequency above 30 GHz.
In this case, the predicted zero-frequency coupling constant is roughly ?0 = 0:028,
which in turn gives CC ? 130 fF.
There are two important implications of the coupled spectra. First, the ex-
istence of transitions from the ground state to three excited states conflrms the
presence of a complete basis for two qubits (j00i, j01i, j10i, j11i).
In addition, the spectra suggest the possibility of constructing two types of
quantum gates. Using the spectra as a guide, the frequency difierence between j10i
and j11i can be calculated as a function of the bias. By applying a microwave
pulse at the appropriate frequency and duration, one half of a period of a coherent
Rabi oscillation will lead to the transitions j10i$j11i. If no other transitions are
excited, this amounts to a controlled-NOT (CNOT) gate, where jAi is the control
bit. However, in an uncoupled system, the energy difierence between j10i and
j11i is the same as that between j00i and j01i, so a microwave pulse will lead to
310
?14
?02
?15
?02
?25
?01
?24
?01
Figure 8.19: Relevant energy levels for a simple controlled-NOT gate. The transition
frequency of the coupled system that most resembles j10i $ j11i is plotted as
dashed lines, while the one that most resembles j00i $ j01i is plotted as dotted
lines. In the absence of coupling, both transition frequencies would be equal to the
solid curve. The choice of coupled states changes at the degenerate points, which
are indicated by vertical lines.
evolution of all of the states, which is unwanted.
The single junction transition frequency !01 of DS2B is plotted as a solid curve
in Fig. 8.19 [using the same parameters as in Figs. 8.17(c) and 8.18(a)]. The bias
currents where degeneracies occur between j10i, j01i, j11i and any other state are
marked with vertical lines. Away from these points, the states of the coupled system
are nearly equal to the uncoupled statesjABi. For example, for small IAb and IBb , j4)
andj1) are roughlyj11iandj10i, respectively. Therefore the appropriate frequency
to apply in this region is !14, where ~!nm is the energy difierence between jn) and
jm). This frequency, required to drive the CNOT gate, is plotted as a dashed line.
The unwanted transition, between j00i and j01i, is at a frequency !02, which is
311
plotted as a dotted line. I have done the same for each of the four regions. The
plot shows that it is easiest to address the desired transition at a degeneracy. For
example, with IAb ? 24:1435 ?A and IBb ? 17:42 ?A, the difierence between the
two transitions is a relatively large 150 MHz (a value that depends on the coupling
strength ?). However at this point, j5) is an equal superposition of j11i and j02i,
so additional manipulations are required to realize a true CNOT gate. Also, the
microwave drive could cause direct pumping to the states j02i and j20i, whose
energy levels I have not considered.
Alternatively, gates can be performed by using the coherent evolution of entan-
gled states, without any potentially disruptive microwave current [112]. The phase
of j11i can be evolved in a controlled way, by bringing the system to the point
where it is degenerate with j02i. This is an example of how the auxiliary states
j02i and j20i of a multi-level system can be useful. Also j10i and j01i may be
evolved into each other by bringing these levels into resonance for a certain amount
of time.17 With the addition of single qubit gates, either of these gates are su?cient
for universal quantum computation.
8.7 Time-Domain Measurement of T1
I have discussed two techniques to experimentally estimate the value of the
relaxation time T1. As described in x7.4, the escape rate of a junction has a T1-
dependent feature at slightly elevated temperature. Secondly, the width of a reso-
nance peak depends on a number of factors, including T1 [see Eq. (3.66)]. In both
cases, T1 leads to a rate which is balanced against other processes, which may be
better understood. Because of this, the experiments may be performed on time
scales much longer than T1. In practice we ramp Ib to make the measurements, but
17In the case of DS2, the two junctions have difierent critical currents, so j01i and j10i are
not degenerate at the same bias that j02i and j20i are. This leads to some complications with
accounting for unwanted evolution between j3) and j4).
312
the sweep is assumed to be stationary, so there is not even an implicit dependence
on time.
The use of a microwave drive allows another method of flnding T1. The idea is
to promote an excited state population and then measure its decay time back to the
ground state. This technique has the advantage of being conceptually simple and a
value can be extracted even if the details of the system are not known. However,
care must be taken that the decay is really due to relaxation of the qubit and not
to, for example, tunneling or the decay of a resonance on the microwave lines.
8.7.1 Microwave Pulse Readout
To measure T1, we performed microwave pulse readout measurements (see
x6.6.2) using the scheme outlined in Fig. 8.20(a). The switching of SQUID DS1 to
the voltage state was continuously monitored while the bias was swept slowly. At
a time deflned to be t = 0, the level spacings of the qubit junction were measured
to be !01=2? = 8:5 GHz and !12=2? = 7:93 GHz. This bias was chosen because
the ground state escape rate ?0 was smaller than 103 1=s. Between t = ?2 ?s and
t = 0, a microwave current with frequency !01 and power P01 was applied to the
junction to excite population into j1i. After an adjustable time delay ?t, a pulse
of microwaves of frequency !12 and power P12 was applied to junction to serve as a
readout.18 If the system was in j1i, this pulse would promote it to j2i, whose escape
rate was large enough to cause immediate tunneling to the voltage state. Thus the
probability to see a tunneling event caused by !12 should be proportional to the
occupation of j1i and will decrease exponentially with increasing ?t.
For each setting of ?t and the pulse power, we measured switching histograms
for 230000 trials. Figure 8.20(b) shows a typical histogram, which has two unex-
18The !01 pulse was applied to the line dedicated to DS1. For convenience, !12 was applied to
another microwave line that was connected to an antenna inside the sample box.
313
(a)
(c) (d)
(e)
(b)
?01 ?12
?t
t0
Figure 8.20: T1 measurement of SQUID DS1 with pulsed readout. (a) A microwave
pulse of frequency !01 (and power P01) creates excited state population which is read
out with a pulse of frequency !12 (and power P12) after a delay ?t. The histograms
were taken with (b) both excitation and readout, (c) the excitation alone, and (d)
the readout alone. (e) The total number of counts, for three sets of pulse powers,
due to the readout pulse (symbols) decays exponentially with ?t, as seen by the flts
(solid lines). All of the data were taken at 20 mK.
314
pected features. For t < 0, there should be essentially no counts, because !01 only
occupies j1i, which is assumed to have a small escape rate. Instead, there are quite
a few counts, which decay at t = 0, when the !01 drive current was turned ofi. After
a delay of ?t = 100 ns, !12 was turned on which lead to a large number of counts.
However, after reaching a minimum at t ? 300 ns, the number of counts begins to
increase again.
In Fig. 8.20(c), histograms taken with only the excitation pulse are shown for
two values of P01. Although the idea was to work deep in the well where ?0 and ?1
were small, the selected bias point was not deep enough. This would have been a
problem for other measurements, but is not a serious one here.
Figure 8.20(d) shows histograms taken with only a readout pulse (with ?t =
0), for two values of P12. Even though the refrigerator was at base temperature and
no !01 pulse was applied, !12 still caused a substantial number of counts. This is
likely to due to relatively large resonance widths. Even ofi resonance and at low
power, the readout pulse promoted the system to a state where it could tunnel. The
increasing number of counts at high t are due to a two-photon 0 ! 2 transition
(whose resonance is at t ? 700 ns).
Ideally, this experiment would be performed at flxed bias and the junction
would only escape to the voltage state when both !01 and !12 were applied. Nonethe-
less, the results can be still be used. I summed the total number of counts be-
tween t = 0 and 400 ns, for each value of ?t and for three sets of excitation and
readout powers. These values, which are proportional to the excited state popula-
tion, are plotted as symbols in Fig. 8.20(e). Based on predictions using Eq. (2.43),
?1 < 105 1=s, so the decay of the excited state should be dominated by relaxation
for T1 < 1 ?s.
As Fig. 8.20(e) shows, the number of counts does not decay to zero. This
happens because !12 forces counts by itself. To account for this, I flt the data to
315
Table 8.1: T1 flt results for SQUID DS1. The power (at the microwave source)
of the excitation and readout pulses are given by P01 and P12. The total number
of switching events due to both pulses as function of the delay ?t, plotted in Fig.
8.20(e), was flt to the functional form y0 +Ae??t=T1, where the best flt parameters
and their uncertainties are listed. The last column gives the reduced chi-square.
P01 (dBm) P12 (dBm) y0 A T1 (ns) ?2?
?50 ?45 58:6?8:0 486?13 44:1?3:1 2.0
?40 ?45 ?36?48 1052?40 59:4?6:4 0.67
?40 ?45 70 980?22 45:8?1:3 1.5
?50 ?35 249? 32 1017?32 41:5?3:9 1.0
y0 + Ae??t=T1; results for the parameters are listed in Table 8.1. The flrst and last
rows give T1 ? 40 ns, with flts of reasonable quality. For P01 = ?40 dBm and
P12 = ?45 dBm, an unrestricted flt returns a negative ofiset y0, which is unphysical
[but which I did plot in Fig. 8.20(e)]. This problem could be solved by taking data
to longer values of ?t. By flxing y0 at 70, which is roughly expected, the best flt
value of T1 ? 46 ns is consistent with the other measurements. Despite the issues of
pulse powers and well depth, this technique appears to provide a relatively simple
way to measure T1 in a time resolved way.
8.7.2 Direct Tunneling Readout
In Fig. 8.20(c), the !01 pulse that created excited state population resulted in
an unintended enhancement in the measured escape rate. After we turned the pulse
ofi at t = 0, the escape rate decays roughly with time constant T1 for t > 0. Thus,
the second readout microwave pulse is not needed if the experiment is performed at
a bias current where ?1 is large. In this case, high powers of the excitation pulse
can be studied. There are reasons not to do this for quantum gate operations, but
as I am most interested in characterizing the devices in as many ways as possible,
316
the direct tunneling readout is an easy way to try to measure T1.
Figure 8.21(a) shows the escape rate of SQUID DS1 when we applied a 25
ns microwave pulse of frequency 7.6 GHz, which was resonant with the 0 ! 1
transition. The three curves, taken at source powers of -26 (circles), -22 (squares),
and -16 (triangles) dBm, are vertically ofiset for clarity. At these high powers, Rabi
oscillations (which I will discuss in the next chapter) are visible for t < 0. The
pulse was shut ofi at t = 0 and the length was chosen long enough for the oscillation
amplitude to decrease, so that the decay for t > 0 did not show any coherence efiects.
For much longer drive pulses, it would not be possible to measure the decay, because
the junction would always switch to the voltage state during the microwave pulse.
This is a drawback of the escape rate measurement that needs to be considered
when working with high microwave power.
For t > 20 ns, the three curves decay exponentially with roughly the same
time constant. Near t = 30 ns, the escape rates increase slightly, indicating the
presence of more than just simple decay processes. The decay is much faster for
earlier times and does not appear to be governed by a single exponential. This is
particularly clear for the highest power, where there is a sharp drop for 0 < t < 2 ns.
For 0 < t < 5 ns, there is a decay with an intermediate time constant, that is again
clearest at -16 dBm (triangles).
I flt the escape rate (for t > 0) with a sum of four exponentials, where three
of them describe the decay. One exponential accounts for ? increasing with time
(with a 475 ns time constant) due to the slow bias ramp used to take the data.
Results for the best flts are shown with solid lines in Fig. 8.21(a). For -16 dBm, the
time constants of the flt are -1.2, -5.8, and -68 ns, while for -22 dBm, they are -3.7,
-4.2, and -59 ns. For -26 dBm, I only used two decaying exponentials, with time
constants of -4.4 and -56 ns. The flts are not of particularly good quality, especially
around the modulation near 30 ns.
317
(a)
(b)
Figure 8.21: T1 measurement of SQUIDs with a direct tunneling readout. (a) Ex-
cited state population was created by a applying a microwave pulse of frequency
7.6 GHz and power -26 (circles), -22 (squares), and -16 (triangles) dBm to DS1. At
t = 0, the pulse was shut ofi, leading to a decay in the measured rates (ofiset for
clarity), which are flt by the solid lines. (b) Similar behavior is seen in DS2B, for
pulses of frequency 6.3 (circles) and 5.6 (triangles) GHz. The decay is much faster
for a 1.4 GHz pulse (squares). These curves are not ofiset. All data were taken at
20 mK.
318
Decay of the escape rates is indicative of a loss of population of a state, due
to a combination of tunneling, dissipation, and perhaps other processes. From mea-
surements using the bias pulse readout described in x6.6.3, it is clear that the fastest
decay corresponds to depopulation of j2i by tunneling at rate ?2; at high powers,
the resonances broaden to the extent that j2i can become populated, even when the
microwaves are at a frequency !01. The time constant is slightly longer than 1 ns,
however, which is unexpectedly long. At the bias current where the measurement
was performed, Eq. (2.43) gives ?2 ? 3?109 1=s. Thus, even in the absence of dis-
sipation, the lifetime of j2i should be less than 0.3 ns. The experimentally observed
decay is likely due to limited time resolution of the switching experiment (both in
detecting a switching event and jitter in triggering the pulse) and the fall time of
the microwave pulse. A direct measurement of the microwave pulse shape at the
device could be very revealing, but is something I never did.
The other two decays are also quite mystifying. The bias pulse measurement
suggests that both of the time constants correspond to a decay of j1i, which has
previously been reported in niobium phase qubits [39]. The longer constant of
? 55 ns is consistent with the microwave pulse measurement of x8.7.1. While that
measurement was done deep in the well, the data of Fig. 8.21(a) were acquired where
?1 ? 3:5 ? 107 1=s. Thus, because of tunneling, the naive assumption is that the
decay of the population in j1i can be no longer than 30 ns. Perhaps, then, T1 is
given by the intermediate decay, which varied between 5 and 15 ns for difierent data
sets. Only the upper end of this range is consistent with the value of 14 ns obtained
from master equation simulations of Fig. 7.13.
A possible explanation for the slow time constant is that the microwave pulse
heated the insulators in the device. Then the system would stay in thermal equi-
librium, as the chip cooled back down to the mixing chamber temperature. This
possibility can be explored by fltting the escape rate to a sum of exponentials. The
319
magnitude of the slow time constant term should be an indicator of the temperature
of the device when the microwave current was shut ofi at t = 0. From other data
on DS1 (not shown), however, I found no correlation between the microwave power
and the contribution to the total escape rate from the slow time constant term. In
addition, small changes in the duration of the pulse lead to large changes in the
magnitude of this term. Therefore the state populations at t = 0 afiect the decay
50 ns later. Another possibility is that the intermediate decay time could be due
to the shape of the microwave pulse (which itself could be a function of frequency),
but it seems implausible that the longest time constant could be connected to this.
Yet another possibility is that the SQUID is not behaving as a three-level
quantum system. Perhaps it is weakly coupled to another degree of freedom that
has a long relaxation time [4]. The tunneling of the qubit junction would just be
a gauge of the population of the external system, without afiecting its dynamics
signiflcantly. Certainly, there are plenty of other systems that could couple to the
qubit junction. The isolation junction is coupled to the junction, but it is unlikely
to have a 50 ns relaxation time, as it is not isolated from the current bias leads.
In addition, the coupling strength between the two junctions can be modulated by
varying the current through the isolation junction, but we never saw a change in
the slow decay for difierent levels of isolation. Alternatively, we could be seeing a
coupling to the two-level systems discussed in x8.5.
We saw similar efiects in SQUID DS2B. The three curves in Fig. 8.21(b)
were taken at difierent values of Ib. The difierent background escape rates (at
t = ?100 ns) are due to this fact and not to any ofiset that I applied in plotting
the data. The solid circles show the escape rate for a resonant 6.3 GHz, -22 dBm
pulse. The signature of three decay constants is present in this curve, but the small
number of counts we took make it di?cult to resolve them clearly. The longest time
constant appears to be about 30 ns.
320
The solid triangles were taken with a resonant 5.6 GHz pulse. Because this
frequency corresponds to a shallower well and thus higher escape rate, we used
a shorter 10 ns pulse and lower source power (-32 dBm), so that the switching
histogram would extend beyond the pulse. The fastest time constant is not present
in this curve, which could be a consequence of the lower power or of j2i being too
broad to be occupied at this high value of the bias. The two decays that do appear
have time constants of about 5 and 50 ns. For the bias conditions at 5.6 GHz, Eq.
(2.43) predicts ?1 ? 2?108. Naively, one should expect the longest time constant
to be no more than 5 ns.
As a check, I also measured the escape rate when a low frequency, ofi-resonant
microwave pulse was used to excite the qubit. The data plotted as open squares in
Fig. 8.21 were taken with a pulse of 1.4 GHz and -18 dBm. The decay of this curve
is very fast and there is no sign of the long time constant. While it is possible that
a multi-photon process was very e?cient in driving the 0 ! 2 transition, it seems
unlikely that a lower order process would not populate j1i at all. In addition, the
same measurement scheme performed over a range of bias currents yielded similar
results. This suggests that the long decay is the result of a resonant process near
the plasma frequency of the junction and not due to simple heating efiects from the
microwave power.
Resolving the origins of the various time constants is an important step in
characterizing the SQUID phase qubit, as this information may dictate how the
design could be improved to yield longer values of T1. Unfortunately, as this section
has shown, considerably more work needs to be done in this area.
8.8 Summary
To measure the energy level spectrum of a qubit, we swept its bias current (and
thus the level spacing) while applying a microwave current of flxed frequency. On
321
resonance, the escape rate increases as excited states gain occupation probability.
In both the LC-isolated (see x8.1) and dc SQUID phase qubits (see x8.2), I found
that the spectrum can be flt to simulations of an ideal current-biased junction. This
applies not just to the 0 ! 1 transition, but to transitions higher in the well and two-
photon transitions (see x8.4) conflrming that the simple junction Hamiltonian is a
good approximation to our more complicated LC-isolated and SQUID phase qubits.
Simple spectroscopy also provides clear evidence that our assumed Hamiltonian is
incomplete, as small splitting appear in the spectra for the SQUIDs (see x8.5). The
microscopic origin of these features and their impact on the system dynamics have
yet to be resolved.
The spectrum of two capacitively-coupled SQUIDs provides evidence that the
qubits are interacting in the expected way and we see avoided level crossings where
there would be degeneracies in the absence of coupling. Such a spectrum gives some
of the information needed to design a two-qubit gate.
The shape of the escape rate enhancement is indicative of the linewidth of
the resonance. The width is due to energy dissipation (on a time scale T1), phase
coherence (on a scale T2), tunneling, and inhomogeneous broadening; it is commonly
characterized by T?2 (see x8.3). We measured a maximum T?2 ? 4 ns for the LC-
isolated qubits and 8 ns for the SQUIDs, suggesting that the broadband isolation
of the SQUIDs is somewhat more efiective, but the times were much shorter than
hoped for. The resonance widths may also be used to extract information about
dissipation and inhomogeneous broadening.
Finally, T1 can be measured by creating excited state population and watching
it decay back to the ground state. The escape rate is a good way to do this, as it
is sensitive to very small population changes. For low excitation powers, the decay
time constant is about 45 ns for the SQUIDs. However, at higher powers, we see two
time constants, a faster decay of less than 10 ns followed by a slower 50 ns decay.
322
It is unclear whether, for example, this is an artifact of the measurement or has a
connection to splittings in the spectra. Both times are much shorter than can be
accounted for from dissipation in the leads.
323
Chapter 9
Coherent Rabi Oscillations
In the chapter, I will show results from various Rabi oscillation experiments
conducted on our qubits. Not only do these manipulations resembles the single qubit
gates required for quantum computation (because they involve simple rotations on
the Bloch sphere), analysis of the data also reveals information about the coherence
time of the devices.
Figure 9.1 shows typical data for SQUID DS1. Several methods of measuring
oscillations have been reported in the literature; we chose to use the simple one
described next [136]. As usual, the escape rate ? of the qubit junction to the
voltage state was measured continuously as the bias was slowly swept to the critical
current. At time t = 0, a microwave pulse of angular frequency !rf was applied to
the junction, which drove any transitions near !rf. Any changes in the occupation
probabilities of the states were re ected in the escape rate. If we were interested
in driving a 0 ! 1 oscillation, then we set t = 0 to occur at the value of the bias
for which !rf = !01, where ~!01 is the energy level spacing between the ground
and flrst excited states. We only performed these experiments on the SQUID phase
qubits, DS1 and DS2. It is possible that oscillations could have been measured in
the LC-isolated qubits if they had been measured with fast, low noise electronics,
as in the SQUID experiments.
For t < 0 in Fig. 9.1, the escape rate is 105 1=s, which is roughly ?0 (where ?n
denotes the escape rate from state jni). As j1i becomes occupied, the escape rate
increases dramatically. The oscillation that follows has a frequency that depends on
the microwave power. The oscillation amplitude decreases with time, due to the loss
of phase coherence; we refer to the decay time of the envelope as T0, which depends
324
Figure 9.1: Rabi oscillations in SQUID DS1. A resonant 7.6 GHz microwave pulse
causes the escape rate to oscillate with a frequency dependent on the pulse power,
-18 dBm at the source in this case. The pulse turns ofi near 30 ns, resulting in a
decay of the excited state population and the escape rate.
on the relaxation time T1 and coherence time T2. For times much longer than T0, the
dynamics can be described by a master equation (see x7.3). The equilibrium value
of the escape rate is simply determined by the various transitions rates (microwave
pumping, dissipation, tunneling) for each level.
The pulse was nominally set to be 35 ns long, but ? decreases quickly at
t = 30 ns, suggesting that the actual pulse was shorter. As discussed in x8.7.2, the
form of this decay reveals information about the state occupancy. If we had instead
turned ofi the microwaves at 4.5 ns, this would resemble a simple quantum NOT
gate, as j0i would have been taken to j1i.
The goal of the experiments described in this chapter is to explain the structure
of Fig. 9.1 and to extract key system parameters and the individual level popula-
tions. We approached this problem by varying the available parameters (microwave
frequency and power, temperature, level spacing) and checking whether the qubit
model we have developed could explain the results. I will begin by showing the power
325
dependence of the oscillations and the resonance widths. Then I will discuss the
information that can be obtained by varying the detuning of the microwaves from
resonance and flnish with density matrix simulations that attempted to reproduce
a full oscillation sequence, such as the one shown in Fig. 9.1.
9.1 Power Dependence
Figure 9.2(a) shows flve Rabi oscillations in SQUID DS1 induced by difierent
nominal powers PS of the microwave source. I have not ofiset the escape rates; at t =
0, ? = 2?105 1=sforallofthecurves. Theescaperatewithoutmicrowavesresembled
the curve shown in Fig. 7.8(b); the resonant pulse began at I1 ? 34:069 ?A, where
!01=2? = 7:6 GHz. The pulse was turned ofi 3 ?s later, so the decay is not visible,
unlike in Fig. 9.1. As in the previous two chapters, I will assume that the energy
levels of the qubit junction of DS1 are described by I!0 = 34:300 ?A and C!J =
4:43 pF, while the escape rates are given by I?0 = 34:308 ?A and C?J = 4:43 pF.
As expected for Rabi oscillations, the oscillation frequency of the curves in Fig.
9.2(a) increases with the microwave power. For each power, the oscillation amplitude
decreases with time, eventually reaching a steady state value. At the highest power,
the qubit almost always switched to the voltage state before t = 20 ns, which leads
to poor statistics at later times. In a two-level system, the steady state value at high
power corresponds to both states being equally populated [see Eq. (3.60)]. Thus,
we would expect that at high powers, the oscillations would become faster, but that
they would be centered about a constant value of ?. Instead, the equilibrium value
increases quite dramatically with power. At the bias where the experiments were
performed, I estimate ?1 to be 3:7?107 1=s using Eq. (2.43). This is the maximum
escape rate for a two-level system, which only occurs if j1i is fully occupied. Instead,
we see escape rates nearly ten times this value, providing strong evidence for the
occupation of j2i. As ?2 is roughly 3 ? 109 1=s, only a 8% population would be
326
(a) (b)
-7
-27
-13
-18
-9
-11
-26
-20
-23
-14
-17
PS (dBm) PS (dBm)
Figure 9.2: Power dependence of Rabi oscillations in SQUID DS1. (a) The symbols
(connected by straight lines to guide the eye) show the escape rate due to a mi-
crowave pulse of the indicated source power. As the curves are not ofiset, it is clear
that the escape rate does not saturate with increasing power. (b) Oscillations at a
difierent set of powers, vertically ofiset for clarity, are flt with the sum of a decaying
sinusoid and a saturating exponential background, drawn as solid lines. All of the
data were taken at 7.6 GHz and 20 mK.
327
required to see such large escape rates.
Ideally, we could compare the data to analytical solutions of Rabi oscillations
for a three-level system. With the presence of tunneling, dissipation, and decoher-
ence, this is di?cult to do accurately. Still, some quantitative information about
the system can be obtained by fltting the escape rates curves to a functional form
that reproduces their main features. Motivated in part by Eq. (3.62), I chose to use
the sum of a decaying sinusoid and an exponentially saturating background [101],
?(t) = g1
?
1?e?(t?t0)=T0
?
cos??R (t?t0)?+g2?1?e?(t?t0)=Te?; (9.1)
where g1 and T0 are the amplitude and time constant of the decay envelope, ?R is
the observed oscillation angular frequency, g2 and Te are the amplitude and time
constant of the background, and t0 is an overall time ofiset. This form does not
reproduce the curvature at small t or the relatively small (but non-zero) value of ?
at t = 0, which is acceptable as I was most interested in the main oscillation.
Figure 9.2(b) shows oscillations for six values of PS, which I have ofiset for
clarity.1 The solid lines are flts to Eq. (9.1). Most of the features of the data are
reproduced, at least qualitatively, including the usual oscillation maxima at high
microwave power. The data were taken while Ib and If were being ramped; in 35
ns, the energy levels change by a small amount, which could account for the falling
? at PS = ?26 dBm as well the oscillation which appears to increase in frequency
with time at PS = ?14 dBm.
Some of the parameters of the flts (for more data than are shown in Fig. 9.2)
are plotted in Fig. 9.3.2 The oscillation frequency, plotted with circles in Fig. 9.3(a)
and its inset, increases fairly smoothly with power, as expected. The one exception
1In calculating the switching histograms, I chose to use large time bins, which accounts for the
lower density of points than in Fig. 9.2(a).
2For PS > ?14 dBm, the flts were done \by eye," so the results are somewhat unreliable.
328
(a)
(b)
(c)
Figure 9.3: Phenomenological flts of Rabi oscillations in SQUID DS1. A decaying
sinusoid with a background was flt to the oscillations in Fig. 9.2 (and others not
shown) to yield the (a) square of the oscillation frequency, (b) decay envelope time
constant, and (c) oscillation amplitude as a function of the microwave pulse power
PS. The solid line in panel (a) is a flt to the two-level theory; the inset shows the
same information for ?R;01. The dashed line is drawn without an ofiset at PS = 0 W.
329
is at PS = ?10 dBm, where the quality of the oscillation was quite low.
For a two-level system driven resonantly, the Rabi frequency should increase
linearly with the microwave current I?w or as the square root of the microwave
power, in the absence of dissipation. As a check on the expected dependence, the
solid line in Fig. 9.3(a) is a linear flt of ??R;01=2??2 as a function of PS (in Watts,
rather than dBm).3 The best-flt line has a slope 1790 MHz2=?W and an ofiset of
1570 MHz2. Ideally, the flt would pass through the origin. The dashed line is drawn
with the same slope, but with zero ofiset; I have not included the dashed line on the
inset, which shows ?R;01 on a linear scale.
There are several possible causes for the flnite ofiset and what appears to be a
slight systematic deviation between the data and flt. In the absence of dissipation,
Eq. (3.12) predicts an oscillation frequency of [(!rf ?!01)=2?]2 at zero power. Thus
the ofiset of the flt could correspond to a detuning of 40 MHz. The full data set did
take 37 hours to acquire (during which no adjustments were made to the timing of
the microwave pulse), but there appeared to be very little drift during this time.
Thedetuning, ratherthanbeingcausedbyimpropertiming, couldbetheresult
of inhomogeneous broadening. Low frequency current noise changes the energy level
spacing, causing an increase in the oscillation frequency, with the most noticeable
efiect at low ?R;01.
Neglecting detuning, dissipation also results in a shift of the oscillation fre-
quency at low power. However, as seen in Eq. (3.64), T1 and T2 result in a negative
ofiset of ?2R;01 vs. PS, whereas the opposite is seen in Fig. 9.3(a).
Finally, from the escape rates of Fig. 9.2, it is clear that we are dealing with a
quantum system of at least three levels. The second excited states acts as a pertur-
bation on 0 ! 1 Rabi oscillations, as discussed in x3.3. However, this phenomenon
3In doing the flt, I made the simple assumption that the uncertainty in ?R;01 was proportional
to the value itself. I also did not include the flve highest powers, as the oscillations at these powers
were not particularly well-deflned. Nonetheless, the flt does approximate their values well.
330
has the largest impact at high power, as I will show in x9.2. Even if the highest pow-
ers are ignored in Fig. 9.3(a), a linear flt still gives an ofiset of roughly 1000 MHz2.
It is likely that all of the efiects are playing some role, but it is unclear which one is
dominant.
Figure 9.3(b) shows the time constant T0 of the decay envelope of the oscilla-
tions in Fig. 9.2. The times increase substantially with power until PS = ?15 dBm,
at which point the flts are unreliable. The maximum value is about 15 ns and the
average is roughly 10 ns, which is what we typically saw on-resonance at other drive
frequencies.
The amplitudes g1 of the oscillations, extracted from the phenomenological flts,
are plotted in Fig. 9.3(c). Again the values increase weakly with power, saturating
at 2:5 ? 107 1=s near PS = ?20 dBm. Returning to Fig. 9.2(a), the amplitude of
the oscillations for PS = ?18 and ?7 dBm are not that difierent, even though the
background is quite a bit bigger in the latter case. Interestingly, the saturation value
is somewhat more than half of the predicted value for ?1 of 3:7?107 1=s, but this
may just be a coincidence.
Itisimportanttounderstandthemechanismforpopulatingj2i, whichamounts
to leakage out of the desired state space for quantum computation. In the spectrum
shown in Fig. 8.8 (taken for a difierent device), the transitions closest to the 0 ! 1
branch are the single photon 1 ! 2 and two-photon 0 ! 2 transitions, both of
which occupy j2i. However, all of the transitions appear to be relatively sharp, as
compared to their spacing in frequency. Two factors cause the impact of well sep-
arated transitions to become signiflcant. The flrst is that our detection technique
of measuring the escape rate is extremely sensitive to population of j2i, due to the
large values of ?2=?1 and ?2=?0. More importantly, there would not be any popu-
lation to detect if the resonances did not broaden to have signiflcant overlap at the
high power at which we perform oscillations.
331
In the rough expression for the resonance width in Eq. (3.66), I did not include
a power broadening term, as we usually measure spectra at very low powers. One
beneflt of analyzing Rabi oscillations is that the flt in Fig. 9.3(a) gives a calibration
of the microwave power. With this information, power broadening can be studied,
which might not only explain the leakage to j2i, but also reveal information about
inhomogeneous broadening, T1, and T2.
In Fig. 8.10(c), I plotted the resonance full width (in terms of the qubit current
I1) of the 0 ! 1 transition of DS1 at 7.6 GHz as a function of the microwave source
power PS. I converted the power4 to ?R;01 and the width to ?! (using spectroscopic
information) and plotted the results in Fig. 9.4 as solid circles.
Equation (3.65), which gives the power dependence of the resonance width
?! = (2=T02)p1+?201T01T02 in terms of the escape rate-dependent parameters T01 =
(1=T1 +?1)?1 and T02 = [1=T2 +(?0 +?1)=2]?1, should describe the data. I flt the
six highest powers, where the broadening is prominent, with the result shown as a
solid line in Fig. 9.4. The efiective time constants are T01 = 3:2 ns and T02 = 4:0 ns.
As the escape rates are predicted to be ?0 = 6:3?104 1=s and ?1 = 3:7?107 1=s
at the bias current where the data were taken, T1 and T2 are predicted to be 3.7
ns and 4.4 ns. With such short times, the escape rates have a negligible efiect.
Remarkably, the saturation value at low power is nearly reproduced by the flt,
even though the spectroscopic measurements of the previous chapter indicated the
presence of signiflcant low frequency noise.
Both T1 and T2 are shorter than my previous measurements would suggest.
The slow bias sweep method in x7.4.2 gave T1 ? 15 ns. With a Rabi decay envelope
of T0 ? 10 ns, Eq. (3.63) predicts T2 ? 7:5 ns. With the inclusion of tunneling,
4I only used the slope of 1790 MHz2=?W in Fig. 9.3(a) in the conversion and dropped the
1570 MHz2 ofiset. If the ofiset is due to detuning, then this is legitimate, because the spectroscopic
widthsweremeasuredbysweeping!01 throughresonance. Iftheofisetwasaresultofanincomplete
description of the junction dynamics, then dropping it results in incorrect values of the Rabi
frequency.
332
Figure 9.4: Power broadening in SQUID DS1. The full width ?! of the 0 ! 1
resonance (solid circles) increases above a certain power of the microwave drive, as
characterized by the Rabi frequency ?R;01. The solid line is a flt to data at high
power (T01 = 3:2 ns and T02 = 4:0 ns), while the dashed line is drawn with values
estimated from other measurements (T01 = 9:7 ns and T02 = 6:6 ns).
the efiective time constants become T01 = 9:7 ns and T02 = 6:6 ns. The dashed
line in Fig. 9.4 shows the resonance width for these parameters. At low power,
the prediction underestimates the data; this can be explained by inhomogeneous
broadening, which would uniformly increase the widths, having the most signiflcant
impact at low power. However, at high power, the prediction is greater than the
data. In our simplifled picture of the system, we might have ignored a process that
would broaden the widths, but it is hard to imagine how we could overestimate the
measured width.
It is unclear how to resolve this critical inconsistency. It is possible that
the calibration of ?R;01 is incorrect. However, it is the most reliable near 100 MHz,
wherethepowerbroadeningdiscrepancyexists. Atlowerpower, Rabioscillationsare
di?cult to resolve and susceptible to detuning efiects. At higher power, perturbation
333
by j2i could be causing signiflcant deviations from two-level behavior. It could be
that decoherence is having a major impact on the oscillation frequency and thus the
calibration of power; comparing the oscillation frequency to theory (without T1 and
T2) is one of the subjects of the next section.
9.2 Detuning and Strong Field Efiects
Before attempting to model the system dynamics in the presence of dissipation
anddecoherence, IwillexaminethebehavioroftheRabioscillationfrequency. While
decoherence has a profound impact on the oscillation amplitude, it should have a
negligible impact on the frequency if T2 is not too short. A careful study of the
oscillation frequency is also a strong test of the junction Hamiltonian, because results
depend on the matrix elements between states in addition to energy levels [83].
A property of Rabi oscillations that can be easily verifled is their frequency
as a function of detuning from resonance. Equation (3.12) gives the efiective Rabi
frequency ?01 =
q
?201 +(!rf ?!01)2 for an ideal two-level system. We choose not
to increase the detuning by changing !rf, for the same reason we did not measure
spectra by sweeping !rf; namely, the frequency response of the microwave lines
could introduce erroneous features. Instead, just as with spectroscopy, we kept
the microwaves at flxed frequency and power and changed the energy levels of the
qubit, through its bias current Ib. This approach, however, does not come without
complications. If a large range of Ib is covered, then several other properties of the
qubit will change along with the energy levels, such as the matrix elements of the
phase difierence ^ and the escape rates. For this reason, comparing two difierent
transitions (for example, 0 ! 1 and 1 ! 2) could be problematic, as they will occur
at difierent values of Ib. An alternate approach, which I did not follow, would be
to carefully calibrate the frequency response of the microwave lines in terms of, for
example, ?01.
334
(a) (b)
(c) (d)
Figure 9.5: Rabi detuning map of SQUID DS2B at elevated temperature. (a) At 110
mK, the measured escape rate enhancement ??=? (circles) at !rf=2? = 6:5 GHz
and PS = ?34 dBm shows four resonance peaks, which are flt well by a sum of
Lorentzians (line). (b) Each horizontal line of the plot is the escape rate due to
a microwave pulse, with a source power of PS = ?15 dBm. (c) The oscillation
frequency is plotted as a function of !01 (circles). The solid line is for a single
photon process in a two-level system. (d) The oscillation frequency for the two-
photon 0 ! 2 transition is reproduced better by a three-level rotating wave solution
(dashed line) than by a simple two-level approximation (solid line).
335
In Fig. 9.5(a), the escape rate enhancement of SQUID DS2B at !rf=2? =
6:5 GHz is plotted as a function of Ib. As the measurement was performed at 100
mK with a relatively high power of PS = ?34 dBm, transitions from j1i are visible,
just as in Fig. 8.10(a). From top to bottom, the peaks correspond to single photon
0 ! 1, two-photon 0 ! 2, single photon 1 ! 2, and two-photon 1 ! 3 transitions.
The solid line is a flt to the sum of four Lorentzians and reproduces the data well,
even though they were taken at elevated temperature.
Figure 9.5(a) was obtained by ramping Ib at a relatively slow rate of 0.026 A/s.
To investigate the efiects of detuning, we measured Rabi oscillations by turning
a microwave pulse on at difierent times during this same ramp. Each of the 98
distinct horizontal lines of Fig. 9.5(b), which I will refer to as a detuning map,
is the escape rate due to a microwave pulse (of frequency !rf=2? = 6:5 GHz and
power PS = ?15 dBm) that started at the value of Ib on the y-axis. I have colored
the entire plot with the same scale, where black represents the highest escape rate
(1:4?108 1=s). The horizontal scale specifles the amount of time from the beginning
of the pulse, during which Ib is also changing. In fact, the lines were taken by
incrementing the start of the pulse by 35 ns during the bias ramp;5 thus, a Rabi
oscillation was measured at increments in Ib of roughly 0.9 nA. For the 30 ns plotted,
the energy level spacing at the end of a particular line is nearly the same as the
spacing at the beginning of the next line. This undesired detuning did not appear
to have any signiflcant efiects.
The 0 ! 1 transition is resonant at Ib = 17:615 ?A, as seen in Fig. 9.5(a).
There is a set of \fringes" in Fig. 9.5(b) centered at the same value of Ib. For larger
Ib (when !rf > !01) or smaller Ib (when !rf < !01), the detuned Rabi oscillations
increase in frequency, resulting in the curvature of the fringes. At Ib = 17:595 ?A,
5There appeared to be a systematic error in the timing of the pulse generator used to gate the
microwaves. I measured the generator?s output (using a SR620 timer), in order to determine t = 0
for each line.
336
the microwaves are resonant with the two-photon 0 ! 2 transition. Another set of
fringes (with a longer period) is visible at this current. The curvature is greater,
as the efiect of detuning is stronger for a two-photon process. Near Ib ? 17:61 ?A,
there appears to be interference between the two processes.
The single photon 1 ! 2 transition is resonant for !rf=2? = 6:5 GHz at
Ib = 17:575 ?A. On this resonance, the Rabi oscillation is quite weak. There
are, perhaps, two lobes in the detuning map, but the amplitude variation is small.
What is particularly puzzling is that the oscillation frequency is lower than the
0 ! 1 transition. For a single photon oscillation between two states, the frequency
should be proportional to the matrix element of ^ that links the states. Then
from Eq. (2.38), the prediction is that ?R;12 ? p2?R;01 for a constant microwave
power, in contrast to what is seen experimentally. It is possible that the small
thermal occupancy of j1i is insu?cient to see a clean two-level oscillation between
j1i and j2i. Better results could be obtained by flrst resonantly populating j1i, and
then attempting to perform a Rabi oscillation. This proved to be a rather di?cult
experiment to carry out and we never obtained good results.
I extracted the oscillation frequency for each horizontal line in Fig. 9.5(b) by
fltting the escape rate with the function in Eq. (9.1). While the decay envelope
was not always well-deflned, particularly at large detuning, the oscillation frequency
generally was. The results are plotted as circles in Fig. 9.5(c) and (d), with error
bars coming from the uncertainty in the flt parameter.
Figure 9.5(c) shows the efiective Rabi frequency for the 0 ! 1 transition. In
making this plot, I converted the current bias Ib to !01, using Eq. (2.44) and I0 =
17:828 ?A and CJ = 4:51 pF, values which came from spectroscopic measurements.
Equation (3.12) predicts that for an ideal two-level system, ?R;01 should reach a
minimum on resonance, at !01=2? = 6:5 GHz. However, the minimum in the data
is clearly at a lower value. While some of the discrepancy could be due to an error in
337
the calculation of !01, the presence of j2i perturbs the 0 ! 1 oscillation, leading to
a shift of resonance at high power (see x3.3). Nonetheless, Eq. (3.12) can be used to
check the expected detuning behavior by allowing !rf to be a fltting parameter, to
take into account the resonance shift. The solid line is such a flt to Eq. (3.12), with
parameters !rf=2? = 6:432 GHz and ?R;01=2? = 358 MHz. With this adjustment,
the oscillation frequency with detuning follows the expected behavior over a wide
range.
The detuning of the two-photon 0 ! 2 transition is shown in Fig. 9.5(d),
where ?R;02 is plotted as a function of !02. Here, the solid line is given by Eq.
(3.23) (which gives the efiective Rabi frequency ?R;02 =
q
?2R;02 +(!02 ?2!rf)2 for
a two-photon transition of an ideal system), with flt parameters !rf=2? = 6:494 GHz
and ?R;02=2? = 239 MHz. While the resonance occurs near the applied microwave
frequency, the agreement between data and theory is quite poor at large detuning.
As a more reflned test, the three-level rotating wave solution ofx3.3 can be used. The
dashed line comes from the numerical solution of Eq. (3.14), for !rf=2? = 6:429 GHz
and I?w = 15:85 nA. In this case, the discrepancy between the flt (6.429 GHz) and
experimental (6.5 GHz) values of !rf is unexpected. Either the calibration of !02 has
a signiflcant error or the rotating wave solution is not fully capturing the behavior
of the two-photon process. Aside from this issue, the dashed line does reproduce
the detuning behavior reasonably well.
We further investigated the resonance shift seen in Fig. 9.5(c) by measuring
this shift as a function of power. Figure 9.6 shows Rabi detuning maps for SQUID
DS2B, using a 5.9 GHz microwave pulse with a source power PS of (a) -27, (b)
-24, (c) -20, (d) -17, (e) -14, and (f) -11 dBm. The bias ramp rate was reduced
to 0.0095 A/s with pulses beginning at 75 ns intervals, to further reduce unwanted
detuned during the pulse. I used a difierent scale to color each map; black represents
2:5?107 1=s for -27 dBm and 5:8?108 1=s for -11 dBm.
338
(a) (d)
(e)
(f )(c)
(b)
Figure
9.6:
Rabi
detuning
map
of
SQUID
DS
2B
at
!rf
=2
?
=
5:9
GHz.
Eac
hhorizon
tal
line
is
the
total
escap
erate
due
to
a
micro
wa
ve
pulse,
with
asource
po
wer
of
(a)
-27,
(b)
-24,
(c)
-20,
(d)
-17,
(e)
-14,
(f)
-11
dBm,
that
starts
at
t=
0.
Eac
hmap
is
colored
with
adifieren
tgra
yscale.
When
Ib
?
17
:66
?A,
!rf
is
near
!01
,while
at
Ib
?
17
:635
?A,
it
is
near
!02
=2.
339
The set of fringes near Ib = 17:66 ?A corresponds to 0 ! 1 Rabi oscillations,
while the set near Ib = 17:635 ?A is for the two-photon 0 ! 2 transition. The data
were taken at the base temperature of the refrigerator, so the 1 ! 2 transition was
di?cult to resolve. As the power increases, the frequency of the 0 ! 1 oscillation
increases, as expected. The fringes also  atten out, because a given detuning !rf ?
!01 will have its largest impact when it is large compared to ?01. A highly detuned
0 ! 1 oscillation is superimposed on the 0 ! 2 at PS = ?17 dBm.
As discussed earlier in this section and in x3.3, the presence of the second
excited state j2i has a two efiects on the 0 ! 1 Rabi oscillation: the minimum
oscillation frequency is suppressed from its two-level value and this resonance occurs
for !rf > !01 [83]. The solid line in Fig. 9.7(a) was generated from the three-level
rotating wave Hamiltonian of Eq. (3.14), by flnding the value of Ib that minimized
?R;01 for a given I?w. The energy levels and matrix elements for this simulation were
calculated for a single junction with I0 = 17:821 ?A and CJ = 4:46 pF.6 The dashed
line is the ideal two-level solution, where ?R;01 / I?w, showing the suppression at
high power. The resonance shift ?!R;01 = !rf ?!01 is plotted as a solid line in Fig.
9.7(b).
To compare these predictions to the data, I flt each horizontal line of Fig.
9.6 to Eq. (9.1) to extract the oscillation frequency and converted Ib to !01 using
Eq. (2.44). Then, as in Fig. 9.5(c), fltting the data to Eq. (3.12) gave the resonant
oscillation frequency ?R;01. It was also necessary to convert the source power PS
to microwave current I?w. As the relationship is expected to be PS / I2?w, the
proportionality constant may be taken as a free fltting parameter. The circles in
Fig. 9.7(a) were plotted using the conversion I?w (nA) = 59:64pPS=mW. The
agreement between the solid line and the circles is good, with the two-level solution
6These junction parameters are slightly difierent than the ones I used for Fig. 9.5. Even though
the two data sets were separated by less than two weeks, what was probably an ofiset in the
trapped  ux resulted in modifled values for the efiective I0 and CJ.
340
(a)
(b)
(c)
Figure 9.7: Strong fleld efiects in SQUID DS2B. The points plotted as circles were
extracted from the data in Fig. 9.6, after converting the source power PS to current
I?w at the junction. The solid lines are calculated from the three-level rotating wave
Hamiltonian. (a) The oscillation frequency of the 0 ! 1 transition deviates from the
two-level solution (dashed) at high current. (b) In addition, the resonance occurs for
!rf > !01; the resonance shift is ?!R;01 = !rf?!01. (c) The two-photon 0 ! 2 Rabi
oscillation frequency has a nearly quadratic dependence on the microwave current.
341
clearly overestimating the oscillation frequency at high power. As the approximate
expression in Eq. (3.21) shows, the suppression of ?R;01 depends on the difierence
!01 ?!12. In fact by fltting the data to that expression, the value of the difierence
is (!01 ?!12)=2? = 940 MHz. The prediction at Ib = 17:66 ?A for I0 = 17:821 ?A
and CJ = 4:46 pF is 910 MHz.
The resonance shift can be extracted from the same flts of ?R;01 vs. !01. This
efiect is barely visible in Fig. 9.6. For PS = ?27 dBm, the longest oscillation period
appears to occur for Ib < 17:66 ?A, while for PS = ?17 dBm, resonance occurs for
Ib > 17:66 ?A. What complicates the situation is that there was a small amount
of drift over the course of taking the difierent maps. It is unclear whether this was
due to, for example, the biasing and detection electronics warming up or the  ux
ofiset of the SQUID changing slightly. Aside from -24 dBm, the drift was less than
1.5 nA, which I would ordinarily ignore. However when looking for small resonance
shifts, the drift can be signiflcant.
I attempted to correct for the drift in two ways. In one method, I chose a
value of I0 and CJ for each map so that the calculated values of !01 as a function
of the background escape rate (no microwaves) would coincide for the six maps.
In a second attempt, I simply added an ofiset in Ib so the background escape rates
would match. Fortunately, both techniques agreed within about 2 MHz. Results are
plotted as circles in Fig. 9.7(b), using the same calibration of I?w as used in (a). At
the lowest powers, ?!R;01 is negative, which is likely indicative of the uncertainty
in the flts and the junction parameters I0 and CJ. The agreement at higher powers
is much better.
The detuning maps provide a reliable method for flnding the resonant oscilla-
tion frequency. In contrast, for Fig. 9.2, all of the pulses were started at the same
value of Ib. If the resonance shifted at high power, then the measured oscillation
frequency would have been greater than ?R;01.
342
(a) (d)
(e)
(f )(c)
(b)
Figure
9.8:
Rabi
detuning
map
of
SQUID
DS
2B
at
!rf
=2
?=
2:95
GHz.
Eac
hhorizon
tal
line
is
the
total
escap
erate
due
to
a
micro
wa
ve
pulse,
with
asource
of
po
wer
(a)
-32,
(b)
-30,
(c)
-28,
(d)
-26,
(e)
-24,
(f)
-22
dBm,
that
starts
at
t=
0.
Eac
hmap
is
colored
with
adifieren
tgra
yscale.
The
applied
micro
wa
ve
frequency
is
roughly
equal
to
!01
=2
for
the
values
of
Ib
sho
wn.
343
Finally, the 0 ! 2 oscillation frequency can be examined. As the detuning
maps of Fig. 9.6 show, it is quite di?cult to see a clean oscillation. In fact for the
three lowest source powers, the frequency appears to be constant; the slight modu-
lation could be unrelated to Rabi oscillations. It is possible that j2i is nearing the
top of the potential well for Ib ? 17:635 ?A, so that the high escape rate would wash
out the oscillation. Nonetheless, the rough values I could extract are plotted in Fig.
9.7(c). The line again comes from Eq. (3.14). In spite of the obvious disagreement at
low power, the agreement at high power is surprisingly good. It is encouraging that
the rotating wave Hamiltonian, derived in the absence of dissipation and evaluated
using properties of a simple current-biased junction, reproduces the Rabi oscillation
frequency well at a range of powers.
Figure 9.8 shows detuning maps of the same device taken at 20 mK for
!rf=2? = 2:95 GHz with source powers of (a) -32, (b) -30, (c) -28, (d) -26, (e) -
24, and (f) -22 dBm. As the microwave frequency is half of that used in Fig. 9.6, the
oscillations are due to a two-photon 0 ! 1 transition. The fringes are not nearly as
symmetric as the corresponding single photon ones and there is a noticeably larger
resonance shift with power.
In principle, the three-level rotating wave solution could describe these data
as well. However, it appears the resonance shift in a two-level system is bigger than
the perturbation due to the third level. A accurate model of this transition would
conflrm the magnitude of the diagonal matrix elements  0;0 and  1;1.
9.3 Density Matrix Simulations
Having shown that the Rabi oscillation frequency is consistent with theory,
I will now turn to describe the full waveform of the oscillations. As discussed in
x3.6, it is practically impossible to write down the full Hamiltonian of the system,
including all of the degrees of freedom that give rise to dissipation and decoherence.
344
A common approach to the problem is to use the density matrix formalism to follow
the evolution of the system [143,144], with phenomenological time constants T1 and
T2.
To simulate the junction dynamics, I numerically integrated the Liouville-
von Neumann equation to flnd the time dependence of the elements of the density
matrix (see x3.8). The matrix that specifles the evolution of the system, whose
structureisgiveninAppendixD, requiresseveralparameters. ForanN levelsystem,
there are N ? 1 energy level difierences ~!nm, N escape rates ?n, N (N ?1)=2
energy dissipation rates Wtnm (see x3.4), N (N ?1) coherence times Tnm2 (see x3.7),
N (N +1) bare Rabi frequencies ?nm, and the temperature T.
While this may seem to be a large number of free parameters, most of which
depend on the current bias Ib as well, we are most interested in verifying the current-
biased junction Hamiltonian, so ideally there are only a few independent choices
to be made. For example, the critical current I0, capacitance CJ, and Ib specify
!nm and ?n. A single value for the relaxation time T1, in addition to T, sets the
values of all of the dissipation rates, assuming the matrix elements  n;m of the
tilted washboard. The same values of  n;m also give all of the ?nm, with a single
conversion factor between source power Ps and, for example, ?01. I also chose to set
all of the coherence times to the same value T2. A strong constraint is that the set of
parameters should be able to fully explain a wide variety of experiments, including
slowly sweeping the bias at elevated temperature, a multi-level Rabi oscillation
caused by a microwave pulse, and a T1 relaxation after the pulse is turned ofi.
I used the density matrix to simulate the Rabi oscillations in Figs. 9.2 and 9.3.
We believe there was signiflcant population in at least three energy level, because
the escape rates saturated at ever increasing equilibrium values as the microwave
power was increased. The simplest test of the simulation was to see if it could
reproduce the equilibrium escape rate values ?eq. This was a valuable test, because
345
?eq depends strongly on the energy levels, escape rates, and matrix elements, but
relatively weakly on T1 and T2, given the large range of powers being considered.
The circles in Fig. 9.9 show ?eq as a function of the measured oscillation
frequency ?R;01 for the curves in Fig. 9.2 and several others not shown. I extracted
both parameters from flts to Eq. (9.1), where ?eq = g1 + g2. The lowest point is
noticeably higher on the log-log scale because the decaying background at this power
lead to a poor flt. At high powers, the large background caused some of the scatter
in resulting values of ?R;01.
As I mentioned in x9.1, I was unable to describe the energy levels and escape
rates of SQUID DS1 with single values of I0 and CJ. Instead, I let the escape
rates be described by I?0 = 34:308 ?A and C?J = 4:43 pF. Evaluating Eq. (2.43)
at I1 = 34:069 ?A, where the experiments were performed, gives ?0 = 6:31?104,
?1 = 3:67?107, ?2 = 3:33?109, and ?3 = 2:80?1010 1=s. The energy levels and
escape rates are specifled by I!0 = 34:300 ?A and C!J = 4:43 pF. At I1 = 34:069 ?A,
!01=2? = 7:60, !12=2? = 6:34, and !23=2? = 5:61 GHz. I used matrix elements
in the cubic approximation; several are given in Eqs. (2.38) and (2.39). As the
refrigerator was at base temperature and we saw no obvious signs of heating due to
the microwave pulse, I usually set T = 0. Based on the experiment in x7.4.2, I set
T1 = 14 ns. The choice of T2 = 7 ns will be motivated below.
With this set of parameters, I evolved the density matrix with a 5 ps time step
to calculate the occupancies ?i. From these results, I flt the escape rate [deflned in
Eq. (3.30)] to Eq. (9.1) in order to extract ?R;01 and ?eq.7 The solid line in Fig. 9.9(a)
shows the results for a two-level system. ?eq saturates at roughly ?1=2 as j0i and j1i
become equally populated at high power (after the system loses phase coherence).
7I ran all of the simulations at Ib = 34:069 ?A (just as with the experiments), even though this
does not correspond to resonance at high power. As a result, ?R;01 < ?01 because the suppression
of ?R;01 outweighs the efiects of detuning. Unlike the data, I held Ib flxed for simulation, on the
assumption that the slow ramp did not result in a signiflcant amount of detuning for the short
times being considered.
346
(a)
(b)
Figure 9.9: Power dependence of ?eq in SQUID DS1. The circles show the equi-
librium escape rate ?eq as a function of the measured oscillation frequency ?R;01,
for the same data set as in Fig. 9.3. (a) Density matrix simulations with two (solid
line), three (dashed), and four (dotted) levels, using junction parameters from in-
dependent measurements, underestimate ?eq at high power. (b) With modifled ?1,
?2, and !12, the agreement for the three-level system is better.
347
Equation (3.60) predicts this should happen for ?01  1=pT1T2 ? 2? ? 16 MHz,
consistent with the simulation.
Since ?eq clearly exceeded ?1=2 at high power, higher levels are needed to
explain the observed escape rates. The second excited state is populate by a two-
photon process with a quadratic power dependence, which explains the rapid in-
crease in ?eq with ?R;01. The dashed line in Fig. 9.9(a) shows simulation results for
a three-level system. While the intermediate powers match relatively well, there is
signiflcant disagreement at the low and high ends. A fourth level could be necessary
at the highest powers. With Ir ? 0:993, the properties of this level will not be
well described by Eqs. (2.43) and (2.44); nonetheless, the dotted line comes from
a four-level simulation with the parameters listed above. Even the extremely high
escape rate of this level cannot account for the observed ?eq at high power. Despite
the nearly 50% disagreement between experiment and theory, the results are still
encouraging, because all of the simulation parameters came from low power (or zero
power) experiments and predictions from the simple junction Hamiltonian.
To proceed, I decided to change some of the parameters \by hand." It seemed
clear that ?1 was too large, so I decreased it from 3:67 ? 107 to 2:6 ? 107 1=s.
This is the value that ?1 takes at I1 = 34:064 ?A, 5 nA smaller than where the
experiments were performed. While there is a redistribution of currents with the
quantum state [73], the shift is only expected to decrease the qubit current by
about 2.5 nA for this device. In addition, the escape rate contribution from j2i was
insu?cient in Fig. 9.9(a). I decided to increase !12=2? from 6.34 to 6.45 GHz (to
bring the 1 ! 2 transition closer to resonance) and increase ?2 from 3:33 ? 109
to 4 ? 109 1=s. The modifled !12 and ?2 occur at I1 = 34:066 and 34.073 ?A,
respectively. The quantum simulations suggest that the qubit current decreases for
j2i by roughly 6 nA, with respect to the value for j0i. This shift would increase !12
(consistent with the modifled value I chose), but decrease ?2 (inconsistent with my
348
choice). Although it is di?cult to justify the new set of parameters, it is interesting
to see how sensitive the results are to such small modiflcations.
The results of two- and three-level simulations with the three modifled param-
eters are shown with a solid and dashed lines in Fig. 9.9(b). The agreement for the
three-level system is good at all but the highest powers, where a fourth level might
be necessary. The same results could have been obtained by, for example, increasing
?2 to 5?109 1=s and leaving !12 untouched. Further work is needed to determine
the most realistic junction parameters and obtain the highly accurate values needed
for these simulations.
The next thing I used the simulations for was to determine what causes the
escape rate in Fig. 9.2 to slowly reach the equilibrium value and why the oscillation
amplitude, or \visibility," is small at high power. While insu?cient time resolution
would decrease the oscillation amplitude, it would not cause the flrst maximum to
occur at a lower ? than the second. Another possibility is that the shape of the
microwave pulse could result in an anomalous Rabi curve. In fact, a pulse that
turns on too quickly can excite population to j2i [113]. That the high power curves
do not decay with time in Fig. 9.2 suggests that this is not a serious problem for
those experiments. A direct measurement of the pulse shape would be very useful in
resolving this matter, but it would have to be done near the device, as there could be
distortion along the coaxial line in the refrigerator. I estimated the time resolution
of the switching measurement in x5.5 to be less than 1 ns. It was di?cult to get an
accurate number as there are so many places where errors were introduced. Thus,
I chose to treat the pulse shape and time resolution as free parameters.
Figure 9.10(a) shows the escape rate generated by a three-level density matrix
with the modifled system parameters and a !rf=2? = 7:6 GHz microwave pulse,
with ?01=2? = 260 MHz. As indicated by the dashed line, I used a square pulse,
i.e. one that turns on and ofi instantaneously, that was on between t = 0 and 23.5
349
(a)
(e)
(d)(c)
(b)
Figure 9.10: Loss of visibility in Rabi oscillations. The density matrix was used to
calculate the total escape rate for Rabi oscillations at ?01=2? = 360 MHz with (a) a
square microwave pulse of frequency !rf=2? = 7:6 GHz and (b) a pulse that rises and
falls with a half-Gaussian proflle; the dashed lines show the functions that provided
the overall scaling for all of the Rabi frequencies. Histograms for the Gaussian pulse
are shown (c) before and (d) after averaging with a time resolution of 1.2 ns. (e) The
resulting ? (solid line) has reduced visibility, but still overestimates the oscillation
amplitude of SQUID DS1 taken at PS = ?12 dBm (circles).
350
ns. During the pulse, the oscillations decay with time constant T0, much like in a
two-level system. The nearly solid band in ? during the pulse is due to a highly
detuned fast oscillation between j1i and j2i. When the pulse turns ofi, j2i quickly
depopulates due to ?2. Following this, j1i empties due to ?1 and T1, leading to slow
decay for t > 24 ns.
In Fig. 9.10(b), I attempted to simulate a realistic pulse by weighting the ?nm
with a time-dependent envelope function, also shown with a dashed line. The pulse
turns on with a half-Gaussian line shape, with a 2.5 ns full width. ?01=2? stays
at 260 MHz between 3.7 and 24 ns and then decays to 0 with a half-Gaussian line
shape of width 2 ns. Based on the manufacturer?s speciflcations, rise and fall times
of this order are to be expected for the microwave source. Of course, re ections in
the microwave lines could lead to a completely difierent rise time. With the modifled
pulse shape, the escape rate has noticeably slower transition edges. In addition, the
second oscillation maximum is slightly higher that the flrst.
Using the inverse of the process described in x6.2, I converted ? to a (normal-
ized) switching histogram, as shown in Fig. 9.10(c). Even though the escape nearly
reaches equilibrium by the end of the pulse, the number of counts in the histogram
decays exponentially.
To mimic the experimental time resolution, I used a Gaussian average of the
histogram. That is, if h(ti) is the number of counts in the bin centered at time ti,
then the number of counts in this bin in the smoothed histogram is
h0(ti) =
P
j h(tj)G(tj ?ti;w)P
j G(tj ?ti;w)
; (9.2)
where G(t;w) is a Gaussian centered at t = 0 with a full width w. Figure 9.10(d)
shows the resulting histogram with w = 1:2 ns. Although this time is longer than
the controlled experiments of x5.5 would suggest, we have never seen an escape rate
351
feature with a time constant faster than 1 ns, so this value does not seem entirely
unrealistic. As a result of the smoothing, the rapid oscillations have been averaged
away and the amplitude of the main 0 ! 1 oscillation is also smaller.
The solid line in Fig. 9.10(e) is the escape rate calculated from the smoothed
histogram. Nearly identical results can be obtained by averaging the escape directly,
but that procedure is slightly less justiflable than working with the histogram. The
oscillations are centered about 108 1=s, just as in Fig. 9.10(a), but their amplitude
is much smaller.
The circles in Fig. 9.10(e) show the measured escape rate of SQUID DS1 due
to a 7.6 GHz microwave pulse of source power PS = ?12 dBm. The measured ?
has the same frequency and equilibrium level ?eq as the simulation, which can also
be seen in Fig. 9.9(b). In addition, the fast decay starting at t = 24 ns matches
the prediction, which is a result of the pulse shape and the smoothing; ?2 by itself
would have lead to a much faster rate. However, the pulse shape has little impact
on the subsequent slower decay that begins near ? = 1:5?107 1=s. The population
in j1i that decays is established during the oscillation, but is hidden by P2?2. By
turning the pulse ofi and letting j2i depopulate, the contribution from j1i is visible.
The agreement here is a good indicator that T1 and ?1 are faithfully reproducing
the junction dynamics. I have not shown that at longer times ? decays with a time
constant of roughly 50 ns, as in Fig. 8.21. The three-level simulation cannot account
for both a 10 ns and a 50 ns decay of j1i.
The critical feature of the data that is not captured by the simulation is the
oscillation amplitude. The flrst oscillation can be brought to a lower level by further
rounding of the pulse shape. However, if ?01 is kept at low levels for a longer time,
the oscillation frequency is noticeably slower, which is not seen in the data. The
visibility could be reduced by degrading the time resolution. This is inconsistent
with the oscillation amplitude we see for very high power. Thus, there appears to
352
be a key element of simulation that has not been included.
I made further tests of the accuracy of the simulations by examining the vari-
ation of the Rabi oscillations with power and time. The symbols in Fig. 9.11 show
measured Rabi oscillations, taken under the same bias conditions as Fig. 9.10(e),
for (a) a long pulse and (b) a 24 ns wide pulse. The curves are ofiset vertically and
labeled with the microwave source power. The solid lines comes from the density
matrix, using the same system parameters and processing shown in Fig. 9.10. I
only varied the overall scaling of the Rabi frequencies ?nm. For source powers of
PS = ?28;?26;?24;?22;?20;?18;?16 dBm, ?01=2? = 65, 75, 89, 112, 137, 174,
216, 270 MHz.
For the intermediate powers, the agreement between data and simulation is
good, both for the main oscillation and the decay. I chose T2 = 7 ns to match the de-
cay envelope of these data. The agreement is good, but not perfect. At high powers,
the maximum visibility of the simulated curves is higher than the measurements, as
discussed earlier. In addition, at the highest powers we measured, the value of ?eq is
not reproduced by the simulations. At the low powers, the simulation overestimates
the measured curves and does not capture the decay correctly, perhaps indicative of
an incorrect ?1. At the lowest power in Fig. 9.11(a), ? decays noticeably. This could
be due to detuning efiects, as the bias was slowly ramped during the oscillation.
Some of the disagreement at low power may be due to low frequency noise in
Ib or I0, which I have not considered. Imagine a simple scenario where during each
cycle of the experiment, Ib took a slightly difierent value. This would mean that
the microwaves would not be on resonance and the efiective Rabi frequency would
increase according to Eq. (3.12). As we extract ? from a large number of trials, this
inhomogeneous broadening could lead to what looks like poor timing resolution. In
x8.3, the full width of the resonance peaks of DS1 were about 4 nA, which places
an upper bound on the current noise. For the conditions of Fig. 9.11, a shift in
353
(a)
(b)
-16
PS (dBm)
PS (dBm)
-20
-24
-28
-14
-18
-22
-26
Figure 9.11: Density matrix simulations of Rabi oscillations in SQUID DS1. The
symbols show the total escape rate (vertically ofiset for clarity) due to a !rf=2? =
7:6 GHz microwave pulse of the indicated source power and a duration of (a) 3 ?s
and (b) 24 ns. The solid lines come from a three-level density matrix simulation,
where only the overall scaling of the bare Rabi frequencies ?nm is varied between
the difierent curves.
354
Ib of 2 nA results in !01=2? changing by less than 30 MHz. Thus for a 200 MHz
Rabi oscillation, this amount of detuning is negligible. However, low frequency noise
could explain the lower than expected escape rates at low power. It could also be
responsible for the shorter Rabi decay envelope times T0 seen in Fig. 9.3(b) at low
PS.
A weakness of the escape rate measurement is that it is only sensitive to the
sum of the tunneling contributions from all of the levels, as ? = Pi Pi?i, where Pi is
the normalized occupation probability of state jii (see x3.5). In order to determine
Pi, at the very least the individual escape rates ?i must be known accurately. Even if
the sum is dominated by one term, for example P2?2 for high power Rabi oscillations,
the di?culty of predicting ?2 results in a large uncertainty in P2. Thus, it is not
easy to compare the populations predicted by the density matrix simulation with
measurements. Some of these issues are addressed in Fig. 9.12. The measured escape
rate of SQUID DS2B due to a resonant 6.2 GHz microwave pulse (with a power of
PS = ?17 dBm and width of 26 ns) as a function of time is plotted with circles in
Fig. 9.12(a).
Under identical conditions, the bias pulse readout method (see x6.6.3) was
used to measure P1 and P2, as plotted with circles in Fig. 9.12(b) and (c). Details
of this readout technique can be found in Ref. [104]. In the absence of a microwave
current, the probability that the junction switched to the voltage state due to a dc
bias pulse added to Ib was determined for a wide range of pulse amplitudes. Then
the same set of dc bias pulses were applied to the junction at a certain time during a
Rabi oscillation, which was due to a microwave pulse. Finally, the state populations
needed to explain the new switching probabilities were extracted. For example, if
the system occupied j1i during the oscillation, then a relatively small bias pulse
would result in a high probability of switching to the voltage state. Knowledge
of the escape rate during a pulse (and not just the resulting switching probability)
355
(a)
(c)
(b)
Figure 9.12: Escape rate and current pulse measurements of Rabi oscillations in
SQUID DS2B. Rabi oscillations due to a resonant 6.2 GHz microwave pulse with
PS = ?17 dBm were readout with two techniques and modeled with a density matrix
simulation. (a) The solid circles show the measured escape rate ?, continuously
measured during the oscillation. The simulation gives the total escape rate (solid
line), as well the contributions from j1i (dotted) and j2i (dashed). As shown with
circles, the bias pulse readout allows direct measurement of (b) P1 and (c) P2.
The solid lines in (b) and (c) come from the same density matrix simulation that
produced the escape rates. The data set was taken at 20 mK by Tauno Palomaki.
356
made it possible to extract P2 as well. This procedure was repeated at several points
along the Rabi oscillation and decay, to map out the populations as a function of
time.
I attempted to reproduce the escape rate and pulsed measurements with a
single density matrix simulation; results are shown with solid lines in Fig. 9.12. The
qubit junction of DS2B was roughly described by single junction parameters I0 =
18:138 ?A and CJ = 4:50 pF. At Ib = 17:955 ?A, Eq. (2.43) gives ?0 = 3:78?103,
?1 = 2:88?106, and ?2 = 5:28?108 1=s, while Eq. (2.44) returns !01=2? = 6:20
and !12=2? = 5:51 GHz. As with SQUID DS1 above, I had to modify some of the
parameters to flnd agreement with the measured escape rate. I chose ?1 = 2?106
and ?2 = 4:2? 108 1=s, and !12=2? = 5:58 GHz. In addition, T1 and T2 were set
to 17 and 8 ns, respectively; relaxation rates cames from Eq. (3.26), evaluated with
matrix elements in the cubic approximation. For the microwave pulse, I used the
same proflles and time averaging given in Fig. 9.10 and a bare Rabi frequency of
?01=2? = 190 MHz.
The agreement between the simulation and the measured escape rate in Fig.
9.12(a) is reasonable, although the decay envelope is not reproduced exactly. In
addition, the data decays more quickly than the simulation when the pulse turns
ofi. It is possible that the time resolution was slightly better than 1.2 ns for this data
set. The simulation also gives P1, which agrees well with the data [see Fig. 9.12(b)],
aside from a slight difierence in the decay rate for t > 30 ns. The population P2 in
the second excited state [see Fig. 9.12(c)] is roughly reproduced by the simulation,
conflrming that a reasonable value of ?2 was used. However, for t > 10 ns, the
simulation predicts a noticeable decay in P2 that is not seen in the data. Oddly,
fairly good agreement is found for T1 = 50 ns, which is the slower time constant we
see in the decay following a high power Rabi oscillation.
The two measurement techniques complement each other well. The escape
357
rate is particularly sensitive to a small population in j2i, while it is relatively easy
to measure the large population in j1i with the pulsed readout. Even though the
density matrix simulation is far from perfect (it only works for a small range of
powers, for example), it is encouraging that it can reproduce the results from both
readouts.
With some confldence that the simulation is generating accurate values, the
total escape rate in Fig. 9.12(a) can be broken into its components; the dashed
line is P2?2 and the dotted is P1?1. This shows that the majority of the measured
? actually comes from j2i, even when we wanted to drive the 0 ! 1 transition.
However, the plot shows that P2 essentially follows P1, because the 1 ! 2 transition
is being driven well ofi resonance. Thus the efiect of the tunneling from j2i is to
amplify P1. Even though we are measuring P2 with the escape rate, we can get an
idea of ?01. This is why the oscillation frequency in Fig. 9.3(a) followed the expected
dependence of a two-level system, even when there was such clear evidence for a third
level.
The density matrix can be used to further examine the efiect of j2i. The solid
lines in Fig. 9.12 show the populations for the simulation in Fig. 9.13. P1 and P0
oscillate out of phase about 48%. The dashed lines were generated with the same
simulation parameters, except ?02, ?12, and ?22 were set to zero, which removed
j2i from the dynamics. Now the oscillation is centered about 50%, but there are no
signiflcant qualitative changes.
It is possible that tunneling to the voltage state is responsible for some of the
observed decoherence; this efiect can be studied with the simulations as well. With
?0 = ?1 = 0 and the second excited state removed, P1 changes by a maximum
of only 0.001 during the oscillation as compared to the case with flnite tunneling.
As 1=?1 is long compared to T1 and T2, tunneling has a negligible impact on the
populations during a Rabi oscillation, but it does make a slightly larger contribution
358
P1
P2
P0
Figure 9.13: Efiect of j2i on 0 ! 1 Rabi oscillations. The three-level simulation
used in Fig. 9.12 gives the occupation probabilities P0, P1, and P2 (solid lines). The
probabilities calculated for a two-level system oscillate about 50% (dashed lines).
to the decay once the microwaves are turned ofi. For the simulations in Fig. 9.11,
the maximum change in P1 due to tunneling is 0.01. Even though ?1 is an order of
magnitude larger for the conditions of that flgure, tunneling has a small efiect on
the Rabi decay envelope.
9.4 Summary
For quantum computation, Rabi oscillations represent a coherent manipulation
of the state of the qubit. In practice, our junction devices are not two-level systems
and it is important to understand the dynamics of the actual system. For example,
we see clear evidence of leakage to the second excited state at the high powers
that are needed to realize fast quantum gates (x9.1). This appears to be due to a
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two-photon process, involving ofi-resonant 0 ! 1 and 1 ! 2 transitions.
The oscillations ofier a way to conflrm our understanding of the Hamiltonian.
The measured oscillation frequency is consistent with a three-level rotating wave
model, which requires the matrix elements and energy levels of the single junction
Hamiltonian (x9.2). The shift of resonance at high power is a particularly sensitive
test of the system parameters.
These measurements also provide estimates for the time constants that de-
termine the quality of the qubit. The observed power broadening of the 0 ! 1
transition corresponds to T1 ? T2 ? 4 ns. Such short values are inconsistent with
T1 measurements of the previous chapters and with the average Rabi decay time
T0 = 10 ns. I used a density matrix simulation to model the dynamics during a
microwave pulse. This was a useful test, because in a single ? curve, the escape rate
during the pulse depends strongly on ?2, energy levels, and the matrix elements,
while the decay after the pulse reveals T1 and ?1. I had to make certain assumptions
about the shape of the microwave pulse and the timing resolution in the experiment,
but the data of SQUID DS1 are consistent with T1 ? 14 ns and T2 ? 7 ns. Further
work is needed to explain the observed loss of visibility at high power.
360
Chapter 10
Conclusions
10.1 The DiVincenzo Criteria Revisited
In this thesis, I have discussed the implementation of Nb/AlOx/Nb Josephson
junction devices as phase qubits for quantum computation. I would flrst like to
summarize the status of the work in view of the DiVincenzo criteria [19{21], given
in x1.1.
Hilbert space control: The dynamics of a current-biased junction are analogous
to a particle (with a mass proportional to the junction capacitance) in the tilted
washboard potential. The two lowest energy levels of this anharmonic potential
can serve as qubit states (x2.3). Care must be taken so as not to occupy the higher
excited states. Microwave spectroscopy provides clear evidence for quantized energy
levels (x8.1 and x8.2) that can be tuned with an external current bias during the
course of an experiment. Additional qubits can be added to the full state space
by capacitively coupling junctions (x8.6). In the future, a variable coupling scheme
may be required [145].
State preparation: Initialization of our qubits is most easily accomplished by
cooling the devices below 50 mK, so that the system occupies the energetic ground
state. Active methods may also work at higher temperatures (x7.5), but we have
not implemented any such technique. For a SQUID phase qubit, the  ux state can
be initialized with  ux shaking (x6.5), a procedure which appears to work for a wide
range of device parameters and for coupled qubits.
Low decoherence: The trade-ofi that comes with the ease of controlling the
energy levels of the qubit is that the bias lines originate at room temperature, leading
to an unacceptable level of dissipation. I have studied two types of isolation: an
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LC fllter that is particularly efiective at high frequencies (x4.2) and a broadband
inductive current divider that results in a dc SQUID (x4.3). In addition, it may be
that junction quality plays an important role in decoherence, as we see evidence for
coupling to microstates that may be due to defects in the barrier (x8.5). For the
SQUID phase qubits I examined, I estimate the coherence time T2 to be under 10 ns.
The time scale for a gate operation is set by the plasma frequency !p=2? ? 5 GHz
of the junction, with a typical two-qubit gate taking roughly 10 ns [112]. Thus, far
better coherence times are required for quantum computation, but studies of error
correction can perhaps begin soon.
Controlled unitary transformations: A microwave current can be used to in-
duce transitions between j0i and j1i, with Rabi oscillations representing a simple
single qubit gate (x9.1). Two-qubit gates may be performed by bringing the junc-
tions in resonance with each other [112]. While I did map out the energy level
structure of two coupled qubits (x8.6), I did not do any experiments where the state
of the coupled system was coherently controlled.
State-speciflc quantum measurements: Measuring the tunneling escape rate of
a junction from the supercurrent to the flnite voltage state is a simple way to perform
state measurement (x6.2). This technique is extremely sensitive to population in the
excited state, but some modeling is required to accurately extract the individual
state occupation probabilities (x9.3). The use of a microwave (x8.7.1) or current
(x6.6.3) pulse will be more suitable for state readout. However, all three of these
techniques are very destructive; not only is there signiflcant heat generated, but the
system leaves the computational space and takes substantial time to reset.
Quantum communication: I did not do any work on the conversion of station-
ary to  ying qubits and this criterion is not of much relevance to solid state qubits
that are coupled by wires.
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10.2 Summary of Experiments
Many of the experiments I performed tested the validity of the Hamiltonian
of the current-biased junction, given in Eq. (2.23). The hope is that a detailed un-
derstanding of the systems will aid in the design of better qubits in the future. The
single-junction Hamiltonian describes a circuit with three main components: a cur-
rent source, an ideal junction, and its shunting capacitance. In contrast, our phase
qubits are made from a multi-layer fabrication process involving difierent materials
and are controlled by a potentially noisy current source through a series of on- and
ofi-chip fllters that allow some coupling to the environment. An additional compli-
cation for the dc SQUID phase qubits is that each of the two junctions represent a
quantum degree of freedom. We attempted to reproduce single junction behavior
by holding the designated isolation junction well out of resonance with the qubit
junction (x6.4).
A strong test of our description of the circuits was reproducing the escape rate
during Rabi oscillations using a density matrix simulation (x9.3). This generic three-
level simulation required many input parameters that were checked independently
through a series of experiments. The escape rate of the ground state was measured
directly (x7.2), while theoretical predictions for the excited states were tested at
elevated temperatures with master equation simulations (x7.4). The 0 ! 1 transi-
tion frequency was mapped out with spectroscopy, which also showed the existence
of levels above the two qubit states and of multi-photon transitions that are the
leading cause of leakage out of the j0i and j1i qubit basis (x8.4). Finally the matrix
elements of the phase operator ^ that characterize transitions were verifled by com-
paring the Rabi oscillation frequency at high powers to a three-level rotating wave
Hamiltonian x9.2.
Another objective of the work was to characterize the quality of the qubits for
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Table 10.1: Summary of characteristic times. The relaxation time T1, coherence
time T2, spectroscopic coherence time T?2, and Rabi decay time T0 were estimated
several ways for the three devices. The flrst line is a purely theoretical prediction;
the rest of the values were extracted from experiments. The measurements were
performed for qubit plasma frequencies 5 GHz < !p=2? < 10 GHz. For the coupled
devices, the measurements were performed on one qubit, while the other was held
well out of resonance.
Time Method LC2 DS1 DS2
T1 Design prediction 5 ?s 200 ns 200 ns
IV curve > 400 ns
Thermal activation 8 ns 10 - 30 ns 10 - 30 ns
Master equation 4 ns 14 ns
Fast ramp < 50 ns < 50 ns
Time domain; ?wave pulse 45 ns
Time domain; tunneling 10 / 50 ns 10 / 50 ns
Power broadening 4 ns
T2 Power broadening 4 ns
Density matrix 7 ns 8 ns
T?2 Spectroscopy 3 ns 4 ns 8 ns
T0 Rabi oscillations 5 - 15 ns 5 - 20 ns
quantum computation. Results are summarized in Table 10.1 for the capacitively-
coupledLC-isolated junctions (device LC2), the single dc SQUID phase qubit (DS1),
andthecapacitively-coupleddcSQUIDphasequbits(DS2). Forthecoupleddevices,
most of the measurements happened to be on LC2B and DS2B, but we never saw
any dramatic difierences with LC2A and DS2A, which they were coupled to. I
will mostly focus on the dc SQUID phase qubits below, as our measurements were
incomplete for the LC-isolated junctions and their performance was not as good.
Although the coherence time T2 is the most relevant parameter for character-
izing a qubit?s utility for quantum computation, the dissipation time T1 serves as
an upper bound on T2 (actually, 2T2 < T1, using the common deflnitions) and is an
364
important measure of the isolation of the qubit. In addition, one of the biggest mys-
teries in the superconducting quantum computing community has been the short
T1 values, regardless of the qubit type, so it is useful to measure dissipation with a
variety of techniques.
Measurement of the IV curve of a junction gives an estimate for the intrinsic
dissipation at low frequency, through the sub-gap resistance and retrapping current.
For our Hypres junctions, the shunting resistance RJ appears to be greater than
100 k?. With a typical junction capacitance of CJ = 4 pF, this would correspond
to T1 = RJCJ > 400 ns. However, this represents a \high power" measurement
and there may be less dissipation at high powers if saturation of two-level systems
occurs.
We performed two types of measurements of T1 at the plasma frequency of
the junction. T1 leads to a dissipation rate that is balanced by other processes
in the junction, such as tunneling and microwave activation. Measurement of the
populations after equilibrium has been established can give estimates for T1; these
experiments can be done on long time scales. Analysis of the escape rate with
classical thermal activation was inconclusive, yielding a possible range between T1 =
10 and 30 ns (x7.1.2). I used a quantum master equation simulation to analyze the
same data, which gave 14 ns (x7.4.2). A related technique is to increase the bias
ramp rate until it moves the system away from dynamic equilibrium (x7.5). As we
never obtained positive results, I can only say T1 < 50 ns. The dependence of the
0 ! 1 resonance width on the applied power depends on T1; the extracted value
was roughly 4 ns (x9.1).
Alternatively, energy relaxation can be measured in the time domain, by creat-
ing an excited state population resonantly, and monitoring the system as it returns
to dynamic equilibrium. This method showed the presence of two time constants
(of roughly 10 and 50 ns) for the decay of population in j1i (x8.7.2). A low power
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measurement only detected the slower of the two (x8.7.1).
The coherence time T2 can also be measured. We measured a maximum spec-
troscopic coherence time T?2 ? 8 ns and this parameter is susceptible to dissipation,
decoherence, and low frequency noise (x8.3). The same power broadening measure-
ment mentioned earlier suggested T2 ? 4 ns. Finally, the Rabi decay time T0 was
about 10 ns at intermediate powers (x9.1), which corresponds to T2 = 7:5 ns for
T1 = 15 ns. Density matrix simulations with T2 ? 7 ns accurately reproduced the
observed decay envelope of Rabi oscillations for a range of powers.
There is a signiflcant uncertainty in the determination of T1, but the values are
far below the value of 200 ns predicted if the noise on the bias lines is characterized
by a 50 ? resistance (x4.3). While it is clear that T1 is not limited by intrinsic
dissipation of the junction at low frequency, high frequency materials properties
may be causing serious problems. It will be crucial to uncover whether the two
decay time constants, splittings in the spectra, and short T1 and T2 are related.
Although there clearly are a great number of important questions to answer
and inconsistencies to resolve at this early stage in the research, superconducting
circuits hold a great deal of potential for realizing a quantum computer.
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Appendix A
Circuit Hamiltonians
This appendix gives the general procedure for deriving the Hamiltonian of
an electrical circuit. I applied this method to several devices in Chapter 2. The
procedure begins by identifying the equations of motion that describe the circuit,
generally obtained from Kirchhofi?s law and other constraints, such as  ux quanti-
zation for superconducting circuits. A generalized coordinate qi in these equations
could be the phase difierence across a junction, the charge on a capacitor, or the
 ux in a loop.
A valid Lagrangian L for the system will generate the equations of motion
from the Euler-Lagrange equations
d
dt
 dL
d_qi
?
= @L@q
i
: (A.1)
With the Lagrangian in hand, the conjugate momenta pi and Hamiltonian H can
be found as
pi = @L@ _q
i
(A.2)
H =
X
pi _qi ?L: (A.3)
Here, H = H(pi;qi) and the generalized velocities _qi may be eliminated using Eq.
(A.2). In addition, H obeys Hamilton?s equations:
_pi = ?@H@q
i
(A.4)
_qi = +@H@p
i
: (A.5)
Proceeding in this formal way will guarantee that the Hamiltonian is expressed
367
in terms of operators that are conjugate pairs ([^qi; ^pi] = i~), which is required for
a quantum mechanical treatment. To obtain Schr?odinger?s equation in coordinate
space, one just makes the substitutions
^pi = ~i @@^q
i
: (A.6)
It is often the case that the Lagrangian for a system can be written as a simple
sum:
L = L1 (qi; _qi)+L2 (qj; _qj)+LC (qi; _qi;qj; _qj); (A.7)
where L1 and L2 are the Lagrangians for two sub-systems and LC characterizes the
coupling between them (and introduces no additional degrees of freedom). Equation
(A.2) gives the momenta
pi = @L1@ _q
i
+ @L2@ _q
i
+ @LC@ _q
i
(A.8)
and Eq. (A.3) gives the Hamiltonian of the full system
H =
X
i
 @L
1
@ _qi _qi +
@LC
@ _qi _qi
?
+
X
j
 @L
2
@ _qj _qj +
@LC
@ _qj _qj
?
?L1 ?L2 ?LC (A.9)
= H1 (qi; _qi)+H2 (qj; _qj)+HC (qi; _qi;qj; _qj); (A.10)
where the contribution from the coupling is
HC =
X
i
@LC
@ _qi _qi +
X
j
@LC
@ _qj _qj ?LC: (A.11)
This expression shows that the system Hamiltonian is the sum of the sub-system
Hamiltonians and coupling contribution, expressed in terms of the generalized coor-
dinates and velocities. Equations (A.8) may be inverted to give H = H(qi;pi;qj;pj).
368
Appendix B
MATLAB Code
The following programs calculate the eigenfunctions, energy levels, and tunnel-
ing rates of a single current-biased junction and two capacitively coupled junctions.
The heart of the algorithms was written by Huizhong Xu; see x2.4 and x3.3.2 of
Ref. [1].
B.1 Solutions of the Junction Hamiltonian
The programs in this section solve the Hamiltonian for a current-biased junc-
tion in the absence of dissipation, given in Eq. (2.23). The nature of these solutions
is discussed in x2.3.3. jjspectrum is the main driver that just collects the solu-
tions returned by jjeigentbc, given below. The diary command creates a flle of
everything that is dumped to the screen, which I found useful for debugging.
function [stuff,wavefn] = ...
jjspectrum(Io, Cj, Iri, Irf, dIr, levelmaxIr, E0, psi0)
% [stuff, wavefn] =
% jjspectrum(Io, Cj, Iri, Irf, dIr, levelmaxIr, E0, psi0)
% This calculates all the energies and wavefunctions for a single
% junction with critical current ?Io? (Amps), junction capacitance
% ?Cj? (Farads), from reduced bias current ?Iri? to ?Irf?, in ?dIr?
% steps. ?levelmaxIr? sets the number of levels to calculate; it?s
% defined in ?keeplevels?. ?E0? and ?psi0? are optional -- they
% specify the initial guesses for all the levels. Everything sent
% back in a big structure.
% calls: hbar, keeplevels, jjeigentbc, wp, plotlevels, xaxis
more off
diary on
global hbar;
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stuff.params.Io = Io;
stuff.params.Cj = Cj;
stuff.params.Iri = Iri;
stuff.params.Ifr = Irf;
stuff.params.dIr = dIr;
stuff.params.levelmaxIr = levelmaxIr;
stuff.params.start = clock;
NIr = floor( (Irf - Iri) / dIr ) + 1;
for Ircount = 1 : NIr
Ir = Iri + (Ircount-1) * dIr;
stuff.Ir(Ircount) = Ir;
disp([?Reduced current ? num2str(Ir)]);
% After the first current, use the previous wavefunction as the
% initial guess. Use the same n, which (at a higher current) will
% give a lower initial guess for the energy.
for levelcount = keeplevels(levelmaxIr, Ir)
disp([?Level ? num2str(levelcount)]);
if Ircount == 1
if nargin == 8
% User supplied energy and wavefunction
solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,...
E0(levelcount+1)/hbar/wp(Ir*Io, Io, Cj) - 0.5, ...
psi0(levelcount+1,:));
elseif nargin == 7
% User supplied energy -- use a random initial wavefunction
solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,...
E0(levelcount+1)/hbar/wp(Ir*Io, Io, Cj) - 0.5);
else
% User didn?t give you anything. Guess the energy and use a
% random psi.
corr = 0.15 - 5 * (1 - Ir - 0.005);
solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,...
levelcount*(1-corr));
end
stuff.params.xleft = solution.x(1);
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stuff.params.dx = solution.dx;
stuff.params.Ngrid = length(solution.x);
else
solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1, ...
n0(levelcount+1), psi(levelcount+1,:));
end
energy = real(solution.E);
gamma = -imag(solution.E) / (hbar/2);
psi(levelcount+1,:) = solution.wavefn;
n0(levelcount+1) = energy / hbar / wp(Ir*Io, Io, Cj) - 0.5;
levstr = num2str(levelcount);
Irstr = num2str(Ircount);
eval([?stuff.energy? levstr ?(? Irstr ?) = energy;?]);
eval([?stuff.gamma? levstr ?(? Irstr ?) = gamma;?]);
eval([?wavefn.level? levstr ?(? Irstr ?,:) = solution.wavefn;?]);
end
if Ircount == 1
Eplot = figure;
end
figure(Eplot);
plotlevels(stuff);
xaxis([stuff.Ir(1) 1]);
shg;
end
stuff.params.stop = clock;
more on
diary off
This is the primary routine that calculates the solutions for a single value of
the bias current, using transmission boundary conditions.
function solution = jjeigentbc(Ib, Io, Cj, nmax, n0, psi0)
% solution = jjeigentbc(Ib, Io, Cj, nmax, n0, psi0)
% Calculates the energy, potential, and wavefunction (on a grid x,
371
% with steps dx) for bias current ?Ib?, critical current ?Io?,
% capacitance ?Cj?, maximum number of levels ?nmax?, and current
% level ?n0? (or the best guess of what it is). ?psi0? is the
% (optional) inital guess for the wavefunction. Uses transmission
% boundary conditions. Results sent back in a structure.
% calls: mj, wp, hbar, jjeigengrid
% Some constants
global hbar;
% Set up a grid to solve Schrodinger?s eq.
[xleft, dx, Ngrid] = jjeigengrid(0.97, 0.999, Io, Cj, nmax);
disp([?xleft = ? num2str(xleft) ? dx = ? num2str(dx) ? ...
Ngrid = ? num2str(Ngrid)]);
% This constant is in front of d2(psi)/dx2. Multiply it over to
% V and E and call them Vp and Ep (p for prime)
m = mj(Cj);
a = 2 * m * (dx / hbar)^2;
Umin = twb(Ib, Io, asin(Ib/Io));
for i = 1 : Ngrid
x(i) = xleft + dx * (i-1);
Utwb(i) = twb(Ib, Io, x(i)) - Umin;
end
Uleft = Utwb(1);
Uright = Utwb(Ngrid);
% The matrix is N-2 x N-2, because the boundary conditions are
% evaluated in the 2 and N-1 equations. Set up H*psi = E*psi.
A(1 : Ngrid-2) = -1;
C(1 : Ngrid-2) = -1;
for i = 1 : Ngrid-2
B(i) = 2 + Utwb(i+1) * a;
end
% Here?s the first guess at the eigenvalue. Start with a random
% wavefunction (if one isn?t provided) and use inverse iteration
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% (Numerical Recipes 11.7) to improve it.
Ep = (n0 + 0.5) * hbar * wp(Ib, Io, Cj) * a;
if nargin == 6
newpsi = psi0(2:end-1);
else
% This is the MATLAB R12 command
% newpsi = random(?unif?, 0, 1, 1, Ngrid-2);
% This is the MATLAB R14 command
newpsi = rand(1, Ngrid-2);
end
newpsi = newpsi / sqrt(sum(newpsi.^2));
% Boundary conditions for first go round.
Btbc = B;
Kleft = sqrt(2 * m * (Uleft - Ep/a)) / hbar;
Btbc(1) = B(1) - exp(-1 * Kleft * dx);
Kright = sqrt(2 * m * (Ep/a - Uright)) / hbar;
Btbc(Ngrid-2) = B(Ngrid-2) - exp(sqrt(-1) * Kright * dx);
% First iterate a couple times, without updating the eigenvalue.
diff = 2; err = 0;
count1 = 0;
while (diff > 1e-6) & (err == 0)
oldpsi = newpsi;
[temppsi, err] = tridiag(A, Btbc - Ep, C, oldpsi);
newpsi = temppsi / sqrt( sum(abs(temppsi).^2) );
diff = max(abs( (abs(newpsi)./abs(oldpsi)).^2 - 1 ));
count1 = count1 + 1;
end
% Now update the energy too
oldEp = Ep;
newEp = oldEp + sum( conj(temppsi) .* oldpsi ) ...
/ sum(abs(temppsi).^2);
diff = 1;
count2 = 0;
373
while((diff > 1e-7) ...
| max(abs( imag(newEp)/imag(oldEp) - 1 )) > 1e-7) & err==0
oldpsi = newpsi;
oldEp = newEp;
Kleft = sqrt(2 * m * (Uleft - oldEp/a)) / hbar;
Btbc(1) = B(1) - exp(-1 * Kleft * dx);
Kright = sqrt(2 * m * (oldEp/a - Uright)) / hbar;
Btbc(Ngrid-2) = B(Ngrid-2) - exp(sqrt(-1) * Kright * dx);
[temppsi, err] = tridiag(A, Btbc - oldEp, C, oldpsi);
newpsi = temppsi / sqrt( sum(abs(temppsi).^2) );
diff = max(abs( (abs(newpsi)./abs(oldpsi)).^2 - 1 ));
newEp = oldEp + sum( conj(temppsi) .* oldpsi ) ...
/ sum(abs(temppsi).^2);
count2 = count2 + 1;
end
% So far, have been normalizing the vector psi. But to make it
% a ?continuous? function on x, do a Riemann sum.
newpsi = -sqrt(-1) * newpsi / sqrt(dx);
wavefn = [newpsi(1)*exp(-1 * Kleft * dx) newpsi ...
newpsi(Ngrid-2)*exp(sqrt(-1) * Kright * dx)];
wavefn = wavefn / sqrt( sum(abs(wavefn).^2) ) / sqrt(dx);
solution.E = newEp/a;
solution.x = x;
solution.Utwb = Utwb;
solution.wavefn = wavefn;
solution.dx = dx;
disp([num2str(count1) ? iterations of first loop; ? ...
num2str(count2) ? iterations of second?]);
This sets up the grid on which the solution is calculated.
function [xleft, dx, Ngrid] = jjeigengrid(Irmin, Irmax, Io, Cj, nmax)
% [xleft, dx, Ngrid] = jjeigengrid(Irmin, Irmax, Io, Cj, nmax)
% This calculates a grid for jjeigentbc. It should select the
% smallest grid compatible for currents between ?Irmin? and ?Irmax?,
374
% critical current ?Io?, capacitance ?Cj?, and maximum quantum level
% ?nmax?. If everything is done on the same grid, then you can take
% inner products and stuff with the wavefunctions later.
% calls: mj, wp, hbar, twb
% Some constants
global hbar;
m = mj(Cj);
% Ideally, you would use the smallest range for a given Irmin/max and
% Cj. However, this is complicated.
% First, you need to find the values of the phase, where the
% washboard hits (again) the local max (to the left) and min (to the
% right) of the first well. The widest range of phase occurs for the
% smallest bias current. Just pick a fixed [0.8, 2.3], which should
% cover down to Ir = 0.95.
% Then, you want enough phase outside of this to capture some
% oscillations (to the right of the well) and the decay (to the
% left). This is set by the constant alpha below. The longest
% spatial scale occurs at the highest current, opposite of the
% previous paragraph -- ignore this. Don?t really know how many of
% these spatial constants to keep. This should be optimized.
wpmin = wp(Irmax*Io, Io, Cj);
alphamin = sqrt(m*wpmin/hbar);
wpmax = wp(Irmin*Io, Io, Cj);
alphamax = sqrt(m*wpmax/hbar);
xleft = 0.8 - 4/alphamin;
xright = 2.3 + 4/alphamin;
% Next get the step size, which is based on the oscillations of the
% highest energy you plan to calculate. These should be evaluated at
% the highest current, where the potential is steep and the energy
% differences are large.
xmin = asin(Irmax);
Umin = twb(Irmax*Io, Io, xmin);
Uleft = twb(Irmax*Io, Io, xleft) - Umin;
Uright = twb(Irmax*Io, Io, xright) - Umin;
Emax = (nmax + 0.5) * hbar * wpmax;
375
lambdal = sqrt(2*m * (Uleft - Emax)) / hbar;
lambdar = sqrt(2*m * (Emax - Uright)) / hbar;
dx = 1 / max([alphamax lambdal lambdar]) / 10;
Ngrid = floor((xright - xleft) / dx) + 1;
Finally, the main M-flles above call several simple routines, given below. In
addition, global variables called hbar and Phio (which, not surprisingly, are equal
to ~ and '0) should be deflned in the workspace.
function levels = keeplevels(levelmaxIr, Ir)
% levels = keeplevels(levelmaxIr, Ir)
% This returns a vector of the levels to keep at a given reduced bias
% current, ?Ir?. ?levelmaxIr?(i) gives the reduced current where the
% (i-1)th state leaves the well (or least where you don?t want it
% anymore). If you should keep it, i-1 is included in ?levels?.
% 0 is the ground state. The number of elements in ?levelmaxIr? sets
% the maximum number of levels to keep.
levels = [];
for i = 1 : length(levelmaxIr)
if Ir <= levelmaxIr(i)
levels = [levels i-1];
end
end
function omegap = wp(Ib, Io, C);
% wp(Ib, Io, C) gives the plasma frequency of a junction
global Phio;
omegap = sqrt(2*pi*Io/C/Phio) .* (1-(Ib./Io).^2).^(1/4);
plotlevels plots the energy levels as the solutions are calculated. Running
the program for a large number of bias currents can be time-taking, so this is a
useful way of spotting trouble early.
376
function plotlevels(eigenstuff)
% plotlevels(eigenstuff)
% This assumes ?eigenstuff? has fields named Ir and energy0, energy1.
colors = ?bgrcmy?;
plotcnt = 0;
fields = fieldnames(eigenstuff);
for i = 1 : length(fields)
if strncmp(fields(i), ?energy?, 6) == 1
data = getfield(eigenstuff, char(fields(i)));
plot(eigenstuff.Ir(1:length(data)), data, ...
colors(mod(plotcnt, 6) + 1));
hold on
plotcnt = plotcnt + 1;
end
end
function xaxis(xbounds)
% xaxis([xmin xmax])
% This replots the current graph, using new x bounds.
graphaxes = axis;
graphaxes(1) = xbounds(1);
graphaxes(2) = xbounds(2);
axis(graphaxes);
function mass = mj(Cj);
% mass = mj(Cj) returns the phase particle mass, given the junction
% capacitance.
% calls: Phio
global Phio;
mass = Cj * (Phio/2/pi)^2;
377
function U = twb(Ib, Io, gamma);
% U = twb(Ib, Io, gamma) returns the tilted washboard potential.
% calls: Phio
global Phio;
U = -Phio/2/pi * (Io * cos(gamma) + Ib * gamma);
B.2 Solutions of the Coupled-Junction Hamiltonian
Equation (2.74) gives the Hamiltonian for two capacitively-coupled current-
biased junctions. Solutions of it can be expressed in terms of the uncoupled junction
basis (which can be calculated with the programs of the previous section). The
Hamiltonian was derived under the assumption that the two junction had the same
capacitance. In the program, the uncoupled states are calculated with potentially
difierent capacitances, while the strength of the coupling depends on the average of
the two capacitances.
function [Nlevel1, EVs, WFs, EVs1, WFs1, EVs2, WFs2] = ...
coupledjj(Io1, Cj1, Io2, Cj2, Cc, Ir1, Ir2, EVs1, EVs2, WFs1, WFs2)
% [Nlevel1, EVs, WFs, EVs1, WFs1, EVs2, WFs2] = ...
% coupledjj(Io1, Cj1, Io2, Cj2, Cc, Ir1, Ir2, EVs1, EVs2, WFs1, WFs2)
% This gives the solutions for two capacitively coupled junctions,
% with critical currents ?Io1? and ?Io2? (in Amps), capacitances
% ?Cj1? and ?Cj2? (in Farads), coupling capacitor ?Cc?, at reduced
% bias currents ?Ir1? and ?Ir2?. All the input parameters are single
% numbers.
%
% ?EVs1?, ?WFs1?, ?EVs2?, and ?WFs2? are optional arguments that give
% the initial guess for the energies and wavefunctions for
% jjeigentbc.m. Pass in both energies or all four.
%
% ?Nlevel1? is the number of levels kept for JJ1. ?EVs? and ?WFs?
% are the complex eigenvalues and wavefunctions (in terms of the
% coupled state basis), both sorted in order of increasing energy.
% The first row is for the ground state. The last four outputs are
378
% for the uncoupled junctions.
%
% calls: hbar, Phio, momentumH, wp
global hbar Phio;
% Get the effective capacitances, from the coupled Hamiltonian.
Ceff1 = Cj1 * (1 + Cc/(Cc+Cj1));
Ceff2 = Cj2 * (1 + Cc/(Cc+Cj2));
% Get the coupling parameter for equivalent junctions; pull out the
% hbars from the momentum terms.
Cavg = (Cj1 + Cj2)/2;
zavg = Cc/(Cc + Cavg);
coupling = -1*hbar^2 * zavg / Cavg / (1+zavg) / (Phio/2/pi)^2;
if nargin == 7
[momH1, EVs1, WFs1] = momentumH(Ir1, Io1, Ceff1);
Nlevel1 = length(EVs1);
[momH2, EVs2, WFs2] = momentumH(Ir2, Io2, Ceff2);
Nlevel2 = length(EVs2);
else
% jjeigentbc wants the level number n, not energy
n0s1 = real(EVs1) / hbar / wp(Ir1*Io1, Io1, Cj1) - 0.5;
n0s2 = real(EVs2) / hbar / wp(Ir2*Io2, Io2, Cj2) - 0.5;
if nargin == 9
[momH1, EVs1, WFs1] = momentumH(Ir1, Io1, Ceff1, n0s1);
[momH2, EVs2, WFs2] = momentumH(Ir2, Io2, Ceff2, n0s2);
else
[momH1, EVs1, WFs1] = momentumH(Ir1, Io1, Ceff1, n0s1, WFs1);
[momH2, EVs2, WFs2] = momentumH(Ir2, Io2, Ceff2, n0s2, WFs2);
end
Nlevel1 = length(EVs1);
Nlevel2 = length(EVs2);
end
% Form the coupled Hamiltonian. The order of the basis vectors
% |jj#1 jj#2>: |0 0>, |0 1>, |0 2>, ..., |0 N2>, |1 0>, |1 1>, ...,
379
% |1 N2>, ..., |N1 N2>. i labels rows; j labels columns.
for i1 = 1 : Nlevel1
for i2 = 1 : Nlevel2
row = (i1-1)*Nlevel2 + i2;
for j1 = 1 : Nlevel1
for j2 = 1 : Nlevel2
col = (j1-1)*Nlevel2 + j2;
coupledH(row,col) = coupling * momH1(i1,j1) * momH2(i2,j2);
if row == col
coupledH(row,row) = coupledH(row,row) + EVs1(i1)+EVs2(i2);
end
end
end
end
end
[WFs, eigenvalm] = eig(coupledH);
for i = 1 : Nlevel1*Nlevel2
EVs(i) = eigenvalm(i,i);
end
energy = real(EVs);
[energy, order] = sort(energy);
EVs = EVs(order);
WFs = WFs(:, order);
WFs = WFs?;
coupledjj calls momentumH, which forms the momentum coupling term in the
Hamiltonian. It calls a function DU, which just returns the tilted washboard barrier.
function [momH, EVs, WFs] = momentumH(Ir, Io, Cj, n0s, WF0s)
% [momH, EVs, WFs] = momentumH(Ir, Io, Cj, n0s, WF0s)
% This calculates the matrix representation ?momH? of the momentum
% operator for a single junction, (apart from h/i) at a reduced
% current ?Ir?. It also returns the complex eigenvalues ?EVs? and
% wavefunctions ?WFs? for all the levels in the well (where the first
% row is the ground state). The junction critical current ?Io? is in
% Amps and its capacitance ?Cj? is in Farads.
%
380
% ?n0s? and ?WF0s? are optional arguments that specify initial
% guesses for the energy level number and wavefunction (for
% jjeigentbc). ?n0s? should be a vector, where the first entry is
% for the ground state. If you want to pass in ?WF0s?, then both
% parameters must be passed in. ?WF0s? should be 2D, where the first
% row is the ground state. They both must be long enough, so this
% only works if you are doing a positive sweep and the number of
% levels can only decrease.
%
% calls: hbar, Phio, DU, wp, jjeigentbc
global hbar Phio;
m = Cj * (Phio/2/pi)^2;
% Overestimate the number of levels in the well, to give a good basis
Nlevel = floor( DU(Ir*Io, Io) / hbar / wp(Ir*Io, Io, Cj) ) + 2;
% Get the energy and wavefunction for each level, with an initial
% energy guess slightly smaller than harmonic states.
corr = 0.10 - 10*(1-Ir-0.007);
for i = 1 : Nlevel
disp([?Level ? num2str(i) ? of ? num2str(Nlevel)]);
if nargin == 3
jjsoln = jjeigentbc(Ir*Io, Io, Cj, Nlevel, (i-1)*(1-corr));
elseif nargin == 4
jjsoln = jjeigentbc(Ir*Io, Io, Cj, Nlevel, n0s(i));
elseif nargin == 5
jjsoln = jjeigentbc(Ir*Io, Io, Cj, Nlevel, n0s(i), WF0s(i, :));
end
U = jjsoln.Utwb;
EVs(i) = jjsoln.E;
Ngrid = length(jjsoln.wavefn);
WFs(i, 1:Ngrid) = jjsoln.wavefn;
dx = jjsoln.dx;
end
381
% Calculate dpsi/dx. The left BC is exponential decay; the right is
% plane wave -- jjeigentbc calculates the wavefunction past the right
% side of the well.
dWFs = WFs;
for i = 1 : Nlevel
dWFs(i, 2:Ngrid-1) = (WFs(i, 3:Ngrid) - WFs(i, 1:Ngrid-2)) / 2/dx;
E = real(EVs(i));
lambdal = sqrt(2*m * (U(1) - E)) / hbar;
dWFs(i,1) = dWFs(i,2) * exp(lambdal*dx);
lambdar = sqrt(2*m * (E - U(Ngrid))) / hbar;
dWFs(i, Ngrid) = dWFs(i,Ngrid-1) * ...
( cos(lambdar*dx) + sqrt(-1)*sin(lambdar*dx) );
end
% Integrate psistar * dpsi
for i = 1 : Nlevel
for j = 1 : i
momH(i,j) = sum( conj(WFs(i,1:Ngrid)) .* dWFs(j,1:Ngrid) ) * dx;
if i ~= j
momH(j,i) = conj(momH(i,j));
end
end
end
382
Appendix C
Three-Level Rotating Wave Approximation
This appendix gives the Hamiltonian of a generic three-level system in the
rotating wave approximation. Details of the derivation are due to Fred Strauch
[2,83].
In matrix form, the Hamiltonian for a three-level system (with orthonormal
basis j0i, j1i, j2i) under an oscillatory perturbation of angular frequency !rf can be
written as
H =
0
BB
BB
@
E0 0 0
0 E1 0
0 0 E2
1
CC
CC
A
+Acos!rft
0
BB
BB
@
y0;0 y0;1 y0;2
y?0;1 y1;1 y1;2
y?0;2 y?1;2 y2;2
1
CC
CC
A
; (C.1)
where the position matrix elements yn;m = hnj^yjmi of the perturbation are di-
mensionless. Let ? = c0j0i + c1j1i + c2j2i be a solution of Schr?odinger?s equa-
tion, H? = i~ _?. Thus for example, i~_c0 = H0;0c0 + H0;1c1 + H0;2c2. Here,
Hn;m = hnjHjmi; for example, H0;0 = E0 +Ay0;0 cos!rft.
Consider the time-dependent transformation eci = ei`i(t)ci, where i = 1;2;3.
Then the state vector e? =ec0j0i+ec1j1i+ec2j2i will be a solution of eHe? = i~_e?. The
coe?cients of j0i in the modifled Schr?odinger?s equation satisfy
i~dec0dt = ?~d`0dt ei`0 c0 +i~ei`0dc0dt (C.2)
=
 
H0;0 ?~d`0dt
?
ec0 +H0;1ec1 ei(`0?`1) +H0;2ec2 ei(`0?`2); (C.3)
from which the identiflcations eH0;0 = H0;0 ?~_`0, eH0;1 = H0;1 ei(`0?`1), eH0;2 =
H0;2 ei(`0?`2) can be made. The remaining elements of eH can be found by examining
383
the coe?cients of j1i and j2i: eH1;1 = H1;1 ?~_`1, eH1;2 = H1;2 ei(`1?`2), eH2;2 =
H2;2 ?~_`2.
The values of `i (t) are chosen to give the rotating frame of interest. For a
single photon transition between j0i and j1i, the appropriate choices are
`0 = E0~ t+ Ay0;0~!
rf
sin!rft (C.4)
`1 =
 E
0
~ +!rf
?
t+ Ay1;1~!
rf
sin!rft (C.5)
`2 =
 E
0
~ +2!rf
?
t+ Ay2;2~!
rf
sin!rft: (C.6)
Then, the time dependence of the diagonal elements of eH vanishes. Each
ofi-diagonal matrix element will contain the exponential of a sine, which can be
expanded as sum of Bessel functions of the flrst kind using the identity
eixsin =
+1X
n=?1
Jn (x)ein : (C.7)
So the matrix element responsible for the 0 ! 1 transition becomes
eH0;1 = Ay0;1 cos!rft e?i!rft exp
?
iA(y0;0 ?y1;1)~!
rf
sin!rft
?
(C.8)
= Ay0;1 cos!rft e?i!rft
+1X
n=?1
Jn
 A(y
0;0 ?y1;1)
~!rf
?
ein!rft (C.9)
? Ay0;12
?
J0
 A(y
0;0 ?y1;1)
~!rf
?
+J2
 A(y
0;0 ?y1;1)
~!rf
??
; (C.10)
where in the last expression, only terms in the inflnite sum with no time dependence
have been kept. This amounts to making the rotating wave approximation. The
remaining elements of eH can be found in a similar way, resulting in the Hamiltonian
given in Eq. (3.14) through Eq. (3.17).
384
Appendix D
Dynamical Evolution Matrices
D.1 Density Matrix
The evolution matrix for the Liouville-von Neumann equation in Eq. (3.68),
for the transitions that we are interested in, can be expressed as
P = ?iE ?G+D?iM cos(!rft); (D.1)
where the structure of each of the matrices is described next.
The convention I will follow is that the elements ?ij of the density matrix
are ordered in a vector, such that i is flxed while cycling through j. For example,
? = [?00; ?01; ?02; ?10; ?11; ?12; ?20; ?21; ?22] for N = 3. Then, ?ij will be element
number fi ? (ij)N +1 in this vector, where the number in parentheses is expressed
in base N and N is the number of levels in the system.1 For N = 3, ?21 is element
number (2?3+1?1+1) = 8.
An element of the energy matrix E depends on the static Hamiltonian and
can be found directly from Eq. (3.35) as i~_?ij = Pm?H0im?mj ??imH0mj?. H0 is
diagonal, so each sum has only one element: i~_?ij = Ei?ij ??ijEj = (Ei ?Ej)?ij.
Therefore, E is diagonal, with Efifi = (!i ?!j).
In the simple way that we have decided to include tunneling, the matrix G is
also diagonal; Gfifi = (?i +?j)=2.
The matrix D accounts for dissipation and decoherence. Each is described by
N (N ?1)=2 time constants, which represent transitions between pairs of states.
As in the previous section, decoherence is assumed to make a diagonal contribution
1In this appendix, Roman letters are quantum state indices and run from 0 to N ? 1, while
Greek letters index the vectors and matrices in Eq. (3.68) and run between 1 and N2.
385
Dfifi = ?1=Tij2 , for i 6= j. Typically, I set all of the times Tij2 to the same value T2,
as I had no physical reason not to.
On the assumption that dissipation afiects only state populations ?ii, only the
elements Dfl? are non-zero, where fl = (ii)N + 1 and ? = (jj)N + 1. For fl < ?,
Dfl? = Wtij = ?ij =?1?e?~!ij=kBT?, to account for gains in jii due to dissipation
from all higher states jji. The spontaneous emission rate ?ij and total thermal rate
Wtij are deflned in x3.4. For fl > ?, Dfl? = Wtij = ?ji =?e~!ji=kBT ?1? to account for
gains in jii due to thermal excitations from all lower states jji. The diagonal term
Dflfl = ?P?6=fl D?fl to account for losses in jii due to both up and down transitions.
The microwave drive is characterized by (N +1)N=2 matrix elements between
each pair of quantum states of the form ?ij ? hij I?w jji=~, with ?ij = ??ji. The
evolution matrix M is largely banded, but I found no obvious pattern to simply
describe it. There are N2 equations of the form _?ij = ?i(?im?mj ??mj?im), which
can be used to flll out M.
For N = 3, the matrices are
E =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
@
0 0 0 0 0 0 0 0 0
0 ?!01 0 0 0 0 0 0 0
0 0 ?!02 0 0 0 0 0 0
0 0 0 !01 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 ?!12 0 0 0
0 0 0 0 0 0 !02 0 0
0 0 0 0 0 0 0 !12 0
0 0 0 0 0 0 0 0 0
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
A
; (D.2)
386
G =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
@
?0 0 0 0 0 0 0 0 0
0 ?0+?12 0 0 0 0 0 0 0
0 0 ?0+?22 0 0 0 0 0 0
0 0 0 ?0+?12 0 0 0 0 0
0 0 0 0 ?1 0 0 0 0
0 0 0 0 0 ?1+?22 0 0 0
0 0 0 0 0 0 ?0+?22 0 0
0 0 0 0 0 0 0 ?1+?22 0
0 0 0 0 0 0 0 0 ?2
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
A
; (D.3)
D =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
B@
?Wtk0 0 0 0 Wt01 0 0 0 Wt02
0 ? 1T01
2
0 0 0 0 0 0 0
0 0 ? 1T02
2
0 0 0 0 0 0
0 0 0 ? 1T01
2
0 0 0 0 0
Wt10 0 0 0 ?Wtk1 0 0 0 Wt12
0 0 0 0 0 ? 1T12
2
0 0 0
0 0 0 0 0 0 ? 1T02
2
0 0
0 0 0 0 0 0 0 ? 1T12
2
0
Wt20 0 0 0 Wt21 0 0 0 ?Wtk2
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CA
;
(D.4)
where Wtk0 ? Wt10 +Wt20, Wtk1 ? Wt01 +Wt21, Wtk2 ? Wt02 +Wt12, and
387
M =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
@
0 ???01 ???02 ?01 0 0 ?02 0 0
??01 0 ???12 0 ?01 0 0 ?02 0
??02 ??12 0 0 0 ?01 0 0 ?02
??01 0 0 0 ??01 ???02 ?12 0 0
0 ??01 0 ??01 0 ???12 0 ?12 0
0 0 ??01 ??02 ??12 0 0 0 ?12
??02 0 0 ??12 0 0 0 ???01 ???02
0 ??02 0 0 ??12 0 ??01 0 ???12
0 0 ??02 0 0 ??12 ??02 ??12 0
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
A
: (D.5)
388
D.2 Master Equations
The evolution matrix p in Eq. (7.9) for solving the master equations is quite
similar to P for the density matrix, except there are no coherence terms. As I only
considered dissipation and tunneling with the master equations,
p = ?g +d: (D.6)
The tunneling matrix g is purely diagonal with gii = ?i. The ofi-diagonal
elements of the dissipation matrix d are dij = Wtij, where these thermal rates are
given by Eqs. (3.26) and (3.27), and the diagonal elements are dii = ?Pj6=i dji.
For N = 3, the matrices are
g =
0
BB
BB
@
?0 0 0
0 ?1 0
0 0 ?2
1
CC
CC
A
(D.7)
and
d =
0
BB
BB
@
?Wt10 ?Wt20 Wt01 Wt02
Wt10 ?Wt01 ?Wt21 Wt12
Wt20 Wt21 ?Wt02 ?Wt12
1
CC
CC
A
: (D.8)
389
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