ABSTRACT
Title of dissertation: TOWARD THE FLUXONIUM
QUANTUM PROCESSOR
Long Bao Nguyen, Doctor of Philosophy, 2020
Dissertation directed by: Professor Vladimir Eduardovich Manucharyan
Department of Physics
This thesis reports recent achievements toward scalable quantum computing with flux-
onium, a superconducting artificial atom with rich energy spectrum and selection rules
similar to those found in natural atoms. We show how such spectral properties can be
harnessed to protect the qubit from energy relaxation and dephasing. At half integer flux
quantum bias, we show that fluxonium?s |0? ? |1? qubit transition has high coherence by
design, with T1, T2 ? 500 ?s in one device, the highest reported in superconducting circuits
so far. Yet, the qubit exhibits the same level of addressability found in more conventional su-
perconducting qubits (Tgate < 50 ns). In addition, a controlled-Z gate can be implemented
by sending a short 2?-pulse at a frequency near the |1? ? |2? transition of the target qubit.
Preliminary results suggest that this gate can be used to entangle two fluxonium qubits with
high fidelity. We also discuss experimental techniques employed to characterize the qubits,
and present a perspective on future fluxonium-based quantum technologies.
TOWARD THE FLUXONIUM
QUANTUM PROCESSOR
by
Long Bao Nguyen
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Dissertation Committee:
Professor Vladimir E. Manucharyan, Advisor
Professor Thomas M. Antonsen, Co-Advisor
Professor Steven M. Anlage
Professor Cheng Gong
Professor Frederick C. Wellstood
Professor Benjamin S. Palmer
Professor Andrew M. Childs, Dean?s Representative
?c Copyright by
Long Bao Nguyen
2020
Acknowledgement
Physics isn?t the most important thing. Love is.
Richard P. Feynman
I would like to begin by expressing gratitude to my advisor, Prof. Vladimir Manucharyan,
for giving me the opportunity to learn about quantum computing and superconducting
circuits under his guidance. I still remember the first time I visited the lab, when it was an
empty space surrounded by four walls, like it was only yesterday. I also recall the first time
I came to see Vlad, he stopped whatever he was doing and started deriving the fluxonium
Hamiltonian on the white board. Together we began building the lab from the ground
up, overcoming broken devices, short coherence times, bad equipment, and administrative
hassles. Being more of a brother figure than a boss, Vlad has also given me useful advice
on teamwork and giving talks.
I owe my enthusiasm in scientific research to my academic advisor, Prof. Thomas An-
tonsen Jr., who also supervised my master thesis work. Tom not only inspires me with his
combination of ingenuity and patience, but also encourages me to get out of my comfort
zone and live my dream. I am especially fond of the few times we played tennis together. I
have not only learned how to solve problems from first principles, but also how to survive
academic life from him. The love he shows toward his students is unparalleled, either in
this universe or another.
I thank Prof. Steven Anlage for teaching me everything I know about superconductivity.
Our characterization of microwave components up to 50 GHz was done using the vector
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network analyzer in his lab. It?s an interesting coincidence that many of my friends from
the same cohort decided to work with Steve, and our discussions throughout the years have
inspired me to take an interest in superconductivity research.
I thank the other committee members, Prof. Benjamin Palmer, Prof. Frederick Wellstood,
Prof. Cheng Gong, and Prof. Andrew Childs for overseeing this dissertation work and the
defense. Comments and suggestions from Ben and Fred helped me tremendously in editing
the manuscript.
The person whom I spent the most amount of time working with is undoubtedly Prof.
Yen-Hsiang Lin, who taught me everything I know about cryogenic engineering, machining,
overcoming difficult people, Asian food, and raising kids. I appreciate his patience and
easy-going temperament. Whenever there was a problem with the fridge, I could count on
Yen-Hsiang to come to the lab and fix it, be it in the middle of a winter storm or at 3 AM in
the morning. Detecting leaks and chatting about life during those times were memorable.
Yen-Hsiang also influenced me with the concept of academic brother, which we now use to
refer to each other.
I am indebted to my academic twin sister Natalia Pankratova (ne?e Solovyeva) for her
love and support. Natalia not only works hard to ensure a friendly working environment,
she also goes the distance to make sure I live a life outside the lab. Thanks to her, I had
the chance to enjoy the cherry blossoms, go to the beach, and attend many school events
with other fellow graduate students. I am sure that the series of birthday parties Natalia
organizes will leave every lab member with happy memories. Our conversations throughout
the years have helped me manage the stress and beyond.
I thank Jonathan San Miguel for making the first half of my time in the lab so enjoyable.
Jon is not only a genius, but also a fun guy to be around with. Every single time I exchanged
an idea with Jon, it bounced back being a better sounding idea. Outside the lab, we became
gym partners and good friends. I also owe Jon my excitement and enthusiasm in coding as
well as in solving riddles and puzzles.
iv
Although he spent only one year at the University of Maryland, Dr. Nathanae?l Cottet
helped us tremendously with our understanding on quantum measurement and detection.
The fluorescence experiment he started has given us the opportunity to learn so much
about open quantum systems. I thank him for all the illuminating discussions about circuits
and readout, and for letting me sleep on the bed during our March Meeting trip. I also
appreciate Nat?s friendship and encouragement throughout the years.
We could measure so many fluxonium devices only because of the hard work from Aaron
Somoroff. Aaron has tirelessly fabricated device after device, made dose test after dose test,
to come up with the right qubits. I also thank Aaron for sharing his life experience outside
the lab, and for cheering me up in time of duress. Partnering up with Aaron and learning
fab from him in the final stretch of my graduate student life was fun and memorable.
The recent achievements in fluorescence shelving readout came from the hard work of
Haonan Xiong. His willingness to learn and discuss science in details is inspiring. I also
thank Haonan for simulating the new cavity and waveguide designs. Haonan took no time
learning about fluxonium, and I have no doubt he will achieve great success with future
fluxonium experiments.
I haven?t been fortunate enough to work in the same project with Dr. Roman Kuzmin,
but his methodological approaches, persistence, and discipline in scientific research have
never failed to impress me. For every problem that I have ever discussed with Roman, he
could always come up with a foolproof solution. Despite his intense focus on research, he
never refused to give us a hand when other problems arose in the lab. I especially miss his
rare but ingenious jokes. In short, Roman is the scientist that I aspire to become one day
I thank Dr. Ivan Pechenezhskiy for his encouragement inside and outside the lab. I
would never have the courage to ski black diamond trails without his instructions and
companionship. Ivan not only taught me so many things about circuits, but also motivated
me to code cleaner python scripts and write better job applications. I owe him many great
advice on financial investments, future career paths, and professional life. Our discussions
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about big pictures in research are especially helpful and memorable.
For a long time, optimizing the readout was not our primary concern, and we only
began to realize its importance at the start of the two-qubit entangling gate experiment.
Fortunately, Dr. Quentin Ficheux arrived and started a revolution on our readout setup just
on time. He has also helped us tremendously with his experience in cQED. I appreciate
the time he spent solving problems from first principles and motivated us to be rigorous.
Quentin has definitely made me become a better scientist.
I haven?t had the chance to work exclusively on the same project with everyone in
the lab, but I appreciate the camaraderie from Nicholas Grabon, Raymond Mencia, Nitish
Mehta, Christiane Ebongue, and Hanho Lee. I will remember dearly the early morning trip
to Dunkin Donuts with Nick, the cookout at Nitish and Ray?s house, going for lunch with
Chris, and discussion about K-pop with Hanho.
Outside the lab, I would like to first thank our theory collaborators Dr. Konstantin
Nesterov and Prof. Maxim Vavilov for teaching me the physics behind multi-fluxonium sys-
tems. I thank Prof. Chen Wang and his students, Dario Rosenstock and Ebru Dogan, for
allowing me to visit their nicely setup lab and learn from them. I am grateful to Prof. Raja-
mani Vijayaraghavan for helping me understand the readout and initialization techniques
in multi-qubit systems, and for discussing the details of our experiments. I have come to
enjoy Vijay?s wonderful insight in life and physics, and I hope to learn more from him in
the future. I thank Prof. William Oliver for giving me the chance to visit his lab and have
fruitful discussion with his group during the crucial stage of the two-qubit project. I thank
Prof. Michel Devoret for his feedback on my talks.
I thank the administrative staff in both the ECE and Physics departments for their help
and assistance throughout the years. From the ECE graduate office, I thank Melanie Prange,
Heather Stewart, Dorothea Brosius, Tracy Chung, Vivian Lu, Bill Churma, and Maria Hoo.
From the Physics department, I thank Lorraine DeSalvo, Paulina Alejandro, Pauline Rirk-
sopa, Margaret Lukomska, Melissa Britton, and Kelly Phillips.
vi
Without my teachers? endless encouragement and belief in me, I would never be able
to get out of my comfort zone and start living the dream. I am especially grateful to Ms.
Huong, Ms. Loan, Mr. Thanh, and Mr. Nghe for their guidance and advice, both inside
and outside the classroom. I thank Prof. Lawrence Overzet and Prof. Daisuke Ogawa for
patiently introducing me to experimental science. I am glad to have received their love
and encouragement despite the fact that I have never been the brightest or most obedient
student in the class. The devotion they give to generations after generations of students has
never failed to inspire me.
The old friends still willing to meetup and listen to my old jokes have made graduate
school more tolerable. From my school days, I thank Khoa, Tung, Nha, Long, Khang, and
Tien for living the past, present, and future all at once with me every time we meet. I
thank Misa for all her love and support when she was alive. Our conversations throughout
the years, from the very first one in the exam room to the very last one in the hospital,
were encouraging and inspiring. Her positive attitude in life was a tour de force that always
motivates me to go further and do better in my ventures. I thank Dai and Kim for dissipating
my nostalgia and creating important retrospection points in my life. From my college years,
I thank Tra and Ngan for visiting and keeping me grounded with their recommendations of
fun things to do.
I am fortunate to have met many amazing friends in graduate school. I especially thank
Dr. Hyun Jung for introducing me to the Korean study group who helped me survive the
qualifying exam and the brutal first year, Dr. Tao Gong for his friendship and inspiration,
Dr. Aydin Cem Keser for motivating me to learn quantum physics and introducing me to
Vlad, Dr. Dmitry Kashin for getting me into distance running, Dr. Tao Cai and Dr. Dakang
Ma for sharing their interest in tennis and photography, and Dr. Setiawan Wenming for
introducing me to the world of Majorana physics. I thank the group of friends including
Monica Gutierrez, Dr. Vikram Un, Dr. Elizabeth Tennyson, Sarthak Subhankar, Sabyasachi
Barik, and Swarnav Banik for hanging out and diversifying my graduate school experience.
vii
Most importantly, I would like to thank my family, especially my mom Loan and my
great-grandma Xinh, for their endless love and encouragement, which have allowed my
curiosity and creativity to thrive. I thank my cousin Anna for visiting and checking on my
well-being. Finally, for accompanying me on this journey, I thank Anna Fin, the love of my
life and the apple of my eye.
Contents
Page
COPYRIGHT i
ACKNOWLEDGEMENT ii
CONTENTS viii
LIST OF FIGURES xiii
LIST OF TABLES xvii
NOMENCLATURE xviii
1 INTRODUCTION 1
1.1 The Second Quantum Revolution . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fluxonium Qubit: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 13
2.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Quantum Evolution and Gates . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
viii
CONTENTS ix
2.2 Spin and Mass On A Spring . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 The Spin System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 The Field Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Coupled Spin-Spring System . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 The Lindblad Master Equation . . . . . . . . . . . . . . . . . . . . . . 43
3 SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 51
3.1 Constructing Superconducting Circuits . . . . . . . . . . . . . . . . . . . . 52
3.1.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 Circuit Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.3 Josephson Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Decoherence in Superconducting Circuits . . . . . . . . . . . . . . . . . . 59
3.2.1 Energy Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Pure Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Quantum Gates and Readout . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.1 Qubit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 FLUXONIUM THEORY 76
4.1 Spectrum and Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.1 Fluxonium with a Single Small Junction . . . . . . . . . . . . . . . . 76
4.1.2 Split-Junction Fluxonium . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Decoherence Mechanisms in Fluxonium . . . . . . . . . . . . . . . . . . . 85
4.2.1 Dielectric Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.2 Stimulated Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.3 Quasiparticle Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 91
CONTENTS x
4.2.4 Thermal Photon Dephasing . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Fluxonium Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Multi-Fluxonium Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4.1 Capacitively Coupled Fluxoniums . . . . . . . . . . . . . . . . . . . . 101
4.4.2 Inductively Coupled Fluxoniums . . . . . . . . . . . . . . . . . . . . 103
4.4.3 Microwave-Activated Controlled-Z Entangling Gate . . . . . . . . . . 105
5 EXPERIMENTAL TECHNIQUES 108
5.1 Cryogenic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1 Material Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.2 Shields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.1.3 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Cavity Simulation and Design . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.1 Resonant Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.2 Cavity-Qubit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.3 External Coupling to Cavity and Qubit . . . . . . . . . . . . . . . . . 124
5.2.4 Cavity Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Microwave Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.1 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.2 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.3 Other Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.3.4 Data Acquisition Software . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4.2 Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6 FLUXONIUM EXPERIMENTS 145
6.1 Cavity Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
CONTENTS xi
6.2 Selection Rules Engineering in Fluxonium . . . . . . . . . . . . . . . . . . 149
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.2 Forbidden Transitions in Fluxonium . . . . . . . . . . . . . . . . . . . 150
6.2.3 Spectral Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.4 Demonstration of Protection . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.5 Metastable Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.3 High Coherence Fluxonium Qubits . . . . . . . . . . . . . . . . . . . . . . 159
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3.3 Discussion of Decoherence Channels . . . . . . . . . . . . . . . . . . 166
6.4 Microwave-Activated Controlled-Z Entangling Gate . . . . . . . . . . . . 171
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.4.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4.3 Single-Shot Joint Readout . . . . . . . . . . . . . . . . . . . . . . . . 174
6.4.4 Single-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.5 Two-Qubit Controlled-Z Gate . . . . . . . . . . . . . . . . . . . . . . 180
6.5 Tune-up Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.5.1 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.5.2 Time Domain Measurements . . . . . . . . . . . . . . . . . . . . . . 191
6.5.3 Preliminary Trace of Quasiparticle . . . . . . . . . . . . . . . . . . . 196
6.5.4 Quantum Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.5.5 Randomized Benchmarking . . . . . . . . . . . . . . . . . . . . . . . 202
7 SUMMARY AND OUTLOOK 205
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.1.1 Selection Rules Engineering . . . . . . . . . . . . . . . . . . . . . . . 205
7.1.2 High Coherence Fluxonium . . . . . . . . . . . . . . . . . . . . . . . 206
CONTENTS xii
7.1.3 Microwave-Activated Two-Qubit Gate . . . . . . . . . . . . . . . . . 208
7.2 Quantum Computing with Fluxoniums . . . . . . . . . . . . . . . . . . . . 208
7.2.1 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2.2 Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2.3 Dispersive Qubit-Cavity Interaction . . . . . . . . . . . . . . . . . . . 210
7.2.4 Flux-Controlled Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2.5 Fixed-Frequency Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2.6 Quantum Adiabatic Optimization . . . . . . . . . . . . . . . . . . . . 211
7.2.7 Optimal Qubit Frequency . . . . . . . . . . . . . . . . . . . . . . . . 212
APPENDICES 214
A NOISE AND ENERGY RELAXATION 215
A.1 Noise Correlation and Spectral Density . . . . . . . . . . . . . . . . . . . 215
A.2 Fermi?s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B EFFECTIVE JOSEPHSON ENERGY IN SQUID 225
C COUPLED CIRCUITS HAMILTONIANS 229
C.1 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
C.2 Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
BIBLIOGRAPHY 233
List of Figures
Figure 1.1 Coherence result highlight . . . . . . . . . . . . . . . . . . . . . . . . 10
Figure 1.2 Two-qubit gate result highlight . . . . . . . . . . . . . . . . . . . . . 11
Figure 2.1 Bloch sphere visualization . . . . . . . . . . . . . . . . . . . . . . . . 15
Figure 2.2 Quantum circuit to create Bell states using CNOT gate . . . . . . . . 23
Figure 2.3 Quantum circuit to create Bell states using CZ gate . . . . . . . . . . 25
Figure 2.4 Qubit rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 2.5 Harmonic oscillator concepts . . . . . . . . . . . . . . . . . . . . . . 31
Figure 2.6 Fock state representation of coherent state . . . . . . . . . . . . . . . 32
Figure 2.7 Visualizations of single-qubit density matrices . . . . . . . . . . . . . 38
Figure 2.8 Visualizations of two-qubit density matrices . . . . . . . . . . . . . . 39
Figure 2.9 Gallery of Hinton diagrams for Bell states . . . . . . . . . . . . . . . 40
Figure 2.10 Decoherence processes . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 2.11 ? matrices for single-qubit gates . . . . . . . . . . . . . . . . . . . . . 49
Figure 2.12 ? matrices for two-qubit gates . . . . . . . . . . . . . . . . . . . . . . 50
Figure 3.1 LC circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 3.2 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Figure 3.3 Window function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Figure 3.4 Exponential versus gaussian decay . . . . . . . . . . . . . . . . . . . 69
Figure 3.5 Effective thermal photon number . . . . . . . . . . . . . . . . . . . . 72
Figure 3.6 Dispersive readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xiii
LIST OF FIGURES xiv
Figure 4.1 Eigenenergies and potential energy for different EJ values . . . . . . 78
Figure 4.2 Eigenenergies and potential energy for different EL values . . . . . . 79
Figure 4.3 Eigenenergies and potential energy for different EC values . . . . . . 80
Figure 4.4 Plasmon and fluxon transitions . . . . . . . . . . . . . . . . . . . . . 81
Figure 4.5 Simulations of matrix elements . . . . . . . . . . . . . . . . . . . . . 82
Figure 4.6 Split-junction fluxonium circuit . . . . . . . . . . . . . . . . . . . . . 83
Figure 4.7 Split-junction fluxonium spectrum . . . . . . . . . . . . . . . . . . . . 84
Figure 4.8 Split-junction fluxonium matrix elements . . . . . . . . . . . . . . . . 85
Figure 4.9 Simulation of T1 limited by dielectric loss . . . . . . . . . . . . . . . . 87
Figure 4.10 Modeling the shunting capacitance of the array . . . . . . . . . . . . 88
Figure 4.11 Effect of temperature on energy relaxation . . . . . . . . . . . . . . . 90
Figure 4.12 Simulation of T1 limited by quasiparticle . . . . . . . . . . . . . . . . 93
Figure 4.13 Matrix elements for quasiparticle relaxation in a split-junction device 94
Figure 4.14 Simulation of T2 limited by thermal photon in cavity . . . . . . . . . 95
Figure 4.15 Simulation of dephasing by photons occupying collective modes . . . 99
Figure 4.16 Simulation of dispersive shift . . . . . . . . . . . . . . . . . . . . . . 100
Figure 4.17 Capacitively coupled fluxoniums . . . . . . . . . . . . . . . . . . . . 102
Figure 4.18 Inductively coupled fluxoniums . . . . . . . . . . . . . . . . . . . . . 104
Figure 4.19 Microwave-activated controlled-Z gate . . . . . . . . . . . . . . . . . 106
Figure 5.1 Cryogenic setup for fluxonium experiments . . . . . . . . . . . . . . 113
Figure 5.2 Performance of cryogenic components . . . . . . . . . . . . . . . . . 116
Figure 5.3 Cryogenic components . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Figure 5.4 Refrigerator input-output transmission . . . . . . . . . . . . . . . . . 119
Figure 5.5 Electric field density of resonant modes in a rectangular cavity . . . . 121
Figure 5.6 Simulation of cavity-qubit coupling . . . . . . . . . . . . . . . . . . . 123
Figure 5.7 Simulation of external coupling to cavity and qubit . . . . . . . . . . 126
LIST OF FIGURES xv
Figure 5.8 Simulation of cavity reflection . . . . . . . . . . . . . . . . . . . . . . 127
Figure 5.9 3D cavity designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure 5.10 Room temperature electronics and microwave lines setup . . . . . . . 131
Figure 5.11 Frequency domain pulse testing . . . . . . . . . . . . . . . . . . . . . 134
Figure 5.12 Time domain pulse testing . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 5.13 Fabrication facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 6.1 1-port cavity reflection measurement . . . . . . . . . . . . . . . . . . 146
Figure 6.2 2-port cavity transmission measurement . . . . . . . . . . . . . . . . 147
Figure 6.3 Change of internal quality factor . . . . . . . . . . . . . . . . . . . . 148
Figure 6.4 Idea for selection rules engineering . . . . . . . . . . . . . . . . . . . 151
Figure 6.5 Spectrum of a split-junction fluxonium . . . . . . . . . . . . . . . . . 153
Figure 6.6 Data around the anti-crossing at 38 mA . . . . . . . . . . . . . . . . . 155
Figure 6.7 Demonstration of protection . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 6.8 Top spectrum of split-junction device . . . . . . . . . . . . . . . . . . 158
Figure 6.9 Metastable regime spectroscopy . . . . . . . . . . . . . . . . . . . . . 159
Figure 6.10 Description of high coherence fluxonium experiment . . . . . . . . . 160
Figure 6.11 Two-tone spectroscopy of a fluxonium qubit . . . . . . . . . . . . . . 163
Figure 6.12 Relaxation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Figure 6.13 Coherence time data . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Figure 6.14 Two-qubit device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Figure 6.15 Spectrum of two capacitively coupled fluxonium qubits . . . . . . . . 175
Figure 6.16 Single-shot joint readout of qubits? thermal states . . . . . . . . . . . 177
Figure 6.17 Initialization by preselection . . . . . . . . . . . . . . . . . . . . . . . 177
Figure 6.18 Control of low frequency qubit transitions . . . . . . . . . . . . . . . 179
Figure 6.19 Benchmarking single-qubit gates . . . . . . . . . . . . . . . . . . . . 181
Figure 6.20 Entangling operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
LIST OF FIGURES xvi
Figure 6.21 Benchmarking two-qubit gate . . . . . . . . . . . . . . . . . . . . . . 184
Figure 6.22 One-tone versus power . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Figure 6.23 One-tone versus flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Figure 6.24 One-tone spectrum of the split-junction device . . . . . . . . . . . . . 188
Figure 6.25 Two-tone spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Figure 6.26 AC Stark shift in spectral domain . . . . . . . . . . . . . . . . . . . . 191
Figure 6.27 Rabi measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Figure 6.28 Ramsey measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Figure 6.29 T1 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Figure 6.30 T echo2 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Figure 6.31 Possible effect from quasiparticles . . . . . . . . . . . . . . . . . . . . 197
Figure A.1 Voltage noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Figure A.2 Circuit dissipation model . . . . . . . . . . . . . . . . . . . . . . . . . 222
Figure C.1 Coupled circuits diagrams . . . . . . . . . . . . . . . . . . . . . . . . 231
List of Tables
Table 2.1 Qubit rotations and expectation values along Z . . . . . . . . . . . . . 20
Table 2.2 Input-output in Bell states circuit using CNOT gate . . . . . . . . . . . 24
Table 2.3 Input-output in Bell states circuit using CZ gate . . . . . . . . . . . . . 25
Table 4.1 Parameters for collective modes simulation . . . . . . . . . . . . . . . 97
Table 5.1 Summary of cavity features . . . . . . . . . . . . . . . . . . . . . . . . 130
Table 6.1 Summary of device parameters . . . . . . . . . . . . . . . . . . . . . . 167
Table 6.2 Two-qubit device parameters . . . . . . . . . . . . . . . . . . . . . . . 174
Table 6.3 Charge matrix elements for computational transitions . . . . . . . . . 174
Table 6.4 Single-qubit gate fidelities . . . . . . . . . . . . . . . . . . . . . . . . . 180
Table 6.5 Gate sequence for single-qubit state tomography . . . . . . . . . . . . 199
Table 6.6 Gate sequence for two-qubit state tomography . . . . . . . . . . . . . 200
xvii
Nomenclatures
Abbreviations
ADC Analog to digital converter
AWG Arbitrary waveform generator
BCS Bardeen-Cooper-Schrieffer
CNOT Controlled-NOT gate
CP Carr-Purcell
CPB Cooper-pair-box
CPMG Carr-Purcell-Meiboom-Gill
CQED Cavity quantum electrodynamics
cQED Circuit quantum electrodynamics
CU Controlled-Phase gate
CW Continuous wave
CZ Controlled-Z gate
DAC Digital to analog converter
xviii
LIST OF TABLES xix
DRAG Derivative removal by adiabatic gate
DUT Device under test
IPA Isopropanol
IQ In-phase, Quadrature
JC Jaynes-Cummings
LO Local oscillator
MLE Maximum likelihood estimation
NMR Nuclear magnetic resonance
RB Randomized benchmarking
RF Radio frequency
RWA Rotating wave approximation
SNR Signal to noise ratio
SQUID Superconducting quantum interference device
SSB Single sideband
TWPA Traveling wave parametric amplifier
VNA Vector network analyzer
Greek Letters
? Coherence state phasor
?ij Readout coefficients
LIST OF TABLES xx
?ij Dispersive shift between states i and j
? Detuning
 Dielectric constant
? Relaxation rate of qubit
? Coupling/relaxation rate of cavity
?? Effective Rabi frequency
? Rabi frequency
? Cyclic frequency
?ij Qubit transition frequency between states i and j
? Density matrix
?I Pauli identity matrix
?X Pauli X matrix
?Y Pauli Y matrix
?Z Pauli Z matrix
Latin Letters
a?, a Creation and annihilation operators
C Capacitance
C? Total qubit capacitance
D Oscillator displacement operator
LIST OF TABLES xxi
EC Electric charging energy
EJ Josephson energy
EL Magnetic inductive energy
f Frequency
g Coupling constant
H Hamiltonian
I Pauli identity matrix
Ic Junction critical current
JC Qubits? capacitive coupling constant
JL Qubits? inductive coupling constant
L Inductance
LJ Josephson inductance
M Measurement operator
N Number of energy quanta
n Charge number operator
ng Gate charge
Q Quality factor
Q(?) Husimi Q function
Qext External quality factor
LIST OF TABLES xxii
Qint Internal quality factor
R?i Rotation operator around axis i with angle ?
T1 Relaxation time of qubit
T2 Coherence time of qubit
T ?2 Coherence time of qubit obtained from Ramsey measurement
T echo2 Coherence time of qubit obtained from echo measurement
X Pauli X matrix
Y Pauli Y matrix
Y (?) Admittance
Z Pauli Z matrix
Z(?) Impedance
Physical Constants
h? Reduced Planck constant, 1.054? 10?34 J ? s
?0 Magnetic flux quantum, 2.067? 10?15 T ?m2
c Speed of light in a vacuum inertial frame, 299792458 m/s
e Electronic charge, 1.602? 10?19 C
h Planck constant, 6.626? 10?34 J ? s
kB Boltzmann constant, 1.38? 10?23 J/K
Citations to Published Work
Parts of Chapters 6 and 7 have appeared in the following papers:
Y.-H. Lin, L. B. Nguyen, N. Grabon, J. F. San Miguel, N. A. Pankratova, V. E. Manucharyan,
?Demonstration of Protection of a Superconducting Qubit from Energy Decay,? Phys. Rev.
Lett. 120, 15, (2018)
L. B. Nguyen, Y.-H. Lin, A. J. Somoroff, R. A. Mencia, N. Grabon, V. E. Manucharyan,
?High coherence fluxonium qubit,? Phys. Rev. X 9, 4, (2019)
xxiii

Chapter 1
Introduction
There are only two ways to live your life.
One is as though nothing is a miracle.
The other is as though everything is a miracle.
Albert Einstein
1.1 The Second Quantum Revolution
In 1620, Francis Bacon argued in his philosophical and scientific work ?The New Instru-
ment? that knowledge gives us power and enables us to do new things. The idea sounds
exceedingly simple, but the connection between science and technological renovation was
in fact revolutionary at the time. Prior to the 16th century, science and technology were
separate fields: new tools were usually discovered by craftsmen instead of scholars, and
hardly any ruler or business person bothered financing research to gain power or wealth.
Through trials and errors, people might have realized the usefulness of the circular wheel,
come up with lightweight paper, or invented bricks to construct buildings. However, the
spread of their usage was slow, innovations were few and far in between, and design and
material used largely remained the same for centuries. The later marriage between science
1
CHAPTER 1. INTRODUCTION 2
and technology gave birth to the Scientific Revolution, which has changed our way of life
forever [1].
The Scientific Revolution is a historical process in which humankind came to believe
in the importance of scientific research, realizing that they could increase their capabilities
through investment in such venture. It started with their willingness to admit their own
ignorance, to realize that there are questions unanswered and problems unsolved. They
then became inquisitive and set out to obtain answers through observation and mathemati-
cal tools, and subsequently gained new insights via their acquired knowledge. Wealthy and
powerful people became willing to invest part of their resources in researches which they
believed would give them the power to gain more resources.
Prior to the 1500s, people were used to the idea that the golden age was in the past, life
was a zero-sum game, and the world was stagnant. For example, the conversion of energy
depended primarily on muscle power for most of human history: the sun and nutrients in
the soil give energy to plants, animals and human gain energy by eating plants or other ani-
mals, and energy is converted from food to work via metabolism. This process is inefficient,
yet hardly anyone thought twice about improving the situation.
Modern culture, on the other hand, embraces the future, in technological innovations
and scientific research. There has been plenty of evidence in recent history to support this
belief in the future. In a short span of time, less than 0.1% of humankind?s entire history,
we have not only figured out how to convert energy much more efficiently, starting with
the Industrial Revolution, but are also learning how to do it in more sustainable fashions.
Within one century, humankind has not only built flying machines but also successfully
commercialized the aviation industry. People can now routinely cross the Atlantic in just a
few hours, a fact that would have startled Christopher Columbus and other early European
explorers.
There is no doubt that the Scientific Revolution has resulted in tremendous progress
CHAPTER 1. INTRODUCTION 3
and unprecedented growth in the modern world. In fact, the modern economy relies sig-
nificantly on technological advances, on unicorn startup companies and ingenious business
ventures of tomorrow. Therefore, there is a constant need for humankind to acquire new
knowledge and transform it to novel technologies.
The fundamental power human has over other species is our ability to store, communi-
cate, and process information. In order to go from nomadic bands of foragers to kingdoms
and empires, human progressively developed new tools and methods to obtain and analyze
necessary data. While societies require the storage and processing of huge amount of infor-
mation such as laws and taxes, the human brain is, unfortunately, not designed by nature to
store and analyze information efficiently. The brain?s capacity is limited, its lifespan is too
short, and it is only adapted to store and process specific types of information. For example,
hunters might only need to remember and identify animals? footprints, whereas gatherers
had to distinguish between edible and poisonous vegetables. The arrival of writing systems,
numbers, and algorithm-like information organization techniques helped change the game.
Laws could be passed down, tax records could be kept and catalogued, large structures
could be built with the help of trigonometry and algebra, and empires started popping up
around the world. To build more powerful countries, stronger economies, and advance
scientific frontiers further, we need more efficient computing devices and information pro-
cessing techniques.
Humankind made a giant leap in the last century with the arrival of the modern com-
puters. While sophisticated computing ideas had been developed at earlier times, with
evidence dating back to as early as c. 1750 B.C., the knowledge was lost in the sands
of time. The modern notion of computing as we know it took form in 1936, when Alan
Turing constructed the model for a programmable computer now known as the Turing ma-
chine. He showed that the machine is universal, meaning that it is able to perform any
computational task computable by any other device. This assertion is known as the Church-
Turing thesis. It is this work that laid much of the foundation for modern computer science.
CHAPTER 1. INTRODUCTION 4
Soon after, the very first computers were built based on John von Neumann?s model for
the Turing machine. Despite the great breakthroughs, those first computing machines were
hardly programmable, and they were bulky, often taking up the entire room. For example,
the ENIAC (Electronic Numerical Integrator and Calculator) took up 1800 square feet, con-
sisted of 18000 vacuum tubes, weighed 50 tons, and was used solely for calculating missiles?
trajectories. Things really took off only after John Bardeen, Walter Brattain, and Williams
Shockley invented the transistor in 1947, and accelerated after Jack Kilby and Robert Noyce
invented the integrated circuit independently in 1958 and 1959, respectively. Computing
power has been growing at an amazing pace ever since, a trend predicted by Gordon Moore
in 1965, and neatly coined into what we now refer to as Moore?s law. It states that comput-
ing power will double every other year at a constant cost. Interestingly, the law has held
true until now. Today, a modern handheld device that fits in one?s palm may wield tremen-
dous power: it can perform billions of calculations per second, store valuable information,
and moreover can connect the user to the internet and cloud services. Much of human
knowledge, ranging from nutrition facts for sriracha sauce to machine learning algorithms,
is just a mouse-click, or more conveniently, a tap on the phone, away.
Yet, there are computational problems that even these powerful machines cannot solve
efficiently. What does ?efficient? mean if the computers are so much faster than us anyway?
Well, computers typically solve problems following sets of instructions or rules called algo-
rithms. In computational complexity language, an efficient algorithm runs in time polyno-
mial to the size of the problem, while an inefficient algorithm runs in time super-polynomial
in time. Roughly speaking, there are problems that no existing algorithm can solve in time
polynominal with respect to the size of the problems. Even if our computers become twice
more powerful every other year, it will never be good enough to solve these hard problems
efficiently! On top of that, Moore?s law cannot hold true forever, and it is widely believed
that we will soon be at the breaking point where it?s impossible to scale the transistors
smaller and pack more of them into a single chip [2].
CHAPTER 1. INTRODUCTION 5
So, can such hard problems be solved more efficiently at all? In the late 1960s and early
1970s, it seemed that a Turing machine was as powerful as any other possible computing
device. This observation was codified as the strong Church-Turing thesis, which states that
any computing process can be performed efficiently using a Turing machine. A major chal-
lenge to the thesis was randomized algorithm, which gives answers in a probabilistic, instead
of deterministic, manner. This motivated David Deutsch to look into quantum mechanical
laws to derive a stronger Church-Turing thesis. This led to the conception of the quantum
circuit model, which is more generally known as gate-based quantum computing. In 1985,
he showed a simple example, demonstrating what is now known as Deutsch algorithm [3],
suggesting that a quantum computer may solve certain problems more efficiently than clas-
sical computers. This small first step was followed by a wave of enthusiastic researchers,
culminating in Peter Shor?s factorization algorithm in 1994 [4], which caused widespread
excitement, first, because finding prime factors of a number could not be done efficiently by
any known classical algorithm, and second, because implementing such algorithm would
have far-reaching applications in modern cryptography and information security [5]. An-
other potential application for a quantum computer is optimization, spear-headed by Lov
Grover. In 1995, he showed, via what is now known as Grover?s algorithm [6], that un-
structured search can be sped up by such computer. Although the algorithm only provides a
quadratic speedup, this is still considerable when the size of the problem is large. A number
of quantum algorithms have since been developed [7, 8].
The most intuitive application of quantum computers is to simulate quantum systems,
as Richard Feynman first suggested in 1982. A system of N qubits interacting with each
other must be described by a state living in a Hilbert space of dimension 2N . Equivalently,
simulating the dynamics of such state involves 2N ? 2N matrices, which is a hard task for
conventional computers. Feynman suggested that a computer that functions according to
the principles of quantum mechanics may allow us to complete this task more efficiently.
CHAPTER 1. INTRODUCTION 6
Following this hint, several researchers began working out approaches to simulate quan-
tum systems using quantum computers and have shown that it is in fact possible to do so
more efficiently than it would be on a classical computer. Ultimately, quantum simulations
may have profound impact on our understanding of complex quantum systems and future
discovery of new medicines and chemicals [9].
Why don?t we have a quantum computer yet?
Let us first recall that the notion of quantum mechanics is not new, and physical prop-
erties resulted from quantum mechanical effects have been understood and utilized exten-
sively in the modern technological world. Our ability to harness these quantum mechanical
effects and use them to our advantage is often referred to as the first quantum revolution.
The quintessential example is the transistor, which has made our experience with the com-
puting world almost seamless. Everyday we use our phones, surf the internet, or write
emails without thinking about the physics inside the transistors which made up the process-
ing unit. Broadly speaking, all electronic equipment around us rely on various integrated
circuits to function.
Quantum mechanics also gave us the laser. Until the 1960s, light was simply made of
random waves. By exploiting stimulated emission in atomic systems, scientists could con-
trol the radiation precisely, being able to select the frequencies and intensity as desired, and
laser technology was born. Lasers are used in a wide range of applications today, from trans-
mitting signal in optical cables over long distances to controlling atomic quantum systems.
With recent breakthroughs in photonic research, it is believed that further technological
advances using lasers will occur [10].
Another important technology enabled by quantum mechanics is Nuclear Magnetic Res-
onance (NMR). The magnetic moments inside the atoms? nuclei, following quantum me-
chanical laws, are quantized. For a spin-half nuclei, the Z-projection of the spin can take
only two different orientations: up or down. This can be controlled by an applied mag-
netic field and radio-frequency pulses. We are probably most familiar with this technology
CHAPTER 1. INTRODUCTION 7
through its application, the Magnetic Resonance Imaging (MRI) technique. During an MRI
scan, a magnetic field is used to align the protons within the tissues of the patient. Radio-
frequency pulses and field gradient are then applied, and the response from the nuclei is
detected. The amount of energy released and the time it takes depend on the density of
the protons and their environment, so the extracted data can be analyzed and used to re-
construct an image of our internal organs. This approach has been used widely to diagnose
certain medical conditions and help countless patients [11]. NMR is important also because
the techniques developed in NMR have been adapted to other quantum systems.
While machines whose working principles are based on quantum mechanical phenom-
ena have been successfully constructed, what we need in order to build a quantum computer
is the ability to coherently control individual quantum systems and their interactions, and
the capability to measure them. To fulfill these challenging requirements, we must satisfy
two seemingly contradicting conditions: keeping the qubits isolated from the noisy envi-
ronment, and at the same time retaining a high degree of controllability of the quantum
degrees of freedom. Moreover, as we scale up to a large number of computing elements,
the level of coherence and addressability must be retained. With the technological advances
in the last two decades, scientists around the world have gradually become more adept at
this. Our demonstrated ability to coherently control quantum systems and their interac-
tions is at the heart of the second quantum revolution, an era in which we seek to harness
our knowledge to construct and operate useful quantum computers. A summary of notable
advances in quantum hardware can be found in ref. [12].
1.2 Fluxonium Qubit: Preview
Among the currently available quantum systems, superconducting circuits based on Joseph-
son tunnel junctions have become a leading platform in the pursuit of quantum computing.
The history of these artificial atoms could be said to have started with the observation of
CHAPTER 1. INTRODUCTION 8
quantized energy levels in a macroscopic circuit in the 1980s [13, 14]. However, coherent
control of Josephson circuit was achieved only in 1999 when Nakamura et al. successfully
performed Rabi oscillation experiment in a Cooper-pair-box [15, 16]. This breakthrough
was followed by various superconducting qubit designs [17, 18].
Superconducting circuits inherit certain advantages from other technologies. They can
be printed on chips in large numbers [19, 20], wired together for strong interactions [21],
and precisely manipulated and readout by radio-frequency electronics [22, 23]. The Joseph-
son tunnel junction provides the necessary non-dissipative non-linearity required to turn
linear electrical circuits into quantum bits and strong circuit-circuit coupling into fast log-
ical operations. They inherit the advances in cryogenic technology and the robustness of
superconductivity [18], both of which have a rich history of innovation. This platform
has ushered in the era of noisy intermediate scale quantum processing [24] and quantum
supremacy over classical computation [25, 26].
However, superconducting circuits also have some disadvantages. Since they are macro-
scopic in size, they can be surrounded by multiple noise sources and impurities [27, 28].
The most persistent problems come from dielectric loss [29, 30] and 1/f flux noise [31, 32,
33], which are due to microscopic two-level defects and the spin-1/2 impurities residing in
the interface oxide layers around the device. These material imperfections persist even at
low temperature and are challenging to get rid of. As a result, superconducting qubits have
relatively short coherence times compared to microscopic quantum systems.
Decoherence is detrimental to quantum operations. It is a process through which the
information about the quantum states is lost to the environment (see section 2.3), and is
a fundamental barrier between the classical and the quantum worlds [34]. In quantum
computing, decoherence introduces errors during gate operations [35] and constrains the
number of qubits that can coherently tunnel in a quantum annealer [36, 37]. Improving co-
herence of superconducting qubits without sacrificing their controllability remains a central
problem in the quest to build complex quantum processors [38, 39, 40, 41, 42, 43].
CHAPTER 1. INTRODUCTION 9
Material imperfections are the major obstacles in improving the coherence of supercon-
ducting qubits. Engineering circuit materials to improve performance is challenging [44,
45, 46, 47, 48]. Alternatively, coherence can be improved by designing noise-insensitive
circuits. The most successful approaches to date all sacrifice qubit anharmonicity: the
transmon qubit [49, 50, 51] and the C-shunt flux qubit [52, 53, 54] are derived from the
Cooper-pair-box [15, 16] and the original flux qubit [55, 56, 57] by shunting the circuit with
a large capacitance. The coherence was improved at the cost of a reduction in anharmonic-
ity to about 5%. Such low anharmonicity adds challenges in scaling up the system. Since
nonlinearity enables logic, reducing it too much will slow down gate operations. For exam-
ple, exciting multiple qubits creates nearly resonant conditions for state leakage outside of
the computational subspace [58], an error which is difficult to correct algorithmically.
This dissertation work describes progress in quantum computing with fluxonium su-
perconducting artificial atoms [59]. The fluxonium qubit is designed to evade decoherence
due to material imperfections without sacrificing non-linearity, flux-tunability, or microwave
controllability [60, 61].
We report long relaxation time T1 in the millisecond regime, enhanced by selection rules
engineering, which is the equivalent of bit-flip quantum error correction at the hardware
level [60]. At the flux symmetric point, coherence time T2 in fluxonium approaches half
a millisecond, still limited by energy relaxation (Fig. 1.1). This is the stepping stone for
high performance quantum computing with fluxonium qubits [61]. Our results indicate
that the current limit is due to our fabrication and material selection, implying that even
longer coherence times are expected upon upgrading our fabrication techniques [30]. This
achievement also sets the stage for further development of more complex superconducting
circuit designs with additional protection from decoherence.
Finally, the thesis discusses our ongoing work on a fluxonium quantum processor con-
structed from two capacitively coupled qubits which can be entangled via a microwave-
activated controlled-Z gate (see Fig. 1.2). We use the gate to create Bell states which are
CHAPTER 1. INTRODUCTION 10
Figure 1.1: Coherence result highlight. Variations of coherence time T2 and relaxation
time T1 of qubits A, C, and I (see table 6.1). The strong correlation suggests that improve-
ment in T1 would further enhance T2.
highly entangled. Benchmarking of the gate shows that the fidelity is approaching the fault
tolerant threshold, with the best fidelity FCZ ? 0.975 ? 0.015 obtained from interleaved
randomized benchmarking. With the tools developed and integrated into current experi-
mental setup, we expect further optimization of the gate. This achievement is a necessary
step toward constructing more complex, high performance fluxonium-based quantum ar-
chitectures.
1.3 Thesis Outline
This thesis is organized as follows.
CHAPTER 1. INTRODUCTION 11
Figure 1.2: Two-qubit gate result highlight. Quantum process tomography ? matrix of
the controlled-Z gate, with process fidelity F ? 0.88.
Chapter 2 introduces fundamental concepts in quantum computing, starting with the
postulates of quantum mechanics in section 2.1. We then discuss in section 2.2 the simple
yet ubiquitous theory of spins and springs, which we respectively use to model ideal two-
level system and harmonic oscillator. We ends the chapter by discussing the formalism
describing open quantum systems and decoherence processes in section 2.3.
Chapter 3 introduces the readers to the field of superconducting circuits. We start by
reviewing in section 3.1 the physical concepts that allow these macroscopic quantum system
to be treated as qubits, namely the phenomenon of superconductivity and the Josephson
effects. Armed with this knowledge, we move on to review circuit quantization formalism.
CHAPTER 1. INTRODUCTION 12
Common decoherence mechanisms in superconducting qubits are discussed in section 3.2.
The chapter ends with a brief description of quantum gates and readout techniques in
superconducting circuits in section 3.3.
The rest of the thesis focuses on fluxonium qubits and the experiments done in the
Superconducting Circuits Lab at the University of Maryland in the past six years. Chapter
4 establishes the theory which the experiments described in Chapter 6 are based on. We
start with a discussion of fluxonium spectrum and selection rules in section 4.1. Specific
decoherence mechanisms in fluxonium are analyzed in section 4.2, and dispersive readout
with fluxonium is described in section 4.3. Relevant properties of coupled fluxonium circuits
are discussed in section 4.4.
The quantum realm can only be explored if great care is taken to (i) isolate a quantum
system from the environment, and at the same time (ii) retain coherent control of quantum
degrees of freedom. We discuss the necessary experimental techniques to fulfill these re-
quirements in Chapter 5, starting with cryogenic engineering concepts employed to setup
the experiments in section 5.1. We follow with a discussion on cavity designs and simula-
tion in section 5.2, and a description of our room temperature electronics in section 5.3.
The chapter ends with an overview of relevant fabrication techniques in section 5.4.
Numerous experiments have been done throughout the course of this dissertation work.
First, we discuss the tune up experiments done with aluminum cavities in section 6.1. The
focus of the thesis are the coherence experiments, discussed in sections 6.2 and 6.3. We re-
port our preliminary results on the microwave-activated controlled-Z gate in section 6.4. We
conclude the chapter with a description of tune-up and auxiliary measurements in section
6.5.
Chapter 7 first discusses the importance of our experimental results in the context of
quantum computing with superconducting circuits in section 7.1. We then present a per-
spective on quantum computing with fluxonium-based architectures in section 7.2.
Chapter 2
Concepts in Quantum Computing
and Quantum Optics
Those who are not shocked when they first
come across quantum theory cannot possibly
have understood it.
Niels Bohr
This chapter serves as the introduction to fundamental and essential concepts that we
will use in the rest of thesis. We first start with concepts from quantum computing and
quantum information [2, 5] in section 2.1. We follow by discussing the simple yet ubiqui-
tous physics of spins and springs [10] in section 2.2 to build our intuition for later chapters.
Section 2.3 discusses the formal approach used to model open quantum systems and deco-
herence, ending with a brief discussion on quantum process tomography [5, 10].
13
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 14
2.1 Fundamental Concepts
2.1.1 State Space
An isolated quantum system is completely described by a state vector. A state vector is a
unit vector that lives in an associated state space, which is a complex vector space, referred
to as the Hilbert space of the system.
The basic building block of quantum computers is the quantum bit, or qubit. It is also
the simplest isolated quantum mechanical system. A qubit lives in a two-dimensional state
space. Suppose, using Dirac?s notation, that the ground state |0? and excited state |1? form
an orthogonal basis for the qubit?s state space, then an arbitrary state vector in the state
space can be written as
|?? = a|0?+ b|1?, (2.1)
where a and b are complex numbers. The condition for the state vector |?? to be a unit
vector is that it satisfies the normalization condition, ??|?? = 1, which requires |a|2+|b|2 = 1.
The states |0? and |1? are analogous to the classical bits 0 and 1 in a digital register. However,
instead of being only in one state or the other, a qubit can be in any superposition state as
generally described by Eq. 2.1. When the qubit is in a superposition state, it cannot be
determined with certainty whether it is in one of its basis states. Instead, it has probability
|a|2 of being in |0? and probability |b|2 of being in |1?. Thus, a and b are referred to as
probability amplitudes. For example, the state
|0??+ |1? (2.2)
2
?
is a superposition of the basis states |0? and |1?, with amplitudes a = b = 1/ 2, which
means there is a 50% probability for the state to be in |0? and a 50% probability for it to be
in |1?.
The state vector of a qubit can be conveniently visualized on a Bloch sphere (Fig. 2.1).
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 15
Figure 2.1: Bloch sphere visualization. (a) Bloch sphere with an arbitrary superposition
state, where the state vector?s end point lies on the surface of the sphere, its direction and
polarization are defined by angles ? and ?. (b) Bloch sphere with superposition states
aligned along the X and Y axes.
It is a sphere with unit radius, and the state vector of a qubit is represented as a unit vector
with the tip lying on the surface of the sphere.
The state vector can then be rewritten in terms of the angles ? and ? as
|?? ? ?= cos |0?+ ei? sin |1?, (2.3)
2 2
which is very useful when we want to describe simple quantum operations on the qubit,
although this visualization is limited to the single-qubit case, as there is no generalization
for multiple qubits.
Note that the normalization condition is automatically satisfied with trigonometric iden-
tity of sine and cosine in Eq. 2.3. The factor 1/2 for angle ? is chosen so that ? = 0 corre-
sponds to |0? state, ? = ? corresponds to |1? state, and ? = 2? corresponds to ?|0? state, so
the qubit is actually 4? periodic. This global phase is, however, not important for the qubit,
as it is not possible to probe it in a single qubit. For a single qubit, the important physical
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 16
quantity is the relative phase between the |0? and |1? components. We will confirm this after
introducing the measurement postulate, and come back to the 4? periodicity in the case
of multiple qubits coupled to each other, where this property allows us to do multi-qubit
entangling gate.
2.1.2 Quantum Evolution and Gates
The evolution of an isolated quantum system from time ti to time tf is described by a unitary
operator U ,
|?(tf )? = U(ti, tf )|?(ti)? (2.4)
The unitary operator U defined above is abstract. Its form depends on the particular system
under consideration. An important practical consideration is how the state varies continu-
ously in time, which is described by the Schro?dinger equation:
d|??
ih? = H|??, (2.5)
dt
where h? is the Planck?s constant, and H is the Hamiltonian of the closed quantum system.
If we know the Hamiltonian of a system, we can then understand its dynamics and predict
its time evolution.
The Hamiltonian is a Hermitian operator with spectral decomposition
?
H = E|E??E|, (2.6)
E
with eigenvalues E and corresponding normalized eigenvectors |E?. Physically, the states
|E? are the energy eigenstates of the system with corresponding eigenenergies E. The
eigenstate with the lowest eigenenergy is conventionally defined as the ground state. Eigen-
states of a closed quantum system are also referred to as stationary states because their
evolution in time is simply a change in phase, which can be readily deduced from Eq. 2.5
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 17
and Eq. 2.6,
|E(t)? = exp(?iEt/h?)|E(t = 0)?. (2.7)
The unitary operator U that acts on a quantum state |?? to evolve it from time ti to time tf
can be formally written in terms of the time-independent Hamiltonian as
(? )iH(tf ? ti)
U(tf , ti) = exp . (2.8)
h?
In quantum computation and quantum information, we often work with abstract uni-
tary operations, leaving the details of the Hamiltonian H to be determined separately. For
example, in the quantum circuit model, the qubit can be manipulated using specific unitary
gate operations which only take into account the initial and final states. To proceed further,
let us introduce the Pauli operators in matrix forms. They are used extensively in quantum
computation and quantum information. There are two equivalent notations frequently used
in literatures written below,
?? ? ? ??1 0?? ??0 1? ? ??I I = , ?X X = ? ,
?0 1? ? ?
1 0 ? (2.9)
?0 ?i? ??Y ? Y = ? , ?Z ? Z = ?1 0 ?? .
i 0 0 ?1
The action of the Pauli matrices on a qubit state is straightforward. The identity operator
I leaves the qubit state unchanged, while the other operators flip it around the respective
axis. The Pauli matrices can further be used to describe arbitrary rotations around the X, Y,
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 18
Z axes via the rotation operators,
R (?) ? e?i?X/2 ?X = cos I ?
?
i sin X, (2.10)
2 2
RY (?) ? e?i?Y/2
? ?
= cos I ? i sin Y, (2.11)
2 2
R (?) ? e?i?Z/2 ? ?Z = cos I ? i sin Z. (2.12)
2 2
Note that we have used the Pauli matrices identity ?i?j = ?ijI + iijk?k after expanding the
exponential in our derivation. In quantum computing, the angles ? = ??/2 are often used
to rotate the qubit onto and away from the X-Y plane, so we often write RX(? = ??/2) and
RY (? = ??/2) as ?X/2 and ?Y/2, respectively.
2.1.3 Measurement
Quantum measurements are described by a set of measurement operators {Mm} acting on
the state space of the system being measured. It is a complete set of operators which satisfies
the completeness relation ?
M ?mMm = I. (2.13)
m
If the state of the system is |?? before the measurement, then after the measurement, the
system is left in
| ? ? Mm|???m = (2.14)
??|M ?mMm|??
with probability
p(m) = ??|M ?mMm|??. (2.15)
F?rom the completeness relation, we can deduce that the probabilities sum up to one,
m p(m) = 1. For many applications, we will be concerned with an important class of
measurements known as projective measurements. In this case, the measurement operators
Mm are orthogonal projectors, MmMm? = ?m,m?Mm. The measurement is then described
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 19
by an observable M with spectral decomposition
?
M = mPm, (2.16)
m
where Pm is the projector onto the eigenspace of M with eigenvalue m. When measuring
state |??, the probability of getting result m is
p(m) = ??|Pm|??. (2.17)
For example, when measuring a qubit along the Z-axis, the observable is ?Z ? Z. We can
write
Z = |0??0| ? |1??1|. (2.18)
?
Thus, if we measure the superposition state |?? = (|0? + |1?)/ 2 along Z, we would get
(+1) with probability ??|0??0|?? = 1/2 and (?1) with probability ??|1??1|?? = 1/2.
We can also measure the qubit along X and Y. From Fig. 2.1, we can infer that we would
get ?X? = ??|X|?? = 1 and ?Y ? = ??|Y |?? = 0. Indeed, we can check by direct matrix
multiplication that ?X? = 1 for superposition state ?. More generally, we can define
| |0?Z + |1?Z |0?Z ? |1?Z0?X = ? , |1?X = ? , (2.19)
2 2
so it?s also possible to write the superposition state |0?Z?+|1?Z in the Z-basis as |0?
2 X
in the
X-basis. While the decomposition of a state vector varies according to the different bases,
the measurement is independent of the choice for the bases.
A practical problem in quantum computing is to determine the observables X and Y
when the system can only be measured along Z. To accomplish this, we simply perform
??/2 rotations right before the projection measurement. This transformation converts what
we measure along Z to the expectation values ?X? and ?Y ?, as shown in Table 2.1.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 20
Table 2.1: Qubit rotations and Z-projection measurements to determine expectation values
of Pauli matrices.
Rotate Measure
I ?Z?
X/2 ?Y ?
Y/2 -?X?
-X/2 -?Y ?
-Y/2 ?X?
We can now examine the importance of the global phase and the relative phase of a
qubit state. Consider two different states |?? and ei?|??. Suppose that we measure it with
an arbitrary observable corresponding to the measurement operator Mm. The expectation
value of such observable for these states are ??|M ?mMm|?? and ??|e?i?M ? i?mMme |??, which
are equivalent. Thus, it is not possible to tell these states apart by any kind of measurement.
We may say that they are equivalent states.
It?s quite a different story with the relative phase. For example, consider the states
| ? |0?Z + |1?Z |0?Z ? |1?Z0 X = ? , |1?X = ? . (2.20)
2 2
The probabilities of the two available expectation values when we measure the states in the
Z basis are the same, and we cannot tell them apart. However, the relative phase difference
in the Z basis is equivalent to a different state in the X basis, and we can measure the
difference when we determine ?X?, with |0?X giving 0 and |1?X giving 1. Thus, the relative
phase is basis-dependent, and the difference between the phases in one basis can result in
physically distinguishable states in another basis. Therefore, states with different relative
phases are not physically equivalent.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 21
2.1.4 Composite Systems
When multiple individual systems are prepared separately and used to construct a compos-
ite system, the state space of such a composite system is the tensor product of the state
spaces of the component systems. The joint state of the total system can be written as the
tensor product of the constituent state vectors
|?? = |?1? ? |?2? ? ...? |?n?. (2.21)
In this case, any manipulation on one part does not affect the others. Unitary operations on
composite system when the operation is applied to only one of the individual systems can
simply be written as the tensor product as well. For example, when we flip the first qubit of
a two-qubit system while leaving the other qubit untouched, the corresponding unitary can
be written as
U = X ? I. (2.22)
However, it is not always possible to write the state vector representing a composite
system as the tensor product of the state vectors of individual systems. For instance, it?s not
possible in the case where two systems are allowed to interact and form an entangled state.
As an example, let us consider the two-qubit state
| ? |00??+ |11?? = . (2.23)
2
We can readily check that there are no single-qubit states |?1?, |?2? such that |?? = |?1? ?
|?2?.
Such entangled states play a crucial role in quantum physics, quantum computation, and
quantum information. Unlike product composite states, when we measure one constituent
qubit, we also project the other qubit onto the respective basis. For example, if we measure
the first qubit in |?? and project it onto |0? (with 50% probability), then the second qubit
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 22
collapses onto |0? with certainty (100% probability).
The state defined in Eq. 2.23 is called a Bell state. There are four of them:
| ? |00??+ |11??00 = ,
2
| ? |00??? |11??01 = ,
2
| ? | ? (2.24)| 01 + 10?10? = ? ,
2
| ? |01??? |10??11 = .
2
Just as there are non-trivial (non-product) composite states, there are also non-trivial
unitary operations which cannot be written as the tensor products of individual unitary
gates of constituent systems. These are called multi-qubit entangling gates, and they are
required to construct universal sets of quantum gates. A universal set of gates is required
to synthesize any arbitrary operation on a quantum computer. A set composed of any two-
qubit entangling gate and all single-qubit gates is universal [2].
A two-qubit gate is an entangling gate if for some input product state, the output after
applying the gate is an entangled state. An important entangling gate is the controlled-NOT
(CNOT) gate, defined by the unitary matrix
? ?
???1 0 0 0? ??0 1 0 0??
UCNOT = ??? ??? . (2.25)?0 0 0 1?
0 0 1 0
We can easily check its action on the basis states: the gate flips the state of the second qubit
if the first qubit state is |1?. Thus, the first qubit is referred to as the control qubit and the
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 23
Figure 2.2: Quantum circuit to create Bell states using CNOT gate. (a) Quantum circuit
diagram showing the creation of Bell states using a single-qubit gate and the controlled-
NOT two-qubit gate. If the initial state |?0? is |00?, then the final state |?2? is the Bell state
|?00?. Other Bell states can be created starting from different initial states (see Table 2.2).
(b) Equivalent representation of the single-qubit gate X/2 and the CNOT gate in a quantum
circuit.
second qubit is referred to as the target qubit,
UCNOT|00? = |00?,
UCNOT|01? = |01?,
(2.26)
UCNOT|10? = |11?,
UCNOT|11? = |10?.
The CNOT gate can be used together with single-qubit gates to generate entangled
states, as shown in Fig. 2.2. The circuit shown is often referred to as quantum circuit
diagram, the input and output are quantum states, and the gates are drawn as rectangular
blocks. Two different representations of the gates are shown. The state of the system is
changed by the gates as follows,
|?0? = | ? | ?
|00??+ |10? | ? |00??+ |11?00 , ?1 = , ?2 = = |?00?. (2.27)
2 2
Other Bell states can be created depending on the initial states, as shown in Table 2.2.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 24
Table 2.2: Creation of Bell states from different initial states using the quantum circuit
diagram in Fig. 2.2.
Initial state |00? |01? |10? |11?
Final state |?00? |?10? |?01? |?11?
Another important two-qubit gate is the controlled-Phase (CU) gate, defined by the
unitary matrix ? ?
???1 0 0 0??
??
?0 1 0 0? ?UCU = ??
?? . (2.28)?0 0 1 0 ?
0 0 0 ei?
When ? = ?, ei? = ?1, the CU gate becomes what we often call the controlled-Z (CZ) gate,
?? ???1 0 0 0? ?0 1 0 0
UCZ = ???? ?
?
??
0 0 1 0 ?? . (2.29)?
0 0 0 ?1
The action of the CU (and CZ) gate is also straightforward. If the two-qubit state is |11?, the
unitary UCU would apply a phase factor ei? to it.
UCU|00? = |00?,
UCU|01? = |01?,
(2.30)
UCU|10? = |10?,
UCU|11? = ei?|11?.
The CZ gate can also be combined with single-qubit gates to create entangled states. For
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 25
Figure 2.3: Quantum circuit to create Bell states using CZ gate. Quantum circuit dia-
gram showing the creation of Bell states using single-qubit gates and controlled-Z two-qubit
gate. Different Bell states can be created starting from different initial states, as summarized
in Table 2.3.
Table 2.3: Creation of Bell states based on different initial states using the quantum circuit
diagram in Fig. 2.3.
Initial state |00? |01? |10? |11?
Final state |?10? |?11? |?01? |?00?
example, we can create different Bell states using the quantum circuit shown in Fig. 2.3.
The circuit?s input-output relations are summarized in Table 2.3.
2.2 Spin and Mass On A Spring
Cavity quantum electrodynamics (CQED) physics is elegant and ubiquitous. Part of its appeal
comes from the simplicity of the model: a two-level system is represented as a spin, and a
quantum field is described as a mass on a spring. Despite its simplicity, the model displays
a wide variety of phenomena, the breadth of which can be explored in references such
as [10]. This section introduces the basic concepts which have been adopted for use in
superconducting circuits to give birth to the field of circuit quantum electrodynamics (cQED)
[22, 62]. We will explore cQED physics in more details in section 3.1.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 26
Realistic quantum systems can often be simplified to include just two states, with com-
plications from other states eliminated by appropriate approximation. The physics involved
is then reduced to two-level system configuration, which can be effectively modeled as a
spin-half, with the spin pointing up representing one state and the spin pointing down
representing the other state. The dynamics of the spin obeys the Bloch equations [10].
When many quanta of energy are involved, a mass on a spring model is better suited to
capture the physics. A harmonic oscillator with its Hamiltonian quadratic with respect to
the system?s conjugate coordinates provides a good description of low lying states in such
system. Note that interactions which are too strong can introduce non-linear phenomena,
and the approximation fails. Therefore, we shall not consider that regime here.
In this section, we start with a description of the two-level system as a spin, and review
how it can be manipulated by an external field. We then move on to analyze the Hamilto-
nian of a quantum harmonic oscillator which models an electromagnetic field, and define
the state of a driven oscillator in equilibrium. Finally, we explore the coupled spin-spring
system in the dispersive limit. This formalism is expanded to explore gates and readout for
superconducting circuits in section 3.3.
2.2.1 The Spin System
A two-level system with eigenstates |0? and |1? is equivalent to a spin one-half aligned
along the Z-axis. Let the difference in its eigenenergies be h??01, then the Hamiltonian for
this system can be written as
Hs = ?
h??01
?Z . (2.31)
2
With this definition, |0? and |1? correspond to the ground and excited states with eigenener-
gies ?h??01/2 and h??01/2, respectively. Let us introduce the raising and lowering operator
??, defined as
?+ = |1??0|, ?? = |0??1|, (2.32)
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 27
which can be written in terms of the Pauli matrices ?X,Y as
1
?? = (?X ? i?Y ). (2.33)
2
We can attempt to drive the two-level system with a classical oscillating field written as
( )
E~ = Ed ~de
?i(?dt+?d) + ~ ?ei(?dt+?d)d , (2.34)
where Ed is the amplitude of the field, ?d is the angular frequency, ?d is the phase, and ~d is
the complex polarization vector. The field can be coupled to the spin via a dipole moment
D~ = d (~ ? + ~ ?s + s ??) , (2.35)
where d is the dipole matrix element and ~s is the spin?s polarization. The coupling Hamil-
tonian can then be written as Hd = E~ ?D~ , with H = Hs +Hd. We can simplify the driving
scheme by letting ~d ? ~ ?d = 1. We can then use the rotating wave approximation in the
interaction representation to get the interaction Hamiltonian of the system rotating at the
field?s frequency ?d [10],
h?? h??
H = ? + (e?i?d? + ei?dI Z + ??)
2 2 (2.36)
h?? h??
= ?Z + (cos(?d)?X + sin(?d)?Y )
2 2
where ? = ?01 ? ?d is the spin-field detuning, and ? = 2dEd/h? is the Rabi frequency of the
driven system, which is intuitively proportional to the matrix element of the spin d and the
amplitude of the field Ed. However, that is not always the exact rotation frequency of the
spin, due to the finite detuning between the drive and the transition frequency. In general,
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 28
we can write the driven Hamiltonian as
h???
HI = ~? ? ~n, (2.37)
2
where ?
?? = ?2 + ?2 (2.38)
is the effective Rabi frequency, and
?~uZ + ? cos?d~uX + ? sin?d~uY
~n = ? (2.39)?
is a unit vector in the Bloch sphere, the direction of which is determined by the ratio be-
tween the detuning ? and the Rabi frequency ?.
The Hamiltonian in Eq. 2.37 thus describes the Larmor precession around an effective
axis along ~n at frequency ??. This is also known as Rabi oscillation. The time evolution of
the spin on the Bloch sphere under the driving field can be visualized by simulating Eq. 2.5
in the rotating frame,
?
ih? |??I = HI |??I . (2.40)
?t
Time evolution of the spin in different detuning scenarios is plotted in Fig. 2.4. For sim-
plicity, we can set ?d = 0, and vary the detuning ?. In the resonant case, the spin vector
rotates around the X-axis. In the detuned case, the axis of rotation is tilted away from the
X-axis by an angle that increases with the detuning. From Eq. 2.39, we can deduce that
the tangent of this angle is equal to ?/?.
Rabi oscillation provides a simple way to prepare the qubit in any arbitrary state. The
most important cases are for rotation angles of ? and ?/2, where a combination of the field
amplitude Ed and the interaction time is tuned to rotate the spin precisely by these angles.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 29
Figure 2.4: Qubit rotation. A qubit rotating in the drive?s frame when the drive is (a)
on resonant with qubit?s frequency ? = 0, (b) somewhat off-resonant ? = ?/2, and (c)
far-detuned from qubit?s frequency ? = 4?. The qubit?s trajectory is described by the blue
arrow tracing the blue line. The red arrow represents the axis of rotation.
2.2.2 The Field Oscillator
Generally speaking, harmonic motion can be characterized by a particle of mass m moving
in a 1D quadratic potential V (x) = m?2x2/2. The motion of the particle in a phase space
of complex variable x + ip is a rotation at angular frequency ?. In quantum physics, the
position and momentum are operators satisfying the commutation relation [x, p] = ih?, and
the Hamiltonian of the system can be written as
m?2x2 p2
H = + . (2.41)
2 2m
Next, we can introduce the natural units for x and p as [10]
? ?
h? m?h?
x0 = , p0 = , (2.42)
2m? 2
then normalize x? = x/2x , p?0 = p/2p0 to get
H = h??(x?2 + p?2). (2.43)
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 30
Quantum harmonic oscillator formalism can be conveniently and elegantly written in
terms of the ladder operators, namely the annihilation operator a = x?+ ip? and its complex
conjugate a?. The commutation relation between x and p leads to [a, a?] = I, and the
spring?s Hamiltonian can be written as
( )
1
H = h?? a?a+ , (2.44)
2
where N ? a?a is the number operator satisfying [a,N ] = a, [a?, N ] = ?a?, and the term
h??/2 represents the vacuum fluctuation energy. We can check that ?x2? = x20 and ?p2? = p20
(or ?x?2? = ?p?2? = 1/4) satisfy the vacuum condition N = 0, H = h??/2. Thus, x0 and p0
are also referred to as zero point position and momentum r.m.s. fluctuations, respectively.
We shall use this formalism in deriving the corresponding zero point coordinates for an LC
oscillator in section 3.1.
Within a constant vacuum energy h??/2, the eigenvalues of the Hamiltonian of an os-
cillator are then h??N , with denoted eigenstates |n?, where n is a non-negative integer.
These are called Fock states of the harmonic oscillator, such that N |n? = n|n?. The ladder
operators? effect on the Fock states is to move the system up or down in the ladder (see
Fig. 2.5(b)),
| ? ?
?
a n = n|n? 1?, a?|n? = n+ 1|n+ 1?. (2.45)
An electric field in a cavity can be expressed as a linear combination of the creation and
annihilation operators,
( )
E~ (?, t) = E f~ae?i?t ? f~?a?ei?t0 , (2.46)
where E0 is the electric field amplitude, and f~ describes the position dependence of the
field, obeying fields equation and boundary conditions.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 31
Figure 2.5: Harmonic oscillator concepts. (a) A harmonic oscillator is represent?ed as a
mass m connected to a spring with spring constant k. Its resonant frequency is ? = k/m.
(b) Energy spectrum ladder of a harmonic oscillator. The levels are equally spaced, with
the lowest state having the vacuum energy of h??/2.
Since the Fock states are equidistant in the energy space, the electric field has an indef-
inite phase and does not correspond to an oscillator excited by a classical field. When an
oscillator is driven by a classical field, it is instead put into a state described by well-defined
amplitude and phase. For example, we can drive a mechanical oscillator into such state by
giving it a momentum kick p, while for an LC oscillator, we can drive it with a current or
voltage source. We call the resulting state coherent state |??, defined as
|?? = D(?)|0?, (2.47)
? ?
where D(?) = e(?a ?? a) is the displacement operator, satisfying D?(?) = D(??) and
D(0) = I. The coherent state can be expressed in terms of Fock states using the expression
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 32
Figure 2.6: Fock state representation of coherent state. Coherent states represented by
the Poisson number distribution (Fock basis) for an average of (a) 1 photon, and (b) 10
photons.
2 ? ?
D(?) = e|?| /2e?a e?? a. Explicitly writing out a?n|0?, we get
? n
|?? = e|?|2/2 ?? |n?. (2.48)
n n!
The square of the probability amplitude gives the probability of finding the excitation num-
ber n as p 2?(n) = |cn| , which has Poisson statistics,
2 |?|2n
p?(n) = e
?|?| . (2.49)
n!
From this, we can deduce that a driven harmonic oscillator is in a superposition of number
s?tates, with the average number of excitations n? = |?|2, and standard deviation ?n =?
?n2? ? ?n?2 = n? [10].
Figure 2.6 shows the probability distribution in the number states basis for 1 and 10
quanta of excitation on average. Note that the relative width of the distribution, ?n/n?
becomes progressively smaller as the amplitude of the field increases. This is consistent
with our definition. The coherent state is ill-defined for a harmonic oscillator in vacuum
state, and becomes more classical as we increase the driving strength.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 33
2.2.3 Coupled Spin-Spring System
From the derivations of spin and field mode previously explored, we can now write the
atom-cavity coupling Hamiltonian as
( )
Hcoupling = ?d (~ ? + ~?? ? ?s + s ?) ? E0 ~ca+ ~ca . (2.50)
Terms like ??a and ? ?+a correspond to highly non-resonant processes and play a minor
role compared to the other two terms ?+a and ??a?, which correspond to photon emission
and absorption processes. Neglecting those non-resonant terms, we can approximate the
coupling Hamiltonian as
Hcoupling = h?g(a?+ + a
???), (2.51)
where g = dE0|~?s ? ~c| is the vacuum Rabi frequency, which represents the strength of
the atom-cavity field coupling. It is proportional to the dipole moment of the atom, the
amplitude of the field across the atom, and the polarization alignment between the atom
and the field. The first term describes the absorption of energy by the atom from the cavity,
and the second term describes the emission of energy from the atom to the cavity. The
Hamiltonian describing the coupled system is called the Jaynes-Cumming Hamiltonian,
( )
? 1 ?sH = h?? a a+ + h? ? ?JC c Z + h?g(a?+ + a ??). (2.52)
2 2
The first term represents the energy of the electromagnetic field inside the cavity, the second
describes a spin one-half, and the last term is the coupling given by Eq. 2.51. Physical
phenomena involving this atom-cavity coupled system has been investigated in atom and
superconducting circuit systems [22, 62, 63, 64].
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 34
Dispersive Case
When the detuning between the atom and the cavity is large, ? = |?c ? ?s|  g, there
is negligible energy exchange between the subsystems. This regime is called the disper-
sive limit of the Jaynes-Cumming model. Expanding the Jaynes-Cumming Hamiltonian in
powers of g/? to second order gives
( )( )
? g
2 1 ?s
H h? ? ?c + ?Z a a+ + h? ?Z (2.53)
? 2 2
The first term describes a harmonic oscillator with resonant frequency dependent on the
state of the atom, ??c = ?c ? g2/?. This frequency shift is called the dispersive shift. It
can be used to measure the state of the spin in a Quantum Non-Demolition (QND) fashion
[10]. This physics has been utilized to realize cavity quantum electrodynamics paradigm
in superconducting circuits [22, 23, 62], where a spring (cavity) is used to probe a spin
(qubit). Note that the dispersive approximation is only valid at sufficiently low photon
number. If the number of cavity excitation exceeds the critical value n = ?2c /4g2, this
approximation breaks down.
The Hamiltonian can also be written to emphasize the dependence of the spin on the
number of excitations in the cavity,
( ) ( )
? ? 1 h? 2g
2 g2
H h?? a a+ + ? ?c s + a a+ ?Z . (2.54)
2 2 ? ?
The spin?s effective frequency is dependent on the Lamb shift g2/? and the Stark shift
2g2n/?. The Stark shift depends on the number of photons in the cavity, which leads to
measurement-induced dephasing in qubits [63, 65]. Moreover, under the dispersive strong
condition where the dispersive shift 2g2/? is larger than the qubit?s spectroscopy linewidth
and dephasing rate, we can use the qubit to query exactly how many photons are in the
cavity [64]. This allows QND measurement of the photon number [66].
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 35
2.3 Open Quantum Systems
So far we have only discussed ideal, isolated systems for which we have complete knowl-
edge. The state of the system is then said to be a pure state. However, that is not always
the case, as the system can have interaction with the outside world, and its coherence can
be destroyed via the leakage of information to the environment. In this case, the system is
in a so-called mixed state.
In this section, we discuss the formalism to describe open quantum systems. We first
introduce the density operator, also known as the density matrix, and its properties. The
formalism is completely equivalent to the state vector approach, and provides a convenient
language to describe a more general quantum state.
2.3.1 The Density Matrix
The density operator language provides a convenient means for describing quantum system
that we do not have complete knowledge about. If we have an ensemble of pure quantum
states |?i? with corresponding probabilities pi, then the density operator for the system can
be defined as ?
? ? pi|?i???i|. (2.55)
i
If the state is known exactly to be in state |??, then the density operator can be simply
written as ? = |????|. This can also be done by rewriting Eq. 2.55 with one of the pi equal
to 1. Otherwise, when pi < 1, the system is in a mixture of states. The decomposition of ?
in Eq. 2.55 also requires ? to be positive and have unit trace,
Tr(?) = 1. (2.56)
?
We can show that Tr(?2) = p2i ? 1, with the equality condition only satisfied when ? is a
pure state.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 36
Other important notions in quantum mechanics can be rephrased in density operators
language in complete analogy to the state vector approach. The evolution of a system
following the unitary operator U obeys
? ?
? = pi|?i???i| ? ?(t) = piU |?i??? ?i|U = U?U ?. (2.57)
i i
A measurement following operator Mm will give result m with probability
p(m) = Tr(M ?mMm?), (2.58)
leaving the system in a state described by a density matrix
M ?mMm?
?m =
Tr(M ?
. (2.59)
mMm?)
If we have multiple quantum systems prepared in state ?i, then the joint state of these
can be described by the density matrix ? = ?1??2? ...??n. Now, suppose we have physical
systems A and B, and the state of the composite system is ?AB = ?A ? ?B. The subsystem
A can be analyzed through the reduced density operator ?A defined via the partial trace
?A ? TrB(?AB), (2.60)
where the partial trace TrB is defined as
TrB(?A ? ?B) = ?ATrB(?B) = ?A. (2.61)
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 37
Density Matrix of a Qubit
Let?s consider the case of a two-level system with basis states |0? and |1?. The density matrix
is then a 2? 2 Hermitian matrix which generally takes the form
? ?
???00 ?01?? = ? . (2.62)
?10 ?11
Its real positive diagonal terms ?00 and ?11, called populations, are the probabilities of
finding the qubit in |0? and |1?, and thus sum up to unity ?00 + ?11 = 1. The off-diagonal
terms ?01 and ?10 = ?01?, called coherences, are important in determining whether the state
is pure or mixed.
The density matrix of a qubit can be expanded onto the Pauli basis as
1
? = (I + ~r ? ~?), (2.63)
2
where ~r is the Bloch vector for state ?. Following the definition of a pure state, Tr(?2) = 1,
we can deduce that ||~r|| ? (r2 1x + r2y + r2z) 2 = 1 only for pure states and ||~r|| < 1 for mixed
states. Thus, the tip of the Bloch vector lies on the surface of the Bloch sphere for pure
states, and inside the Bloch sphere for mixed states. The center of the Bloch sphere would
then correspond to the maximally mixed state ? = I/2.
While the Bloch sphere is a great way to visualize the state of a qubit, it is not possible
to generalize it to multiple qubits. An alternative approach is to plot the density matrix
of the system in a matrix histogram, as shown in Fig. 2.7 for single-qubit and Fig. 2.8 for
two-qubit states. Hinton diagrams are also used to describe matrices in the literature. We
show some examples of Hinton diagrams in Fig. 2.9, where the density matrices correspond
to the four Bell states.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 38
Figure 2.7: Visualizations of single-qubit density matrices. Amplitude and phase of
density matrix?s entries are presented as the height and color of the bars.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 39
Figure 2.8: Visualizations of two-qubit density matrices. Amplitude and phase of density
matrix?s entries are presented as the height and color of the bars.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 40
Figure 2.9: Gallery of Hinton diagrams for Bell states. Visualization of qubit states using
Hinton diagrams. The opaqueness and colors represent the amplitude and phase of the
density matrix? entries. Hinton plots are used interchangeably with the matrix bar plots in
the literature.
Quantum State Tomography
Quantum state tomography is the procedure of determining the state of an unknown quan-
tum system via the construction of its density matrix ?. If we are given just a single copy of
? then it would be impossible to determine it because we can only measure along one axis
at a given time. On the other hand, if we prepare a large ensemble of identical matrices
??s, then we can perform different measurements along orthogonal axes and determine the
ensemble averages of the relevant observables.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 41
For example, we can rewrite the density matrix for the two-level system given in Eq.
2.63 as
1
? = (I + ?X?X + ?Y ?Y + ?Z?Z) , (2.64)
2
where the expectation values of observables X, Y , Z are obtained by repeating the mea-
surement along the respective axis, given that ??? = Tr(??). In principle, the density matrix
of a two-level system has four entries, with one constraint Tr(?) = 1, so we have to perform
three different measurements to reconstruct ?.
This formalism can be extended to an n-qubit system,
1 ?
? = (?~?1 ? ~?2 ? ? ? ? ? ~?n?~?1 ? ~?2 ? ? ? ? ? ~?n). (2.65)
2n
~?
For example, the two-qubit density matrix can be written as
1
? = (II + ?IX?IX + ?IY ?IY + ?IZ?IZ
4
+ ?XI?XI + ?XX?XX + ?XY ?XY + ?XZ?XZ
(2.66)
+ ?Y I?Y I + ?Y X?Y X + ?Y Y ?Y Y + ?Y Z?Y Z
+ ?ZI?ZI + ?ZX?ZX + ?ZY ?ZY + ?ZZ?ZZ).
In practice, the measured expectation values are converted from physical measurements,
and at least 15 rotations on the pair of qubits must be performed, together with the con-
straint Tr(?) = 1, or ?II? = 1, to reconstruct the two-qubit density matrix. The experimental
procedure to reconstruct density matrices is discussed in section 6.4.
Generally, ideal tomography plots, such as those shown in Fig. 2.7 and Fig. 2.8, can be
used to compare with experimental data. However, a more rigorous approach is needed to
check how close a prepared state and the ideal state are [5]. The first number developed to
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 42
measure this closeness between two quantum states ? and ? is the trace distance, defined as
1
D(?, ?) = Tr|?? ?|, (2.67)
2
which is ideally zero for two identical states. The second measure is the fidelity. For two
quantum states ? and ?, the fidelity is defined as
?
F (?, ?) = Tr ?1/2??1/2. (2.68)
We note that it is more common in the literature to write the fidelity as
( ? )2
F (?, ?) = Tr ?1/2??1/2 . (2.69)
It is therefore a good idea to clearly state the formula used wherever a fidelity value is
quoted. In either case above, we can check that the fidelity for two identical states is 1, and
for two orthogonal states is 0.
Eq. 2.68 can be simplified if one of the states is a pure state, which is usually the case
when we compute the fidelity. The fidelity between a pure state |?? and an arbitrary state
? can be written as
?
F (?, |??) = ??|?|??. (2.70)
Finally, it follows that the trace distance and the fidelity between two quantum states
are related,
?
1? F (?, ?) ? D(?, ?) ? 1? F (?, ?)2. (2.71)
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 43
For the special case when one of the states is pure, we get
1? F (|??, ?)2 ? D(|??, ?). (2.72)
2.3.2 The Lindblad Master Equation
To describe the evolution of a quantum system described by the density matrix ? in the
presence of a noisy environment, we use the Lindblad master equation [5, 10]:
( )
d? ? i
? 1 1
= [H, ?] + L??L
? ?
? ? L?L??? ?L??L? , (2.73)dt h? 2 2
?
where H is the Hamiltonian describing the quantum system in isolation, and L? is the
operator describing the effect of the environment on the system.
The derivation of the master equation is based on the Kraus-sum formulation of quan-
tum maps, and the Markov approximation [10]. The Markovian limit assumes that the
evolution is local in time, ??(t)?(t + dt)? = 0. Let the characteristic timescale of the envi-
ronment be ?c, the typical time after which the environment ?forgets? about the information
that it has acquired from the system, and subsequent information exchanged between the
system and the environment is considered uncorrelated. The other time scale of importance
is the long time Tr during which the system evolves. Markovian approximation is then valid
for time step ? in which ?c  ?  Tr. In other words, the equation is valid during a time
coarse enough for the evolution of the system to be insensitive to any fast environmental
fluctuations, but fine enough to describe a quantum evolution over the long time scale of
interest.
Equation 2.73 provides a convenient method to simulate and understand the effect of
relaxation and decoherence mechanisms on the evolution of a quantum system. First, let?s
examine a two level system undergoing relaxation from the excited state to the ground at
?
rate ?1 = 1/T1. The corresponding jump operator is L1 = ?1??. The master equation
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 44
thus becomes
d?
= ? ?01 ?1i [?Z , ?] + (2????+ ? ?+???? ??+??) . (2.74)
dt 2 2
Inspecting individual matrix elements of ?, we find
d?00 d?11 ? d?10 ? ? ?1= ?1?11, = ?1?11, = i?01?10 ?10. (2.75)
dt dt dt 2
Hence, during the relaxation process, the qubit?s population transfers from the excited state
to the ground state at rate ?1, and the off-diagonal coherences decay at rate ?1/2. The
decay rate ?1 can be obtained via Fermi?s golden rule, which is discussed in more details in
Appendix A.
In general, a qubit can undergo both emission and absorption processes in a thermal
environment, both of which induce relaxation to equilibrium, which is a mixed state of |0?
and |1? with populations in each state determine?d by the effective te?mperature of the bath.
The jump operators can be modified as L? = ???? and L? = ???+ to describe the
relaxation by emission and excitation by absorption processes, respectively. The rates ??
and ?? must obey the detailed balance relation,
?? = ?
?h??
?e 01
/kBTeff . (2.76)
We can check that the resulting master equation now includes two jump processes, one
corresponding to relaxation with rate ?? described by Eq. 2.74, and the other correspond
to excita(tion process with) rate ??. Combination of these gives a decay rate ?1 = ?? +
? = ? 1 + e?h??01/kBT? ? eff , the rate at which an excited qubit returns back to equilibrium.
The relaxation rate ?1 corresponds to depolarization along the qubit quantization axis, as
depicted in Fig. 2.10(a).
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 45
Figure 2.10: Decoherence processes. (a) Relaxation is the process in which the state of
the qubit is flipped. At finite temperature, the qubit effectively relaxes to an equilibrium
mixture of |0? and |1?. (b) Pure dephasing is the process during which we lose track of
the phase of the state vector due to the fluctuation of the qubit?s parameters. The resulted
averaged state vector after dephasing will have amplitude less than unity, its tip lying inside
the Bloch sphere. (c) Rabi oscillation when there is decoherence.
The qubit?may also undergo a pure dephasing process, corresponding to the jump op-
erator L? = ??/2?Z . As depicted in Fig. 2.10(b), this process happens because we lose
track of the qubit?s information over a certain time scale at rate ??, due to the fluctuations
of the qubit transition frequency in time. When we average the fluctuations, the resulting
Bloch vector will have a reduced amplitude with its tip lying inside the Bloch sphere. The
master equation for this case can be written as
d? ? ?01 ??= i [?Z , ?] + (?Z??Z ? ?) , (2.77)
dt 2 2
resulting in
d?01,10
= ?i?01?01,10 ? ???01,10. (2.78)
dt
The coherence of the qubit thus decays at rate ??. Under both relaxation and pure de-
phasing, the resulting decoherence rate can be written as the sum of both processes, ?2 =
?1/2 + ??.
The effect of decoherence on gate operations can be visualized as follows. As shown in
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 46
Fig. 2.10(c), due to our losing track of the qubit?s trajectory, the applied drive does not rotate
the qubit in a circle anymore. Rather, the state vector?s amplitude becomes progressively
smaller, until all the information about it is lost, and the state becomes completely mixed.
That?s primarily how decoherence introduces error during gate operation. Dephasing of
qubits in the rotating frame is discussed in ref. [67].
Quantum Process Tomography
An important task in experimental quantum physics is to characterize quantum processes.
Assume that we are given information on the input and output states coming in and out
of a quantum circuit, how should we proceed to determine what has been done inside
this circuit? The answer can be obtained following a procedure called quantum process
tomography [68, 69].
A quantum operation is a linear map E describing the change of the quantum system
such that
?? E(?). (2.79)
In the operator-sum representation, the quantum map E can be written as
?
E(?) = Ei?E?i , (2.80)
i
?
where {Ei} completely takes in account the state changes, with ?iEiEi = I. To relate {Ei}
to measurable parameters, we can consider an equivalent description of E using?{E?i} which
form a basis set of operators. {Ei} can be expanded on the {E?i} basis, Ei = m eimE?m,
and we can write
?
E(?) = E? ?E??m n?mn, (2.81)
? mn
where ?mn = ?i eimein are complex entries of a matrix, simply called the ? matrix. The
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 47
central task we must accomplish in quantum process tomography is to determine ?. The
reason is because the quantum map E in the operator basis {E?i} can be fully described by
this ? matrix.
A general linear map between two d?d complex matrices is described by d4?independent
parameters. However, there are d2 constraints from the completeness relation ?iEiEi = I,
so ? contains d4 ? d2 independent parameters (d = 2n, where n is the number of qubits).
Mathematically, ? can be computed for any unitary operation using Eq. 2.81. We can
simply expand the operation on the operator basis, and find ? using the complex coefficients
eim. Experimentally, we determine ? by first preparing a set of linearly independent input
states, {?j}, then applying the quantum map E on the input states to get E(? ?j) = ?j .
Finally, we can determine the output states ??j using quantum state tomography. Then, we
can reconstruct ? using linear algebra techniques.
The first step is to expand ??j as a linear combination of the basis states,
?
??j = ?jk?k, (2.82)
k
from which the coefficients ?jk can be extracted experimentally, ?jk = Tr(?k??j). Next, we
may write ?
E?m? E? = ?
mn
j n jk ?k, (2.83)
k
where ?mnjk are complex numbers which can be extracted using the trace operation. We can
then write ?
?mnjk ?mn = ?jk, (2.84)
mn
from which the ? matrix can be reconstructed. We can compute ? as a vector, ? as a d4? d4
matrix with columns indexed by mn and rows by jk, then invert ? = ??1 to get ? = ? ? ?.
This gives ? as a vector, which, in the final step, can be reshaped into a d2 ? d2 matrix.
Ideal ?matrices for single-qubit gates based on the {I,X, Y, Z} operator basis are shown
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 48
in Fig. 2.11(a, b). In addition to unitary operation, we can simulate the output density
matrix resulted from energy relaxation and pure dephasing by utilizing the master equation
(Eq. 2.73). The results are shown in Fig. 2.11(c-f).
Process tomography ? matrices for two-qubit CNOT and CZ entangling gates based
on the {{I,X, Y, Z}?2} operator basis are shown in Fig. 2.12. To reconstruct ? for two-
qubit gates, a set of 16 input states are prepared, and each resulting output state can be
determined by quantum state tomography. Linear inversion of Eq. 2.84 results in a ? vector
with length equal to d2 = 256, which is then reshaped into a 16? 16 matrix.
Similar to state fidelity, we can write the process fidelity based on the experimentally
obtained ? matrix and the expected ?ideal matrix as
( ? )2
F (?, ?ideal) = Tr ?1/2? ?1/2ideal . (2.85)
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 49
Figure 2.11: ? matrices for single-qubit gates. Quantum process tomography ? matrix
corresponding to (a) identity gate, (b) RZ(?/2) gate, (c) relaxation process at t = T1, (d)
relaxation process at t = 10T1, (e) pure dephasing process at t = T?, (f) pure dephasing at
t = 10T?. The ? matrices are based on the {I,X,Y,Z} operator basis.
CHAPTER 2. CONCEPTS IN QUANTUM COMPUTING AND QUANTUM OPTICS 50
Figure 2.12: ? matrices for two-qubit gates. (a) ? matrix for CNOT gate. (b) ? matrix
for CZ gate. Both are based on the {{I,X, Y, Z}?2} operator basis.
Chapter 3
Superconducting Circuits:
Macroscopic Artificial Atoms
Many years later, as he faced the firing squad,
Colonel Aureliano Buendia was to remember
that distant afternoon when his father took
him to discover ice.
Gabriel Garcia Marquez
In the previous chapter, we introduced cavity quantum electrodynamics in an abstract
manner. The formalism can be applied to construct powerful quantum machines whose
degrees of freedom are described by quantum operators. Now, we are going to explore spe-
cific realization of atoms and photons. Our artificial atoms are made of macroscopic circuit
elements , inductors and capacitors, plus a special ingredient: the Josephson junction.
Superconducting circuits are much larger than the usual microscopic atoms, with cir-
cuits? dimensions in the ?m to mm scale, compared to the ytterbium atom?s Van der Waals
radius of 228 pm. Although these circuits are usually visible to the naked eye, their cur-
rents and voltages, behave like quantum mechanical variables [70]. These circuits can be
51
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 52
designed and fabricated at will to create different kinds of resonators that store microwave
photons and artificial atoms that have specific quantized energy levels. With breakthroughs
in circuit designs, lithography techniques, and clever engineering in microwave readout
and control, quantum processors based on superconducting circuits have been successfully
constructed [71, 72] and their performances, quantified by quantum volume [73], have
been continuously improved. Recently, a claim was made that quantum supremacy has been
achieved in a programmable superconducting quantum processor [42, 43, 26].
The chapter is organized as follows. We first review the basics of superconductivity
and the Josephson effects in section 3.1. We discuss common decoherence processes in
superconducting circuits in section 3.2. The chapter ends with a brief discussion of quantum
gates and readout in section 3.3.
3.1 Constructing Superconducting Circuits
3.1.1 Superconductivity
Upon thinking about building electrical circuits, the first element that may come to one?s
mind is the resistor. However, with resistance comes dissipation, which would cause energy
relaxation and destroy the coherence property of qubits. In order to reveal the quantum
nature of electrical circuits, we have to get rid of resistance in the circuits as much as
possible.
Fortunately, superconductivity was discovered at the beginning of the last century, a
phenomenon in which the resistance of certain metals vanishes at low temperatures. In
1911, while studying the temperature dependence of mercury?s resistivity, the research
group leaded by Heike Kamerlingh Onnes at Leiden Laboratory discovered that at a cer-
tain temperature Tc ? 4 K, the resistance of the mercury sample dropped to zero and
could not be measured at any temperature below Tc. At first, it was thought to be a fluke
due to some broken equipment, but repeated measurements yielded consistent results, and
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 53
the same phenomenon was found in other metals soon after: superconductivity was real.
Materials exhibiting superconductivity are called superconductors [74].
It was found later that superconductivity depends not only on the temperature of the
sample, but also on the magnetic field around it, the critical value of which is denoted as Hc.
When the field amplitude is higher than Hc, superconductivity is destroyed. Interestingly,
when a magnetic field with an amplitude less than the critical value is applied to a bulk
type I superconductor cooled down to T < Tc, the said field is expelled from the sample,
regardless of the field condition in which the sample is cooled down. This is the Meissner-
Ochsenfeld effect [75].
There is yet another profound magnetic field effect associated with superconductors.
Consider a superconducting ring cooled down through its critical temperature in the pres-
ence of a magnetic field. Since the current can persist in an infinitely long time, it can
sustain a permanent magnetic flux. If we switch off the magnetic field after, such current
can keep the field going through the loop at the initial level. However, it turns out that the
resulting magnetic flux can only take on values that are an integer multiples of the magnetic
flux quantum ?0 = h/2e ? 2.067 ? 10?15 T ?m2. The physical quantities h and e are the
Planck?s constant and electron charge, respectively.
Explaining these phenomena proved to be challenging. Early phenomenological equa-
tions proposed by the London brothers could describe perfect conductivity and the Meiss-
ner effect [76]. However, later attempts to explain superconductivity at a fundamental level
were unsuccessful, including those by prominent scientists [77]. The most successful theory
so far is the BCS theory, named after John Bardeen, Leon Cooper, and John Schrieffer [78].
It shows that weak attractive interaction between electrons mediated by electron-phonon
interaction results in the formation of bound pairs of electrons having equal and opposite
momentum and spin. These so-called Cooper pairs are the long-anticipated superconduct-
ing charge carriers [75]. The BCS theory explains superconductivity in aluminum which is
largely used to fabricate superconducting qubits.
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 54
3.1.2 Circuit Quantization
With resistance in the circuit naively taken care of by superconductivity, let us consider
a circuit constructed from an inductor and a capacitor in parallel. We can assume the
lumped element limit, i.e. the physical size of the circuit (sub-mm) is much smaller than
the wavelength of electromagnetic waves of interest, namely of the resonant modes of the
circuit under consideration (cm).
Figure 3.1: LC circuit diagram. The two dependent degrees of freedom are the electric
charge accumulated on the capacitor and the magnetic flux threading the inductor.
The classical current-flux relation for an inductor and voltage-charge relation for a ca-
pacitor are
LI = ?, CV = Q. (3.1)
Also note that the time derivative of flux and charge are voltage and current, respectively,
?? = V, Q? = I. (3.2)
The magnetic inductive energyEL stored inside an inductor with inductance L can be found
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 55
from the electrical power P ,
? ?
dEL = Pdt = V Idt = ?? dt = d?, (3.3)
L L
so by integration in flux, we get E = ?2L /2L. Similarly, for a capacitor with capacitance C,
Q Q
dEC = Q?dt = dQ, (3.4)
C C
SO the electric charging energy stored is EC = Q2/2C = C??2/2. For now, we relate the
electric energy and magnetic energy in an LC oscillator to the kinetic energy and potential
energy in a mechanical oscillator, respectively. The convenience of this choice will become
clear later. We can then write the Lagrangian of the oscillator as
C??2 2L = ? ? . (3.5)
2 2L
Comparing Eq. 3.5 with the Lagrangian of a mechanical oscillator,
mx?2 2Lmech = ?
kx
, (3.6)
2 2
we can interpret the capacitance C as an effective mass m, and the inductor L as the
inverse of an effective spring constant k. The classical Hamiltonian can be derived using
the Legendre transformation,
?L ?L Q2 2
Q = = C??, H = ?? ? L ?= + . (3.7)
??? ??? 2C 2L
To describe a quantum LC oscillator, we have to promote the charge and flux variables to
quantum operators. By analogy, p?? Q?, x?? ??, the proper commutation relation is [70]
[ ]
??, Q? = ih?. (3.8)
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 56
We can also check that the classical coordinates satisfy the Poisson bracket {?, Q} = 1.
In the rest of the thesis, charge and flux are quantum operators, and the hat above the
notations is omitted for simplicity.
For convenience, we define here the unit-less quantum operators reduced flux ? ?
2??/?0 and charge number n = Q/2e. We proceed to write the charging energy as
E 2C = e /2C and the inductive energy as E = (? )2L 0 /L, where ?0 = ?0/2? = h?/2e is
the reduced flux quantum. The new operators satisfy the commutation relation [?, n] = i.
The Hamiltonian then takes the form
1
H = 4ECn
2 + E 2L? . (3.9)
2
Following the procedure introduced in section 2.2, we can write the Hamiltonian and
operators of an LC oscillator in terms of the creation and annihilation operators. Let us
introduce
? ( ) 1
h?Z 2EC 4
?ZPF = ? = ?o2 ( EL ) , (3.10)1
h? EL 4
QZPF = = e .
2Z 2EC
One can check that ?0|?2|0? = ?2 2 2ZPF, ?0|Q |0? = QZPF, and ?ZPFQZPF = h?/2. We can write
( )
? = ? (a?ZPF + a ),
Q = iQ a?ZPF ? a (, ) (3.11)
Q2 ?2 1
H = + = h?? a?a+ .
2C 2L 2
Since we will primarily use the reduced flux ? and charge number n operators expressions,
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 57
it is useful to write them explicitly as well,
( ) 1
?1 8EC
4
? = ?
2 ( EL ) (a + a), (3.12)1
i EL 4
n = ? (a? ? a).
2 8EC
Although the circuit is quantum, when we drive such a harmonic oscillator with a field
?
on resonant with ? = 1/ LC, the oscillator is put in a superposition of number-basis states
called the coherent state, no matter how small the field amplitude is, as shown in section
2.2. In order to realize a two level system and control a single quantum of energy, some
non-linear element is needed to build up anharmonicity in the circuit.
3.1.3 Josephson Effects
1962 was a remarkable year, witnessing the birth of Walmart, Kmart, Spiderman, Motel 6,
Evander Holyfield, Demi Moore, Jodie Foster, and others. Within a short time span of 12
months, the first men were put into space, Launcher IV crashed on the moon surface, Brazil
won the World Cup the second time in a row, and plastic surgeons performed silicone breast
implant for the first time. However, in our opinion, the most significant event taking place
that year was the submission of a theoretical paper by Brian Josephson to Physics Letters
[79]. The paper predicted the Josephson effects, which we briefly review below.
Consider a thin layer of insulator sandwiched between two slabs of superconductors. A
supercurrent can flow through the junction, given by
I(?) = Ic sin(?), (3.13)
where Ic is the maximum current the junction can sustain, and ? is defined formally as
the gauge-invariant superconducting phase difference between the two superconducting
islands. For all practical purposes, it can be interpreted as the normalized flux we defined
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 58
previously, hence the same notation. This is the DC Josephson effect [79].
If a voltage V is maintained across the junction, the phase difference ? would obey the
relation
?0 d?
V = , (3.14)
2? dt
This is the AC Josephson effect. By taking the time derivative of Eq. 3.13 and expressing
d?/dt in terms of V using Eq. 3.14, we can show that the junction acts as if it has an
inductance,
V ?0 LJ
L(?) = = = . (3.15)
dI/dt 2?Ic cos? cos?
Thus, the Josephson junction can be modeled as a non-linear inductor which varies as a
function of ? with amplitude LJ = ?0/2?Ic. The energy associated with this circuit element
can be found as if it is associated with an inductor,
? ?
Ic?0
E = IV dt = sin?d? = ?EJ cos?. (3.16)
t 2? ?
In Eq. 3.16, we have omitted the constant term after the integration step. In practice, this
constant energy is not important because we can only probe the difference between energy
levels. Here, E Ic?0J = 2? is called as the Josephson energy.
In a circuit, the Josephson junction is visualized as shown in Fig. 3.2(b). Note that a
physical junction has a self-shunting capacitance due to electric field in the tunnel oxide, so
it is more often interpreted as a parallel circuit consisting of a capacitor and a Josephson
element discussed above.
We can write the Hamiltonian for a circuit consisting of an inductor, a capacitor, and a
Josephson junction in parallel as
1
H = 4E n2C + EL?
2 ? EJ cos(?+ ?ext), (3.17)
2
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 59
Figure 3.2: Josephson junction. (a) A tunneling junction is constructed from a weak
link sandwiched between two superconductors. (b) Circuit representation of a Josephson
junction: a cross surrounded by a box. (c) In practice, a Josephson junction has a small
self-capacitance, so it can be interpreted as a parallel circuit consisting of a capacitor with
a Josephson element.
where ?ext = 2??ext? is the reduced external flux achieved by threading the circuit with a0
magnetic field. It is of course only available in circuits with a closed loop. The three parame-
ters EC , EL, EJ can be adjusted by changing the geometry of the circuit elements. Different
combinations of these parameters correspond to various species of superconducting qubits
with distinct properties [18, 80].
3.2 Decoherence in Superconducting Circuits
Superconducting circuits are macroscopic in size, and hence they are typically surrounded
by more impurities and coupled to noise sources more strongly compared to microscopic
ions and atoms, leading to relatively lower coherence times. This section discusses common
decoherence mechanisms in superconducting circuits and how to model their effects on
qubit performances. We also briefly review relevant methods to mitigate 1/f noise [81, 54].
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 60
3.2.1 Energy Relaxation
Dielectric Loss
Dielectric loss is attributed to material imperfections in the dielectric materials near the
qubit. It is believed that the relaxation process occurs via the qubit?s coupling to two level
fluctuators present in the oxide surface interfaces between different layers of the circuit
[30]. This material imperfection can be modeled by introducing a capacitor with complex
dielectric constant  = ? + i??. A quantity called dielectric loss tangent can be defined as
tan ? = ??/?d . For example, an electromagnetic wave that propagates along x in a lossy
material with finite loss tangent has a complex wave number k = kr + iki. The wave decays
as e?kix = e?kr tan ?dx, so the smaller the loss tangent, the further it can propagate.
A cavity resonator filled with a lossy material having permittivity  = ? + i??, and
suffering no other loss, would have a quality factor equal to the inverse of the dielectric
loss tangent, Q = tan?1 ?d [82]. If the cavity is only partially filled with a lossy medium,
then only the electric field inside the material contributes to energy loss. This is the case
for a resonator or qubit with a thin layer of oxide on top of the metal surface. The effective
quality factor Qdiel is then [30]
1 ? pi
= , (3.18)
Qdiel Qi i
where pi is the participation ratio of material i with quality factor Qi = 1tan ? . The relaxationi
time T1 is directly related to the effective quality factor Qdiel, so we shall only consider this
quantity in our general analysis of dielectric loss.
Relaxation process corresponding to a lossy shunting admittance model [70, 83] is de-
rived using Fermi?s golden rule in Appendix A,
??
1 2
ij = 2 |?j|O?|i?| S?(?ij),h?
where O? is the quantum operator that couples the system to the noise, and S?(?ij) is the
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 61
noise spectral density of noise ? that causes the decay at qubit transition frequency ?ij ,
[ ( ) ]
h??
S?(?) = h??ReY?(?) coth + 1 . (3.19)
2kBT
To model the admittance Ydiel associated with dielectric loss, consider a capacitor filled
with lossy dielectric medium. The complex dielectric constant gives rise to a complex effec-
tive capacitance Ceff = C(1+i tan ?d), with the capacitance C related to the charging energy,
EC = e
2/2C. The admittance of the capacitor can be written as Ydiel = i?C(1 + i tan ?d),
from which we have
?C
ReYdiel = . (3.20)
Qdiel
The decay rate of transition i? j at zero temperature T = 0 can thus be written as
?2 C
?diel
2
ij = | ? | | ?|2
ij
?o j ? i . (3.21)
h? Qdiel
Flux Noise Dissipation
It has been reported that flux noise with 1/f spectrum is responsible for energy relaxation
in flux qubits at low frequencies [54, 84]. The noise couples to the qubit via the persistent
current Ip = ?/L and would cause the qubit to decay at a rate [84]
?? 2ij = |?j|Ip|i?| S?(?ij)
(3.22)
1 1
= |?j|?|i?|2S?(?ij).
2e2 L2
Since the inductance in fluxonium is three orders of magnitude higher than in flux qubits
(L = 100? 200 nH compared to L ? 0.5 nH) , this loss mechanism has negligible effect on
fluxonium.
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 62
Quasiparticle Loss
Relaxation in superconducting qubit can be induced by quasiparticles tunneling across the
Josephson junction, absorbing energy from the qubit. Quasiparticle dynamics in supercon-
ducting qubit have been actively investigated [85, 86, 87, 88, 89]. Below we follow refs.
[90, 91] to summarize the general theory for a single junction.
From Fermi?s golden rule, we can write the the relaxation rate corresponding to quasi-
particle tunneling across a junction as
? ?2
?qp ?? ? ??ij = ??j| sin |i?? Sqp(?ij), (3.23)2
where the noise spectral density Sqp describes the quasiparticle environment,
Sqp(?) = ?ReYqp(?)/?gK , (3.24)
with ( )
1 2? 3/2
ReYqp(?) = xqpg , (3.25)
2 h??
where nxqp = qp2? ? is the quasiparticle density normalized by the density of Cooper pairs, g iso
the junction conductance. We can writeEJ = g?/8gK , where gK = e2/h is the conductance
quantum and ? is the superconducting gap.
So for a single junction, the quasiparticle spectral density is inversely proportional to
the square root of frequency. We can then write the relaxation rate explicitly as
? ? ?2
?qp ??? ? ?ij = ?j| sin |i??? ? 8EJ 2?xqp (3.26)2 ?h? h??ij
Although BCS theory predicts that the quasiparticle density would vanish at the temper-
ature of the mixing chamber in a dilution fridge, many experiments have confirmed a finite
xqp value [86, 87]. The effect are thus believed to be due to non-equilibrium quasiparticles
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 63
[88]. Recent experimental results indicate that these non-equilibrium quasiparticles may
come from high energy noise reaching the qubit inside the dilution fridge [89, 92].
3.2.2 Pure Dephasing
Modeling Noise and Dephasing
Pure dephasing can be viewed semi-classically as a process in which the qubit transition
fluctuates in time, and a random phase accumulates between states |0? and |1? (see sec-
tion 2.3), destroying the coherence. The frequency of this process is much lower than the
transition frequency, resulted from the slow fluctuation in time of qubit parameters.
To understand pure dephasing, we may consider a simple two-level system with an
effective Hamiltonian that fluctuates in time,
1
H = (?eg + ??eg(t))?Z , (3.27)
2
where ??(t) is a fluctuation caused by classical noise. Such function can be treated stochas-
tically, where we consider an ensemble of individual fluctuations and assign a probability
and distribution function to ??. Each run of the experiment is affected by one part of this
ensemble. We can further look at the system in the rotating frame with respect to ?eg with
HI = ??(t)?Z/2, considering only the variation of the transition energy in time.
Assuming the fluctuation is small, we can Taylor expand the change in frequency as
? ??eg
??eg(t) = ??(t) +O(??
2), (3.28)
??
{?}
where {?} is the set of qubit parameters, and ??eg?? can be viewed as the sensitivity of the
qubit transition to the respective parameter ?. The accumulated frequency drift at time T
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 64
to first order in ?? is then
? ??? Teg
??eg(T ) = dt??(t). (3.29)
??
{?} 0
We make a further simplifying assumption that the noise is Gaussian distributed, such
that the parameter fluctuation has zero average, ???(t)? = 0, and can be described by a
correlation function ???(t)??(t?)? = G?(t? t?). The free induction decay from fluctuation of
2
parameter ? follows the exponential of the mean-squared phase noise, ?ei??eg? = e????eg?/2,
with ( )2?? T ? T ?
? ??eg??2eg?? = dt dt???(t)??(t?) . (3.30)?? 0 0
Using relation A.9, we can write
( )2 ? ?
? 2 ? ??eg d???eg ? = S?(?)WT (?), (3.31)?? ?? 2?
where WT (?) is the window function,
????? ?T ???2 2?i?t sin (?T/2)WT (?) = dte = . (3.32)
0 (?/2)
2
For noise tha?t is roughly constant with amplitude S? in the interval [0, 2?/?c], we can2
use the relation ? dx sin x?? 2 = ? to write ?ei??eg? = e??2T wherex
( )
?? 2eg S?
?2 = . (3.33)
?? 2
Generally speaking, in order to define a ?dephasing rate?, we must consider an evolution
that is ?filtered? in the time window of the measurement [93].
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 65
Dynamical Decoupling
There are certain techniques developed within the NMR community to mitigate noise that
has long correlation time [81]. One such strategy is to use the spin echo sequence [94], in
which a short ?-pulse is used to flip the spin in mid-evolution. Then, the effect of slow-drift
in the time interval [0, T/2] is reversed and canceled out in the second half of the evolution
[T/2, T ]. This modifies the window function as
????? ?2? T/2
? T ?
W (?) = dte?i?t ? dte?i?t?T ? ?0 ? T/2 ? ?1 ? 2= 2ei?T/2 ? ei?T ? ?12 ? (3.34)?
2
= tan2
sin (?T/2)
(?T/4) .
(?/2)2
From Eq. 3.34, we see that the spin echo sequence filters out the low frequency part
of the noise spectrum S?(?) via the factor tan2(?T/4), so we can think of WT (?) as a
filter function which depends specifically on the sequence of pulses we use to manipulate
the qubit. As shown in Fig. 3.3, the spin echo technique cancels out the very slow drift
accumulated during the long measurement. Therefore, coherence time measured using
echo is the most important T2 time.
The spin echo technique can be generalized to include an odd-number 2N?1 of ?-pulses
during the otherwise free induction decay. By inspecting the 2N integrals and canceling out
the repeating terms, we can generalize Eq. 3.34 to
1 ??? ?2W (?) = 2ei?T/2N i?T ?T ? e ? 12 ??
2 (3.35)
= tan2
sin (?T/2)
(?T/4N) .
(?/2)2
If the correction pulses are in the same quadrature as the ?/2-pulses, the sequence is
called Carr-Purcell (CP) [95], and if they are in the quadrature orthogonal to the ?/2-pulses,
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 66
Figure 3.3: Window function. WT=100 ?s(?) for free induction (N=0), echo (N=1), and
CPMG with 3, 5, and 7 refocusing pulses.
the sequence is called Carr-Purcell-Meiboom-Gill (CPMG) [96]. These techniques have
been applied to flux qubits to enhance coherence times limited by slow drift [81, 54]. In
practice, the pulses have finite length and are imperfect, hence beyond an optimal number,
the refocusing pulses may contribute to the qubit?s dephasing.
1/f Noise
Various noise sources in solid state systems display a somewhat similar 1/f behavior [97,
32, 98, 31, 33]. Prediction of decoherence due to this type of noise is complicated due to its
long correlation time. The measurement frequency band is limited, for instance by the finite
measurement time tm, so during the measurement, the noise source may not reach equi-
librium. In addition, 1/f noise cannot be generally assumed to be stochastically Gaussian.
Some deviations from this assumption have been reported in refs. [99, 100]. Analytical
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 67
model regarding this type of noise typically involves the ?spin fluctuator? approach, which
can be solved exactly under general conditions. From this model, ?weakly? coupled fluc-
tuators can be approximated as a Gaussian process. Averaging over an ensemble of such
fluctuators yields a 1/f spectral density. Fluctuators with frequency < 1/tm are considered
strong and slow, and hence not effective. We will work within this approximation, and
consider the noise spectral density for 1/f flux noise as
2?A2
S(?) = | | , (3.36)? ?
with noise amplitude A = 10?6 ? 10?5?o [32, 33] and power 0.8 < ? < 1.3 [54].
The 1/f law does not need to be valid spanning across the entire frequency spectrum. We
can merely assume it holds in the range of relevant frequency in our experiments ?low < ? <
?high. The limiting frequencies depend on the specific architecture and experimental setup.
In our measurements, ?low is defined by the total measurement time ?low = 2?/Tmeas ?
2??0.1 rad Hz, and ?high by the total free induction time ?high = 2?/TFI ? 2?[106?2?104]
rad?Hz . The mean square phase variation is then
( )2 ? ?
? 2 ? ??eg 2 d???eg ? = A | | W? T (?). (3.37)?? ?? ?
For free induction decay WT , we can write
( ) ?
?? 2 ?? 2 ? eg d???eg ? = 2 A2T 2 sinc2(?T/2). (3.38)?? 0 ??
In the limit T  1/?, we can use the static approximation sinc(?T/2) ? 1. This happens
when the noise spectrum has a peak at low frequency. In the simple case ? = 1, the
2
qubit phase decays following a Gaussian profile in time, e?(??T ) , with a dephasing rate ??
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 68
corresponding to noise source ?,
( )?
??eg
?? = S
?, (3.39)
?? ?
where the noise spectral density can be approximated as
? ?high
S? ? 2 d? ?high? A = A2ln . (3.40)
? ? ?low low
On the other hand, for a wideband 1/f noise at T > 2?/?high, we replace the higher limit
with 1/T and get ? 1/T
? ? 2 d? 2?S? A = A2ln . (3.41)
? ? ? Tlow low ?
The pure dephasing time can be defined as T? = 1/??, where ?? = ? ??.
Equation 3.41 shows that if we want to obtain a nice signal for a Ramsey (free induc-
tion) decay by increasing the averaging time, we will get a logarithmically shorter coherence
time. Empirically, averaging many decaying oscillations at different frequencies would re-
sult in a faster decay. Hence, a long Ramsey measurement shows a time scale that is actually
lower than the realistic coherence time T2.
As shown in Fig. 3.4, a Gaussian curve seems to decay faster in time compared to an
exponential. Hence, fitting the data that follows a Gaussian profile to an exponential would
give an incorrect dephasing time.
The use of echo decoupling sequence introduces a tan term to the window function
[98], hence we get
( )2 ?
? ??
?
eg
??2 ? = 2 A2T 2 d(?T/4)eg ? ( ) sin
3(?T/4)
?? 0 (?T/4)
3
(3.42)
?? 2eg
= 2 A2T 2 ln 2,
??
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 69
Figure 3.4: Exponential versus Gaussian decay. For the same time constant T? = 10 ?s,
the Gaussian curve shows a faster decay compared to an exponential function. We should
therefore choose the right function to fit the data to extract the correct coherence time.
and the dephasing rate can be written analytically as
( )
?? ?eg
?? = A ln 2. (3.43)
??
This formula is exact and can be conveniently used to estimate flux noise amplitude when
the dominant dephasing source is 1/f noise.
1/f Noise: Second Order Effect
At the symmetry bias point, for example when the flux threading the loop in fluxonium
is at half integer flux quantum, superconducting qubits are first-order insensitive to the
respective noise [33]. Such bias point is referred to as sweet spot. However, the qubit is
still sensitive to higher order noise effect. Here we consider how the second-order noise
may dephase the qubit.
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 70
The second order term that was left out of Eq. 3.28 can be written as
? 2 ? T
??(2)
1 ? ?eg
eg (T ) = dT (??(T ))
2, (3.44)
2 ??2
{?} 0
and the dephasing can be approximated in the low (?low < ? < 2?/T ) and high (?high >
(2)
? > 2?/T ) frequency limits using linked-cluster series expansion [98, 101], e?i??eg (T )? =
f lffhf . The contribution from the low frequency part is given as
f lf
1
? (T ) = ? . (3.45)1? 2i(?2?eg/??2)A2T ln(2?/?lowT )
For the high frequency part at long times, we can approximate T  [(?2?eg/??2)A2/2]?1
and find ( )
hf ? ?
2?eg
f 2? (T ) = exp ? A T . (3.46)2 ??2
So the resulting decay follows the combination of a power law at short time and an ex-
ponential law at long time. Note that only the real part of f lf should be considered in
practice.
For the echo decay, assuming a sharp cutoff at ?high, we get
1
fE(T ) = ? . (3.47)
1 + (1/8)(?2? 2 2 4 2 4eg/?? ) A ?high ln(1/?lowT )T
These equations seem daunting, but there is a timescale that we can prescribe as the coher-
ence time due to second order noise effect (2)T? at long time. The corresponding dephasing
rate in Eq. 3.46 can be written as
2
(2) ? ? ?eg? 2?,1/f A . (3.48)??2
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 71
It is a good approximation in our system to treat the time decay in a typical echo mea-
surement to be exponential, assuming the lower frequency part is efficiently filtered out
by a ?-pulse. The easiest way to check if this is a good approximation is to compare the
frequency dependence of the window function for the echo characterized by T = 100 ?s
with (2)??,1/f .
More importantly, we can estimate the dephasing time limit corresponding to second-
order noise using Eq. 3.48. To accomplish this, we have to compute the second-order
?2sensitivity ?eg2 , and extract the noise amplitude A by measuring the coherence time away??
from the sweet spot (see Fig. 6.13(a), for instance).
Photon Noise
When dispersive readout is used in cQED [22, 62], the number of photons at the cavity fre-
quency can shift the effective qubit frequency. For example, consider the Jaynes-Cumming
Hamiltonian in the dispersive limit ?  g (Eq. 2.54). In this case, the qubit transition
frequency is shifted by n?, where n is the cavity photon number and ? = 2g2/? is the
dispersive shift. As n fluctuates in time, it changes the qubit frequency randomly, and hence
dephasing occurs.
In an ideal scenario with the cavity in thermal equilibrium at 10 mK, we expect the
average number of thermal photons at the resonant frequency fr ? 10 GHz to be very small
(see Fig. 3.5), following the Boltzmann distribution
1
n ?21th = = O(10 ). (3.49)
eh??r/kBT ? 1
So, we may naively expect almost no photons in the the cavity at 10 mK. However, due
to a combination of imperfect shielding, isolation, and thermalization, there exists a finite
photon population which can dephase the qubit.
Following refs. [102, 103], the thermal photon dephasing rate for cavity-qubit system
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 72
Figure 3.5: Effective thermal photon number. Thermal photon number nth as a function
of frequency (in GHz) at 100 K, 5 K, 1 K, and 100 mK.
with dispersive shift ?01 is given by??( ) ?
? 2 efftot ? i?01 4i?01n?th = Re 1 + + th ? 1? . (3.50)
2 ?tot ?tot
In the small photon number limit, the rate is reduced to
neff 2
? = th
??01
th 2 . (3.51)?2 + ?01
For large ?tot/? ratio, Eq. 3.50 simplifies to
?2 neff ( )
? = 01 th neffth th + 1 , (3.52)?tot
and for large ?/?tot, it saturates to
?th = ?totn
eff
th . (3.53)
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 73
Critical Current Noise
Finally, since superconducting qubits are built from artificial Josephson junctions, certain
properties within the junctions themselves can change over time, resulting in dephasing
[104, 105, 106]. For instance, some change in the tunneling conductance directly causes
fluctuation in the critical current, and thus EJ and EL.
Following ref. [106], the critical current noise spectrum obeys the phenomenological
relation
SE SJ R= n
SI
= c ? 1 T 12 10?13Hz?1, (3.54)E R2 2 2J n Ic A/?m 1K f
exhibiting a 1/f dependency, scaling proportionally with temperature and inversely with
junction area. However, their measurement is limited to the range around 1 Hz and tem-
perature around 315 mK.
At the time of writing of this manuscript, little is understood about critical current noise
at low temperatures, and contradicting results have been reported for circuits at kelvin-
range temperatures [104, 105, 106]. Future experiments in which qubit coherence is not
limited by other mechanisms may open the door to more rigorous investigation on junction
noise.
3.3 Quantum Gates and Readout
3.3.1 Qubit Control
From Eq. 2.36, we can come up with a convenient method to perform X and Y control of
superconducting qubits. Denoting I = d cos(?d) as the in-phase and Q = d sin(?d) as
the quadrature components of the applied microwave field with amplitude d, the resonant
(? = 0) driving Hamiltonian in the rotating frame takes the form
h??
Hd = (I?X +Q?Y ), (3.55)
2
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 74
such that an in-phase signal rotates the qubit around the X-axis, and a quadrature signal
rotates the qubit around the Y -axis. In microwave electronics, this is accomplished by using
an IQ mixer to multiply the carrier signal at the qubit frequency and the pulse generated by
an arbitrary waveform generator (AWG). Note that the choice of absolute phases of I and Q
components is arbitrary, and only the phase difference between them is important.
Since superconducting circuits are not perfect two-level systems, other eigenstates can
be excited by the applied microwave signal. An important technique to mitigate this prob-
lem is to turn on the applying field slowly by using an I or Q pulse with an envelope s(t),
which is typically Gaussian. This effectively reduces the spectral width of the pulse (see
Fig. 5.11(a)). The effective rotating angle is thus
? t
?(t) = ? s(t?)dt? (3.56)
0
A Rabi experiment (section 6.5) is performed by varying either the pulse?s width or am-
plitude to find the correct rotation angles. Another experiment to tune the gate parameters
can be done by measuring the qubit after applying a sequence consisting of an odd-number
of ?/2-pulses on the qubit. This allows us to determine which is the best gate configura-
tion that minimizes over- and under-rotation. When short pulses are desired, pulse shaping
techniques such as Derivative Removal by Adiabatic Gate (DRAG) [107, 108, 58] can be
used to minimize gate errors from other energy levels.
Z-control of a qubit can be done using a sequence of gates, either by applying an off-
resonant X or Y pulses, or by implementing a virtual-Z gate as shown in ref. [109].
3.3.2 Readout
In this thesis, we use the standard dispersive readout technique using a cavity coupled to
a qubit system [22, 62, 63]. The dispersive shifts depend on the specific qubits, but the
principle is essentially described by Eq. 2.53. For a general discussion here, we assign
CHAPTER 3. SUPERCONDUCTING CIRCUITS: MACROSCOPIC ARTIFICIAL ATOMS 75
Figure 3.6: Dispersive readout. (a) Transmission amplitude and (b) phase of cavity signal
across the resonance. The cavity is shifted by ?0 = ??/2 when qubit is in ground state |0?,
and by ?1 = ?/2 when qubit is in excited state |1?.
?0 = ??/2 and ?1 = ?/2, such that ?01 = ?. The readout scheme is illustrated in Fig. 3.6.
The qubit shifts the cavity resonance differently when it is in the ground or the excited
states. By probing the cavity at a frequency near its resonance, we can infer the state of the
qubit.
Generally speaking, a multi-level system shifts the cavity by ?i when it is in level i, such
that the cavity resonance is modified as ?c ? ?c + ?i. The different shifts can be used to
readout different states of a single qubit [110] or multiple qubits [111, 112]. As discussed
in section 6.4, we have experimentally demonstrated this principle with a two-qubit device.
Rigorous description of superconducting qubit readout and amplification can be found
in refs. [83, 113]. We describe our IQ demodulation technique in more details in sec-
tion 5.3.
Chapter 4
Fluxonium Theory
Nothing is ever so expensive as what is offered for free.
Viet Thanh Nguyen
This chapter provides a theoretical background for the experimental discussion in Chap-
ter 6. First, in section 4.1, we discuss the range of parameters EJ , EC , EL that defines
fluxonium and inspect the typical spectra and matrix elements. We then follow up with
decoherence analysis in section 4.2 and dispersive readout using cavity-fluxonium system
in section 4.3. Finally, we discuss coupled fluxonium qubits and the microwave-activated
controlled-Z gate in section 4.4.
4.1 Spectrum and Selection Rules
4.1.1 Fluxonium with a Single Small Junction
Fluxonium circuit consists of a Josephson junction with energy EJ shunted by a capacitance
with charging energy EC and inductance with inductive energy EL. Let us recall that the
76
CHAPTER 4. FLUXONIUM THEORY 77
Hamiltonian for this circuit (see Eq. 3.17) can be written as
H = 4E n2
1 2
C + EL? ? EJ cos(?+ ?ext)
2
where n and ? are the charge number (or normalized charge) and phase (or normalized
flux) operators, respectively. In the harmonic oscillator basis, they can be written as (see
Eq. 3.12) ( ) 1
4
? = ?1 8EC (a?( ) + a),2 EL 1
?i EL
4
n = (a? ? a).
2 8EC
The parameters must satisfy EJ/EL  1 and, opposed to the previous experiments
[59, 85], EJ/EC ? [1, 10]. The energies ratios distinguish fluxonium from other induc-
tively shunted junction devices. To satisfy the requirement on the small inductive energy,
a superinductance with inductance per unit length of 104 ?0 constructed from a chain of
N ? 102 Josephson junctions with area of approximately 1 ?m2 is used. One can thus
interpret fluxonium as an inductively shunted charge qubit which is insentitive to charge
noise, or a generalized flux qubit with decreased sensitivity to flux noise, where the first-
and second- order coupling to flux noise is suppressed as 1/N and 1/N2, respectively.
There are certain physical constraints on the circuit parameters. First, it is hard to
make the charging energy EC larger than 10 GHz because of the stray capacitance from
the junction leads. Likewise, it is hard to reduce the Josephson energy EJ below 2 GHz,
since we cannot fabricate junctions with arbitrarily small area reliably. In our experiments,
the small junction?s size goes from 90 nm ? 90 nm to 150 nm ? 150 nm, corresponding to
EJ ranging from 2 to 7 GHz. Note that EJ also depends on the oxidation conditions. The
typical inductive energy EL in high coherence fluxonium is around 0.5? 1 GHz.
In this section, the superinductor is treated as a lumped circuit element [59, 114]. With
this assumption, the circuit is analogous to a particle in a box, where EC sets the magnitude
CHAPTER 4. FLUXONIUM THEORY 78
Figure 4.1: Eigenenergies and potential energy for different EJ values. Adjusted
eigenenergies (horizontal lines) and potential energy (thick lines) for different EJ values,
showing the more characteristic fluxonium properties at high EJ values: high anharmonic-
ity, low 0? 1 splitting at half flux quantum bias.
CHAPTER 4. FLUXONIUM THEORY 79
Figure 4.2: Eigenenergies and potential energy for different EL values. Adjusted
eigenenergies (horizontal lines) and potential energy (thick lines) for different EL values.
of the kinetic energy, while EJ and EL shape the potential energy, with the ratio EJ/EL
determining the number of localized wells. In addition, the external flux can be used to
tune the relative position of the neighboring wells. We can gain intuition about the system
by inspecting the potential wells and energy levels when the circuit is biased at integer and
half-integer flux quantum. The results here were simulated numerically using QuTip [115,
116]. A more rigorous approach with analytical formulas can be found in refs. [117, 118].
We first consider the impact of the Josephson energy EJ . Since it is the non-linear term,
the higher EJ , the more anharmonic the system becomes, and the higher the potential
wall between the wells is. This subsequently makes the energy splitting between otherwise
degenerate states decrease with increasing EJ , as shown in Fig. 4.1.
The inductive energy EL, on the other hand, decreases the splitting at higher values,
as shown in Fig. 4.2. This suggests that the ratio EJ/EL plays a more important role than
CHAPTER 4. FLUXONIUM THEORY 80
Figure 4.3: Eigenenergies and potential energy for different EC values. Small EC
makes fluxonium similar to transmon at integer flux, and to flux qubit at half-integer flux.
each energy parameter?s absolute value.
Finally, the charging energyEC sets the overall scale for the kinetic energy. It determines
the plasma frequency scale ?p, affects the tunneling rate across the potential well [118],
and also sets the anharmonicity of the system at integer flux [49]. On one hand, the small
transition frequency between the two lowest states depends strongly on the EJ/EC ratio,
because the tunneling rate is exponentially sensitive to this ratio, as predicted by the WKB
approximation. On the other hand, the higher levels are separated by the plasmon gap,
?
roughly given by 8EJEC , and reside at much higher energies.
In general, there are two types of transitions in fluxonium when we move across the
spectrum in flux, as shown in Fig. 4.4. The first type is called plasmon, since it is simply a
plasma oscillation between states localized in the same potential well, and is thus associated
with a large matrix element. The second type is called fluxon, because it is associated with
CHAPTER 4. FLUXONIUM THEORY 81
Figure 4.4: Plasmon and fluxon transitions. (a) Typical transition frequency spectrum
of a device with EJ = 5 GHz, EC = 1 GHz, and EL = 1 GHz. Plasmon transitions
are relatively flux-independent, while fluxon transitions are strongly flux-dependent. (b)
Wavefunctions and potential energy showing plasmon (blue to blue and red to red) transi-
tions and fluxon transition (blue to red or vice versa). A plasmon is just plasma oscillation
between states localized in the same well, hence it is allowed, whereas a fluxon transition
happens through the tunneling across the potential barrier, and is thus almost forbidden.
flux tunneling between states localized in adjacent wells. For EJ  EC , the tunneling
amplitude is exponentially small, and the fluxon transition is nearly forbidden away from
the half-integer flux bias.
We simulate the charge and phase matrix elements and plot the results in Fig. 4.5. Note
the vanishing of the 0 ? 2 matrix elements at the symmetric flux bias (interger and half-
integer flux quantum). This is due to symmetry of the potential seen by the ? variable.
In fact, any even-number quanta of energy exchange is forbidden at the symmetry point,
including any |i? ? |i + 2?, qubit-cavity sideband transitions, and qubit-qubit double exci-
tations such as |00? ? |11?. It is possible to use higher order process, such as two-photon
excitation or Raman driving, to manipulate these forbidden transitions.
CHAPTER 4. FLUXONIUM THEORY 82
Figure 4.5: Simulations of matrix elements. (a) Charge number matrix elements for
0? 1, 1? 2, and 0? 2 transitions in a fluxonium circuit with EJ = 5 GHz, EC = 1 GHz,
EL = 1 GHz. The 0? 1 transition dipole is flat for the plasmon region, declines sharply in
the fluxon region, and reaches a constant around the symmetric flux point at half-integer
flux quantum. The 0 ? 2 dipole vanishes at the symmetry flux bias due to selection rules.
(b) Phase matrix elements for the same circuit. The 0 ? 1 phase matrix element also
declines sharply around the fluxon region, but reaches a maximum at the half-integer flux
quantum bias, due to strong overlapping at low frequency between the hybridized states.
The 0 ? 2 phase matrix elements also vanish at symmetric flux due to selection rules. (c)
0 ? 1 charge matrix elements in the same circuit but for different EJ values. The dipole
moment for the fluxon is smaller for higher EJ . The plasmon, however, has a larger charge
matrix element for higher EJ values. (d) 0 ? 1 phase matrix element in the same circuit
but for different EJ values. The fluxon transition matrix element again becomes smaller for
higher EJ . The phase matrix element rises sharply at the symmetry flux point for higher EJ
due to stronger hybridization.
CHAPTER 4. FLUXONIUM THEORY 83
Figure 4.6: Split-junction fluxonium circuit. Circuit diagram showing a fluxonium with
two parallel tunnel junctions. The phases across them are labeled ?1 and ?2, whereas the
phase across the inductor is labeled ?.
The small fluxon dipole moments produced by increasing EJ is intriguing. It suggests
an experiment where we can simply tune EJ and see if the matrix elements vanish. This can
be accomplished by introducing a SQUID loop (split-junction) [49]. The experiment [60]
is discussed in section 6.2. We develop the theory for the split-junction fluxonium below.
4.1.2 Split-Junction Fluxonium
It turns out that we can employ a SQUID loop in fluxonium to effectively tune the Joseph-
son energy EJ . The circuit now consists of two loops, as depicted in Fig. 4.6. Let ?1,2
be the phase differences across the right and left junctions respectively, we can write the
Hamiltonian as
H = 4E n2
1
+ E ?2C L ? EJ1 cos(?1)? EJ2 cos(?2). (4.1)2
The magnetic fluxes ?s and ?ext threading the SQUID loop and the main circuit loop
impose the constraints on the phase differences accordingly,
?s
?1 ? ?2 = 2? = ?s,
?o (4.2)
? ?ext? ?1 = 2? = ?ext.
?o
CHAPTER 4. FLUXONIUM THEORY 84
Figure 4.7: Split-junction fluxonium spectrum. Transition energy spectrum of a split-
junction fluxonium circuit with EJ? = 10 GHz, EC = 1 GHz, EL = 1 GHz, EJ1 ? EJ2 =
1 GHz, and the flux through the SQUID is 10 times smaller than the flux through the main
loop at any total flux bias, ?ext = 10??s. Note that the anti-crossing between plasmon and
fluxon becomes larger as the flux through the SQUID reduces the effective EJ .
Figure 4.7 shows an example spectrum of split-junction fluxonium circuit with EJ? =
EJ1 + EJ2 = 10 GHz, EC = 1 GHz, EL = 1 GHz, EJ1 ? EJ2 = 1 GHz. In this circuit, the
flux through the small loop is 10 times smaller than the flux through the big loop at any
total flux bias, ?ext = 10? ?s. We can observe that the anti-crossings between plasmon and
fluxon become larger when the flux through the SQUID reduces the effective EJ .
The corresponding charge number and phase matrix elements are plotted in Fig. 4.8.
CHAPTER 4. FLUXONIUM THEORY 85
Figure 4.8: Split-junction fluxonium matrix elements. (a) The charge number and (b)
phase matrix elements of the 0? 1 transitions in the split-junction device. As the effective
Josephson energy becomes smaller toward the symmetry flux point, both the charge and
flux couplings for the fluxon transition become larger. The charge coupling for the plasmon
transition also becomes smaller, consistent with Fig. 4.5(c).
When the effective Josephson energy EJ becomes smaller, the fluxon?s matrix elements
become larger as the transition becomes less forbidden. On the other hand, the plasmon
charge matrix element becomes smaller, consistent with the single tunneling junction flux-
onium analysis (see Fig. 4.5(c)).
4.2 Decoherence Mechanisms in Fluxonium
In this section, we develop the theoretical tools to model different decoherence effects in
fluxonium qubits, starting with relaxation mechanisms. Flux noise is discussed separately in
section 3.2. We end the section by describing briefly relevant parameters in our experiments
related to thermal photon noise.
CHAPTER 4. FLUXONIUM THEORY 86
4.2.1 Dielectric Loss
We begin by applying Fermi?s golden rule (see Appendix A) to a superconducting circuit at
finite temperature T and write the relaxation rate as
?
h??2 ? ( ) ??
?diel
ij 2 ? h??ij ?
ij (?ij) = |?i|?|j?| ?coth + 1? , (4.3)4ECQdiel(?ij) 2kBT
where Qdiel(?ij) = 1/ tan ?C is the effective quality factor of the lossy capacitance, and EC
is the charging energy of the circuit.
There are two obvious ways to reduce the dissipation rate: first, by de-coupling it from
the noise source, which we demonstrate in section 6.2, and second, by increasing the effec-
tive dielectric quality factor. To reduce tan ?C , we can turn to either material engineering
[44, 46], or antenna design [30]. Another approach is to tune the qubit to work in the low
frequency regime where relaxation is slowed down, which we demonstrate in section 6.3.
Quantitatively, this comes from the quadratic frequency term in Eq. 4.3. We simulate T1
limited by dielectric loss and show the results in Fig. 4.9.
It?s also worth noting that at low frequency, the dielectric quality factor is likely bet-
ter [119], which gives additional advantage to low frequency qubits if the loss is dominated
by dielectric loss. To account for this effect, we model the quality factor to have a small
frequency dependence accordingly,
( )
tan ? C(?) ?
? = , (4.4)tan ?C(? = 2? 6 GHz) 2? ? 6 GHz
where the larger the  power, the better it is for fluxonium.
Chain Dielectric Loss
An additional feature of fluxonium circuit compared to other superconducting qubits is the
inductor, which we have so far treated as a lumped circuit element. In our devices, the
CHAPTER 4. FLUXONIUM THEORY 87
Figure 4.9: Simulation of T1 limited by dielectric loss. Relaxation time limited by di-
electric loss simulation results for (a) different EL values which mostly affect the qubit?s
frequency, and (b) different effective dielectric quality factors.
inductor is a chain of junctions with metal parts covered with dielectric, so it inevitably
introduces additional shunting capacitances with finite loss tangent. Below we consider
two types of possibly lossy spurious capacitors associated with the chain, and discuss how
they may contribute to relaxation.
First, we can approximate the loss from the self-capacitance of a chain of junctions by
applying Eq. 4.3 to the junctions in the chain. Denoting the array using subscript A, we can
write
?N 2 ? ( ) ?
diel h??A ij
? ?
?ij (?ij) = |?
?
i| |1?|2 ? h??ijcoth + 1?
4ECAQA(?ij) N?? (
? )2kBTi 2 ?? ?
?
(4.5)
h??ij | |2 h??ij
?
= ?01 coth + 1?? ,4NECAQA(?ij) 2kBT
which is inversely proportional to the number of junctions in the chain. In our fluxonium
devices, all the chain junctions have dimensions 0.4 ?m? 2 ?m, giving CJA ? 36 fF, using
the empirical formula C 2J ? 45 fF/?m . From our relaxation data, we can extract the bound
for loss tangent of the junction oxide layer, which we assume to be tan ?A = tan ?C/N .
These values are much smaller than the value reported in literature (tan ? ?3AlOx ? 10 )
CHAPTER 4. FLUXONIUM THEORY 88
Figure 4.10: Modeling the shunting capacitance of the array. (a) Infinite transmis-
sion line terminated by a short is a good model of the array of Josephson junctions. (b)
Equivalent circuit of an inductor in parallel with a capacitor. The capacitor C ? effectively
contributes to the total shunting capacitance of fluxonium circuit.
(see Table 6.1), indicating that the junction oxide is not as lossy as commonly believed at
frequencies < 1 GHz.
Another dielectric loss channel in the Josephson junction array comes from the ground
capacitance between the parallel conductors. To model this, let?s first consider an infinite
and lossless transmission line terminated by a short (see Fig. 4.10(a)), which roughly mod-
els the array. A transmission line having inductance an?d capacitan?ce per unit length Ll and
Cl, respectively, has a characteristic impedance Zo = Ll/Cl = L/C, where L = Ll ? l,
C = Cl ? l. The input impedance of the transmission line is then
Zin = jZo tan(?l), (4.6)
? 3
where ? ? ? LlCl is the wave number. Expanding tanx = x + x3 + O(x5) in Eq. 4.6, we
get
( )
?2LC
Zin ? j?L 1 + . (4.7)
3
CHAPTER 4. FLUXONIUM THEORY 89
Next, we consider the simple effective circuit in Fig. 4.10(b), where the shunting capaci-
tiance is C ?. The input impedance for this circuit is
i?L?
Z ? 2 ? ?in = ? ? ? ? j?L (1 + ? L C ), (4.8)1 ?2L C
where we assume the circuit?s resonance is much higher than the frequency range of inter-
est, i.e. ?2  1/L?C ?. From Eq. 4.6 and Eq. 4.8, we get L? = L and C ? = C/3. In our
fluxonium circuits, the chain is made up of N junctions divided into two parallel lines (see
Fig. 6.10), with the shunting capacitance per array junction Cg ? 36 aF, determined from
measurement of chain modes [120, 121]. The effective total capacitance is thus C = NCg/2
for the transmission line, and C ? = NCg/6 for the effective circuit model. For a typical ar-
ray consisting of 100 junctions, the estimated shunting capacitance from the chain is thus
C ? ? 0.6 fF.
In our fluxonium designs, Cshunt ? 20 fF C ?, where Cshunt is the shunting capacitance
of the antenna. This tells us that the chain has a very small effect on the total capacitance.
The ratio C ?/Cshunt gives us a limit on the effective dielectric quality factor associated with
the array. While the exact quantitative value of the array?s participation ratio has not been
determined, we can safely conclude that (i) the chain?s effective dielectric loss must be
> 20 times worse than the antenna?s to degrade the circuit?s performance, (ii) improving the
circuit material should also increase the array?s effective quality factor, and (iii) contribution
from the array can be mitigated by constructing fewer junctions or making wider chain
which has lower Cg.
4.2.2 Stimulated Emission
When the qubit transition frequency is low, we have to take the effect of temperature into
account. As discussed in Appendix A, a resistor at finite temperature can produce stimulated
emission and absorption, namely, it can relax and excite the qubit. In other words, it not
CHAPTER 4. FLUXONIUM THEORY 90
Figure 4.11: Effect of temperature on energy relaxation. The decay enhancement factor
Eth in Eq. 4.9 is plotted for (a) Different qubit frequencies as a function of temperature, and
(b) Different temperatures as a function of qubit frequency. As long as the qubit frequeny
is in the 0.5-1 GHz range and the temperature is kept reasonably low, thermal effects on
relaxation rate remain minimal.
only increases the relaxation rate, but also introduces an excitation rate. Both excitation
and relaxation contribute to the decay to equilibrium ?1 = ?? + ??. A hot qubit that has a
low transition frequency will thus be a bad qubit.
We can examine the effect of temperature by considering the quantity
(???? ( ) ? ? ( ) ?)E h??ij ??? ??? ?h??ij ?th = coth + 1 + coth + 12k T 2k T ?? /2, (4.9)B B
where the factor of 2 corresponds to the case when temperature is zero. This can be con-
sidered as the enhancement factor. It is plotted in Fig. 4.11. This shows that if the qubit
frequency is not too low (0.5 GHz is the typical threshold), the adverse effect from a finite
temperature of 20 mK is minimal, and we can gain a quadratic improvement in relaxation
time T1 by operating at low frequency.
To account for temperature effects, the relaxation rate in Eq. 4.3 is analyzed at temper-
ature T = 20 mK. It?s likely that the effective temperature of the qubit is higher. Note that
this can be compensated in our analysis by increasing the power factor  of the dielectric
CHAPTER 4. FLUXONIUM THEORY 91
loss tangent in Eq. 4.4.
4.2.3 Quasiparticle Tunneling
Quasiparticles have been reported to affect the coherence of various types of superconduct-
ing circuits [87, 85, 88, 89, 54]. For the quasiparticles tunneling across the small junction,
the corresponding relaxation rate using Eq. 3.23 is
? ? ?2
qp ??? ?? | ? | ??? ? 8EJ 2??ij = j sin i xqp . (4.10)2 ?h? h??ij
Notably, the matrix element coupling the qubit to quasiparticle tunneling is suppressed at
the half-integer flux quantum bias, and fluxonium qubit becomes insensitive to this quasi-
particle tunneling across the small junction (Fig. 4.12(a)).
We can extend Eq. 4.10 to account for quasiparticle tunneling across the chain junctions.
Consider the general case in which M + 1 junctions separating M + 1 islands form a loop.
When the inductance from the lead is much smaller than LJ , the phases across the junctions
obey the flux quantization constraint:
M?+1 ?e
?? = 2?k , (4.11)
?o
?=0
where k takes non-zero integer values. The transition rate then follows from Eq. 3.23:
M?+1 ???? ( ) ?2?qp ?ij = ?j| ??sin |i??? Sqp(EJ , ?ij), (4.12)2 ?
?=0
In fluxonium, an array of identical junctions of large Josephson energy EJ1 = EJ2 =
... = EJ is connected to a weaker junction EJ0  EJ1 . This gives EM+1 L ? EJ1/M . Here,
we use the convention that ? = 0 corresponds to the small tunneling junction. It is safe
to assume that the phases across the array junctions are equal ??=6 0 = (?o ? 2??e/?o)/M .
CHAPTER 4. FLUXONIUM THEORY 92
Since individual phases across large array junctions are small, we can linearize the sin func-
tion for ?? 6=0. We further approximate that the quasiparticle density on the array junctions
are identical, so the relaxation rate due to quasiparticles in the array is
?? ?? ?2
?qp,array = ??ij ?j| |i??2 ? Sqp(EL, ?ij), (4.13)
with Sqp(EL, ?ij) calculated by replacing EJ with EL, starting from Eq. 4.10. In deriving
Eq. 4.14, we use the property ?j|2??e/?o|i? = 0 for i 6= j. We can write the explicit formula
as
? ?2 ?
?qp,array
?? ?
ij = ??j|? | ??? 8EL 2?i xqp . (4.14)2 ?h? h??01
In contrast to a single junction, the relaxation from quasiparticles in the array scales
quadratically with the matrix element ?ij , which is maximal at half-integer flux quantum
for the 0? 1 transition (Fig. 4.12(c)). We note that for the same T1 limit, the quasiparticle
density xqp in the array has to be much less than that around the small junction.
Interestingly, the 0 ? 2 transition is maximally sensitive to quasiparticles tunneling
across the small junction (Fig. 4.12(b)) but insensitive to quasiparticle effect in the array
(Fig. 4.12(d)). In addition, we can recall that 0 ? 2 linear decay is suppressed due to
selection rules. Thus, measuring T1 for both 0? 1 and 0? 2 simultaneously in fluxonium
would allow us to test for quasiparticle effects more rigorously.
?
A double loop fluxonium is an excellent testing device for quasiparticles. The sin {1,2}2
matrix elements never reach vanishing values simultaneously when we sweep the flux due
to the finite value of flux through the small loop, as shown in Fig. 4.13. Interestingly, when
the flux through the small loop reaches half-integer flux quantum value, the phases across
the two small junctions are different by ?, and one matrix element reaches a maximum,
while the other reaches a minimum. As we move away from this flux bias, the situation
CHAPTER 4. FLUXONIUM THEORY 93
Figure 4.12: Simulation of T1 limited by quasiparticle. Simulations performed using
circuit parameters EC = 1 GHz, EL = 1 GHz, and EJ = 3 GHz. (a) T 0?11 limited by quasi-
particles tunneling across the small junction. (b) T 0?21 limited by quasiparticles tunneling
across the small junction. (c) T 0?11 limited by quasiparticles tunneling in the chain. (d)
T 0?21 limited by quasiparticles tunneling in the chain. (e) T
0?1
1 limited by both quasiparti-
cles processes in (a) and (c). (f) T 0?21 limited by both quasiparticles processes in (b) and
(d).
CHAPTER 4. FLUXONIUM THEORY 94
Figure 4.13: Matrix elements for quasiparticle relaxation in a split-junction device.
?
Simulation of the sin {1,2}2 matrix elements in the split-junction device with the same pa-
rameters discussed in section 4.1.2. Due to the finite flux through the SQUID, the matrix
elements never have the same vanishing values at the same flux. When the flux through the
small loop reaches half-integer flux quantum value, one of the matrix elements reaches its
maximum, making the circuit very sensitive to quasiparticles.
reverses. Therefore, if quasiparticles limit our relaxation time T1, we should be able to ob-
serve the cross-overs near the small loop?s half-integer flux quantum region. As reported in
section 6.2, it is not the case, so we conclude that the device is not sensitive to quasiparti-
cles, assuming that the theoretical model above is correct.
4.2.4 Thermal Photon Dephasing
Cavity Modes
It has been shown that in a cQED system, residual thermal photons inside the cavity can
dephase the qubit by dispersively modifying its transition frequency [122, 54, 123]. Similar
to any other superconducting circuit coupled to a resonator, fluxonium is also susceptible
CHAPTER 4. FLUXONIUM THEORY 95
Figure 4.14: Simulation of T2 limited by thermal photon in cavity. Coherence time limit
from Eq. 3.51 for different average thermal photon numbers as a function of ?/?, with
?/2? = 5 MHz.
to this dephasing mechanism. In high coherence devices reported in section 6.3, the small
dispersive shifts (see Table 6.1) result in a very high coherence limit. This confirmed the
approached used in previous studies to enhance coherence time limited by photon noise
[102, 103]. However, we should consider strong dispersive case in future experiments.
Specifically, from Eq. 3.51, the dephasing rate is
neffth??
2
? = 01th .
?2 + ?201
We can study the coherence limit of fluxonium with strong dispersive interaction by
varying the dispersive shift ?01 and thermal photon numbers neffth . The result corresponding
to different average thermal photon numbers nth is plotted in Fig. 4.14 for cavity coupling
?/2? = 5 MHz, which is the lower limit in our experiments. While the dephasing time is
obviously bad for high photon numbers, large dispersive shift does not necessarily limit T2
CHAPTER 4. FLUXONIUM THEORY 96
if we take care to lower the effective cavity photon temperature, which has been achieved
in previous cQED experiments [124, 123, 125].
Recent experiments with fluxonium in our lab typically involve ?/? ? 1, and the coher-
ence time is high, T2 > 100 ?s. This suggests that the effective thermal photon number is
nth ? 0.001 in our latest experiments.
Collective Modes
In fluxonium, the chain modes which couple to the qubit can be populated by thermal pho-
tons, therefore may contribute to decoherence. As discussed previously, the array of junc-
tions can be considered a transmission line. In our fluxonium design, it is shorted at one
end, and thus acts as a quarter-wavelength resonator [120, 121] with resonant microwave
modes. These modes can be divided into even (superscripted as e) modes which inter-
act with the qubit, and odd modes (superscripted as o) which interact among themselves.
Changes in the occupation of the even modes can thus dephase the qubit. In addition, the
nonlinearity of the array junctions gives rise to the dependence of the inductive energy EL
on the occupations of collective modes, creating another dephasing channel. We follow
the formalism developed in ref. [126] to compute the possible dephasing rate due to the
collective modes below.
The coupling coefficient of mode ? is given by
E? e4 CEC,? c2?
g2? = ? 2 , (4.15)2N Eag s2?
where N is the number of junctions in the chain, c2? = cos
??
N , s2? = sin
??
N , and E?
?
C is
the qubit mode?s charging energy normalized by the chain and ground capacitances (sub-
scripted as g), and
( )
1 1 1 1 2N + 1
= +
? Eb NEa
+ 1? ? . (4.16)
E C 4E 3 NC C t
CHAPTER 4. FLUXONIUM THEORY 97
Table 4.1: Parameters for collective modes simulation.
EbC EL E
b
J E
a
J C
a
g Ca Cc gq ?/2? N
1 GHz 1 GHz 3 GHz 150 GHz 36 aF 36 fF 1 fF 0.1 GHz 5 MHz 100
Here, the total capacitance that determines Et is given as
Ct = 2(C
b
g + Cc) + (N ? 1)Cag , (4.17)
and the dimensionless parameter ? is defined as
(N ? 1)Cag (N ? 1)Et
? = = . (4.18)
Ct Eag
The normalized array junction charging energy for the even mode ? is defined via
1 1 1
Ee
= a + . (4.19)
C,? EC 4E
as2g ?
These modes have frequencies ?
?? = 8Ee EaC,? J , (4.20)
and would have dispersive shifts
? ? ?
1 EaJ 2 ? ?|? | | ?|2 2? ? ? ?10 |? | | ?|2 l0 |? | | ?|2 ?l1?? = g 0 n? 1 + 0 n? l + 1 n? l ?
2 8Ee ?
.
C,? ?
2
10 ? ?2? ?2l0 ? ?2? ?2 ? ?2l?2 l?2 l1 ?
(4.21)
In addition, if we take into account the nonlinearity of the junctions array, we would see
a dependence of the inductive energy EL on the collective modes? occupation. This leads
to variation of the qubit frequency, and consequently qubit dephasing. We can model this
CHAPTER 4. FLUXONIUM THEORY 98
frequency dependence as a dispersive shift ??EL? ,
E l2
?E 1 ?? LL 10 ??? = . (4.22)2 ?EL 2N
The photons occupying the collective modes can decay via the coupling of the qubit to
the cavity mode. Considering the emission of the qubit excitation to the bath, we can use
Fermi?s golden rule to derive the Purcell decay rate,
?g2q
?q = ? , (4.23)(?q ?c)2
where ? is the coupling rate of the cavity, and gq is the cavity-qubit coupling,
E?c
gq = 2eV c ?0|n|1?. (4.24)EC
The collective modes decay rate in the dispersive limit is given by
[ ]2
? eV 1
?? = ? g?? . (4.25)(? 2? ?c) 4EcC 2l?
Using Eq. 4.24, we can write Eq. 4.25 in terms of gq, g?, and other known parameters.
We simulate the coupling coefficients g?, mode frequencies ??, decay rates ??, disper-
sive shifts ?? for even modes, and ??EL? resulted from nonlinear effect using the parameters
provided in Table 4.1. Each mode would dephase the qubit at a rate given by Eq. 3.50. As-
suming the effective occupation number of each mode to be equal, n? = 0.01, we estimate
the coherence time T2 to be of order ? 100 ms. This long time scale is mostly due to the
slow decay rate ?? of the collective modes. We note that the rate given by Eq. 4.25 cor-
responds only to Purcell effect, and the photons may decay faster through other channels.
Our simulation results suggest that the thermal photons in the chain modes at higher fre-
quencies would unlikely contribute to dephasing in fluxonium. However, proper shielding
CHAPTER 4. FLUXONIUM THEORY 99
Figure 4.15: Simulation of dephasing by photons occupying collective modes. (a)
Coupling coefficient g2? for even modes. (b) Frequencies ?2? of even modes. (c) Decay rate
of collective modes. (d) Dispersive shifts of even modes and changes in EL resulted from
array-junction nonlinearities.
and filtering techniques should still be employed to keep n? low.
4.3 Fluxonium Readout
We discuss dispersive readout of fluxonium in this section. The dispersive shift between
two different energy eigenstates |i? and |j? in fluxonium can be derived using second order
CHAPTER 4. FLUXONIUM THEORY 100
Figure 4.16: Simulation of dispersive shift. (a) Transition energy spectrum of a fluxonium
circuit with EC = 1 GHz, EL = 1 GHz, and EJ = 5 GHz. Transitions from |0? are plotted
in solid, and transitions from |1? are plotted in dashed lines. (b) Dispersive shift of the
0? 1 transition as a function of resonator frequency at the half-integer flux quantum bias.
Whenever a higher qubit transition crosses the cavity resonance, the dispersive shift given
by Eq. 4.26 diverges.
perturbation [127],
??? ? ?2 | 2 2fik 2fjk?ij = gqR nik| 2 ? ? |n |2 ?jk 2 ? , (4.26)f f2 f f2
k=6 i ik r k 6=j jk r
where fr is the resonator?s resonant frequency. We can immediately deduce from this
analytic formula that direct interaction between the transition |i? ? |j? and the cavity is not
important. In fact, the best strategy to enhance dispersive shifts in fluxonium circuits is to
engineer the qubit to have a transition frequency fik near the cavity resonant frequency, as
illustrated in Fig. 4.16 for a fluxonium circuit with parameters EC = 1 GHz, EL = 1 GHz,
and EJ = 5 GHz. This strategy was used to enhance the readout in two-qubit devices where
strong joint readout is required.
Increasing the coupling coefficient gqR is the more straightforward strategy, however, as
we cannot always engineer the spectrum of the qubit or choose an arbitrary cavity resonant
CHAPTER 4. FLUXONIUM THEORY 101
frequency. Design and simulation of 3D cavity and the cavity-qubit coupling are discussed
in more details in section 5.2.
4.4 Multi-Fluxonium Circuits
4.4.1 Capacitively Coupled Fluxoniums
Two fluxonium circuits capacitively coupled via a capacitor CM can be described by the
Hamiltonian
H = HA +HB + JCnAnB, (4.27)
where n? is the reduced charge on capacitor C?, JC = 4e2CM/(CACB) in the limit CM 
CA, CB, and HA, HB are the uncoupled fluxonium Hamiltonians. We note that CM also
slightly renormalizes the charging energy E?C . Our analysis focuses on the half-integer flux
quantum bias where the qubit transition 0? 1 has high coherence.
From Fig. 4.5(a), we see that n?0?1 is smaller than the charge matrix elements of other
transitions n?i?j at the flux sweet spot. This reduces the interaction between the compu-
tational states compared to that of higher states. This can be illustrated by inspecting the
energy levels of two fluxonium circuits with varying coupling constant JC . For this purpose,
we study two capacitively coupled fluxonium qubits (Fig. 4.17(a)) with EAC = E
B
C = 1 GHz,
EAL = 1 GHz, E
B
L = 1.5 GHz, E
A
J = 5 GHz, and E
B
J = 5.5 GHz. We follow the notation
convention |kl?0 = |kA?|lB? for bare state and |kl? for dressed state.
In Fig. 4.17(b), we see that due to the small charge matrix elements associated with
computational transitions, the computational states |00?, |10?, |01?, |11? are minimally dis-
turbed. Because n1?2  n0?1, the states |12? and |21? push each other strongly with
increasing JC . States |02? and |20? are relatively unperturbed because the uncoupled ma-
trix elements n0?2 and n2?2 vanish at half-integer flux quantum. Here we emphasize that
only eigenstates with similar eigenenergies would repel each other. For example, the pair
CHAPTER 4. FLUXONIUM THEORY 102
Figure 4.17: Capacitively coupled fluxoniums. (a) Circuit diagram of two capacitively
coupled fluxonium qubits. In the small coupling limit, the ratio CM/(CACB) determines
the coupling strength. (b) Eigenenergies of the coupled system versus the coupling strength
JC at the half-integer flux bias. The computational states are minimally disturbed, while
the |12? and |21? states push each other strongly. (c) Matrix elements of computational
transitions versus the coupling strength JC . nA?s are plotted in solid, and nB ?s are plotted
in dashed lines. nB00?10 becomes non-zero due to mixing of |00? and |10? with higher states.
nA00?01 turns on much slower because the corresponding interference of all contributions is
negative. (d) Matrix elements of higher transitions versus coupling strength JC . States |12?
and |21? interact strongly with increasing JC , leading to the renormalization of nA,B11?12.
of states |20? and |13? are not close, therefore do not repel each other despite having large
matrix element between them.
Interestingly, due to the dressing of states, some matrix elements are renormalized. As
shown in Fig. 4.17(c), the charge matrix element nB00?10 becomes finite as the coupling
CHAPTER 4. FLUXONIUM THEORY 103
strength increases, due to the mixing of computational states. Roughly speaking, we can
write
|00? = a00|00?0 + a11|11?0 + a31|31?0 + a13|13?0 + . . . ,
|10? = b10|10?0 + b01|01?0 + b03|03?0 + b21|21?0 + . . . ,
(4.28)
|01? = c01|01?0 + c10|10?0 + c30|30?0 + c12|12?0 + . . . ,
|11? = d11|11?0 + d00|00?0 + d20|20?0 + d02|02?0 + . . . ,
which are perturbative to first order in JC , with a00 = b10 = c01 = d11 = 1, and other
coefficients depend linearly on JC . As a result, |00? mixes with |11?0 and |13?0, while |10?
mixes with |01?0 and |03?0 at finite JC . All these contributions make ?00|nB|10? increase
with JC . On the other hand, ?00|nA|01? is relatively unaffected because the interference
between the contributions is negative.
In Fig. 4.17(d), the direct interaction between |12? and |21? causes them to hybridize
strongly and renormalizes nA B11?12 and n11?12. In addition, n
B
10?20 also becomes finite when
the coupling is turned on, due to the dressing of |10? and |20?.
4.4.2 Inductively Coupled Fluxoniums
In a similar fashion, two fluxonium qubits coupled inductively via a mutual inductor LM
(Fig. 4.5(a)) can be described by the Hamiltonian
H = HA +HB ? JL?A?B, (4.29)
( )2
where ?? is the reduced flux across the inductor L , J = h? LM? L 2e L L is the qubit-qubitA B
inductive coupling constant in the limit LM  LA, LB, and HA, HB are the uncoupled
fluxonium Hamiltonians. We note that LM slightly modifies EL?.
From Fig. 4.5(b), we see that ?0?1 is large at half flux quantum. This causes strong
mixing of the computational states. We can confirm this by inspecting the eigenenergies
of two inductively coupled fluxonium qubits biased at half flux quantum with parameters
CHAPTER 4. FLUXONIUM THEORY 104
Figure 4.18: Inductively coupled fluxoniums. (a) Circuit model of two inductively cou-
pled fluxonium qubits. In the small coupling limit, the ratio LM/(LALB) determines the
coupling strength. (b) Eigenenergies of the coupled system versus the coupling strength
JL at the half-integer flux bias. Due to the strong 0 ? 1 phase matrix element, the com-
putational eigenstates become strongly hybridized. (c) Charge matrix elements of com-
putational transitions versus the coupling strength J AL. n ?s are plotted in solid, and nB ?s
are plotted in dashed lines. The strong mixing of the computational states results in large
changes in the matrix elements with respect to JL. (d) Charge matrix elements of higher
transitions versus coupling strength JL.
EAC = E
B
C = 1 GHz, E
A
L = 1 GHz, E
B
L = 1.5 GHz, E
A
J = 5 GHz, E
B
J = 5.5 GHz, and varying
JL (Fig. 4.18(b)). We observe that the |01? and |10? states push each other strongly even at
small coupling value JL ? 50 MHz.
The mixing of these computational states leads to strong renormalization of the corre-
sponding matrix elements, for example, nA B00?10 and n00?10, as shown in Fig. 4.18(c). The
CHAPTER 4. FLUXONIUM THEORY 105
mixing of higher states is not as dramatic, but still results in strongly dressed states at a
modest coupling value JL ? 100 MHz (Fig. 4.18(b)). Consequently, nA,B11?12 are renormal-
ized rapidly with the coupling coefficient JL. On the other hand, the negative interference
between the relevant states leaves nB10?20 relatively unchanged (Fig. 4.18(d)).
4.4.3 Microwave-Activated Controlled-Z Entangling Gate
In atomic systems, qubit states are protected from environmental noise and have long co-
herence, whereas quantum gates and readout are performed via transitions outside of the
computational subspace. This separation of states in quantum information storage and pro-
cessing allows both long coherence and fast operation. Since fluxonium has energy spectra
and selection rules similar to natural atoms, recent experimental achievements in atomic
systems motivate us to consider using higher levels to perform a two-qubit entangling oper-
ation. Simulations of two capacitively coupled fluxonium showing strong mixing of higher
states and minimal perturbation to the computational space, illustrated in Fig. 4.17, further
strengthens this idea.
Following this direction, a recent paper [128] proposes a controlled-Z gate for two flux-
oniums with an always-on interaction. The gate is activated by a microwave pulse near the
|1? ? |2? transition of either qubit. Numerical simulations in the paper shows promising
fidelity, above 99.9% for gate time below 100 ns when decoherence is neglected. Here, we
focus on the capacitive coupling case which has been investigated in the lab. Preliminary
experimental results are reported in section 6.4.
The main idea of the CZ gate is as follows. When there is no coupling, ?01?02 =
? B11?12 = ?1?2, and we cannot selectively drive |11? ? |12?. As the interaction JC is turned
on, |12? and |21? push each other apart due to the high single-qubit matrix element n1?2,
while |02? and |20? remain relatively unperturbed. This lifts the degeneracy above. The CZ
gate can then be realized by selectively driving |11? ? |12? (or |11? ? |21?). One oscillation
CHAPTER 4. FLUXONIUM THEORY 106
Figure 4.19: Microwave-activated controlled-Z gate. (a) Microwave-activated
controlled-Z gate scheme. The large matrix elements nA,B1?2 push the dressed states |12?
and |21? apart, lifting the degeneracy. This allows us to selectively drive either |11?? |21? or
|11? ? |12? (magenta) to realize the CZ unitary. (b) Gate rate and spurious ZZ coupling ver-
sus JC . The gate rate (solid) is typically much higher than the spurious coupling (dashed).
We note that the ZZ coupling turns on rapidly with JC , compared to the gate rate, so op-
timal configuration may be obtained in the region where JC < 200 MHz for this pair of
qubits.
|11? ? |12? ? |11? (or |11? ? |21? ? |11?) results in a phase factor ei? for state |11?. This
operation is thus equivalent to applying the unitary operator UCZ = diag(1, 1, 1,?1) to the
computational subspace. The idea is illustrated in Fig. 4.19(a).
We can define a gate rate [129], which is characterized by the frequency mismatch
? A Bgate = f01?02 ? f11?12. It increases with decreasing ? = |f1?2 ? f1?2| and increasing JC .
This implies that smaller ? generally leads to better performing CZ gate.
On the other hand, the always on interaction introduces a spurious ZZ coupling ?ZZ =
|f01?11?f00?10|, which creates error during the entangling operation. The ZZ coupling also
affects our single-qubit gate fidelity [130]. This can be interpreted as the dispersive shift
between two qubits: the transition frequency of qubit A depends on the state of qubit B,
and vice versa. We note that the ZZ coupling itself can be employed to realize an entangling
gate. To examine the difference between the gate rate and the ZZ coupling rate, we simulate
CHAPTER 4. FLUXONIUM THEORY 107
?gate and ?ZZ using the parameters from Fig. 4.17, and plot the results in Fig. 4.19(b).
To minimize this spurious effect, we can straightforwardly decrease the coupling strength
JC at the expense of slowing down the CZ gate rate. We note that for the pair of qubits
being examined, the ZZ coupling turns on rapidly with JC , compared to the gate rate, so
having JC < 200 MHz is preferable in this case. Another option is to engineer the precise
spectrum of two fluxonium circuits to cancel out this effect, which will be the focus of future
research.
Chapter 5
Experimental Techniques
He stepped down, trying not to look long at her, as if she were
the sun, yet he saw her, like the sun, even without looking.
Leo Tolstoy
Exciting quantum phenomena can only be revealed once excellent engineering steps
are taken to design and set up the experiments. This chapter describes our philosophies
and approaches in cryogenic experimental setup (section 5.1), simulation and design of
3D cavities (section 5.2), practical considerations regarding microwave electronics (section
5.3), and fabrication techniques (section 5.4).
5.1 Cryogenic Setup
Cryogenic engineering lies at the heart of superconducting qubit experiments. In order
to observe quantum mechanical behaviors in these macroscopic systems, it is necessary
to get rid of thermal effects. For example, a body with mass m at temperature T radiates
electromagnetic photon flux following Planck?s law of black body radiation, which can easily
smear out any quantum effect if T is high. A qubit with transition frequency of a few
GHz is constantly excited to a mixed equilibrium state at room temperature. In order for
108
CHAPTER 5. EXPERIMENTAL TECHNIQUES 109
the system to display quantum behaviors, it is necessary to cool it down sufficiently and
isolate it from external environment. In this section, we will review how to set up the
experiment in a dilution refrigerator and the approaches we use to isolate our devices from
the environment.
The ideal quantum experiment would take place at the absolute temperature of 0 kelvin,
but it?s not possible to reach this temperature with current technology. However, ad-
vances in cryogenic engineering currently allow construction of modern dilution refrig-
erators which can cool down to below 10 mK, two orders of magnitude colder than the
interstellar vacuum space. The base plate of such a fridge is where the quantum phenom-
ena in superconducting circuits occur.
5.1.1 Material Consideration
Setting up a cryogenic experiment requires the knowledge of relevant properties of mate-
rials at low temperatures, in particular their thermal expansion, thermal conductivity, and
electrical conductivity.
The thermal vibration of atoms changes with temperature, resulting in a difference in
these atoms? average separations. Materials subsequently expand at higher temperature
and contract at lower temperature. Thermal expansion or contraction must be taken into
account when choosing materials for parts in the experiment, especially for the nuts and
screws used to connect different components together. Tight screw joints may become
loose at low temperature if materials are chosen poorly. Thermal contact can generally be
improved after cooldown by the addition of a washer with low thermal expansion coefficient
under the screw/nut with higher expansion coefficient.
In cryogenic experiments, we typically use four different metals: copper, brass, stainless
steel, and aluminum. In general, brass has a higher expansion coefficient than copper and
stainless steel, so we typically use brass screws and stainless steel washer if possible. As
CHAPTER 5. EXPERIMENTAL TECHNIQUES 110
the temperature drops, brass screws contract more than the stainless steel or copper parts,
and can create a pressing force to keep the parts together. Brass is also a decent thermal
conductor, so using a brass screw to establish a direct thermal link between the fridge and
the electrical component is another good approach to thermalize the components.
In solids, heat can be transported via two mechanisms. In insulators, only lattice vibra-
tions are there to transport heat, while in conductors, free electrons from the conduction
band near the Fermi energy can also carry heat. The characteristic velocity of phonons is
the sound velocity, which is about three order of magnitudes lower than the Fermi velocity
of free electrons. Therefore, insulators are generally poor thermal conductors at low tem-
perature (T < 1 K), and metals are generally good thermal conductors [131]. However, we
should keep in mind that certain metals can become superconducting below their critical
temperatures. In the superconducting state, the free electrons form Cooper pairs which
cannot carry heat. The thermal conductivity of a superconductor below Tc is thus only as
good as that of an insulator.
For thermal conduction, copper is considered the king of all materials in cryogenic en-
vironment due to its high thermal conductivity at low temperature and its relatively low
cost. Stainless steel is a poor thermal conductor, so it can be used when thermal isolation
is desired. Alumninum becomes superconducting below Tc = 1.2 K, so it is a bad thermal
conductor at mK range. Our general approach is to use copper where thermal conduction
is desired, and stainless steel where thermal isolation is needed. Using insulators such as
plastic tapes inside the fridge should generally be avoided, because they cool down very
slowly
5.1.2 Shields
Another path through which heat can be carried inside the fridge is via black body radia-
tion, where thermal photons are emitted from a stage at higher temperature and absorbed
CHAPTER 5. EXPERIMENTAL TECHNIQUES 111
by components at lower temperatures. This effect is small compared to heat transferred
by solid or gas. While residual thermal photons may not heat up the fridge, they can af-
fect qubits and cavities, so protection against thermal photons is crucial in achieving high
coherence [132].
In practice, several metal cans are put together successively at each stage of the refrig-
erator to block the radiation from parts of the fridge at higher temperatures. In addition, a
gold plated copper radiation shield can be custom-ordered from BlueFors to further protect
the bottom part of the fridge. Despite the protection provided by the metal cans discussed
above, hot photons may leak through crevasses between metal joints, and high energy ra-
diation may pass through thin metal shields as well. We seal those small openings with
copper tape before each cooldown.
In addition to the metal shields, a mixture of Stycast 2850 and carbon powder can be
used to paint the inside of the radiation shield and absorb stray infrared photons [133].
We initially implemented this type of black paint but abandoned it after a few cooldowns
to make room for more cryogenic components. Its effect on fluxonium?s coherence thus
remains inconclusive. A single layer of cryoperm magnetic shield surrounding the cold
finger at 10 mK provides further protection against spacial electromagnetic radiation.
In recent experiments starting from Summer 2019, we have been using several layers
of ?-metal shields at room temperature to minimize stray magnetic field from neighboring
labs. Recent investigation has concluded that high energy radiation may be the source that
generates quasiparticles, and room temperature shielding helps protect superconducting
circuit systems [92]. Interestingly, this result is consistent with our best coherence data
obtained from the setup with additional room temperature shields.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 112
5.1.3 Filters
The experimental lines connecting the device under test (DUT) inside the fridge to the clas-
sical electronics equipment at room temperature may allow thermal photons to propagate
and disturb the system. These measurement lines must therefore allow passage of applied
microwave signal at a few GHZ range and at the same time block out the rest of the elec-
tromagnetic spectrum. For this purpose, we employ separate techniques for the input and
output lines.
The complete setup of the fridge is shown in Fig 5.1. We change the specific placement
of the components from time to time to accommodate new components and measurement
lines.
Input Line
For the input line, since we can control the power of the drive signal to overcome dissipation
along the way, the best general strategy in reducing hot photons from room temperature is
to use commercial XMA cryogenic attenuators. This approach is generally cheap and rea-
sonably effective. The total attenuation should be low enough to allow RF signal generated
by electrical equipment at room temperature to reach the base plate with sufficient power.
Three important details determine how well we actually isolate the system from thermal
photons at higher temperatures: the placement of the attenuators, their thermalization,
cooling powers, and working frequencies.
Electrically speaking, attenuators are simply resistors (or networks of resistors). Thus,
they attenuate input photons, and also generate thermal photons corresponding to the tem-
peratures of their bodies. The effective temperature of the output of an attenuator at a fixed
frequency f is written as
Tout = Ta +A? Tin, (5.1)
where Tin,out are the effective temperatures of the input and output photons, Ta is the
CHAPTER 5. EXPERIMENTAL TECHNIQUES 113
Figure 5.1: Cryogenic setup. (a) Typical experimental setup for high coherence fluxonium
qubits. Recent addition of the TWPA at the base plate comes with an additional double-
junction circulator, also in 4-8 GHz range. (b) Dilution fridge. (c) Base plate setup.
temperature of the attenuator, and A = 1/G is the attenuation factor1. If the fridge had
infinite cooling power at 10 mK, it would then be ideal to place all the attenuators there.
Unfortunately, this is not the case, and the cooling power at different stages of the fridge has
to be taken into account. For example, the cooling power at the cold plate of the BlueFors
LD 250 fridge is around 300 ?W while it is only around 14 ?W at the base plate. As a
result, it is better to distribute the attenuation. We can approximate the effective noise
1The attenuation A typically depends on frequency f .
CHAPTER 5. EXPERIMENTAL TECHNIQUES 114
temperature of the input line at the mixing chamber with an arbitrary configuration of
attenuation distribution using
Seff = S(10 mK) +A10 mK(S(100 mK) +A10 mK(S(4 K) +A4 KS(300 K))), (5.2)
where S(T ) is the noise spectral density of a resistor at temperature T .
In our setup, 20 dB XMA cryogenic attenuators are placed at 4 K, 100 mK, and 10
mK plates, which should naively reduce the hot flux of photons at 300 K to a sufficiently
low amount at the mixing plate [122, 123]. This approximation is only accurate if (i)
the attenuators are thermalized to equilibrium at the expected temperatures, and (ii) the
attenuators work at all frequencies. A broken and hot attenuator sitting next to the device
being measured at 10 mK can easily become an experimentalist?s nightmare.
To thermalize these hexagonal prism attenuators, we bolt two copper bars using brass
screws and stainless steel washers to clamp copper strips to two opposite sides of the atten-
uators. The strips are then bolted to the plates of the fridge. The attenuators are positioned
underneath the respective plates to avoid radiation from the higher temperature stage above
them. The outer body of the XMA attenuators are made of stainless steel, while its resistive
part is made of thin films, so it is unclear if they can be thermalized well. Recent studies
have proposed novel cryogenic attenuators that thermalize better [124, 125].
Another weak point of these commercial attenuators is their working bandwidth. As
shown in Fig. 5.2(a), the XMA attenuator may not work at high frequencies above 25 GHz.
To block out radiation above this threshold, we employ a commercial low pass filter with
cutoff frequency at 12 GHz from K&L, which works well up to 40 GHz (see Fig. 5.2(b)).
To protect the experiment from noise having frequencies above 40 GHz, we use a home-
made Eccosorb CR110 low pass filter that gradually cutoffs at higher frequencies, as shown
in Fig. 5.2(c). The design of our Eccosorb filter basically consists of SMA connectors at-
tached to a section of Eccosorb CR110 transmission line, as shown in Fig. 5.3(a). The
CHAPTER 5. EXPERIMENTAL TECHNIQUES 115
dimensions of this section are designed such?that it has 50 ? characteristic impedance.
Eccosorb CR110 is engineered to have Z0 = ?/ = 1, and to attenuate heavily at high
frequency, > 100GHz range, while its insertion loss is small at 10 GHz [134, 135]. The
attenuation per unit length of our homemade filters at 8 GHz is roughly 1 dB/cm, so a short
section can even be placed on the output line. Using the same principle of spacing out the
attenuators above, we place one Eccosorb filter at 4 K and one at 10 mK. High coherence
results were achieved with the Eccosorb filters in place.
To transmit the signal through the input line from one stage to another, we use stainless
steel coax cables, due to their excellent thermal isolation. Soldering stainless steel can be
tricky, but it is possible if necessary steps are taken. In our lab, we mostly use direct-plug
SMA connectors and solder the cables ourselves.
Output Line
The design philosophy for the output line is different. Resistors attenuate signals both ways,
which does not work if we want to measure the small signal coming from the device. To
allow the one-way transmission of the signal at the designated frequency, we can use RF
cryogenic circulators or isolators.
A circulator consists of a strongly magnetic ferrite component, which directs the flow
of RF signal inside. This interaction creates a magnetic field vortex inside the component,
making the flow of signal possible only in one direction. This is analogous to a ship sailing
around a whirlpool: it is prohibitively hard to go in the opposite direction of the flow.
When one of the three ports of a circulator is terminated with a 50 ? resistor, it becomes an
isolator. Since this resistor also acts as a black body, it is important to thermalize it well.
In fluxonium experiments, we typically use a combination of three circulators/isolators
from QuinStar, or a triple junctions circulators/isolators from Low Noise Factory. The perfor-
mances of some circulators at liquid nitrogen temperature (77 K) are shown in Fig. 5.2(d).
All circulators are placed at the 10 mK plate.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 116
Figure 5.2: Performance of cryogenic components. (a) Transmission through commer-
cial XMA attenuators. They do not work well above 25 GHz. (b) Transmission through K&L
filter with cutoff at 12 GHz. It does not work well above 40 GHz. (c) Transmission through
homemade Eccosorb CR110 filters which continue to attenuate signal at high frequencies.
They are supposed to work well up to THz range. (d) Isolation measurement of QuinStar
circulators. They only work well within the specified frequency range.
The bandwidth of the circulators has to cover the frequency of the output signal. Since
it is relatively narrow compared to that of other components, the bandwidth of the cir-
culator determines the bandwidth of the measurement line. We typically use components
with 4-8 GHz or 8-12 GHz bandwidths. Some experiments requiring broad bandwidth are
accommodated with 4-12 GHz circulators/isolators.
We recently added a traveling wave parametric amplifier (TWPA) [136] to the output,
which enabled us to perform single-shot measurement of two-qubit systems. In the new
CHAPTER 5. EXPERIMENTAL TECHNIQUES 117
Figure 5.3: Cryogenic components. (a) Homemade Eccosorb CR110 low pass filter con-
sisting of a transmission line section having Z0 = 50 ? sandwiched between two SMA
connectors. (b) The copper cavity is mounted on a cold finger. The input and output ports
are connected to non-magnetic SMA connectors. The magnetic coil can be attached to the
cold finger or directly to the cavity.
setup (not shown), the TWPA is sandwiched between a set of circulators in the 4-8 GHz
range, three on the input, and two on the output side.
Unfortunately, these commercial circulators don?t isolate well outside their bandwidth
(see Fig. 5.2(c)). To cover the rest of the electromagnetic spectrum, we use a selection of
K&L low pass filters and Eccosorb CR110 filters. The cryogenic amplifier is believed to be a
good filter of thermal photons above 4 K, so we did not use any additional filters above the
4 K stage. To optimize the SNR of the output signal, the Eccosorb filter on the output line is
small, with insertion loss less than 0.5 dB. Low frequency noise in the line is considered less
important because the cold 3D cavity filters it out quite well (see Fig. 6.2). At the time of
writing of this manuscript, we are designing both high pass and band pass filters to provide
better protection at frequencies below 4 GHz.
As in the driving line, all components in the readout line are thermalized either by being
attached directly to a cold finger, or via copper strips firmly clamped to their bodies to make
as much metal-to-metal contact as possible. Apiezon cryogenic thermal grease was used in
CHAPTER 5. EXPERIMENTAL TECHNIQUES 118
a few cooldowns to thermalize components without flat surfaces, or to attach the chip to
the cavity, but there was no apparent difference in the qubits? coherence. We stopped using
this grease in 2018 because it increasingly made things messy after repeated cooldowns.
To transmit the signal from the device under test to the output port of the fridge, we
initially used stainless steel coax cable as well, but then switched to CuNi in 2017, sacrificing
thermal isolation for lower insertion loss. Starting with the two-qubit experiment, we used
commercially-assembled superconducting NbTi cables below 4 K to minimize the signal
loss. Small sections of copper coax are used to connect cryogenic components at the mixing
chamber stage of the fridge, since thermal isolation between components sitting at the
same temperature is undesirable. On the other hand, this last cable must be as cold as
possible to minimize the effective thermal photon noise. Copper with its excellent thermal
conductivity is thus a good material here. All connectors inside the cryoperm shield is made
of non-magnetic material to achieve magnetic field hygiene around the sample, as shown
in Fig. 5.3(b).
We tested the refrigerator input-output line by measuring the transmission signal at
room temperature, during cooldown, and when the base plate reached 10 mK. We repeated
the measurement after the installation of a cryogenic amplifier. The results are shown in
Fig. 5.4. From this data, we could calibrate the total attenuation, which includes 50 dB
of attenuation from the XMA commercial attenuators2 and 10 dB from the coax cables. In
addition, we could also check the performance of the cryogenic amplifier and the bandwidth
of isolators.
Flux Bias Line
An important part of a fluxonium experimental setup is the flux bias line used to apply a
magnetic field to the circuit. To construct the magnetic coil, we designed a brass hollow
2This measurement was performed in 2016. We added an additional 10 dB at the base plate in later experi-
ments.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 119
Figure 5.4: Refrigerator input-output transmission. Performance of the fridge line at
various temperatures, and with the addition of a cryo-amplifier. At the time of the mea-
surement, there were 50 dB attenuation from XMA commercial attenuators and 10 dB
attenuation from the coax cables. Per manufacturer?s specification, the cryo-amplifier pro-
vides 40 dB of gain from 4 to 16 GHz, which is consistent with our data.
cylinder to fit the body of the copper cavity and cold finger. It could be attached to the body
of the cavity or the cold finger interchangibly, depending on how strong and homogeneous
we would like the magnetic field at the chip to be (see Fig. 5.3(b)). The coil was made of
NbTi, a superconducting material. The wire was coiled around the brass body using a drill.
We then fixed the wire to the body using Stycast 2850. The superconducting coil eliminates
dissipation caused by the applied current, and also helps with thermal isolation.
For input/output lead connecting to the coil, we used manganin wire. We thermalized
this wire by attaching it to metal posts positioned at the 50 K and 4 K plates using GE
CHAPTER 5. EXPERIMENTAL TECHNIQUES 120
varnish. The manganin wire and the NbTi coil were soldered together at the 4 K stage.
We noticed that the addition of the coil corresponds to a rise in the temperature of the
fridge?s base plate, which implies that our thermalization of the wire is not perfect. This
could be due to the copper cladding on the outside of the wire. In addition, whenever
a large current (>100 mA) is applied to the coil, we see a sharp increase in the fridge?s
temperature, which can be due to the Joule heating of the manganin wire or the solder
contact.
5.2 Cavity Simulation and Design
The purpose of the 3D cavity in fluxonium experiments is to readout the state of the
qubit(s). In designing the cavity, three important properties have to be considered: the res-
onant modes of the cavity, the cavity-qubit coupling, and its coupling to the input/output
transmission line.
5.2.1 Resonant Modes
The cavity?s geometry must allow good thermal contact with a copper cold finger, mounting
of a magnetic coil directly on the cavity?s body, and it must fit inside a cylindrical cryoperm
magnetic shield. Taking into account these conditions, we chose rectangular geometry and
copper as the material for the body. A few aluminum cavities were used to test the shielding
inside the dilution fridge (see section 6.1).
To protect the qubit?s lower spectrum from decay via the cavity modes, we chose the
fundamental mode TE101 for readout purpose [137, 51]. Since fluxonium has many levels
which may interact with the cavity?s higher modes, it is important to take them into account.
The resonant frequencies of a rectangular 3D cavity follows [131]
?
c (m?)2 ( ( )? n?)2 l? 2fmnl = + + , (5.3)
2? ?rr a b d
CHAPTER 5. EXPERIMENTAL TECHNIQUES 121
Figure 5.5: Electric field density of resonant modes in a rectangular cavity. Electric
field density for higher modes of the 7.5 GHz cavity. Note that not all modes couple to the
qubit, and higher modes would be occupied by fewer thermal photons, therefore contribut-
ing much less to the dephasing of the qubit.
where a, b, d (with a > d > b) are dimensions of the cavity, and m, n, l are integers, no
more than one of which can be 0. They are used to identify the mode number of the cavity.
For example, TE101 mode corresponds to m = 1, n = 0, l = 1.
We designed the cavity by first using Eq. 5.3 and then by checking the resonances and
the corresponding electromagnetic fields on ANSYS Electronics Desktop. The dimensions
of the cavity were chosen to give the fundamental resonance within the bandwidth of the
measurement line. The lowest TE101 mode of a rectangular cavity and integration of a
qubit has been studied in refs. [51, 138]. Here, we focus on the simulation of some higher
modes up to 20 GHz. Their electric field density distributions on the surface of the cavity
are shown in Fig. 5.5.
It is apparent that if we place the qubit along the electric field of the fundamental mode
CHAPTER 5. EXPERIMENTAL TECHNIQUES 122
in the middle of the cavity, we can protect it from coupling to most of the higher modes. On
the other hand, the modes at higher frequencies are less populated by thermal photons due
to Bose-Einstein statistics, so they should not be severely problematic, provided that our
filters work well. Regardless, we should still take care to design qubits to have transition
frequencies away from, for instance, the mode at 13.1 GHz. Direct excitation from the
computational subspace via this mode may still dephase the qubit.
5.2.2 Cavity-Qubit Coupling
The height of the cavity (the smallest dimension, see Fig. 5.5) does not dictate the cavity?s
resonant frequencies, but it is important for the coupling between the cavity and the qubit.
The smaller the height dimension, the larger the electric field across the qubit?s antenna,
and consequently the higher the coupling. This cavity-qubit coupling also depends on the
size and shape of the dipole antenna. The shunting capacitance, which contributes signif-
icantly to the charging energy EC of the qubit, comes from the antenna as well [51]. We
simulated different cavity and qubit geometries on Ansys Desktop to search for the desired
configuration.
To simulate the cavity-qubit coupling, we used the eigenmode solver. We assigned an LC
boundary condition to the virtual circuit between the antenna. By varying the inductance
value of this circuit and finding the resonant frequencies, we extracted the amplitude of the
avoided crossing between the circuit mode and the cavity?s fundamental mode that we used
for readout, as shown in Fig. 5.6(b). Note that the coupling here has unit of Hz.
If desired, we could also check the LC circuit?s coupling to higher modes, which is either
zero or small compared to the coupling to the TE101 mode. The size and shape of the
dipole antenna can be varied to meet our requirements for the coupling strength g and the
shunting capacitance.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 123
Figure 5.6: Simulation of cavity-qubit coupling. (a) Geometry of antenna and couplers.
(b) Eigenmode simulation. The anti-crossing amplitude is twice the coupling constant g/2?.
(c) Coupling constants simulation of two type of antennas. The length of the antenna is
varied, resulting in different coupling. (d) Shunting capacitance simulation of two types
of antennas. (e) Comparison of shunting capacitance versus coupling constant for both
antenna geometries, showing their correlation. (f) Using the couplers allows us to enhance
the coupling significantly with much less increase in the shunting capacitance, as compared
to using only the symmetric dipole antenna.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 124
We concluded that regardless of the size and shape of the antenna, increasing the cou-
pling correlates with increasing the shunting capacitance, which is equivalent to decreasing
the charging energy EC (see Fig. 5.6(c-e)). This is unavoidable in any symmetric dipole
antenna. To overcome this relation, we designed a pair of couplers to increase the cou-
pling without increasing the shunting capacitiance (see Fig. 5.6(a)). The idea was to create
an equipotential conducting element to concentrate the electric field corresponding to the
TE101 mode of the cavity such that the voltage drop across the qubit?s antenna is enhanced.
These couplers are particularly useful in experiments requiring a strong coupling of qubit
to cavity mode, or qubit to a traveling wave mode, without affecting the charging energy
EC . The simulation result for the couplers is shown in Fig. 5.6(f).
5.2.3 External Coupling to Cavity and Qubit
Next, we designed and engineered the coupling of the cavity to the 50 ? transmission line,
which involves a transition between a TE101 standing wave in the cavity and a TEM mode
in the coax cable. The loss of energy from the cavity is characterized by the rate ?, where
Q = ?0/? is the total quality factor of the cavity. The loss can be due to intrinsic loss inside
the cavity, or due to the coupling of the cavity to the external transmission line.
In appendix A, we show that loss can be modeled as a shunting admittance with non
vanishing real part. To simplify here for simulation, we can model the real part of the
admittance as a pure conductor/resistor. The quality factor is then Q = ?cRC. When the
loss is mostly intrinsic, Qint < Qext, we say that the cavity is under-coupled, and when the
loss is mostly extrinsic via coupling to input/output connection Qint > Qext, we say that
the cavity is over-coupled. In designing the input-output coupling, it is necessary for the
signal to go to the output port instead of being lost inside the cavity or going back through
the input. Thus, in a reflection measurement, we designed the port to be over-coupled,
and in transmission measurement, we designed the input and output coupling such that
CHAPTER 5. EXPERIMENTAL TECHNIQUES 125
Qin, Qint  Qout.
The capacitive external coupling of the cavity to the transmission line can be controlled
by changing the length of the pin going through the ports. This can be verified by finite
element method simulation. First, we assigned a boundary condition on the port Rport =
50 ? and varied the length of the conductor going inside the cavity to check for Qext. Then,
we chose the length corresponding to a desired quality factor, and ran the simulation again
with varying Rport to check for consistency. The results are shown in Fig. 5.7(a) and (b) for
input and output ports, respectively.
Experimentally, the coupling was tuned using a feedback loop of measuring the external
quality factor of the cavity and cutting the pin shorter until the desired coupling is at-
tained. Fig. 5.8 shows the amplitude and phase of the reflection S11 parameter with various
Qext/Qint, following
2i(f ? f0)? f0/Qext + f0/Qint
S11(f) = . (5.4)
2i(f ? f0) + f0/Qext + f0/Qint
The linewidth depends strongly on this ratio, and we can deduce that to get an accurate
fit, it is better to bring the ratio closer to unity, which corresponds to a narrow linewidth.
Typical cavity reflection measurement data is shown in Fig. 6.1.
One notable feature in our cavity design is the placement of the input port. Both the
input and output ports were initially placed in symmetric positions along the diagonal of
the cavity to minimize the perturbation to the surface current of the cavity?s TE101 mode, as
shown in Fig. 5.9(a). However, the microwave control of the qubit?s transition becomes very
slow due to the screening of the cavity in this design. We realized that at frequency far below
the fundamental mode, the direct capacitive coupling between the qubit and the input pin
becomes dominant. We subsequently designed the input port to be as close to the chip as
possible, as shown in Fig. 5.9(a). Such placement of the input pin provides a substantial
coupling enhancement between the microwave drive and the qubit, allowing us to drive
CHAPTER 5. EXPERIMENTAL TECHNIQUES 126
Figure 5.7: Simulation of external coupling to cavity and qubit. (a) and (b): Simulation
of cavity?s external coupling to the transmission line for the input and output ports, respec-
tively. (c) Simulation of the capacitive coupling between the pin and the qubit, which is
modeled as a lumped port. This shows the importance of the placement of the connector.
As the connector is placed closer to the chip, the coupling is enhanced significantly. (d)
Simulation of input pin - qubit coupling in a broad frequency range. Position of the port is
not so important at frequencies higher than the cavity?s fundamental resonance. This also
explains how stray radiation can couple strongly to the qubit, despite the protection from a
3D cavity.
the low qubit transition frequency sufficiently fast. The simulation of this effect is shown
in Fig. 5.7(c). Note that the cavity still filters out noise very well below its fundamental
mode. We also measured qubit A coherence (see Table 6.1) using both types of cavities,
and observed no difference in 0? 1 transition?s T1 and T2.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 127
Figure 5.8: Simulation of cavity reflection. Amplitude and phase of reflection S11 given
by Eq. 5.4. The linewidth depends strongly on the ratio between external and internal
quality factor Qe/Qi. In order to get an accurate fit of high internal quality factor, we can
tune the external coupling to be around the expected internal loss rate.
While the placement of the port strongly influences the direct capacitive coupling be-
tween the pin and the qubit at low frequency in the 0 ? 6 GHz range, it has little effect at
high frequencies above the fundamental resonant mode (see Fig. 5.7(d)). This implies that
stray radiation photons at high frequencies may couple to the qubit and create unwanted
effect regardless of the position of the port and despite the protection from the cavity. This
confirms that the position of the input port does not compromise qubit coherence, and
shows the importance of shielding and filtering in our setup to protect the device under test
from high frequency noise.
5.2.4 Cavity Preparation
Fig 5.9 shows the schematics of our cavity designs. The first few cavities were machined
in the lab, and later ones were made by professional machine shops. We first designed the
cavity to have high internal quality factor, which comes into play in our experiment with
CHAPTER 5. EXPERIMENTAL TECHNIQUES 128
Figure 5.9: 3D cavity designs. These designs allow mounting of a magnetic coil onto the
body of the cavity. (a) Design A has only one port place at in the middle of the diagonal
line across the cavity. This placement minimally disturbs the surface current and results in
higher internal quality factor. The sample groove was machined across the cavity?s width,
allowing customization of the chip?s position. (b) Design B has asymmetric input/output
port for transmission measurement. The input port is placed close to the chip to allow
strong driving at low frequency. The third dimension of the cavity is decreased to enhance
qubit-cavity coupling. The surface area of the sample groove is increased to allow better
thermalization of the chip.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 129
high-Q aluminum cavities (see section 6.1). The rest of the cavities are made from copper.
The sample groove was made long, allowing us to customize the position of the chip. By
moving the chip horizontally, we could tune the coupling between the qubit and cavity, as
well as between qubit and driving port.
The later design prioritizes qubit?s driving at low frequency, so the input port was made
to be as close to the chip as possible. This allows us to apply fast gate on transition frequen-
cies as low as 50 MHz. The output port is overcoupled to the readout line. Since the output
pin is a little longer, this port is positioned further away to minimize direct coupling of low
frequency noise to the qubit.
Both designs have through holes on the side for direct attachment to the cold finger on
the base plate, and tapped holes for direct mounting of the coil. Alternatively, this tapped
hole can be drilled through, and the coil can be mounted using a long screw. Table 5.1
shows a summary of different types of cavities mentioned in this thesis work.
After machining, the cavities are further polished with sandpapers, starting from a
coarse grit, CAMI standard of 600, to a finer grit, CAMI standard of 1500. The copper
cavities are then washed with soap and dipped in a solution of citric acid, commercially
available as Citranox, for ten minutes. Finally, they are rinsed with DI water and sonicated
with IPA for another ten minutes. When measured at 10 mK, the copper cavity has internal
quality factor Qint = 25000. The interior of the cavity may contain impurity from machining
which may not cooldown well and create thermal photons which can be detrimental to the
qubit?s coherence. The cleaning step is therefore essential, even if it does not result in a
higher internal quality factor.
For reflection/hanger experiment setup, we used a single port with correspondingQout ?
1000. For transmission experiments, we set Qout ? 600, Qin ? 3000, so the linewidth
for a 7.5 GHz cavity is approximately 15 MHz. Recent two-qubits experiments have in-
creased quality factors to improve the dispersive shift over coupling rate ?01/? ratio, with
Qout ? 2000, Qin ? 6000, corresponding to a linewidth of around 5 MHz.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 130
Table 5.1: Summary of cavity features.
Cavity # Material Design feature
1 6061 Al Design A
2 4N6 Al Design A
3 C101 Cu Design A
4 C101 Cu Design B
Aluminum cavities become superconducting at low temperature so its internal quality
factors can reach millions [139, 140]. To determine the Qint more precisely in such case,
we tuned the external quality factor Qext in aluminum cavities to at least 2 ? 105. We also
performed electro-polishing of these cavities to get the best possible quality factor. This
allows us to test our filter and shielding, as discussed in section 6.1.
The qubit is mounted onto the cavity with pressed indium wire. Too much applied
pressure can break the chip, so the best strategy is to apply just enough indium, then use
the other half of the cavity to push on the chip. We usually perform a final check to see if
the chip sticks well to the copper surface.
5.3 Microwave Electronics
The microwave electronic wiring, optical table, physical signal connections, and typical
instruments are shown in Fig. 5.10. We will go into the details in this section. We dis-
cuss control and readout separately, as the former involves signal synthesis, and the later
involves signal analysis.
5.3.1 Control
The pulses for controlling qubits and performing cavity readout are generated by an arbi-
trary waveform generator (AWG). In the lab, we mostly use a Tektronix 5014C AWG, which
CHAPTER 5. EXPERIMENTAL TECHNIQUES 131
Figure 5.10: Room temperature electronics and microwave lines setup. (a) Wiring
diagram. (b) Optical table setup. (c) IQ RF sources and Keysight AWG + digitizer PXI
modules.
has 4 analog channels, each comes with 2 digital marker channels. These digital channels
are useful for generating pulses to readout the cavity, gate the IQ generators to decrease
carrier leakage, or to trigger other instruments. A single Tektronix AWG is usually good
enough to perform most experiments.
We recently incorporated a more capable Keysight PXI 3202 AWG, which can be ex-
tended to have as many as 32 analog channels. However, it doesn?t include digital channels,
so we usually have to use the analog channels to generate simple square pulses to trigger
the digitizer or to gate the IQ generators, which is not cost effective. The Keysight tends to
crash less often than the Tektronix.
The Tektronix 5014C has maximum sampling rate of 1.2 GSa/s, whereas the Keysight
sampling rate is 1 GSa/s. Both can reliably generate pulses with maximum resolution of
CHAPTER 5. EXPERIMENTAL TECHNIQUES 132
around 2 ns. Beyond that, there is distortion in the synthesized pulses which can be seen
with a fast oscilloscope. The Tektronix can also generate RF pulses with frequency up to
250 MHz in normal mode, or up to 350 MHz in direct output mode at amplitudes below
0.5 V. We note that the practical performance of the AWGs we tested is often not as good as
specified by the manufacturers. For example, there is distortion in the sinusoidal wave at
300 MHz, and there are also spurious modes coming from the AWG at higher frequencies,
which can be seen in Fig. 5.11(e) and (f).
The cavity tone is generated by a Rohde & Schwarz SMB100A RF source in pulse mod-
ulation mode, with frequency range from 100 kHz up to 20 GHz, and on/off ratio higher
than 100 dB.
We use Rohde & Schwarz SGS100A SGMA IQ vector signal generators to drive qubits.
These generators can perform IQ modulation in a wide frequency range, from 80 MHz up to
12.75 GHz, and can output CW tone from 1 MHz to 12.75 GHz, so we can use them to scan
the spectrum of fluxonium circuits conveniently. These small boxes can be stacked snugly
on any shelving unit, so many of them can be easily added if any experiment demands
additional drives. They may produce distortions when the input IQ pulses? amplitudes
are too high, as shown in Fig. 5.12(b). When high fidelity operation is desired, it is best
to double check the RF pulses both in frequency and time domain to confirm that the
instruments generate the signal we expect.
For local oscillator, we use refurbished Agilent HP 8671a sources together with Mini-
Circuits amplifiers at their outputs to increase the power needed to mix the signal down.
We typically use a Mini-Circuits ZX05-153 frequency mixer which requires the LO power to
be at least 10 dBm.
To generate signal at higher frequencies, we employ a Rohde & Schwarz SMW200A
vector signal generator which can output RF signal at frequency up to 40 GHz. This instru-
ment typically has better specifications, with up to 2 GHz IQ modulation bandwidth, higher
on/off ratio, and better spectral purity. It can be programmed to work as an AWG as well,
CHAPTER 5. EXPERIMENTAL TECHNIQUES 133
although we have not utilized this function.
Microwave electronics are generally quite reliable, but we have to make sure to follow
the specifications listed by the manufacturers. Occasionally, the experiment stops suddenly,
and the first reasonable response is to check the microwave electronics to see if they output
the signal with correct frequency, power, and modulation options or not. If there is indeed
a problem with an RF source, it can often be fixed by simply power cycling the instrument
and performing internal adjustment and calibration. In rare occasions, broken instruments
have to be sent back to get fixed, in which case having an extra RF source on standby is not
a bad idea.
When testing the instruments, we note the importance of the linewidth of the Gausssian
pulses. Sharp pulses are wide spectrally, potentially creating leakages to other transitions,
as shown in Fig 5.11(a). This can be verified by takin?g the Fourier transform of a Gausssian2 2
function. For example, for f(t) = e??t , F (?) = ? e?? /4?? , so the linewidths in time
domain and frequency domain are inversely proportional. A normal modulated Gaussian
pulse is shown in Fig. 5.12(a). When short pulses are desired for a crowded spectrum,
DRAG can be used to mitigate the problem [107, 108]. Note that there are some distortions
in the spectrum of a short pulse at low frequency, due to the limitation in IQ modulation
capability of the generator, as shown in Fig 5.11(b).
A common problem in RF electronics is the harmonics generated by the instruments,
usually at twice the carrier frequency. A low pass filter with an appropriate cutoff frequency
can be used to mitigate this problem. Other common problems include carrier leakage and
asymmetric IQ modulation, shown in Fig 5.11(c).
When the frequencies are beyond the specification of the instrument, IF pulse leak-
age also appears. As shown in Fig 5.11(d), when we configured an IQ modulation with
?IF/2? = 100 MHz, ?RF/2? = 180 MHz using the Rohde & Schwarz SGS100A SGMA
source, the IF tone leaked through and could be detected by the spectrum analyzer.
If the qubit transition frequencies are too low, beyond the range of the IQ source (below
CHAPTER 5. EXPERIMENTAL TECHNIQUES 134
Figure 5.11: Frequency domain pulse testing. (a) The Gausssian pulses are generated
by the AWG and feed into the IQ RF boxes. We see that the shorter the pulse, the wider
the spectral width becomes. (b) There can be distortion at low frequency if the pulse
profile exceeds the IQ RF source?s specification. (c) IQ modulation imperfection, including
carrier leakage and asymmetric sidebands. (d) When the carrier frequency is lower than
specification, IF leakage is present. (e, f) Errors in AWG?s generated pulses, namely noises
at higher frequency, and lower pulse amplitude when both Channel 1 and Channel 2 are
used.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 135
Figure 5.12: Time domain pulse testing. (a) A good Gausssian enveloped pulse. (b) A
bad pulse caused by the input Gausssian pulse having an amplitude higher than the IQ RF
source can handle. (c) Time delay checking. The blue pulse is about 5 ns ahead of the
orange pulse. (d) Phase checking. The signals are fitted to a model and the phases can be
compared. In this case, the phase difference between the I and Q signals is good.
80 MHz), we can apply an RF pulse directly from the AWG. We noted one subtle problem
with the AWG used in fluxonium experiments. When there is a waveform loaded into
channel 2, channel 1?s pulse amplitude changes (see Fig. 5.11(f)). So if both channel 1 and
channel 2 are used simultaneously, we have to calibrate the pulses when both channels are
loaded with some waveform.
Note that pulse modulation usually well work only when the external pulse from the
AWG has high amplitude. On the other hand, IQ modulation does not work well when the
pulse?s amplitude is too high, as shown in Fig. 5.12(b). Delays between pulses should also
CHAPTER 5. EXPERIMENTAL TECHNIQUES 136
be checked and compensated for if we do not want the pulses to overlap (see Fig. 5.12(c)).
Ideally, the IQ signal is generated such that its components are mathematically 90o out
of phase, but distortion can disturb this phase difference. Therefore, it is best to check the
pulses with a fast oscilloscope and compensate for any mismatch (see Fig. 5.12(d)).
5.3.2 Readout
In the experiments reported in this thesis, we use heterodyne readout scheme as described
in ref. [141]. The main signal going through the fridge and the reference signal going
directly to the digitizer?s other input are generated by the same RF source. Since electrical
analog-to-digital converter (ADC) can only digitize at a maximum rate of a few giga-samples
per second, the signal must be mixed down to intermediate frequency, which is 50 MHz in
our experiments. We use a commercial RF mixer from Mini-Circuits for this purpose. The
resulting signal S(t) is then digitized by an analog-digital-converter (ADC) electronics card,
and post-processed in IQ fashion,
?
1 t+TIF
I(t) = cos(? ? ? ?IFt )S(t )dt ,
TIF ?t (5.5)
1 t+TIF
Q(t) = sin(? ? ?IFt )S(t )dt
?,
TIF t
which give us the IQ signals averaged over the demodulation period for both the main
signal and the reference, denoted as A and B, respectively. The demodulated IQ signals
can be written as
IA(t) = AA(t) cos(?A(t) + ?0(t)),
QA(t) = AA(t) sin(?A(t) + ?0(t)),
(5.6)
IB(t) = AB(t) cos(?B(t) + ?0(t)),
QB(t) = AB(t) sin(?B(t) + ?0(t)),
CHAPTER 5. EXPERIMENTAL TECHNIQUES 137
from which we can cancel out the phase instability of the RF source ?0, and record the phase
difference betwe?en the signals, ?(t) ? ?A(t)? ?B(t), along with the amplitude of the main
signal, AA(t) = I2A +Q
2
A. Alternatively, heterodyne demodulation using single-sideband
balanced IQ mixers can be used, which technically gives better SNR [80].
We noticed that the ADCs have large leakage from the trigger and the 10 MHz reference
clock. This is usually not a big problem because (i) we use a reference signal and the setup
is only sensitive to the phase difference between the signal coming from the fridge and the
reference, and (ii) we only care about the change in the signal when the qubit shifts the
cavity?s resonance.
We may bypass these intrinsic RF noise by choosing the right configuration for the de-
modulation. First, the trigger duration should be multiples of the clock?s period Tclk =
100 ns to avoid jitters in the demodulated signal. Second, we should choose an optimal
IF frequency. In choosing this, we have to take into consideration the bandwidth of the
digitizer, the timescale of dynamics we want to measure, noise spectrum from electronics
at the card?s input. To bypass the clock reference leakage, a frequency that is multiple of 10
MHz is preferable. In the end, we choose 50 MHz, although other frequencies would also
work.
A small phase drift is often observed in long measurements. We speculated that this
drift is due to the fluctuations inside the room temperature amplifier, since readout lines
with different amplifiers consistently correspond to different amount of drift, and the mea-
surement utilizing the best room temperature amplifier (LNF-LNR4-8C) has the smallest
drift.
Depending on the specific dispersive shift of the cavity induced by the different qubit
states, the shift can be clearer either in phase or magnitude. In general, one must check
both at different cavity frequencies to find the optimal readout setting. For most of the
experiments involved here, the dispersive shifts are substantially smaller than the cavity?s
linewidth, so we mainly readout the phase of the cavity. In recent experiments where the
CHAPTER 5. EXPERIMENTAL TECHNIQUES 138
dispersive shift is comparable to cavity?s coupling rate, ?01 ? ?, we typically project the
signal along the axis with highest separation in the IQ plane.
We use two different types of digitizers in the lab. The first one is the Alazar 9870, and
the second one is the Keysight 3102. The advantage of the Alazar is its fast data transfer rate
to the computer?s memory at 1.6 GB/s with very good buffer management, lower noise at
the input of the channel, and a faster digitization rate at 1 GSa/s. The Keysight lags behind
in these areas with a slower data transfer rate at 200 MB/s, unknown noise at channel 1 and
channel 4 inputs which can be due to internal leakage, and a slower 500 MSa/s sampling
rate. However, the Keysight has 4 input channels, 2 GB of on-board RAM, and can be used
together with other Keysight PXI cards to do FPGA-based feedback operations. Both of them
exhibit noise from the 10 MHz reference clock and the trigger input mentioned above.
5.3.3 Other Equipment
We use a Yokogawa GS200 programmable DC voltage/current source to bias the magnetic
flux. We initially used the instrument in voltage bias mode, but then noticed that the
applied current fluctuated and drifted in time, causing significant flux instability. We have
since then used the source in current bias mode. The instrument has a feedback loop that
stabilizes the output current with high precision in this mode, ?I = ?0.003% in one day.
For S-parameters measurement, we extensively use a Rohde & Schwarz ZNB20 vector
network analyzer (VNA) that works up to 20 GHz. The VNA is also used to perform continu-
ous wave (CW) spectroscopy. It generally takes data faster than a heterodyne setup because
we only need to set the starting and stopping frequencies, together with the number of
points to scan. The instrument?s control then communicates with the generator internally,
which is more efficient than the full heterodyne experimental scheme.
To check the spectral components of RF signals, we use a Rohde & Schwarz FSV30
signal analyzer that functions as both a spectrum and an IQ analyzer. It can analyze signal
CHAPTER 5. EXPERIMENTAL TECHNIQUES 139
at frequencies ranging from 10 Hz up to 30 GHz, and be configured to work as an ad-hoc
oscilloscope of high frequency pulse as well.
We use a Rohde & Schwarz HMO2024 oscilloscope to check the pulses in time domain.
It has a maximum sampling rate of 2 GSa/s and a bandwidth between 100 and 300 MHz,
which is good for low frequency signals. Since the specification for the IQ RF source is
generally better at frequencies above 1 GHz, there is generally no need to check for per-
formance above 1 GHz if the signal purity is as good as expected below 1 GHz. We also
use a Picoscope 9301 to check the instruments? performance at higher frequencies in time
domain.
5.3.4 Data Acquisition Software
A few different generations of control softwares have been used in the course of this dis-
sertation work as the lab evolved from its infancy. We started by using LabView to measure
cavities and the first few fluxonium devices. In 2016, we developed the first control scripts
using Python and a database to store data. We also developed a sequencer to transfer com-
plex sequences to the Tektronix AWG more efficiently. Later that year, we began using the
Python-based HQC software package developed at ENS Paris and combined it with our se-
quencing technique. We gradually switched to Labber starting in 2018. Certain capabilities
of existing Labber drivers, such as the multi-qubit pulse generator, have been modified to
meet our needs.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 140
5.4 Fabrication
5.4.1 General Considerations
After constructing the cryogenic setup, connecting the microwave electronics, and machin-
ing the cavities, it comes time to prepare the device. Superconducting circuits can be de-
signed and fabricated using standard lithography techniques. The fabricated devices must
satisfy two demands: (i) the circuit parameters must be close to the values desired, and (ii)
the surface of the chip must be as clean as possible to avoid unwanted dissipation in the
form of dielectric loss.
In the lab, we choose silicon as the substrate and aluminum as the superconducting
material. Silicon is currently the work horse in the semiconductor industry, and circuits
based on silicon fabrication have been proven to have low loss [48]. In addition, silicon can
be etched by plasma-chemical etching, which opens up new possibility for more complex
fabrication techniques. Inspired by the long relaxation time of a borrowed transmon on
sapphire sample, we recently started fabricating qubits on sapphire.
Aluminum is not only one of the most conventional superconductors, its behavior and
properties well predicted by theory, but its native oxide can also be conveniently utilized to
make Josephson junctions. Aluminum is thus far the best metal to build superconducting
circuits with.
Our fabrication follows the Dolan Bridge technique [19], which allows small overlap-
ping structures to be made with high accuracy and reproducibility using electron beam
lithography.
First, a double layer of polymer called resist is spun on top of the substrate. Then, the
sample is exposed to high energy electrons beam with a pre-defined design pattern. The
exposed regions become soluble in a developer, and they are then washed away, yielding a
mask. The two layers of polymer are chosen such that the lower layer is much thicker and
softer than the top one, leading to the creation of undercut in a structure.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 141
Figure 5.13: Fabrication facilities. Our fabrication facilities include (a) a chemical film
hood, (b) a Plassys evaporator, and (c) an Elionix e-beam machine.
The effect of undercut is then used to form hanging bridges in the top layer. The metal
film is then deposited in a metal evaporator. The exposed region allows patterning of the
metal film onto the substrate surface, whereas the rest of the mask with the metal film on
top is then removed using acetone.
The fabrication procedure essentially consists of six steps: chip preparation, resist appli-
cation, electron beam writing, development, deposition, and liftoff. It is important that the
details of the recipe are followed in an exact manner every time for consistency. The recipe
is thus included below.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 142
5.4.2 Fabrication Recipe
Chip Preparation
We first cut out the chip using a cleave such that its dimension must fit the cavity on which
it is mounted. For fluxonium experiment, we use high resistivity undoped silicon wafer. We
begin by spraying isopropanol (IPA) onto the wafer and blow dry it with nitrogen gas (N2).
We use a glass slide and a scribe to scratch out the part we want to cleave. We repeat the
cleaning with IPA and N2 between every step to ensure that the surface of the chip is free
of debris, which is mostly silicon dust in this case. Once the cut on the chip is aligned with
the cleaver, we can gently press it down to cleanly cut the chip away from the wafer. We
repeat this as many times as needed to prepare the final chip with the exact dimensions.
The cutout chips are cleaned through solutions of acetone and IPA to get rid of any
residue or dust on the surface. The chips are typically sonicated in said solution for 3
minutes each. We have to make sure that the chip is dropped from acetone into IPA solution
right away to prevent acetone drying, otherwise some acetone residue may stick on. After
removal from IPA, the chip is again blown dry with N2. We then inspect the chip under the
microscope for possible dust or defects and decide to proceed to the next step or prepare
another chip.
Resist Application
Two layers of resists have to be put on the chip?s surface, MMA EL 13 and 950 PMMA A3.
First, we set the hot plate to 180o C, then spin MMA EL 13 at 5000 RPM on the spinner for
1 minute. Once the spin is completed, we remove the chip and bake it on the hot plate at
180o C for 1 minute. We then spin the 950 PMMA A3 on the spinner at 4000 RPM for 1
minute, and bake it on the plate for 30 minutes, also at 180o C. We do chip inspection after
every step to make sure the surface is clean and the resist spreads evenly across the area
where the pattern will be written.
CHAPTER 5. EXPERIMENTAL TECHNIQUES 143
Electron Beam Writing
Electron beam writing was first done in the clean room using a Raith and then a Joel
electron beam machine. We started using the Elionix ELS G100 electron beam lithography
system in 2016 and subsequently increased the fabrication productivity. All the devices
reported in this thesis, except for the double loop device, were fabricated using the Elionix.
The Auto CAD design file is first converted to a schedule file and loaded into the system.
Specific settings must be chosen to calculate the right doses for the lithography. After all
the parameteres have been set, the chip can be loaded into the machine. We then focus the
beam and set the correct current (1 nA in this case) and execute the write. We check the
pattern and parameters again after every step to make sure that nothing may go wrong.
The big features are written with 100 nA current.
Development
Once the electron beam writing has been finished, the chip is dipped into a solution called
the developer, which is 3:1 IPA:DI by volume. The solution is kept at 6o C at all time.
To make the development step more consistent, we use an ice bath with an exact ratio of
water to ice every time, and the developer?s beaker is immersed in the ice bath until the
temperature reaches equilibrium, which typically takes 10 minutes. The chip is held by
a pair of reverse-action tweezers and wiggled back and forth inside the developer for 2
minutes at a frequency of approximately 2 Hz. The chip is taken out immediately after the
timer goes off and blown dry with N2. The cold bath is a key feature of the development
process.
Deposition
We load the chip into the Plassys evaporator and pump the loadlock for 20 hours before the
deposition to get rid of possible contaminants. We check that the loadlock pressure reaches
CHAPTER 5. EXPERIMENTAL TECHNIQUES 144
1.3 ? 10?7 mbar before beginning deposition. The deposition process include the follow-
ing steps: (i) 20 s descum with argon at each deposition angle (?23.83o). (ii) Titanium
evaporation at a rate of 0.1 nm/s for 2 min with the shutter closed to clean the chamber.
(iii) First Al deposition at a rate of 1 nm/s at an angle of 23.83o, resulting in approximately
20 nm of aluminum. (iv) 10 minutes static/dynamic oxidation at 100 mbar. (v) Second Al
deposition at a rate of 1 nm/s at -23.83o. Approximately 40 nm of aluminum is deposited.
(vi) 20 minutes oxidation at 10 mbar.
Liftoff
We place the chip in acetone and put the beaker on a hot plate at 60o C for 3 hours. We
then sonicate it for 5 s and sonicate again in IPA for 10 s. Finally, we blow dry the chip with
N2
The device is generally considered completed at this stage. However, significant aging
occurs in the first few hours after the chip is taken out of the evaporator, and the junctions
continue to age for at least a month or two, but at an exponentially slower rate. There-
fore, we usually fabricate the desired device about two weeks in advance of the intended
cooldown to ensure that the parameters have settled down by the time they are extracted
from measurement. That way, we can design our next device to meet the required specifi-
cations based on the previously measured ones.
Chapter 6
Fluxonium Experiments
Whatever it is you?re seeking won?t
come in the form you?re expecting.
Haruki Murakami
This chapter discusses the experiments with fluxonium superconducting artificial atoms
in our lab at the University of Maryland. We start with rudimentary cavity characterization
in section 6.1, namely determining the cavity resonance and quality factor. Then we focus
on our primary results, showing the demonstration of protection from energy relaxation in
section 6.2 and high coherence results in section 6.3. We report current progress on the
microwave-activated controlled-Z entangling gate in two capacitively coupled fluxonium
qubits in section 6.4. Finally, we review auxiliary experimental details in section 6.5.
6.1 Cavity Characterization
Before cooling down and measuring the qubit, the nominal first step is to characterize the
cavity, determine the input/output coupling, and check for the resonant frequencies. We
measured cavities in both 1-port reflection and 2-port transmission configurations using the
145
CHAPTER 6. FLUXONIUM EXPERIMENTS 146
Figure 6.1: 1-port cavity reflection measurement. Fitting (solid orange) of the reflection
S11 data (blue) using Eq. 5.4 for both (a) amplitude and (b) phase.
vector network analyzer. Typical results at room temperature are shown in Fig. 6.1 and
Fig. 6.2.
In reflection measurements, both magnitude and phase data are used to extract the
quality factors. In order to find the individual quality factor of each port, we use a port-
less part for the other half of the cavity (see Fig. 5.9). For most copper cavities used for
fluxonium readout, the input port Qin is around 1000, and the output port Qout is around
3000. As explained in section 5.2, the fit is more accurate when the external and internal
quality factors are close, Qint ? Qext.
We also performed transmission measurement of our 3D cavities, first to check the total
quality factor, and second, to record the presence of other resonances. The result in Fig. 6.2
shows the small transmission magnitude in the low frequency region of the 3D cavities.
We deduced that this was due to direct capacitive coupling between the ports. The data
suggests the importance of the pins? placement, and simulation has shown that placing the
pin closer to the qubit enhances the coupling between the input port and the qubit mode at
low frequencies (see Fig. 5.7(c) and (d)). This increased capacitive coupling to the dipole
antenna allows fast gates for the low transition frequencies in our devices.
After setting up and testing the cryogenic lines, the very first experiment we performed
CHAPTER 6. FLUXONIUM EXPERIMENTS 147
Figure 6.2: 2-port cavity transmission measurement. Transmission measurement result
of cavities with resonant frequencies at 7.5 GHz and 11 GHz. Note that the transmission
profile of a 3D cavity differs from that of an ideal LC oscillator away from resonance.
in the lab was measuring the quality factors of aluminum cavities across different cool-
downs and shielding configurations to find the optimal setup. The changes in Qint are sum-
marized in Fig. 6.3. Both high-power (blue squares) and low-power (orange circles) probe
tones were used, with higher power typically resulted in higher quality factors (Fig. 6.3,
inset).
We started with a cavity made of 6061 aluminum alloy, the same material used in aircraft
and yacht construction due to its good mechanical properties. The cavity was made in-
house using a micro milling machine. Then we switched to a 4N6 aluminum cavity, also
machined in-house. The design is shown in Fig. 5.9(a). The purer 4N6 aluminum cavity
has higher quality factor. Since the cavities are superconducting below 1.2 K, their internal
quality factor should naively approach infinity at 10 mK, but the data deviates from this
expectation.
We first improved Qint of the 4N6 aluminum cavity by etching it via electrochemical
CHAPTER 6. FLUXONIUM EXPERIMENTS 148
Figure 6.3: Change of internal quality factor. First, the cavity material was changed from
6061 Al to 4N6 Al. Etching process then improved the quality factor by a factor of two. The
best improvement in Qint was made by installing a gold-plated radiation shield underneath
the base plate. Further increase in Qint was made by applying a small magnetic field to
the cavity (see text). (Inset) Variation of internal quality factor of the 4N6 cavity inside the
extra radiation shield at 11 mK with respect to input power from the VNA.
polishing. The slow etching process [142] helped improve Qint by more than a factor of
two. The most notable change happened after we installed a gold-plated radiation shield
underneath the base plate. This could be due to the lower radiation level which may create
non-equilibrium quasiparticles in the superconducting cavity.
Further increase in Qint was made by applying a small magnetic field to the cavity.
The field was generated by a permanent magnet placed under the dilution fridge. We
CHAPTER 6. FLUXONIUM EXPERIMENTS 149
varied the distance between its center of mass and the bottom of the fridge, then measured
the corresponding cavity quality factors. When the magnet was placed right below the
fridge, the quality factor dropped. As we removed the magnet completely, the quality factor
returned to its initial value after we installed the radiation shield. While experiments with
magnetic fields were not rigorous, the result shows that the presence of such a field can
mitigate dissipation, which is likely due to residual non-equilibrium quasiparticles. This
suggests that further filtering and shielding should be employed in our initial setup.
6.2 Selection Rules Engineering in Fluxonium
6.2.1 Introduction
The principle behind protection of a quantum system from energy relaxation is as follows
[143, 144]. The two qubit states involved must have weakly overlapping wave functions,
such that the transition matrix element of all local operator, called transition dipoles in
atomic physics, is suppressed. This mechanism plays a crucial role in atomic clock [145],
but it is not clear if this can help extend the lifetime of macroscopic superconducting sys-
tems, since cQED in the metastable regime involving forbidden transitions has not been
explored experimentally. Fluxonium is especially suited to demonstrate this selection rules,
since there are both forbidden transitions with small dipole associated with suppressed flux
tunneling, and allowed transitions with large dipole associated with plasma oscillations in
the circuit.
This section shows the accomplishment of this key step in a superconducting artificial
atom. We designed circuit parameters to combine both forbidden and allowed transitions.
The transition dipoles are flux tunable while the transition frequencies are minimally mod-
ified. The forbidden transitions can still dispersively interact with a cavity mode even if its
transition dipole vanishes, due to the virtual excitations of the allowed transitions to higher
states in the circuit.
CHAPTER 6. FLUXONIUM EXPERIMENTS 150
6.2.2 Forbidden Transitions in Fluxonium
To achieve the necessary tunability, we replace the single weak Josephson junction with
a split-junction, also referred to as SQUID-junction in literature. The circuit is coupled
capacitively to a 3D copper cavity via a large dipole antenna. The resulting circuit, shown
in Fig. 6.4 (a) and (b), can be viewed as a 3D split transmon shunted by a large linear
inductance. The circuit Hamiltonian is defined by the inductive energy of the chain EL,
the the charging energy of the total capacitance EC , and the variable Josephson energy
of the split-junction EJ(?1). The circuit can be tuned by the fluxes ?1 and ?2 threading
the two loops. The reduced flux ? in the inductance is the position-like quantum degree
of freedom of the circuit. It moves in an effective potential given by U(?) = EL?2/2 ?
EJ(?1) cos(? + ?J(?1) ? ?2, where EJ(?1) and ?J(?1) are derived in Appendix B (see
Eq. B.13 and Eq. B.14). The kinetic energy is given by 4ECn2 where n is the momentum-
like continuous variable conjugate to ?.
In the regime where EJ/EL  1 and EJ/EC ? 10, there are two distinct types of
transitions in fluxonium circuit. The former condition ensures that the potential consists of
multiple Josephson wells whose depth and tilt are controlled by the fluxes ?1 and ?2. The
latter condition weakens quantum tunneling between states localized in different wells. The
intrawell transitions are called plasmons by analogy with plasma oscillations in junctions,
similar to transitions found in a transmon [49]. The difference here is that a plasmon
is insensitive to charge noise due to the inductive shunt [118]. The interwell transitions
are called fluxons. They are accompanied by a 2? phase twist in the superconducting
phase along the circuit?s main loop. A fluxon is analogous to the transition of a flux qubit
[146, 55, 54], but with two to three orders decrease in sensitivity to flux noise due to
the large number of junctions in the loop [114]. When the two wells are offset against
each other, the two localized states have vanishing overlap. Thus, a fluxon is a forbidden
transition with an exponentially small matrix element for sufficiently large EJ/EC ratio
CHAPTER 6. FLUXONIUM EXPERIMENTS 151
Figure 6.4: Idea for selection rules engineering. (a) Images of a double-loop fluxonium
device in a 3D cavity. The antenna is directly connected to the split-junction. (b) Circuit
models of a double-loop fluxonium atom and its coupling to a 3D copper cavity. (c) Lowest
four energy levels of the atom accurately positioned in the double-well potential profile
U(?), along with their calculated wave functions. (d) The interwell fluxon transition 0-1
(magenta) has a vanishing transition dipole and hence is forbidden. The intrawell plasmon
transition 0-2 and 1-3 (blue and red) by contrast have transition dipoles of the order of
unity and are thus allowed.
[147]. By contrast, plasmons are allowed because their transition dipoles are near unity
across a broad flux range (see Fig. 4.5).
As mentioned earlier, readout of the forbidden fluxon transition is possible through the
virtual plasmon excitations (Eq. 4.26). The origin of the nonzero dispersive shift can be
understood as follows. When the qubit is in state |0?, the cavity is shifted by an amount ?0
due to its interaction with plasmon modes of the circuit. This ground state dispersive shift
value is large as the plasmon frequency approaches the cavity resonance, similar to the case
in transmon qubits [49]. State |1? localized in the other potential well shifts the cavity via
the same mechanism. The difference in the plasmon frequencies, blue vs. red in Fig. 6.4(c),
thus results in a non zero dispersive shift ?01 = ?1??0. Quantitatively, the dispersive shifts
are found by taking into account all the contributions from other states starting from |0? or
|1?, and is in general non-vanishing [127].
CHAPTER 6. FLUXONIUM EXPERIMENTS 152
6.2.3 Spectral Features
Two-tone spectroscopy as a function of coil current and frequency reveals the spectrum of
our device (Fig. 6.5). The cavity is irradiated near resonance at 10 GHz. The coil current can
be converted to the total flux through the circuit in a linear fashion, ?1,2 = ?1,2Icoil, where
Icoil is the coil current and ?1,2 are the flux coupling constants. The observed resonances
vary with the coil current in a quasi-periodic manner. They fit remarkably well to the
analytical model using only six adjustable parameters: EC , EL, EJ1, EJ2, ?1, ?2 (see Eq. 4.1
and Eq. 4.2). From this excellent fit, we can confidently calculate the corresponding matrix
elements of charge and flux operators at every flux point.
The spectrum shown in Fig. 6.5 contains interesting features. First, the quasiperiodicity
of the spectrum as a function of the coil current corresponds to changing the external flux
in the big loop by a flux quantum, ?2 ? ?2 + 2?. The SQUID loop has a much smaller area,
and hence a much larger flux period. The point of inversion symmetry of the spectrum at
Icoil ? 45.5 mA corresponds to biasing the SQUID loop with a half flux quantum, ?1 = ?,
and the effective Josephson energy reaches its minimum. Note that the minimum value is
nonzero due to the asymmetry of the split-junction, EJ1 6= EJ2.
The separation of the spectrum into plasmons and fluxons is apparent in the region
38 mA < Icoil < 42 mA. The weakly flux-independent transition with multiple sweet spots
is the lower-well plasmon. Because of the presence of the inductive shunt, the plasmon?s
frequency is not a monotonic function of EJ , although it reduces with EJ on average (see
Eq. B.13). The transition that changes linearly with the coil current in a zigzag pattern is a
fluxon. The avoided crossings correspond to a full hybridization of a fluxon with a plasmon
(Fig. 6.5, left inset). The frequency splitting quantifies the strength of interwell transitions,
varying from 100 MHz at Icoil = 38.56 mA, where a fluxon is well-defined, to over 1 GHz
near the spectral inversion point, where fluxon-plasmon strongly mix. Fig. 6.5?s top right in-
set illustrates that at sufficiently large EJ value, the logarithm of the anti-crossing frequency
CHAPTER 6. FLUXONIUM EXPERIMENTS 153
Figure 6.5: Spectrum of a split-junction fluxonium. Transmission signal near cavity res-
onance as a function of the coil current and spectroscopy tone frequency. Cavity resonance
is off the graph at about 10 GHz (Fig. 6.24). Dashed lines represent a fit to the circuit
model in Fig. 6.4 using Eq. 4.1. (Bottom inset) Enlargement of the smallest fluxon-plasmon
splitting region. (Top inset) measured values of the seven splittings visible in the plot vs the
extracted value of 1/2EJ in frequency units (see Eq. B.13).
scales as 1/2EJ , in agreement with the WKB description of tunneling [141].
6.2.4 Demonstration of Protection
Controlled inhibition of energy decay in our circuit is most clearly demonstrated by mea-
suring the lifetime of the lowest |0? ? |1? transition as we tune the coil current through
the plasmon-fluxon anti-crossing, shown in the lower inset in Fig. 6.5. Indeed, we observe
a drastic enhancement of energy relaxation time T1 time from T1 < 10 ?s at the plasmon
side to T1 > 1 ms at the fluxon side (Fig. 6.6(a)). To interpret the data quantitatively, we
CHAPTER 6. FLUXONIUM EXPERIMENTS 154
turned to the model of dielectric loss, common in superconducting circuits [18]. This model
is consistent with the observed enhancement of T1 and requires the bounds on the effec-
tive loss tangent tan ? of the total capacitance shunting the qubit to be between 2 ? 10?5
and 2 ? 10?6. For the same range of coil currents, we plot the calculated dispersive shift
?, which remarkably does not drop significantly at the fluxon side of the anticrossing (see
Fig. 6.6(b)). The measured dispersive shift, extracted from the Rabi oscillation amplitude,
agrees well with a theoretical expression without adjustable parameters. In sharp contrast,
the dispersive shift calculated using only states |0? and |1? drops rapidly with the enhance-
ment in T1, emphasizing the importance of plasmons in creating the finite dispersive shift
for fluxons.
Rabi oscillations along with echo experiments demonstrate that a fluxon remains coher-
ent even when its transition dipole is suppressed to the extent that T1 > 1.5 ms (Fig. 6.6(c)).
The coherence time T2, given by the characeteristic decay time of the echo signal, is given
by T2 ? 4 ?s and is limited by the first-order flux noise with amplitude of approximately
10?
?
6?0/ Hz at 1 Hz. The decay of the readout signal following a ? pulse on a qubit fits the
exponential function well, and repeated experiments did not produce more than a factor of
two variations of T1 within about one hour (Fig. 6.6(c)). This leads us to believe that the
fluctuation of T1 values in Fig. 6.6(a) occur on a longer than a one-hour timescale.
The central feature of our work is the collection of energy decay times for the qubit
transition at a number of values of the coil current such that the transition dipole, given by
d01 = ?0|?|1?, vastly varies, while the transition frequency is confined to a narrow interval
of 3.5-4.5 GHz. These transitions lose energy to essentially identical environmental factors.
Therefore, Fermi?s golden rule predicts that at zero temperature, the relaxation rate scales
quadratically with the transition dipole for an arbitrary linearly coupled environment. Our
data obey this simple scaling for the values of T1 spanning a remarkable range of over 2
orders of magnitude (Fig. 6.7(a)). Despite some fluctuations, the data clearly show that the
dramatic enhancement of T1 of a qubit occurs solely due to the reduction of its transition
CHAPTER 6. FLUXONIUM EXPERIMENTS 155
Figure 6.6: Data around the anti-crossing at 38 mA. (a) The measured lifetime T1 of
the |0? ? |1? transition (square marker) and its frequency (solid line). Dotted lines are
the dielectric loss prediction corresponding to the loss tangent values of 2 ? 10?5 and 2 ?
10?6. (b) Measured dispersive shift (square marker) and calculated dispersive shifts, taking
into account all transitions to higher levels (solid line) and only states 0 and 1 (dashed
line). Note that ? decreases much slower than T1 grows. (c) Examples of time-domain
measurement data: Rabi oscillation trace (red), echo trace (green), and energy relaxation
trace (blue). The inset shows the variation of relaxation time T1 over a period of about one
hour.
CHAPTER 6. FLUXONIUM EXPERIMENTS 156
Figure 6.7: Demonstration of protection. Measured energy decay times for transitions
|0? ? |1? (blue) and |1? ? |2? (red) taken from a narrow range of frequencies plotted
against the calculated values of the corresponding transition dipole squared. Dashed lines
illustrate Fermi?s golden rule predictions for dielectric loss (a, b) and quasiparticle loss (c, d)
using the expressions for their respective quality factors (Eq. 4.3 and Eq. 4.12, respectively).
dipole. Because the suppression of the matrix element in the fluxon regime has no classi-
cal analog, the observed scaling evidences that the energy decay occurs by a spontaneous
emission rather than by a thermal activation. Data for the 2.5-3.5 GHz frequency range,
including relaxation of the |1? ? |2? transition confirms our conclusion (Fig. 6.7(b)).
The only known nonlinear loss mechanism in which coupling to the bath cannot be de-
scribed using the matrix elements of ? is quasiparticle tunneling across the small junction
CHAPTER 6. FLUXONIUM EXPERIMENTS 157
[90, 91]. The effective transition dipole dqpij corresponding to the coupling to quasiparti-
cles involves a complex function of ? and depends on external flux. However, replotting
the energy decay times against this quantity (Fig. 6.7(c) and (d)) shows that coupling to
quasiparticles is suppressed in similar fashion. This is simply due to the fact that the fluxon
transition is forbidden from any local bath by the vanishing overlap of the qubit states
wavefunctions.
6.2.5 Metastable Regime
Additionally, by increasing the energy ratio EJ/EC , we reached the metastable regime,
where the fluxon state?s lifetime becomes immeasurably long. The spectrum corresponding
to max EJ/EC in the device is shown in Fig. 6.8. The main feature here is the plasmon
transition spectrum. Because tunneling is strongly suppressed, we could not coherently
manipulate fluxon transitions. Even at strong qubit tone power which is enough to clearly
reveal the |0? ? |2? two photon plasmon transitions, the fluxon line is still obscure.
While it was not possible to address the fluxon transition directly, we could observe state
|1? being populated by an incoherent pumping process, shown in Fig. 6.9. The cavity tone
effectively becomes a pump at high power, transferring the qubit?s population to the first
excited state which is localized in the higher potential well (Fig. 6.5(c)). This mechanism is
possibly similar to the measurement induced transition discussed in ref. [148]. When the
population in the first excited state is higher than in the ground state, the |1? ? |3? spectral
line appears with better contrast than the |0??|2? line. As we turned down the cavity power,
the population reverted back to the ground state.
CHAPTER 6. FLUXONIUM EXPERIMENTS 158
Figure 6.8: Top spectrum of split-junction device. Spectrum corresponding to the highest
EJ range in the split-junction device (device C in Table 6.1). Fluxon transitions are strongly
forbidden here, and only plasmon transitions are visible. The data still fits remarkably
well to the simple model. Using a high power qubit tone, we could reveal the two-photon
plasmon transitions at half of the |0? ? |2? transition frequencies (fitted to dashed lines),
which allows extraction of the anharmonicity. Another special feature visible here is the
|1? ? |3? transition in blue lines.
CHAPTER 6. FLUXONIUM EXPERIMENTS 159
Figure 6.9: Metastable regime spectroscopy. (a) Zoomed-in of the spectrum showing the
visible |0? ? |2? and |1? ? |3? transitions. (b) Qubit?s effective temperature depends on
cavity tone power. At low cavity power, the qubit is predominantly is the ground state, thus
|0? ? |2? transition is visible. At high cavity power, qubit localizes more in the excited state,
thus |1? ? |3? transition is visible, while |0? ? |2? transition becomes obscure.
6.3 High Coherence Fluxonium Qubits
6.3.1 Introduction
In this section, we describe fluxonium superconducting qubits designed to evade decoher-
ence due to dielectric loss and flux noise without sacrificing anharmonicity, flux-tuning
range, or controllable interactions. The geometry of the qubit is shown in Fig. 6.10(a), and
its circuit diagram is drawn in Fig. 6.10(b).
In this experiment, we operate the qubit near the ?ext = ? ?sweet spot?, where the
first order sensitivity of the transition frequency to flux vanishes by symmetry (Fig. 6.10e).
The qubit states |0? and |1? correspond to the tunnel splitting of the two-fold degenerate
classical ground state. The circuit parameters are such that the qubit transition frequency
is about an order of magnitude lower than that at ?ext = 0. The non-computational states
|2?, |3?, etc., are separated by a plasmon gap (Fig. 6.10e) and they form an anharmonic
spectrum with a rich selection rule structure.
CHAPTER 6. FLUXONIUM EXPERIMENTS 160
(a) (b)  (c)
coupling antenna
EJ EC EL AlO
100 ?m x
small junction ext
1 ?m
1 ?m
(d) (e)
1 ?m ext = 0 ext = ?
readout  |3i
resonator | i |3i2
| |2i1i
plasmon 
gap
SMA |1i
in/out port |0i |0i
10 ?m device chip
 
Figure 6.10: Description of high coherence fluxonium experiment. (a) Images of a
single-junction fluxonium device indicating the antenna, the loop, the small junction, and
the chip mounted in a copper resonator. (b) The three-element circuit model of fluxonium.
(c) Implementation of a large-value inductance L using a linear chain of Josephson junc-
tions. (d, e) The particle-in-a-box potential energy, the spectrum, and the eigenstates for
the circuit model in (b) in the cases ?ext = 0 (d) and ?ext = ? (e), which is the focus of the
present work.
By reducing qubit frequency tenfold from typical 4 ? 8 GHz range, the energy relax-
ation rate is expected to naturally slow down without the need for improving materials
(see Eq. 4.3). One might think this trick would inevitably slow down interactions too, as
it happens with weakly anhamonic oscillators. Fortunately, this is not the case with fluxo-
niums. For example, the dispersive coupling to a cavity mode can be as large as it is in a
typical transmon setup owing to the high frequencies and dipoles of transitions outside the
computational space. Furthermore, inductive coupling can be made large even within the
computational subspace, as it was experimentally demonstrated by the molecular binding
spectra of two fluxoniums with shared chain junctions [149].
6.3.2 Results
Similar to the original fluxonium design, here we attach an external capacitance in the form
of a simple dipolar antenna directly to the small junction. The capacitance is mainly due
Energy
Energy
CHAPTER 6. FLUXONIUM EXPERIMENTS 161
to this antenna. Device C is used to demonstrate protection in the previous experiment
described in section 6.2. It contains a split-junction (DC superconducting quantum inter-
ference device). It is now measured at the simultaneous sweet spot (the so-called double
sweet-spot) of both loops, which were fabricated to have commensurate areas. In all other
devices we used a single junction for simplicity. The qubits were capacitively coupled to a
3D copper box readout mode with a frequency of 7.5 GHz and a linewidth ?/2? ? 15 MHz
(this is equivalent to a rather low external quality factor Q = 500). The state of the qubit
was monitored in a basic two-port cavity transmission measurement. For consistency, all
devices were measured at a small external magnetic field. The precise effect of this field
remains inconclusive at this stage.
Cryogenic setups are described in section 5.1. The robust Dolan bridge technique is
used in fabrication. It yields the smallest possible stray capacitance in a planar geometry.
We can fabricate up to 40000-junction long chains without any fault. The advantage of
such chain geometry is the lack of need for resist mask?s precision tuning. The substrate is
high-resistivity silicon covered by native oxide. Our fabrication procedure is discussed in
more details in section 5.4.
The measured spectrum of device A together with the fitted theory lines is shown in
Fig. 6.11. The data was obtained by conventional two-tone spectroscopy [63]. The theory
is a result of numerical diagonalization of the simple Hamiltonian (Eq. 3.17) with EJ , EC ,
EL, and flux to coil current conversion as fit parameters. Note that in addition to fitting
the two lowest frequency transitions, the theory precisely matches the red sideband of the
readout mode with the transition between states |0? and |4?. This indicates that there are no
stray chain modes at frequencies below 10 GHz and the simple Hamiltonian is an accurate
model of our complex device. At ?ext = 0 the qubit transition (labeled ?0 ? 1?) corresponds
to anharmonic oscillations in the central Josephson well. The nature of this transition,
along with its frequency at about 4.5 GHz and the value of the transition dipole ?0|?|1?
is similar to that of a typical transmon qubit. The difference here is that the Josephson
CHAPTER 6. FLUXONIUM EXPERIMENTS 162
well is slightly deformed by the shunting inductance (Fig. 6.10(c)) and the anharmonicity,
approximately given by EC ? 0.8 GHz, is considerably enhanced owing to the reduced
value of the shunting capacitance.
As the flux is tuned toward the sweet-spot at ?ext = ?, the qubit transition monotonically
drops to about 800 MHz. Already with the naked eye it is evident that the sensitivity of the
qubit transition to flux does not exceed about 20 GHz per flux quantum, while the frequency
is tuned by over two octaves. Due to the thermal occupation of the |1?-state one can see the
transition ?1 ? 2? in the small vicinity of ?ext = ?. The transition ?0 ? 2? is parity-forbidden
exactly at ?ext = ?, which is correctly reflected by the continuous reduction of its power-
broadened linewidth (in a fixed power experiment) upon tuning the flux towards the sweet
spot. The large anharmonicity of the qubit at the sweet spot can be characterized by the
ratio ?12/?01 ? 3? 10 for our typical circuit parameters. Finally, we note that no two-level
defects were spotted in the spectrum near ?ext = ? with anticrossings larger than a few
MHz.
The frequency dependence of energy relaxation time T1, covering several frequency oc-
taves, was measured by flux-tuning the qubit transition between ?ext = 0 and ?ext = ?.
The T1 values were obtained by a standard time-domain experiment recording the evolu-
tion of the cavity transmission following a ?-pulse to a qubit. The majority of relaxation
signals fit well to an exponential function and the characteristic decay time is quoted as T1.
We observed infrequent instances of a double-exponential decay, and those cases were not
included in extracting the qubits? T1 values.
To extract the noise spectral density S?(?), we normalized the measured relaxation
lifetimes by the square of the matrix element and plotted the quantity T1 ? |?0|?|1?|2 as a
function of qubit frequency ?01/2? (Fig. 6.12, top). According to Fermi?s golden rule, this
quantity is inversely proportional to the spectral density of the noise coupled to the phase
variable ? at the qubit frequency ?.
At frequencies above 1 GHz, the data follows a simple model S?(?) = h?C?2 tan ?C ,
CHAPTER 6. FLUXONIUM EXPERIMENTS 163
Figure 6.11: Two-tone spectroscopy of a fluxonium qubit. Two-tone spectroscopy trans-
mission signal (arbitrary units) as a function of spectroscopy frequency and flux through
the loop for device A. The contrast was optimized piecewise to maximize the visibility of
the resonances. Lines indicate a fit to the spectrum of the Hamiltonian. The extra resonance
line crossing (not anticrossing) the qubit transition 0 ? 1 is the red sideband of the 0 ? 4
transition and the readout at 7.5 GHz
CHAPTER 6. FLUXONIUM EXPERIMENTS 164
103
102
101
0.5 1.0 5.0
F I
B H
3 C10 D
G A E
102
0.2 0.5 1.0
Transition frequency (GHz)
Figure 6.12: Relaxation data. (top) Normalized energy relaxation time (a quantity in-
versely proportional to the spectral density of noise coupled to the ?-variable) as a function
of qubit transition frequency, measured by tuning flux in devices A, B, G, H. (bottom) Same
quantity, including repeated in time measurements, for all devices A-I biased at their half-
integer flux sweet-spots. Dashed lines represent a dielectric loss theory (see text).
assuming an effective frequency-independent dielectric loss tangent tan ?C of the total ca-
pacitance C across the weak junction. At lower frequencies we have to account for the stim-
ulated emission by thermal photons in the environmental modes and a possible frequency
dependence of the dielectric loss tangent (Eq. 4.3). The latter is modeled empirically as
tan ?C ? ?,  > 0. Using T = 20 mK (base temperature of the dilution refrigerator),
 = 0.15, and an interval tan ?C(? = 2? ? 6 GHz) = (2.0 ? 3.6) ? 10?6, we obtain a good
agreement between theory and the measured T1 times for all devices biased at their sweet
spots (Fig. 6.12, bottom).
The coherence time measurement was performed using a standard single ?-pulse echo
T1 ? |h0||1i|2 (?s)
CHAPTER 6. FLUXONIUM EXPERIMENTS 165
(a) (b) (c)
0 1 qubit A
ext/2? = 0.46
Time delay ( ?   s )
qubit C
ext/2? = 0.50 qubit I
ext/2? Time delay ( ?   s ) Consecutive attempt index
Figure 6.13: Coherence time data. (a) Coherence time T2 (markers) and the qubit fre-
quency (dashed line) as a function of flux. Solid line indicates a prediction for the first-order
coupling to a 1/f flux noise. (b) Gaussian echo signal away from the sweet spot (top) and
an exponential echo signal at the sweet spot (bottom).(c) Interleaved measurement of tem-
poral fluctuations of T1 (blue markers) and T2 (green markers) at ?ext = ? over a time
interval of approximately 12 hours.
sequence [33] in order to eliminate the uninformative low-frequency (minutes-scale) drifts
in the setup. This protocol choice was also dictated by the relatively low readout efficiency
in this particular setup, which required minutes of averaging time. The measured coherence
times T2 as a function of flux for device A are shown in Fig. 6.13(a). Away from the
sweet spot, where the qubit transition is maximally sensitive to flux, we typically measure
T2 ? 3 ? 6 ?s and the echo signal has a distinct Gaussian shape (Fig. 6.13(b), top). This
confirms that decoherence is due to pure dephasing. Tuning the flux towards the sweet
spot, we observe a monotonic rise in the coherence time to about T2 ? 100 ?s ? T1. Here
the echo signal is exponential, which is consistent with relaxation-dominated decoherence
(Fig. 6.13(b), bottom). The flux dependence of T2 agrees with the first-order coupling to
?
the 1/f flux noise with the amplitude 2? 10?6(h/2e)/ Hz at 1 Hz.
To understand the robustness of measured T2 values at the sweet spot, we have per-
formed repeated interleaved measurements of times T1 and T2. The results for the repre-
sentative devices A, C, and I are shown in Fig. 6.13(c). Each attempt took approximately
T2 (?s)
Cavity  phase shift (deg)
T1, T2 (?s)
Qubit transition frequency (GHz)
CHAPTER 6. FLUXONIUM EXPERIMENTS 166
30 min of data acquisition. Both the values of T1 and T2 drift in time, but their variation
typically does not exceed a factor of two over about a half-day. In some attempts we have
observed T2 ? 2T1, but a typical situation is T1 < T2 < 2T1. We believe that the difference
between 2T1 and T2 may be caused by a combination of imperfect tuning of echo pulses,
possible flux drifts due to imperfect magnetic shielding, and long averaging times. How-
ever, fluctuations in pure dephasing time, formally defined as 1/(1/T2 ? 1/2T1) could not
be fully excluded at this stage. A setup involving a quantum-limited amplifier would clarify
the details of the temporal drift of coherence times in future experiments. However, it is
clear that the coherence is already largely limited by energy relaxation.
The longest reliably measured values of T2 together with the accompanying value of
T1 are quoted for all devices in Table 6.1. Indeed, the worst (best) coherence device D
(I) has T2 = 90 ?s (T2 > 400 ?s) and corresponds to a slightly lower (higher) than usual
value of T1 (Table 6.1). Interestingly, the second longest coherence time (T2 > 300 ?s) was
observed in the device C, which contains a spit-junction, and it is thus sensitive to flux noise
in two loops. This strengthens the conclusion that flux-noise is not a limiting decoherence
mechanism in our devices at the sweet spot.
6.3.3 Discussion of Decoherence Channels
The summary of measured coherence times along with the extracted device parameters
is given in Table 6.1. Using this information we can place important bounds on various
decoherence mechanisms, which are summarized in the table and discussed below.
Flux Noise
Decoherence times measured away from the sweet spot as a function of flux agree with a
1/f flux noise model with the amplitude approximately equal to 2 ? 10?6(h/2e)/Hz1/2 at
1 Hz (see Eq. 3.43), which is similar to the noise level in flux qubits. However, here the
CHAPTER 6. FLUXONIUM EXPERIMENTS 167
Table 6.1: Summary of relevant device parameters. The values of EJ , EC , EL were ob-
tained from spectroscopy fits; loss mechanism limit estimates are discussed in the text.
? denotes effective EJ of the split-junction device biased at the double sweet-spot.
Qubit EJ EC EL N T1 T ?01 ?12 ?012 2? ? 2? tan ?C tan ?AlOx x tan ?L01
GHz GHz GHz - ?s ?s GHz - MHz ?10?6 ?10?4 ?10?8 ?10?8
A 3 0.84 1 100 110 160 0.78 3.4 0.27 1.7 1.1 3.84 15.4
B 4.86 0.84 1.14 136 250 150 0.32 11.1 0.57 1.5 1.3 0.52 2.03
C 2.2? 0.55 0.72 102 260 350 0.48 3.8 0.08 1.23 0.9 1.77 5.75
D 2.2 0.83 0.52 196 70 90 0.56 4.1 0.1 3.12 4.0 7 28.25
E 1.6 0.86 0.5 100 108 140 0.83 2.5 0.05 1.52 1.0 7.8 30.22
F 3.4 0.8 0.41 348 270 165 0.17 18.3 0.28 1.94 4.5 0.63 2.1
G 1.65 1.14 0.19 400 110 140 0.55 4.1 0.03 2 3.8 8.65 34.9
H 4.43 1 0.79 100 230 235 0.32 11.8 0.1 1.68 0.9 0.72 2.85
I 3.43 1 0.58 144 500 510 0.395 8.2 0.08 0.63 0.5 0.53 2.11
off-sweet-spot coherence time is 2-3 orders of magnitude longer (a few microseconds), due
to the large number of junctions N and hence the proportionally reduced first-order flux
sensitivity, whose maximal value is given by 2?EL ? 2e/h ? 1/N . Having measured both
the transition frequency vs. flux and the flux noise amplitude, we can estimate the limit on
coherence time at the sweet-spot due to the second order coupling. The coupling can be
approximated by (2?)4E2L/f01?(2e/h)2, which gives a range of coherence times 10?100 ms
for the typical devices presented here. Such long times are possible entirely due to the 1/N2
scaling of the second-order flux sensitivity.
Interestingly, the data shows no signatures of flux noise induced energy relaxation, re-
ported in recent experiments on flux qubits. Formally, the relaxation rate is given by the
familiar Fermi?s golden rule expression where the spectral density of the bath is replaced
by that of the flux noise, such that it grows at low frequencies (see Eq. 3.22). In the case
of a fluxonium, the relaxation rate scales as E2L ? 1/N2 (the square of the energy matrix
CHAPTER 6. FLUXONIUM EXPERIMENTS 168
element in the Fermi?s golden rule). This gives a protection against the suggested energy
relaxation by a factor 102 ? 104 in comparison to flux qubits. Such a large protection factor
may explain the dramatic difference between the T1 vs qubit frequency data reported in
and that shown here.
Quasiparticles
A qubit can relax by emitting a photon which is absorbed by an unpaired quasiparticle
tunneling across a junction. Assuming that the T1 values in Table 6.1 are limited by the
tunneling of out-of-equilibrium quasiparticles across the chain junctions, we conclude that
their normalized density xqp can be below 10?8. Given the device dimensions, this number
corresponds to less than one quasiparticle in the entire chain.
According to theory, the tunneling across the weak junction is coherently suppressed
at ?ext = ? by the destructive electron-hole interference [90, 91]. The interpretation of a
previous fluxonium experiment in terms of this effect implied a two orders of magnitude
difference between the values of x for the weak junction and for the chain junctions [85],
which seems unlikely. In our experiment, an estimate of the quasiparticle density near
the small junction can be obtained from device C which has a double-loop. There, the
coherent suppression of tunneling is absent even at the double sweet spot of both loops
because one of the two weak junctions is always away from the ?-phase bias. Yet, the
measured relaxation rate is similar to those of the single-loop devices. Its value requires
the absence of quasiparticles near the weak junction, consistent with the above conclusion
regarding the absence of quasiparticles in the chains. We cautiously speculate that vortices
may be efficient at trapping quasiparticles in our specific device geometry, although further
experimental results are needed to verify this hypothesis.
CHAPTER 6. FLUXONIUM EXPERIMENTS 169
Thermal Photons
Every qubit undergoing a dispersive readout will experience dephasing due to photon shot
noise if the readout mode is not properly thermalized [63] (Eq. 3.51). In this work we
deliberately avoided this issue by making the readout mode linewidth ? much larger than
the dispersive shift ?. As Table 6.1 illustrates, there is no obvious correlation between ?
and T2 which is evidence that thermal photons in the readout mode are indeed irrelevant.
Of more importance is the photon shot noise in theN?1 collective modes of the Joseph-
son chain. Their frequencies bunch near the junction?s plasma frequency, which in our case
is around 20 GHz. Because of the non-linearity of the plasma modes, occupation of one such
mode by a single photon introduces a dispersive shift of about 10?4 of the qubit frequency.
Such a shift is much larger than the qubit?s natural linewidth. Hence, in order for the qubit
to have a coherence time T2, the average time for the absence of an out-of-equilibrium pho-
ton excitation in each mode must be longer than N ? T2. Given numbers in Table 6.1, we
estimate this time to be longer than 50 ms, which means that the chain is practically empty
of out-of-equilibrium photons. The thermalization of plasma modes in our chains is intrigu-
ing because the microwave environment at such high frequencies is poorly characterized.
On one hand, with the external quality factor Q ? 500, our cavity is an order of magni-
tude more open to the readout line than those typically used in high-coherence transmon
experiments. Hence the circuit modes are well exposed to the environment of the readout
line. On the other hand, losses in coaxial components in this line are substantially higher at
20 GHz and this might lead to a better thermalization of the high-frequency modes.
Dielectric Loss
Because our devices have capacitive antennas, they are exposed to surface loss in the same
way as any other capacitively-shunted junction qubit. Small temporal fluctuations of T1
may be consistent with the recent data on the X-mon qubits, explained by the drifts in the
CHAPTER 6. FLUXONIUM EXPERIMENTS 170
value of tan ?C due to the dynamics of the weakly-coupled two-level defects [150]. Fluctu-
ations aside, the relaxation time T1 grows upon reducing the qubit frequency in agreement
with the model of a weakly frequency-dependent dielectric loss tangent tan ?C ? ? of the
total capacitance C across the weak junction. Let us first comment on this frequency depen-
dence. Unlike most energy relaxation measurements of superconducting qubits, the data
in Fig. 6.12 spans a remarkable 6 octaves of frequency, which makes it more sensitive to
the possible frequency-dependence of tan ?C . We used the values  = 0.15 and T = 20 mK
(the base temperature of our dilution refrigerator) to match theory and data in Fig. 6.12.
However, it is likely that the device temperature is higher and this would require a larger
value of . More accurate experiments are required to establish bounds on the value of 
to clarify the underlying mechanism. Note that having  > 0 is a fortunate circumstance
because it helps to increase the energy relaxation time towards lower frequencies.
The absolute values of T1 at the sweet spot can be explained assuming a narrow range
of tan ?C ? (2.0 ? 3.6) ? 10?6 taken at the frequency of 6 GHz (Fig. 6.12). Importantly,
these numbers are nearly an order of magnitude larger than those extracted for optimally
designed 3D-transmons [30], where tan ?C reaches down to tan ?C ? 3? 10?7. Given that
our antenna has a similar geometry, we expect it to have a similar surface participation
ratio. Therefore, our relaxation data strongly suggest that losses in the interface layers of
our devices are significantly stronger than those of optimally-fabricated transmons. This is
not unexpected, given that our simplistic Al on Si fabrication procedure does not involve any
advance surface preparation steps. In fact, the production of device I, which corresponds to
a notably smaller value of tan ?C (Fig. 6.12 (bottom) and Table 6.1), included an extra step
of dipping the chip into buffer oxide etching solution prior to spinning resist (see 5.4). This
is an additional evidence of surface loss dominating the energy relaxation.
Dielectric loss in the tunnel oxide of the chain junctions is another potentially important
energy relaxation mechanism. Assuming that each junction?s capacitance has a non-zero
loss tangent, we can estimate its average value as tan ? < 10?3AlOx . Fortunately, the large
CHAPTER 6. FLUXONIUM EXPERIMENTS 171
number of junctions in the chain helps to reduce the relaxation rate proportionally to 1/N .
This is because the alternating voltage across the antenna is divided by N for each junction
of the chain [114]. Our estimate on the loss tangent of AlOx (10?4) is about an order of
magnitude smaller than the previously reported bulk value [151]. It is possible that the area
of each chain junction (about 1 ?m2) is sufficiently low to make encountering a strongly-
coupled charge defect statistically unlikely. This effect can be explored in future devices
upon varying the chain junction area.
Inductive Loss
By analogy with the effective dielectric loss tangent tan ?C of the capacitance C we can
introduce an empirical inductive loss tangent tan ?L = Im[L]/Re[L] for the inductance L.
One can show that the two loss tangents leading to the same value of T1 are related accord-
ing to tan ?L/ tan ?C = ?201LC. For a harmonic oscillator, the ratio above is a unity, which
means that T1 is determined by the mechanism with the largest loss tangent. By contrast,
our devices are much more sensitive to the absolute value of the inductive loss tangent
rather than to an absolute value of the capacitive loss tangent, because for our parameters
?201LC ? 10?1 ? 10?2. Given that for the quasiparticles loss tan ?L ? x, one can qualita-
tively understand the nearly two orders of magnitude difference between the bounds on
tan ?C and x in Table 6.1.
6.4 Microwave-Activated Controlled-Z Entangling Gate
6.4.1 Introduction
A crucial step toward constructing a fault tolerant quantum computer is the implementa-
tion of a universal set of high-fidelity single- and multi-qubit gates. In a typical fluxonium
circuit with high coherence and high anharmonicity [61], single-qubit gates are expected
CHAPTER 6. FLUXONIUM EXPERIMENTS 172
to be nearly perfect. The evolution of fluxonium-based quantum processing technology has
therefore reached the second stage, where multi-qubit gates are explored and optimized.
In fluxonium-based architectures, a microwave-activated entangling gate has multiple
advantages. First, the qubits can be fixed at their flux sweet spots, so they remain highly
coherent throughout the entire operation. Second, the microwave control is analogous to
that of single-qubit gates, where an RF tone at a certain frequency is applied for a spe-
cific amount of time. Moreover, the spectra of fluxonium devices can be engineered with
more freedom since the Hamiltonian involves three energy parameters EC , EJ , and EL.
As discussed in section 4.1, fluxonium?s spectrum at half-integer flux is less sensitive to
fluctuations in EJ and EL than to changes in the ratio EJ/EL. This robustness against
fabrication errors makes scalable optimization of the microwave-activated interaction in
fluxonium practical.
In this section, we describe the experimental progress in implementing the controlled-Z
two-qubit entangling gate introduced in section 4.4 and ref. [128]. Due to the small matrix
element in the computational space, n0?1  n1?2, we can implement a fast entangling
operation and at the same time limit the unwanted interaction in the computational space
to achieve a high-fidelity two-qubit gate. To realize the gate, we employed two capacitively
coupled fluxonium qubits. To bias both qubits simultaneously at their sweet spots using the
magnetic coil shown in Fig. 5.3, we placed the two loops close to each other. To satisfy the
requirements on the qubit-qubit capacitive coupling constant JC , the qubit-cavity couling
constant g, and the charging energy EC , we designed the dipole antenna to be asymmetric
as shown in Fig. 6.14(a). The qubits are mounted on a copper cavity and readout jointly
[112, 111].
CHAPTER 6. FLUXONIUM EXPERIMENTS 173
Figure 6.14: Two-qubit device. (a) Optical image of two capacitively coupled fluxoniums.
(b) Joint readout setup using a copper cavity with linewidth ?/2? ? 4.5 MHz.
6.4.2 Spectrum
To characterize the two-qubit device, we first performed a two-tone spectroscopy scan to get
its spectrum, which was then fitted using the simple model to extract the qubits? parameters.
The fitting procedure consisted of two steps. First, we fitted the 0? 1 and 0? 2 transitions
of each qubit, assuming they are uncoupled. This allowed us to obtain the initial guess,
and to identify the coupled transition energies for the next step. Then, the spectrum was
fitted to a capacitively coupled fluxonium model (Eq. 4.27) utilizing up to seven transitions
across two flux periods. The extracted parameters can be used to simulate the spectrum to
CHAPTER 6. FLUXONIUM EXPERIMENTS 174
Table 6.2: Extracted parameters for two capacitively coupled fluxonium circuits.
EAC E
A A B B B
L EJ EC EL EJ JC g unit
0.972 0.456 5.899 1.027 0.683 5.768 0.224 0.1 GHz
Table 6.3: Charge matrix elements for computational transitions.
|00? ? |10? |00? ? |01? |10? ? |11? |01? ? |11?
nA -0.0252j -0.001j 0.0015j 0.0252j
nB 0.0007j 0.0446j -0.0446j -0.0006j
compare with the data as shown in Fig. 6.15(a).
For the specific two-qubit device reported in this thesis, the parameters including the
qubit-cavity coupling constant g are listed in Table 6.2. The spectrum contains many fea-
tures, but we only have to pay attention to the half integer flux quantum region. The ex-
tracted computational transition frequencies are: f00?01 = 136.3 MHz, f00?10 = 72.4 MHz,
and the gate transition frequencies are f11?12 = 4.902 GHz, f11?21 = 5.198 GHz. We
extracted the ZZ coupling rate ?ZZ ? 28 kHz and the gate rates f11?21 ? f10?20 = 22
MHz, f01?02 ? f11?12 = 15 MHz. The reduced charge matrix elements corresponding to
transitions within the computational space are computed and shown in Table 6.3.
Although the applied external fluxes for the two qubits were not perfectly aligned, we
could still choose a flux bias close to the two sweet spots which gave us a decent coherence
time T echo2 ? 40 ?s and relaxation time T1 ? 300 ?s for both qubits. The rest of this section
describes experiments done at this flux bias.
6.4.3 Single-Shot Joint Readout
We used a TWPA [136] in this experiment to improve the SNR of the cavity signal. This
allowed us to readout the qubit states in single-shot manner. To obtain the data shown in
CHAPTER 6. FLUXONIUM EXPERIMENTS 175
Figure 6.15: Spectrum of two capacitively coupled fluxonium qubits. (a) Spectrum
with fit from the simple model. (b) Zoomed-in spectrum showing the computational sub-
space with ?00?10/2? = 72.4 MHz, ?00?01/2? = 136.3 MHz. The ZZ coupling rate is
?ZZ = 28 MHz. (c) Zoomed-in spectrum showing the CZ gate subspace, with ?11?12/2? =
4.902 GHz, ?11?21/2? = 5.198 GHz. The gate rates are f01?02 ? f11?12 = 15 MHz,
f11?21 ? f10?20 = 22 MHz.
CHAPTER 6. FLUXONIUM EXPERIMENTS 176
Fig. 6.16(a), we first optimized the TWPA pump parameters, namely the pump frequency
and power, then swept the cavity power and frequency until the four blobs corresponding
to the computational states |00?, |10?, |01?, and |11? became well-resolved in a 2D IQ his-
togram plot. We also swept the integration time separately to find the optimal readout con-
figuration. The blobs were identified at a later stage by Rabi driving the higher transitions
|11? ? |12?, |01? ? |02?, |11? ? |21?, |10? ? |20? after preselection. Additional preselected
Rabi experiments for |00? ? |01? and |00? ? |10? were used to check for consistency.
We fitted a sum of four Gaussian functions to the projected single-shot histogram data
and extracted the populations of the computational states, as shown in Fig. 6.16(b). We
estimated the effective temperatures for the qubit A and qubit B assuming Boltzmann statis-
tics (see Eq. 2.76). For example, we made the assumption that the ground and excited state
populations of qubit A are P00 + P01 and P10 + P11, respectively. We found that these effec-
tive temperatures depended strongly on the cavity and TWPA pump tones? power. We do
not understand this effect fully at the moment, and further research is required to unveil
the underlying physics behind this behavior. With the optimal readout configuration, the
extracted populations are P|00? = 0.3, P|10? = 0.35, P|01? = 0.11, P|11? = 0.24. These cor-
respond to effective temperatures TA ? ?9 mK, TB ? 10 mK for qubit A and qubit B. We
emphasize again that these are effective, rather than physical, temperatures.
Our ability to readout the qubits in single-shot manner allowed initialization by pres-
election technique [152]. This was achieved by including in the pulse sequence another
pulse at the cavity frequency referred to as the heralding pulse, and a delay time ? = 5/? af-
ter the heralding pulse. The delay allows the cavity photons to decay before any qubit gate
operations. The result in Fig. 6.17 shows that we could initialize the qubit to the |00? state
with fidelity ? 0.9, corresponding to the effective temperatures TA ? 1.3 mK, TB ? 2.1 mK.
The initialization error comes from the fact that the readout is not perfectly QND.
CHAPTER 6. FLUXONIUM EXPERIMENTS 177
Figure 6.16: Single-shot joint readout of qubits? thermal states (a) 2D histogram of cav-
ity signal showing the four blobs corresponding to the qubit in four computational states.
(b) Projection of data in plot (a) onto the I-axis. Fitting of the 1D histogram to four Gaus-
sian curves allowed us to determine the populations, and subsequently estimate the tem-
peratures of the qubits: TA ? ?9 mK, TB ? 10 mK.
Figure 6.17: Initialization by preselection. (a) 2D histogram of cavity signal after pre-
selection showing only one blob corresponding to the ground state |00?. (b) Projected
data from (a) onto the I-axis. The temperatures after initialization were TA ? 1.3 mK,
TB ? 2.1 mK.
CHAPTER 6. FLUXONIUM EXPERIMENTS 178
6.4.4 Single-Qubit Gates
Although the qubits? transition frequencies are low and the computational states? thermal
populations are substantial, the high fidelity initialization procedure allowed us to improve
the signal contrast significantly, and we could readout the states of the qubits much more
reliably. Qubits having low transition frequency pose yet another problem: no commercial
IQ mixer that we are aware of works below 80 MHz. Therefore, to control the qubits at
such low frequencies, we resorted to using the AWG?s ability to synthesize RF signal up to
300 MHz.
We observed Rabi oscillations in both qubits by driving them directly with the AWG
(Fig. 6.18(a) and (b)). The qubit frequencies were found with Ramsey experiments (see
section 6.5). The pulses have Gaussian shape with width ? = 150 ns and cutoff boundary at
4?. We further tuned up the gates by applying a sequence consisting of an odd number of
?/2 pulses in either quadrature with varying pulse amplitude, and selecting the amplitude
corresponding to the curve that varied the least with respect to the number of pulses [153].
Using this pulse configuration, we prepared different single-qubit states and performed
quantum state tomography. The single-qubit density matrices were reconstructed using
linear inversion (see section 2.3). The results are shown in Fig. 6.18(c), with the average
single-qubit state fidelity F = 0.98 using Eq. 2.69.
We measured the single-qubit gates? fidelities using interleaved randomized benchmark-
ing (RB) [154], which works as follows. First, a regular RB measurement is performed by
applying a random sequence of m Clifford gates Ci, chosen randomly from the Clifford
group Cliff1 (Fig. 6.19(a), top). The (m + 1)th gate is chosen to effectively turn the entire
sequence into identity operation. The measurement is randomized K = 11 times, and the
CHAPTER 6. FLUXONIUM EXPERIMENTS 179
Figure 6.18: Control of low frequency qubit transitions. (a) Rabi oscillation between
|00? and |10?. (b) Rabi oscillation between |00? and |01?. Both qubits were driven using
the AWG?s direct outputs. (c) Single-qubit state tomography results showing the density
matrices of the (from left to right) ground, superposition, and excited states of qubits A
and B.
averaged result is fitted to either the zeroth or first order model:
F (0)(m) = A0p
m +B0,
(6.1)
F (1)(m) = A1p
m + C m?21(m? 1)p +B1,
to extract the depolarizing parameter p. The average error rate over the entire Clifford gate
sequence is given by
1? p
r = (6.2)
2
Then, an interleaved RB measurement is performed by inserting an additional gate of
choice C after every Clifford gate Ci, up to the mth random gate (Fig. 6.19(a), bottom).
Again, the (m+ 1)th gate is chosen to be the inverse of the combination of the previous m
gates. The measurement is repeated to realized K = 11 randomizations, and the averaged
CHAPTER 6. FLUXONIUM EXPERIMENTS 180
Table 6.4: Single-qubit gate fidelities.
Gate Cliff1 I X -X X/2 - X/2 Y -Y Y/2 -Y/2
A, 0th 0.976 0.989 0.982 0.982 0.987 0.988 0.982 0.982 0.987 0.985
B, 0th 0.987 0.9925 0.9886 0.9885 0.9927 0.9921 0.9889 0.9881 0.9927 0.9915
A, 1st 0.954 0.981 0.961 0.962 0.973 0.973 0.959 0.961 0.972 0.969
B, 1st 0.993 0.994 0.9924 0.9924 0.9948 0.9941 0.9924 0.9918 0.9949 0.994
result is fitted to Eq. 6.1 to extract the depolarizing parameter pC . The error rate rC of the
interleaved gate C is estimated as
1? pC/p
rC = . (6.3)
2
Randomized benchmarking results for single-qubit gates are shown in Fig. 6.19(b) and
(c) for qubits A and B, respectively. The fidelities are summarized in Table 6.4 for both
zeroth and first orders fit. Gate errors are likely due to a combination of decoherence, spu-
rious ZZ coupling, and microwave cross-talks. Further improvements would involve shorter
gate times, composite pulses to cancel out the crosstalks, and pulse shaping techniques such
as DRAG [107, 108]. Future 2D integration of individual microwave controls should make
the crosstalks negligible.
6.4.5 Two-Qubit Controlled-Z Gate
Characterization of the controlled-Z gate is discussed below. First, we performed two-tone
spectroscopy to locate the gate transition frequency and Rabi experiments to find the right
pulse parameters. To avoid state leakage, we simply used a Gaussian pulse with width ? =
150 ns and total duration equal to 4?. The gate transition has T1 ? 10 ?s Rabi oscillation
between |11? ? |21? and |11? ? |21? are shown in Fig. 6.20(a) and (b), respectively. We
used a Gaussian pulse with total duration equal to 600 ns.
CHAPTER 6. FLUXONIUM EXPERIMENTS 181
Figure 6.19: Benchmarking single-qubit gates. (a) Randomized benchmarking sequence
(top), and interleaved randomized benchmarking sequence (bottom). Normal RB sequence
gives us the average fidelity for the entire set of Clifford gates. The interleaved RB sequence
gives us the error for specific gates. The target gate C (light blue) is interleaved with random
gates (dark blue) chosen from the Clifford group. In both cases, a final gate (orange) is used
to make the sequence equivalent to the identity operation. (b) Interleaved randomized
benchmarking results for qubit A. (c) Interleaved randomized benchmarking results for
qubit B.
CHAPTER 6. FLUXONIUM EXPERIMENTS 182
After tuning up the gate, we used the sequence described in Fig. 2.3 to create Bell states
and then characterized them using quantum state tomography. To ensure that the resulting
density matrices are physical, we used maximum likelihood estimation (MLE) to determine
the output state (see Eq. 6.8). Since we measured starting with a thermal state and could
initialize to any computational state using preselection, we could essentially create four
different Bell states in a single measurement. Results for |?10? and |?00? are plotted in
Fig. 6.20 (c-d).
We computed the concurrences and fidelities for these states. Concurrence is a simple
entanglement monotone measure [155]. The procedure to determine it is as follows. First,
we compute the product R = ?(Y ? Y )??(Y ? Y ), then take the eigenvalues of R and
arrange them in decreasing order as {?1, ?2, ?3, ?4}. The concurrence is then defined as
? ? ? ?
C(?) = max(0, ?1 ? ?2 ? ?3 ? ?4). We found C = 0.915 for |?10? and C = 0.88
for |?00?, indicating that the states we created are highly entangled. The state fidelities
computed using Eq. 2.69 are F|? = 0.956 and F = 0.915.10? |?00?
We next performed quantum process tomography for the CZ gate by preparing 16 inde-
pendent input states, applying the gate, and utilizing quantum state tomography via MLE
to determine each output state. The resulting ? matrix based on the {I,X, Y, Z}?2 operator
set was computed using linear inversion (see Eq. 2.84) and shown in Fig. 6.21(a). The cor-
responding process fidelity computed using Eq. 2.85 is F = 0.88, which is likely dominated
by SPAM errors.
Finally, we extracted the CZ gate fidelity using interleaved RB. The procedure for two-
qubit case is similar to the single-qubit one. First, the regular RB result is fitted to Eq. 6.1 to
extract the depolarlizing parameter p2QB, and the average error for the two-qubit Clifford
group is
3
r = (1? p). (6.4)
4
The interleaved RB result is then fitted to extract pCZ, which gives us the error of the gate
CHAPTER 6. FLUXONIUM EXPERIMENTS 183
Figure 6.20: Entangling operation. (a) Rabi oscillation between |11? and |21?, realized by
applying a Gaussian-shaped pulse at the exact transition frequency and varying the pulse
amplitude. (b) Rabi oscillation between |11? and |12?, also realized by varying the Gaussian
pulse?s amplitude. (c) Bell state |?10? created by using the sequence in Fig. 2.3 and |00? as
initial state. (d) Bell state |?00? created by using the sequence in Fig. 2.3 and |11? as initial
state.
as ( )
3 ? pCZrCZ = 1 . (6.5)
4 p2QB
We typically observed FCZ = 1? rCZ = 0.96? 0.02 (Fig. 6.21(b)), and the best fidelity
achieved was 0.975 ? 0.015. The current result is most likely limited by a combination
of ZZ coupling, dephasing, and AC Stark shifts of nearby transitions. The uncertainty in
characterizing the two-qubit gate is mostly due to errors from single-qubit gates. Further
CHAPTER 6. FLUXONIUM EXPERIMENTS 184
Figure 6.21: Benchmarking two-qubit gate. (a) Quantum process tomography of the CZ
entangling gate, with process fidelity F = 0.88. The infidelity is likely dominated by SPAM
errors. (b) Interleaved RB result with FCZ = 0.96? 0.02.
CHAPTER 6. FLUXONIUM EXPERIMENTS 185
improvement in single-qubit gates would reduce this uncertainty. Alternatively, using itera-
tive RB [153] should allow us to extract the small gate error more confidently. Improving
the CZ gate should be possible by using shorter gate with pulse shaping techniques such as
DRAG [107, 108]. Optimization techniques such as one proposed in ref. [156] would be
necessary to find the optimal pulse shape and detuning.
6.5 Tune-up Experiments
This section describes typical fluxonium measurements we performed in the lab. We focus
on discussing the techniques and experimental details which are essentials to obtaining the
results in the previous sections.
6.5.1 Spectroscopy
Cavity: One-Tone Spectroscopy
One may be tempted to measure the cavity at room temperature when the qubit has been
mounted, but care must be taken because a high-amplitude electric field across the small
Josephson junctions can kill it. To avoid destruction from electrostatic discharge, we should
wear anti-static straps upon handling the chip or cavity. A 50 ? load resistor is connected
to the cavity port to avoid accidentally shocking the device. It is usually safe to check the
cavity with the VNA after mounting it onto the cold finger and connecting the coax cables,
since there is now at least 60 dB of attenuation between the cavity and the VNA.
After cooling-down, the very first measurement is to check the cavity resonance. Next,
we must determine at which power we should readout the cavity and qubit system, so a one-
tone experiment with power variation must be done. The result typically shows a change
in cavity resonance across a threshold value, as shown in Fig. 6.22. The cavity power must
generally be chosen to be lower than the threshold for the qubit to shift the resonance.
CHAPTER 6. FLUXONIUM EXPERIMENTS 186
Figure 6.22: One-tone versus power. Cavity resonance versus probe tone power with
data along the y-axis normalized to the mean value. The power should be lower than
the threshold (black vertical line) where the cavity resonance starts going back to its bare
value. (a) The dispersive shift is large, so the change in cavity resonance is obvious. (b)
The dispersive shift is small, so we can do a finer scan to find the threshold.
Choosing a cavity tone power that is too high may make the cavity insensitive to the state
of the fluxonium qubit, preventing us from reading it out. On the other hand, if we choose
a power that is too low, the SNR would be too small, and the measurement would take a
long time to average out the noise. Therefore, the best strategy is to pick the power right
below or at the threshold value (black vertical line in Fig. 6.22).
After choosing the appropriate power, we can perform a quick measurement where we
sweep the flux and keep the cavity tone frequency fixed. This helps us to quickly check if the
qubit is alive because a typical fluxonium qubit modulates the cavity resonance with flux.
This also helps us determine the flux periodicity of the device, as shown in Fig. 6.23(a). A
full one tone spectroscopy can now be started using the appropriate power, frequency, and
flux range. The cavity-qubit anti-crossing can be fitted to extract the coupling coefficient.
The fitting, however, is best done after we have acquired some knowledge about the qubit
spectrum, and identified which transitions actually cross the cavity (see Fig. 6.23 (b)).
In the case when the anti-crossing is too small, such as when the matrix elements of
the corresponding transition is small due to large EJ/EC ratio, the small modulation of
CHAPTER 6. FLUXONIUM EXPERIMENTS 187
Figure 6.23: One-tone versus flux. (a) Cavity resonance versus flux. The measurement
helps us confirm if the qubit is alive, and allows the extraction of flux periodicity. (b) One-
tone spectroscopy showing the cavity anticrossing with qubit transition. The data can be
used to extract the qubit-cavity coupling coefficient.
CHAPTER 6. FLUXONIUM EXPERIMENTS 188
Figure 6.24: One-tone spectrum of the split-junction device. The cavity resonance is
modulated slightly by the qubit as the plasmon changes with EJ . In this case, the coupling
coefficient can also be extracted by fitting the cavity resonance across the entire flux period.
The vertical black lines are fitted theoretical lines corresponding to the fluxon transitions
(data not visible).
the cavity by the plasmon in flux can also be fitted to extract coupling coefficient g. For
example, in the selection rules experiment discussed in section 6.2, the variation in EJ with
flux effectively changes the cavity frequency. There is a large change over the small loop?s
flux period, but also a small change in cavity frequency over the big loop?s flux period, as
shown in Fig. 6.24. This fitting procedure is more efficient if the qubit spectrum has been
measured and the qubit?s parameters have been extracted, so the coupling coefficient is the
only fitting parameter.
CHAPTER 6. FLUXONIUM EXPERIMENTS 189
Qubit: Two-Tone Spectroscopy
After measuring the cavity, we proceed to scan the qubit spectrum by performing two-tone
spectroscopy. This is typically done in pulse mode, where the qubit pulse is first applied,
and the cavity readout pulse is applied right after. If the qubit tone excites the qubit, the
cavity resonance would be shifted, with the value of the shift dependent on the dispersive
shift at that particular flux bias. By measuring the cavity before the qubit decays back to its
equilibrium state, we can record the qubit frequency. Then, we can get the qubit spectrum
by sweeping the coil current.
Since fluxonium has transition frequencies across a very broad range, while the 3D
cavity filters out the input signal at low frequencies, we have to perform spectroscopy using
different qubit tone powers in different frequency ranges. The rule of thumb is to use
qubit tone power at -30 dB less than cavity power in the region ?3 GHz away from cavity
resonance. If time allows, a power sweep can be done to check for the best option. The
power should be increased approximately 5 dB per GHz as we move lower in frequency,
depending on the distance between the chip and the input port. We used this strategy to
obtain the spectroscopy data in Fig. 6.25 for a representative fluxonium sample. We can
fit spectroscopy data to the Hamiltonian model (Eq. 3.17) to extract the parameters and
calculate relevant properties of the qubit. Further interpretation of spectroscopy data is
discussed in section 6.3.
As shown in the previous chapter (see Fig. 5.11(a)), a pulse with sharp rising time has
a wide spectral bandwidth, and may excite additional transitions, blurring the main line.
To avoid this, we often use a Gaussian pulse for qubit spectroscopy. In general, a wider
Gaussian pulse in time domain results in cleaner spectral features. While pulse spectroscopy
is typically used to obtain a clean spectrum, it requires synchronization between the qubit
and cavity RF tones, which is most conveniently done using an AWG.
When there is no AWG available, it is also possible to perform a continuous wave (CW)
CHAPTER 6. FLUXONIUM EXPERIMENTS 190
Figure 6.25: Two-tone spectrum. Two-tone spectroscopy result of a typical fluxonium
sample showing the 0?1, 0?2, 1?2 transition energies. The scan is typically completed in
3 stages. First, the plasmon region is scanned using 10 MHz frequency step and 10 ?A flux
step. Then, the fluxon region is scanned using 10 times finer resolution in both frequency
and flux. Finally, the half flux spectrum region is scanned at a higher power or longer qubit
pulse.
measurement where both cavity and qubit tones are applied simultaneously. This results in
faster measurement time at the cost of a spectrum with many lines corresponding to higher
order processes. While these lines may also appear in pulse spectroscopy when high qubit
tone power is used, their number is fewer. CW is most conveniently used in a VNA setup,
where two ports of the VNA are used to measure the S parameter of the cavity, and an
additional port or an external RF source is used as the qubit tone.
In CW spectroscopy, the qubit is measured in the presence of cavity photons, so its
frequency may not be that of the bare qubit, due to the AC Stark shift introduced in section
CHAPTER 6. FLUXONIUM EXPERIMENTS 191
2.2. On the other hand, this shift helps us determine the number of photons present in the
cavity. For this experiment, we can measure the qubit spectrum at a fixed flux bias and vary
the cavity power. An example is shown in Fig. 6.26(a). If the dispersive shift per photon
has been determined, it is then straightforward to calibrate the photon number based on
the frequency shift. At the same time, the fluctuation of the photon number scales with
its square root, so more photons in the cavity would result in faster dephasing, effectively
broadening the qubit?s linewidth. This effect is shown in Fig. 6.26(b).
Figure 6.26: AC Stark shift in spectral domain. (a) As the cavity power increases, the
qubit frequency is shifted by photons occupying the cavity. (b) The qubit linewidth broadens
as the number of cavity photons increases, due to the measurement back action dephasing.
At low cavity power, the linewidth is not well resolved due to the finite frequency step.
6.5.2 Time Domain Measurements
Rabi Oscillations
The qubit is not proven quantum until we can successfully perform a Rabi oscillation exper-
iment on it, and the linear dependence of the Rabi frequency on the applied power validates
the two-level system approximation. In fact, the birth of the field of superconducting cir-
cuits is sometimes considered to be the demonstration of coherent oscillation in the charge
qubit in 1999 [16], almost two decades after macroscopic tunneling and discrete energy
CHAPTER 6. FLUXONIUM EXPERIMENTS 192
Figure 6.27: Rabi measurement. Rabi sequence (a) and data (b). The experiment was
perform by varying the qubit pulse?s width.
levels were observed in Josephson junction devices [13, 14].
To perform a Rabi experiment, we first apply a pulse with varying amplitude or duration
at the qubit frequency, and then readout. Those two techniques are referred to as amplitude-
Rabi and time-Rabi, respectively. The projection of the rotating state vector on the Bloch
sphere along the Z-axis results in a sinusoidal signal, as shown in Fig. 6.27.
Rabi oscillations can be used to measure relevant noise in the rotating frame [67].
To determine the dephasing rate in the rotating frame, a time-Rabi must be used. More
importantly, Rabi experiments let us determine the parameters for the microwave gates we
need to perform further experiments such as T1, Ramsey fringes, and T2 echo, as well as to
do quantum computation.
To get high resolution Rabi data, amplitude-Rabi is usually a better choise, since the
AWG?s analog channel has 14-bit amplitude resolution. Another advantage of amplitude-
Rabi is at the software level. Measurement software?s pulse generator driver generates the
?/2 pulse by dividing the ? pulse amplitude in half, so the amplitude-Rabi data can be used
to check if this approach is valid.
CHAPTER 6. FLUXONIUM EXPERIMENTS 193
Ramsey Fringes
After the parameters for qubit gates have been determined, we can perform Ramsey ex-
periment. The sequence is shown in Fig.6.28(a). It involves two ?/2 pulses applied at a
frequency a little bit detuned from the qubit, and their separation in time is varied. The
first pulse rotates the qubit state vector to the Bloch sphere?s XY-plane, where it precesses
around Z (in the rotating frame) at the detuning frequency during the delay time between
the pulses. The second ?/2 pulse projects the qubit to an axis whose direction depends on
the angle around Z that the qubit has rotated. The qubit is measured along Z right after,
resulting in a sinusoidal signal whose frequency is the difference between the applied RF
tone and the qubit transition frequency. Moreover, decoherence would make us lose track
of the qubit?s rotation angle, and averaging an ensemble of measurement data would result
in a decaying envelope for the oscillations.
Ramsey data gives us two important parameters. First, it reveals very precisely how
large the detuning between the applied RF tone and the qubit transition is. This property of
the Ramsey technique has made it the gold standard in finding atomic systems? frequencies.
Second, it gives us the dephasing time of the qubit, often referred to as the free induction
decay time. As argued in section 3.2, because the qubit transition frequency can fluctuate or
simply drift over a long time scale during which we perform the measurement, the Ramsey
decay time we achieve depends on the measurement time and long time scale fluctuation
statistics. Thus, the coherence time T ?2 measured using Ramsey is not the actual time scale
we are usually interested in. We confirmed this by varying the averaging time, and for
longer measurement time, the resulting Ramsey decay becomes shorter. We note that when
measured with a TWPA, the Ramsey time of qubit I was about two times smaller than the
echo time, whereas it was three to five times smaller than the echo time in experiments that
needed long averaging time to resolve the qubit states.
CHAPTER 6. FLUXONIUM EXPERIMENTS 194
Figure 6.28: Ramsey measurement. (a) Ramsey sequence. (b) Ramsey fringe data show-
ing a decaying sinusoidal. The oscillation frequency is equal to the difference between
the applied RF tone and the qubit transition frequency, while the decay envelope gives us
information about the free induction dephasing time T ?2 ? 140 ?s.
Relaxation Time T1
To measure the relaxation time, a ? pulse is first applied at the qubit frequency to drive it
to the excited state. This is followed by a varied delay time during which the qubit decays
back to the ground state, or more precisely, to equilibrium. The state of the qubit is then
measured by applying a readout pulse at the cavity frequency. The typical T1 sequence and
decay data are shown in Fig. 6.29.
Alternatively, a long saturation pulse can be used instead of a ?-pulse to simply drive
the qubit out of equilibrium to a mixed state. However, we notice that sometimes the
Rabi oscillation does not decay to its median point. In addition, T1 measured with the
saturation pulse is usually longer than the T1 measured with the ?-pulse. We speculate that
the longer T1 may be due to the effective decay from other states being pumped by the
long saturation pulse. Other possible reasons for the effect include quasiparticle dynamics
around the small junctions and two-levels defects causing dielectric loss. Quasiparticle
density can be changed, and two-level defects can be saturated by such long pulses. For
CHAPTER 6. FLUXONIUM EXPERIMENTS 195
Figure 6.29: T1 measurement. (a) T1 sequence. (b) Data showing the relaxation rate of
qubit excited state to equilibrium with T1 ? 238 ?s.
these reasons, we only consider T1 data measured using ?-pulse.
Dynamical Decoupling
As explained in section 3.2, we can partially cancel out dephasing from the low frequency
noise affecting long measurement time by applying a ? pulse between the ?/2 pulses, effec-
tively shifting the window function to a higher frequency. The pulse sequence and typical
result for echo are shown in Fig. 6.30.
The echo technique can be visualized in an intuitive manner. After the applied ?/2
pulse, the qubit is projected on the XY-plane, and its precession trajectory can be affected
by the environment via certain noise channels. If after time ?/2, we flip the qubit by ?,
and let the same noise channels reverse the qubit trajectory in another time period ?/2, we
can effectively cancel out the previous phase drift. The ? pulse inserted in between is thus
often called the refocusing pulse, and the name ?echo? comes from the refocused output
appearing after time ? [94].
This technique can be generalized to one involving an odd number of ? pulses in be-
tween the ?/2 pulses. The technique is called CP (Carr-Purcell) if the refocusing pulses flip
CHAPTER 6. FLUXONIUM EXPERIMENTS 196
Figure 6.30: T echo2 measurement. (a) T
echo
2 sequence. The ?-pulse is in the same quadra-
ture as the ?/2-pulses in our measurements. (b) Data showing the exponential decay used
to extract decoherence time T2 ? 205 ?s.
the qubit around the same axis as the ?/2 pulses do, and CPMG (Carr-Purcell-Meiboom-Gill)
if the refocusing pulses flip the qubit around the perpendicular axis. It has been demon-
strated that performing the CPMG technique effectively increases the qubit coherece time if
the dominant dephasing mechanism is due to 1/f flux noise [81].
There are other techniques to probe the noise spectrum affecting coherence times, such
as the strong Rabi driving technique [67], and the spin-locking technique [157]. Both
of them give information about the noise spectrum at the rotating frequencies. Future
experiments utilizing these techniques to study noise in fluxonium qubits would be helpful
in obtaining further information about noise in multi-junction circuits.
6.5.3 Preliminary Trace of Quasiparticle
At the start of the coherence study described in section 6.3, we measured T1 along the 0? 1
spectrum of device A in Table 6.1 across the first flux period, and at an applied magnetic
field of around 2 gauss, corresponding to the 20th flux period. The data are shown in
Fig. 6.31. T1 away from the sweet spot is longer at a higher magnetic field. Theoretical
CHAPTER 6. FLUXONIUM EXPERIMENTS 197
Figure 6.31: Possible effect from quasiparticles. T1 relaxation data (blue square) mea-
sured along the 0? 1 spectrum of device A at (a) zero magnetic field and (b) approximately
2 G. Solid line corresponds to theoretical dielectric loss limit, and dash lines correspond to
addition of dissipation due to quasiparticles tunneling across the small junction (see text).
lines are plotted for comparison in these plots. They were computed as follows. First, a
dielectric loss limit with corresponding loss tangent tan ?C = 2 ? 10?6 (see Eq. 4.3) was
assumed (blue solid line). We then added to this limit the dissipation due to quasiparticle
tunneling across the small junction (Eq. 4.10), using different normalized quasiparticle
density xqp?s (dashed lines).
The filtering and shielding setup was minimal at the time of this measurement. In
later experiments, we made several improvements in our cryogenic setup, and observed no
noticeable enhancement in T1 near integer flux when a magnetic field of the same amplitude
? 2 G was applied.
CHAPTER 6. FLUXONIUM EXPERIMENTS 198
6.5.4 Quantum Tomography
Quantum State Tomography
We briefly outline the procedure to mathematically reconstruct a density matrix describing
a two-level system and extend it to a two-qubit system in section 2.3. In practice, what we
measure is the signal in units of voltage, and we have to post-process the data into expec-
tation values of Pauli operators. We describe our experimental quantum state tomography
procedures for single-qubit and two-qubit systems below.
The measurement of a two-level system in the ground state can be described by a mea-
surement operator M = |0??0| = 12 (I + Z). We get ?M? = 1 for qubit in the ground state,
and ?M? = 0 for qubit in the excited state. To convert the measured data in physical unit to
expectation values, we use the ?-coefficients and write M = ?II + ?ZZ, where ?I and ?Z
have units of voltage. The conversion between voltage and expectation values then involves
precise calibration of these coefficients. Note that the factor 1/2 has been absorbed into the
??s coefficients, which is valid as long as our calibration and post-processing consistently
take this into account.
If the measurement is ideal and strong enough to give us single-shot readout, the ?-
coefficients can be determined most directly from the IQ single-shot histogram data. How-
ever, in cases when the measurement is not strong enough, or not QND, we can calibrate
these coefficients by measuring the signal after preparing the qubit in the ground and ex-
cited state to get VI = ?I + ?Z and VZ = ?I ? ?Z , respectively. This helps us bypass
systematic state preparation and measurement errors, with the dominant errors in the re-
constructed states originating from imperfect gates. This approach has been employed to
benchmark gates in superconducting qubits [112, 111].
After calibrating the ?-coefficients, we can perform qubit rotations before reading out
the cavity to determine the corresponding expectation values, as outlined in Table 6.5. Since
the density matrix has 4 entries and must satisfy 1 constraint Tr(?) = 1, we need at least
CHAPTER 6. FLUXONIUM EXPERIMENTS 199
Table 6.5: Gate sequence for single-qubit state tomography.
Measurement # Rotation Outcome
1 I ?I + ?Z?Z?
2 X/2 ?I + ?Z?Y ?
3 ?Y/2 ?I + ?Z?X?
3 different measurements. Instead of the unit trace constraint, we use ?I? = 1. Following
Eq. 2.64, the density matrix can then be reconstructed from the expectation values ?X?,
?Y ?, and ?Z?.
Two-qubit quantum state tomography follows the same procedure. To calibrate the
?-coefficients in M = ?IIII + ?ZIZI + ?IZIZ + ?ZZZZ, we measure the cavity signals
VII = ?II + ?ZI + ?IZ + ?ZZ , VZI = ?II ? ?ZI + ?IZ ? ?ZZ , VIZ = ?II + ?ZI ? ?IZ ? ?ZZ ,
and VZZ = ?II ? ?ZI ? ?IZ + ?ZZ , where VIZ , for example, is the measured signal after
flipping the second qubit to excited state while keeping the first qubit in the ground state.
Once the ?-coefficients calibration has been completed, we can extract the expectation
values for all the relevant Pauli operators. With the density matrix describing a two-qubit
system having 16 entries and satisfying 1 constraint, we need at least 15 measurements in
two-qubit state tomography. By performing 15 different combinations of rotations on the
qubits and using the constraint ?II? = 1, we can extract the necessary expectation values
via linear algebra. The measurement outcomes, as shown in Table 6.6, are organized into
matrix form, and a simple matrix inversion can be used to compute the expectation values.
For two-qubit state tomography, we found that the reconstructed matrix is often not
physical, since our linear reconstruction neglect the Hermiticity and positivity properties of
a density matrix. To account for these constraints, we use MLE technique [158, 159]. It
works as follows. First, we note that the density matrix can be decomposed into Cholesky
CHAPTER 6. FLUXONIUM EXPERIMENTS 200
Table 6.6: Gate sequence for two-qubit state tomography.
Measurement # Rotation on A Rotation on B Outcome
1 I I ?II + ?ZI?ZI?+ ?IZ?IZ?+ ?ZZ?ZZ?
2 X I ?II ? ?ZI?ZI?+ ?IZ?IZ? ? ?ZZ?ZZ?
3 I X ?II + ?ZI?ZI? ? ?IZ?IZ? ? ?ZZ?ZZ?
4 X/2 I ?II + ?ZI?Y I?+ ?IZ?IZ?+ ?ZZ?Y Z?
5 X/2 X/2 ?II + ?ZI?Y I?+ ?IZ?IY ?+ ?ZZ?Y Y ?
6 X/2 Y/2 ?II + ?ZI?Y I? ? ?IZ?IX? ? ?ZZ?Y X?
7 X/2 X ?II + ?ZI?Y I? ? ?IZ?IZ? ? ?ZZ?Y Z?
8 Y/2 I ?II ? ?ZI?XI?+ ?IZ?IZ? ? ?ZZ?XZ?
9 Y/2 X/2 ?II ? ?ZI?XI?+ ?IZ?IY ? ? ?ZZ?XY ?
10 Y/2 Y/2 ?II ? ?ZI?XI? ? ?IZ?IX?+ ?ZZ?XX?
11 Y/2 X ?II ? ?ZI?XI? ? ?IZ?IZ?+ ?ZZ?XZ?
12 I X/2 ?II + ?ZI?ZI?+ ?IZ?IY ?+ ?ZZ?ZY ?
13 X X/2 ?II ? ?ZI?ZI?+ ?IZ?IY ? ? ?ZZ?ZY ?
14 I Y/2 ?II + ?ZI?ZI? ? ?IZ?IX? ? ?ZZ?ZX?
15 X Y/2 ?II ? ?ZI?ZI? ? ?IZ?IX?+ ?ZZ?ZX?
form
T ?T
? =
Tr(T ?
, (6.6)
T )
where the upper triangular matrix T for a two-qubit system is
?? ???t1 t5 + it6 t11 + it12 t15 + it16? ????0 t ?2 t7 + it8 t13 + it14T = ? ? . (6.7)? ??0 0 t3 t9 + it10??
0 0 0 t4
CHAPTER 6. FLUXONIUM EXPERIMENTS 201
The matrix entries {ti} can be found from minimization of the likelihood function
?15
L = (mi ? Tr(M 2i?)) , (6.8)
i=1
where mi and Mi are respectively the measured signal and measurement operator corre-
sponding to measurement i. By keeping track of the list of rotations, and thus the measure-
ment operators, we can post-process the mi data to reconstruct the density matrix. Unlike
the linear reconstruction approach, we may want to use an over-complete set of measure-
ments to estimate the density matrix more precisely using MLE. For example, a set of 30 or
36 measurements can be extended from the one shown in Table 6.6 by using negative-angle
rotations such as ?X/2 and ?Y/2.
Quantum Process Tomography
With quantum state tomography procedure described, we now move on to discuss quantum
process tomography, the procedure to identify the quantum operation map E [68, 160]. The
technical outline is described at the end of section 2.3. Below we discuss the experimental
procedure to find the ? matrix.
First, we prepare a set of pulse sequences, each one comprises (i) a pre-pulse preparing
the initial state ?i, (ii) the intended gate(s) we use to realize the intended quantum opera-
tion, and finally (iii) a post-pulse used for state tomography. For single-qubit process tomog-
raphy, we use the pre-pulse set {I, X, Y/2, ?X/2}, and the post-pulse set shown in Table
6.5, resulting in 12 different measurements. For two-qubit process tomography, we use the
pre-pulse set {I?I, I?X, I?Y/2, I??X/2, X?I, X?X, X?Y/2, X??X/2, Y/2?
I, Y/2?X, Y/2?Y/2, Y/2??X/2, ?X/2?I,?X/2?X, ?X/2?Y/2, ?X/2??X/2},
and the post-pulse set shown in Table 6.6, resulting in 240 different measurements.
Then, quantum states tomography is used to reconstruct the 16 output density matrices
{?j}, preferably by using maximum-likelihood estimation. By keeping track of the pulse
CHAPTER 6. FLUXONIUM EXPERIMENTS 202
sequence order, and thus the initial states, we can reconstruct the ? matrix based on the
{I, X, Y, Z}?2 operator basis using Eq. 2.84. Note that this linear conversion produces ?
as a vector, and we must reshape it into a d2?d2 matrix. Alternative to ?, the Pauli transfer
matrix R, which maps an input Pauli state vector to an output Pauli state vector, can be
used to characterize a quantum process as well [130]. We may obtain both maps and check
for consistency, for example, by comparing the fidelities extracted from the two different
methods.
6.5.5 Randomized Benchmarking
Although quantum process tomography is the standard technique for characterizing a quan-
tum operation, it is subjected to two drawbacks. First, the procedure is increasingly tedious
for large systems comprising of multiple qubits. The number of measurements scales as
d4 ? d2 where d = 2n is the dimension of the Hilbert space, and n is the number of qubits.
As outlined above, while process tomography involves only 12 measurements for a single
qubit, that number goes up to 240 for two qubits, and would be 4032 for three qubits.
Second, the result is sensitive to SPAM errors. For instance, our current single-qubit gates
are not much better than the entangling gate, so the extracted fidelity is prominently due
to single-qubit gate errors.
One method for characterizing qubit gates that overcomes these shortcomings is ran-
domized benchmarking (RB) [161, 154]. This technique involves the generation of random
sequence of gates, with the last gate computed to make the total sequence equivalent to an
identity operation. Averaging over different randomized sequence results give us a decay
that scales with the average fidelity of our gates, which we can fit to a simple model. The
process is independent of SPAM errors.
We describe the RB and interleaved RB protocols below, following ref. [154]. They are
restricted to gates within the Clifford group, denoted as Clifn for n qubits. For a single
CHAPTER 6. FLUXONIUM EXPERIMENTS 203
qubit, the Clif1 group consists of 24 elements, each Clifford gate has an average length of
1.875 pulses. For two qubits, the Clif2 group consists of 11520 elements, with each Clifford
gate containing on average 1.5 CZ gates and 8.25 single-qubit gates [35, 162]. The gate
sequences are shown in Fig. 6.19(a). An interleaved RB sequence includes an additional
gate C of interest after each random gate Ci, in a sequence of m gates. The regular RB
decay is fitted to either the zeroth or first order model in Eq. 6.1 to extract the depolarizing
parameter p. The average error rate over all Clifford gates is given by
r = (d? 1)(1? p)/d, (6.9)
where d = 2n is the dimension of the system.
In interleaved RB, the decay is fitted to extract the depolarizing parameters pC . Because
of addition of the interleaved gate, the fidelity curve decays faster, corresponding to pC < p.
The average error of this interleaved gate is estimated by
(d? 1)(1? pC/p)
rC = , (6.10)
d
and must be within the range [rC ? E, rC + E], where E = min{(d ? 1)(|p ? pC/p| + (1 ?
? ?
p))/d, 2(d2 ? 1)(1? p)/pd2 + 4 1? p d2 ? 1/p}.
In experiments where the change in qubit populations can be described by the projection
of the demodulated signal onto its I or Q quadrature (or magnitude/phase), the change in
the signal can be used to fit the decay. When a joint readout is used for two-qubit devices,
cross-correlation between the measurements of qubitA and qubitB may make the projected
signal to scale non-linearly with the populations. In this case, single-shot histogram readout
data should be used to extract the populations {Pi} of all the states after each randomized
sequence m. The change in the populations can then be used to fit the RB data. For single-
qubit RB in a two-qubit device, we fit the data set P|00?(m) + P|01?(m) for qubit A, and
CHAPTER 6. FLUXONIUM EXPERIMENTS 204
P|00?(m) + P|10?(m) for qubit B, where |AB? denotes the qubit state. For two-qubit RB, we
fit the data set P|00?(m).
Chapter 7
Summary and Outlook
I dream of painting and then I paint my dream.
Vincent van Gogh
7.1 Summary of Results
7.1.1 Selection Rules Engineering
Our experiment described in section 6.2 demonstrated for the first time that the energy
decay time in a macroscopic artificial atom can be extended into the millisecond range by
a controlled suppression of the overlap of the qubit state wave functions. We measured the
energy decay time T1 of a qubit and observed a textbook enhancement of T1 as a function
of dipole moment. The energy decay quality factor Q1 reached the value of 4? 107, corre-
sponding to T1 = 2 ms for qubit frequency of 3.3 GHz, without sign of saturation. We have
also shown evidence of the metastable fluxonium dynamics. The result was confirmed in
an independent experiment by Earnest et al. [163].
205
CHAPTER 7. SUMMARY AND OUTLOOK 206
7.1.2 High Coherence Fluxonium
In the experiments reported in section 6.3, we presented a specific design of fluxonium
qubits which repeatedly yielded high coherence times T2 at the half-integer flux bias, with
the best device satisfying T2 > 400 ?s. To our knowledge, this is the longest coherence
time found in a superconducting qubit today. Compared to a typical flux qubit, the effect
of 1/f flux noise is practically eliminated by the large loop inductance (L ? 102 nH) of the
Josephson chain. The qubit transition frequency can still be flux-tuned by many octaves
while keeping coherence time above a few microseconds, limited by first-order coupling
to flux noise. Our fluxoniums are compatible with transmon-based scaling architectures,
which require connecting an external coupling capacitance (antenna) to the small junction.
This connection comes at the price of energy relaxation induced by the surface loss in the
antenna. The surface loss problem was largely (but not completely) bypassed by reducing
the qubit frequency to around ?01/2? ? 500 MHz. Because the spectral density of noise
associated with surface loss drops rapidly with frequency, the relaxation time of our qubits
exceeded that of the best capacitively-shunted junction circuits despite a sub-optimal Al on
Si fabrication procedure.
Several design features possibly contributed to the dramatic improvement of coherence
compared to previous fluxonium experiments. The original design was severely limited by
the surface loss (T2 ? T1 ? 10 ?s) in the finger capacitors attached to the small junc-
tion [59, 114]. In a subsequent series of experiments this capacitance was removed and the
readout was done through the inductive coupling to a low quality factor chain mode dressed
by an external capacitance [85, 87, 149, 164]. This expectedly eliminated the surface loss
and dramatically improved T1 but not T2. In the present work we went back to the original
design [59, 114], including the Dolan bridge fabrication and (i) used a better geometry of
the antenna to minimize the surface participation ratio the way it is done in modern trans-
mons [30] and (ii) used the fundamental mode of a simple copper box for readout [102].
CHAPTER 7. SUMMARY AND OUTLOOK 207
Finally, the geometry of our Josephson chain might be favorable for trapping quasiparticles
in the background magnetic field.
Importantly, the approximately tenfold reduction of the qubit frequency compared to
the traditional 5 GHz value does not prevent fast gate operations or strong interactions.
This is because the spectrum of a typical fluxonium has a rich structure of excited states
reminiscent of the fine structure in the spectrum of conventional atoms. Transitions outside
the computational subspace belong to a traditional frequency and transition dipole range,
which can be utilized for creating fast flux- or microwave- activated quantum gates. More-
over, the inductive connection of fluxoniums via shared junctions can make the exchange
coupling comparable to the qubit frequency without leaving the computational subspace.
These large exchange couplings combined with the exceptional coherence times can be
especially useful for constructing coherent quantum annealers. The next steps towards
quantum computing with fluxoniums consist of demonstrating high coherence in combina-
tion with fast single-shot readout, fast flux tuning, and fast two-qubit gates, which we are
currently working on.
Following the common viewpoint of minimizing the number of junctions per qubit, one
may be tempted to replace the discrete chain by a patterned film of a highly-disordered
superconductor with a comparable kinetic inductance. It is important to realize that the ef-
fective loss tangent of the inductance must be in the 10?7?10?8 range in order to reach the
coherence times reported in this experiment. Whether such a low loss can be reached with
dirty superconductors is an interesting question [165, 166, 167, 168]. Our specific Joseph-
son tunnel junction chain design was primarily motivated by the maximal simplicity of the
qubit fabrication procedure: it is single step and does not require high-contrast lithography.
The high coherence was made possible, in part, by the remarkably good thermalization of
both quasiparticles and collective modes in the chains. Understanding this effect in future
experiments may have high impact on quantum circuit design. In the meantime, our exper-
iment demonstrated for the first time that coherence time of superconducting qubits can be
CHAPTER 7. SUMMARY AND OUTLOOK 208
extended beyond the state-of-the art by increasing circuit complexity.
7.1.3 Microwave-Activated Two-Qubit Gate
We reported current progress on the microwave-activated entangling gate in section 6.4,
showing that two high coherence fluxonium qubits can be entangled with high fidelity. In
one single device, we successfully demonstrated single-shot readout capability, initialization
by preselection, precise single-qubit controls at very low frequencies, and high fidelity two-
qubit gate. The computational states are at around 100 MHz and have high coherence, yet
the multi-qubit interaction is quite fast, especially compared to the decoherence time and
spurious ZZ coupling.
Our preliminary results show that fluxonium is a promising qubit in the quest to build
quantum computers. The experimental procedures described can be readily applied to study
other microwave-activated gate such as the cross-resonance gate [169, 130] or the bSWAP
gate [170] using fluxonium qubits. Improving the gate past fault-tolerance threshold is cur-
rently being explored. This includes utilizing composite pulses [107, 108] and numerical
?last-mile? optimization [156]. The next step in scaling up fluxonium devices would involve
the development of 2D architecture along with local microwave control and readout inte-
gration, similar to the progress already made on transmon qubits (see refs. [35, 26], for
example).
7.2 Quantum Computing with Fluxoniums
Here we outline how fluxoniums can be integrated into existing schemes of scalable quan-
tum computing. This outlook discussion is largely based on already published theoretical
and experimental work on interacting fluxoniums.
CHAPTER 7. SUMMARY AND OUTLOOK 209
7.2.1 Capacitive Coupling
Viewing fluxoniums as inductively-shunted charge qubits, one can understand the effect of
connecting two such devices by a mutual capacitance CC using the charge qubit expres-
sions. Assuming that CC  C, the effective interaction term is given by Hint = JCn1n2,
where JC = 2EC ? (CC/C) and n1,2 are the charge operators of the two devices. Given
that in our design EC ? 1 GHz, and choosing a moderate ratio CC/C = 10 one can readily
get an exchange constant JC ? 200 MHz, which is similar to what is possible with capac-
itive coupling of transmons. Note that the repulsion of qubit frequencies by the term Hint
is significantly reduced by the small value of the charge matrix elements ?0|n1,2|1?. Fortu-
nately, this is not the case for transitions outside the computational subspace, where matrix
elements are large and one can use them to perform, e.g. microwave-activated two-qubit
gates [128].
7.2.2 Inductive Coupling
In contrast with capacitive coupling, a proper inductive coupling of fluxoniums allows for
extremely strong interactions already within the computational subspace. Viewing fluxoni-
ums as superconducting loops with a weak link, two such devices can be inductively coupled
by sharing one or several junctions between the two loops. Assuming that the fraction of
the shared junctions is m  1, the interaction term is given by Hint = JL(?1/?)(?2/?),
where JL = m?2EL and ?1,2 are the phase operators of the two devices. Normalization of
the phase operators by ? is convenient because ?0|?1,2|1? ? ? at the sweet spot. Therefore,
even for a modest m = 0.1 (about 10-20 shared junctions), we get the frequency repulsion
of qubit transitions on the order of JL ? 0.5 GHz, i.e. it is comparable to transition frequen-
cies. In fact, by making m ? 1, a molecular-type binding of two fluxoniums has already
been experimentally demonstrated [149].
CHAPTER 7. SUMMARY AND OUTLOOK 210
7.2.3 Dispersive Qubit-Cavity Interaction
Circuit quantum electrodynamics with fluxonium qubits was described in Ref 31. The most
useful strong dispersive regime of cQED corresponds to ? ?. Introducing the dimension-
less photon creation (annihilation) operator a?(a), the coupling term is H ?int = gCni(a? a )
for capacitive coupling and H ?int = gL?(a + a ) for inductive coupling. Given the discus-
sion of inductive and capacitive coupling above, it is straightforward to achieve the values
gC/2?, gL/2? ? 100 MHz, typical of conventional qubits [164, 60]. However, here the qubit
frequency (typically 500 MHz) is far detuned from the photon frequency (typically above
5 GHz). Nevertheless, the shifts ? can be large due to the virtual transitions connecting
states |0? and |1? to the non-computational states [127]. In fact, it was shown experimen-
tally [60] that dispersive shift is non-zero even for a vanishing qubit transition dipole, i.e.
?0|?, n|1? ? 0.
7.2.4 Flux-Controlled Gates
Since fluxonium?s spectrum can be tuned by flux it is tempting to consider flux-controlled
gate operations. Perhaps the simplest such gate is the analog of a C-phase gate for trans-
mons [71], relying on the repulsion of the two-qubit states |11? and |20? or |02?. The states
repulsion can be generated by a direct capacitive or inductive connection of fluxoniums. In
fact, one may expect an enhancement of gate fidelity because fluxoniums maintain a rela-
tively high coherence time T2 ? 5 ?s while biased outside the sweet spot during the gate
operation.
7.2.5 Fixed-Frequency Gates
In section 4.4, we describe a fast CZ gate between two capacitively fluxoniums obtained by
applying a 2?-pulse at a frequency near the |1? ? |2? of the target qubit [128]. In general,
the quantum state leakage during such gate operations was shown to be remarkably low
CHAPTER 7. SUMMARY AND OUTLOOK 211
owing to the large anharmonicity of the non-computational part of the spectrum. Although
microwave-activated gates still require static flux-biasing of qubits at their sweet spots,
their advantage is that they are compatible with the 3D circuit QED architecture used in
this thesis.
The experimental results shown in section (6.4) are promising, and further steps should
be taken to minimize spurious interactions and optimize the gate fidelity. For example,
the unwanted ZZ coupling is particularly large in a few devices, which is linked to the
interactions between higher levels.
Other microwave-activated gates, such as the cross-resonance gate [169] and bSWAP
gate [170] can also be used to entangle two fluxoniums by directly driving transitions re-
siding within the computational subspace. The advantage of these gates compared to the
CZ gate comes from the long coherence of the |0? and |1? computational states. Preliminary
simulation results are promising for both, and future theoretical description of these gates
in fluxonium-based systems will be available soon.
7.2.6 Quantum Adiabatic Optimization
A network of interconnected fluxoniums, after the projection to the computational sub-
space, can implement a generic quantum spin-1/2 Hamiltonian:
?
H = hZ X XXi ?Zi + hi ?Xi ? Ji,j ?Xi?Xj . (7.1)
i,j
Here the field hZ is the qubit transition frequency at the sweet spot, field hX is the detuning
from the sweet spot, and JXX is the nearest neighbor coupling constant. The field hZ can
also be tuned independently from hX by replacing a single weak junction by a split junction,
as it was done in device C. Such a Hamiltonian is typically implemented using a system of
semi-classical SQUID circuits to explore quantum annealing algorithms [171].
Our devices can provide a previously unavailable realization of this model. (i) Owing
CHAPTER 7. SUMMARY AND OUTLOOK 212
to the extremely large anharmonicity, ?12/?01  1, a network of fluxoniums remains in
its computational subspace even in the presence of multiple spin-flips, i.e. projecting to
the computational subspace remains valid. (ii) Even far away from the sweet spot, we get
T2 ? 5 ?s, which translates into the level broadening of about 30 kHz. This number in prin-
ciple allows resolving the many-body level spacing in a system of 10 locally coupled spins.
(iii) Inductive connection allows a local coupling to multiple neighbors with the condition
JXX ? hZ . These three conditions are simultaneously required for exploring the most
intriguing scenarios of quantum many-body physics of spin systems. Fluxoniums are there-
fore well positioned for constructing the next generation of quantum annealers operating
in a highly-coherent regime where quantum speed up is expected from theory [37].
7.2.7 Optimal Qubit Frequency
It is interesting to discuss the choice of the optimal qubit frequency as the design, in prin-
ciple, allows to reduce it to an arbitrary low value. We believe that the presented qubit
frequency range around 500 MHz is currently the optimal for a number of reasons.
The first concern is the finite temperature of the qubit. Already at 500 MHz, which
translates to a temperature of 25 mK, a significant population of state |1? is expected. In
principle, this is not a problem for a quantum processor as long as the energy relaxation
time T1 is sufficiently long. The qubits anyway need to be initialized with a high-fidelity.
However, it is convenient to be able to characterize low-frequency devices without the need
to do so, hence keeping the qubit frequency not far below the temperature is advantageous.
More importantly, for h??  kBT , the relaxation time T1 must be rescaled compared to its
zero-temperature value due to the stimulated emission factor T1 ? T1(T = 0) ? (kBT/h??)
(see Eq. 4.9 and Fig. 4.11).
From a technical viewpoint, the chosen frequency range appears particularly convenient
for scaling: there is room to frequency-resolve neighboring qubits by spreading them by a
CHAPTER 7. SUMMARY AND OUTLOOK 213
few hundred MHz; Rabi-driving with a frequency up to 50 MHz can be applied even within
the 3D circuit QED architectures, which can provide 10 ns-long single-qubit pulses; cross-
talks are in general expected to be reduced at lower frequencies. Last but not least, qubit
pulses can be done using cheaper digital electronics which can significantly reduce scaling
costs.
Appendices
214
Appendix A
Noise and Energy Relaxation
Winter is coming.
George R.R. Martin
In this appendix, we connect noise described by the noise spectral density to energy
relaxation via Fermi?s golden rule.
A.1 Noise Correlation and Spectral Density
First, we explore how to characterize noise in quantum systems, following ref. [83]. Con-
sider a classical voltage noise signal v(t) that fluctuates randomly in time and have zero
mean value, ?v(t)? = 0. The frequency spectrum of this noise is defined via the Fourier
transform ? ?
V (?) = v(t)ei?tdt. (A.1)
??
The cycle averaged intensity of the noise at frequency ? is proportional to
? ? ? ?
?| ?V (?)|2? = dt dt??v(t)v(t?)?ei?(t?t ). (A.2)
?? ??
215
APPENDIX A. NOISE AND ENERGY RELAXATION 216
In a real experiment, the data is taken over a period of time T , so the relations above should
be redefined via a ?windowed? Fourier transform :
?
1 T
V (?) = ?? v?(t)e
i?tdt, (A.3)
T 0
T T
?| 1 ?V (?)|2? = dt dt??v(t)v(t?)?ei?(t?t ). (A.4)
T 0 0
?
Note that the unit of the noise in frequency domain is volts/ Hz. This allows us to normal-
ize V (?) to be independent of the sampling time T . The spectral density of the noise v(t) is
defined to be
SV (?) ? lim ?|V (?)|2?. (A.5)
T??
In the time domain, the noise v(t) can be characterized by its correlation function
?
1 T
Gv(?) = ?v(t)v(t? ?)? = v(t)v(t? ?)dt, (A.6)
T 0
which tells us how the voltage fluctuations are correlated within the time scale ? = t ? t?.
The correlation function can be assumed to decay to zero in a finite characteristic correlation
time ?c.
We now work to establish the relation between SV (?) and Gv(?). We first rewrite Eq.
A.4,
? T ? T
?| 1 ?V (?)|2? = ? dt? dt
??v(t)v(t?)?ei?(t?t )
T 0 0 (A.7)
1 T T
= dt d??v(t)v(t? ?)?ei?? .
T 0 ?T
APPENDIX A. NOISE AND ENERGY RELAXATION 217
In the limit T  ?c, we can approximate
?
1 T
? ?
SV (?) = ? dt d??v(t)v(t? ?)?ei??T 0 ??? (A.8)
= d?Gv(?)e
i?? ,
??
which tells us that the spectral density is simply the Fourier transform of the correlation
function. This gives the intensity of the noise at certain frequencies. The inverse transform
subsequently gives ?
1 ?
G (?) = d?S (?)e?i??V V . (A.9)
2? ??
We note that noise with short correlation time is broadband in frequency. For example, the
white noise correlation function is
GV (t) = ??(t), (A.10)
which corresponds to a flat spectrum
SV (?) = ?. (A.11)
Noise having correlation time ?c  T corresponds to Markovian processes, where the dis-
turbance from the noise is random, uncorrelated, and independent of each other. Noise not
satisfying this condition is classified as non-Markovian.
Fluctuation-Dissipation Theorem
At first, the Hamiltonian formalism seems to be powerless in treating dissipation which is
an irreversible process. Our approach in modeling dissipation quantum-mechanically is to
consider an infinite collection of harmonic oscillators which consists of inductors and ca-
pacitors with purely imaginary admittances. The real part of the total admittance accounts
APPENDIX A. NOISE AND ENERGY RELAXATION 218
for noise that dissipates the qubit?s energy [70].
For instance, a resistor can be modeled as a semi-infinite transmission line.?The impedance
seen by the qubit is precisely the characteristic impedance of the line Zo = l/c. In elec-
trical engineering language, the energy absorbed by this line travels in one direction away
from the qubit and is never reflected back. We can then write the associated Lagrangian
and define the noise spectral density for such dissipative element.
First consider a single lossless LC resonant mode with Hamiltonian
Q2 ?2
H = + .
2C 2L
Introducing the creation and annihilation operators as usual with commutation relation
[a, a?] = 1, we have
?
h?Z
? o? = (a? + a),2
h?
Q = i (a? ? a), (A.12)
2Zo
h??o 1
H = (a?a+ ),
2 2
and the charge at time t is
Q(t) = eiHt/h?Q(0)e?iHt/h?. (A.13)
The current correlation function for a mode at frequency ?o = (LC)?1/2 is
2
?i(t)i(0)? = ? d ? h??oQ(t)Q(0)? = (?a?a?ei?ot + ?aa??e?i?ot). (A.14)
dt2 2L
APPENDIX A. NOISE AND ENERGY RELAXATION 219
At thermal equilibrium, the distribution of photons follows Bose-Einstein statistics,
[ ( ) ]
? 1 1 h??oa?a? = = coth ? 1 ,
eh??o/kBT ? 1 [ 2 ( 2k)BT ] (A.15)
?aa?? = ?a?a? 1 h??o+ 1 = coth + 1 ,
2 2kBT
from which we get
[ ( ) ]
? h??o h??oi(t)i(0)? = coth cos(?ot)? i sin(?ot) . (A.16)
2L 2kBT
Next, consider a frequency-dependent admittance Y (?) with Fourier transform
? ?
Y (?) = y(t)ei?tdt, (A.17)
?
which can be extended to the complex plane as
? ?
Y (? + i?) = y(t)ei(?+i?)tdt. (A.18)
?
We can then define the generalized admittance function
Y (?) = lim Y (? + i?). (A.19)
??0
The asymptotic expression for the nth oscillator?s admittance is
{ [ ( ) ( )]}
? i ?n ?n
Yn(?) = yn ?n [?(? ? ?n) + ?(? + ?n)] + p.p
2 2 ? ? + p.p ,?n ? + ?n
? (A.20)?
where yn = Cn/Ln and ?n = 1/ LnCn. The generalized admittance has both real and
imaginary parts, and the smooth ReY(?) function is replaced by an infinitely dense comb
of ? functions. Using the Caldeira-Leggett representation, we can add up all the modes
to obtain the correlation of the noise. Note that the model does not specify the internal
APPENDIX A. NOISE AND ENERGY RELAXATION 220
working of a dissipative element.
We can then rewrite Eq. A.16 for a collection of harmonic oscillators as
? ? [ ( ) ]
? h? h??oi(t)i(0)? = ?d? coth + 1 ReY(?)e?i?t. (A.21)
2? ?? 2kBT
?
And with the definition of the spectral density ?SI(?) = ?? dt?i(t)i(0)?ei?t, we get
[ ( ) ]
h??
SI(?) = h??ReY(?) coth + 1 . (A.22)
2kBT
On the other hand, we can also define the voltage noise spectral density as
[ ( ) ]
h??
SV (?) = h??ReZ(?) coth + 1 . (A.23)
2kBT
For example, the spectral density of voltage fluctuations from a resistor R following the
equation above is
2Rh??
SV (?) =
1? . (A.24)e?h??/kBT
In the limit kBT  h??, the noise reaches the classical limit
SV (?) = 2RkBT. (A.25)
In the quantum limit, which is achieved by freezing the resistor to absolute zero, the noise
spectral property becomes
SV (?) = 2Rh???(?), (A.26)
where ?(?) = 1 for ? > 0 and ?(?) = 0, otherwise.
Let us connect the plot shown in Fig. A.1 to Bose-Einstein statistics. The positve fre-
quency part of the noise spectral density corresponds to the absorption of energy by the
resistor, and the negative part to its emission of energy. From this, we learn that in the
APPENDIX A. NOISE AND ENERGY RELAXATION 221
Figure A.1: Voltage noise. Voltage noise spectral density from a resistor at T = 10 mK.
The classical noise at 10mK and the quantum noise are shown for comparison.
classical limit, the resistor emits as much as it absorbs, whereas in the quantum limit, the
resistor only absorbs energy from the quantum system it connects to. We will see next that
this also explains why we cannot observe quantum behavior in the classical limit where the
noise is symmetric, ?? = ??, and in the quantum limit, only decay process happens.
A.2 Fermi?s Golden Rule
Consider a two-level quantum system with transition energy Eeg = h??eg, which is perturbed
by a noise source with amplitude f(t) and coupling efficiency A
H1(t) = Af(t)?X . (A.27)
APPENDIX A. NOISE AND ENERGY RELAXATION 222
Figure A.2: Circuit dissipation model. Dissipation in superconducting circuits can be
modeled using a complex shunting admittance Y (?) which has a finite real part [70].
We assume for now that the perturbation is small and linearize it to first order. In the
interaction picture the state of the system evolves as
?
| i
T
?I(T )? = |?(0)? ? dtH1,I(t)|?(0)?, (A.28)
h? 0
which can be cast into matrix form as [172]:
?? ? ? ?? ??c?g?? ?? 0 ei?get?ih? = Af(t) ???cg?? . (A.29)
c? ei?egte 0 ce
If we prepare the system in the ground state, the excited state coefficient at time T is
?
iA T
ce(T ) = ? dtei?egtf(t), (A.30)
h? 0
and the probability of the system to be in the excited state is
2 ? T ? T
|c (T )|2 A= dtdt?e?i?eg(t?t?)e 2 f(t)f(t?) (A.31)h? 0 0
Taking an ensemble average with time long compared to noise correlation time, T  ?c,
APPENDIX A. NOISE AND ENERGY RELAXATION 223
and changing variables ? = t? t?, we have
2 ?A T ? ??pe(T )? = 2 dt d?e?i?eg? ?f(t)f(t? ?)?. (A.32)h? 0 ??
Writing the noise spectral density describing process f(?) as
? ?
Sf (?) = d?e
i?eg? ?f(t)f(t? ?)?, (A.33)
??
and using time translational invariance to change the average brackets quantity to ?f(?)f(0)?,
we can write the probability as a linear function of time T ,
2
? Ape(T )? = 2 Sf (??eg)T. (A.34)h?
The time derivative of the averaged probability gives the transition rate
A2
?? = 2 Sf (??eg). (A.35)h?
If the system is prepared in the excited state, the same algebraic steps give us the decay rate
A2
?? = 2 Sf (+?eg). (A.36)h?
In thermal equilibrium, the transition rates obey detailed balance relation ??/?? =
eh??eg/kBT . This implies
Sf (+?eg) = e
h??eg/kBTSf (??eg). (A.37)
We see again that in the classical limit, h??eg/kBT ? 0, the noise is symmetric, and the two
transition rates are equal. This effectively blurs all quantum effects. In the quantum limit
T = 0, ?? and Sf (??eg) would vanish, and no energy is transferred to the system from the
bath.
APPENDIX A. NOISE AND ENERGY RELAXATION 224
The relaxation rate from state i to j caused by source ? follows the Fermi Golden rule
derived above as
??
1
ij = 2 |?j|O?|i?|2S?(?ij). (A.38)h?
where S?(?ij) is the noise spectral density of source ? that causes the decay at transition
frequency ?ij . If the noise source is modeled as a shunting admittance, then the relevant
operator is flux, corresponding to the matrix element ?o?j|?|i?, with ?o = h?/2e, and the
noise spectral density is given as
[ ( ) ]
h??
S?(?) = h??ReY?(?) coth + 1 . (A.39)
2kBT
Appendix B
Effective Josephson Energy in SQUID
One Ring to rule them all, One Ring to find them,
One Ring to bring them all, and in the darkness bind them.
John R. R. Tolkien
First, let us consider a SQUID circuit consisting of two Josephson junctions. The physics
is similar to that of the Cooper-pair-box since the Josephson junctions form islands on each
side. The flux threading the SQUID loop may however effectively change the Josephson
energy. The Hamiltonian of the circuit is
H = 4ECn
2 ? EJ1 cos?1 ? EJ2 cos?2, (B.1)
where ?1,2 are the respective gauge invariant phase differences across the two junctions.
The relation between N individual phase differences around the loop and the magnetic flux
?s threading the loop must obey Kirchoff?s voltage law and flux quantization constraint
[49],
?N ( )?ext
?i = 2? N + ? ?ext, (B.2)
?
i o
225
APPENDIX B. EFFECTIVE JOSEPHSON ENERGY IN SQUID 226
where N is an integer, ?o = h/2e is the flux quantum, and ?ext is the reduced external
magnetic flux. If the phases ?1,2 are defined with respect to ground , with ?2 going against
the direction of the external flux, Eq. B.2 can be used to write
?1 ? ?2 = ?ext. (B.3)
For a SQUID circuit, replace notation ?ext with ?s, we can write the Hamiltonian in Eq. B.1
as
H = 4ECn
2 ? EJ1 cos?1 ? EJ2 cos(?1 ? ?s). (B.4)
Therefore, there is only one degree of freedom in the circuit. To derive the effective Joseph-
son energy, we define the sum and difference between the individual Josephson energies
as
EJ = EJ1 + EJ2 ,?
E ? E (B.5)J J
d = 1 2 ,
EJ?
which are equivalent to
1
EJ1 = EJ (1 + d),2 ? (B.6)
1
EJ2 = EJ (1? d).2 ?
The potential term in the Hamiltonian can then be written
[ ( ) ( ) ( ) ( )]
?1 + ?2 ?s ?1 + ?2 ?s
VJ = EJ cos? ( ) cos( )[? d sin ( )sin ( )] (B.7)2 2 2 2
?1 + ?2 ?s ?1 + ?2 ?s
= EJ cos cos 1? d tan tan . (B.8)? 2 2 2 2
We can redefine the phase degree of freedom (?1 + ?2)/2 = ?, effectively changing the
degree of freedom to ?, and use a parameter ? such that tan ? = d tan (?ext/2). Simplifying
APPENDIX B. EFFECTIVE JOSEPHSON ENERGY IN SQUID 227
the potential further, we get
( )
cos ?s2 cos (?+ ?)
VJ = EJ? ( c)os?? (B.9)?s
= E cos 1 + tan2J ? cos (?+ ?) .? 2
We can redefine the phase degree of freedom again ? + ? ? ? and write the effective
Josephson energy EJ as
( )? ( )
EJ eff ?
?s ?
E 2 2
s
J cos 1 + d tan . (B.10)? 2 2
The Hamiltonian can thus be written as
H = 4E n2C ? EJ cos?. (B.11)
Note that for the Hamiltonian given by Eq. B.11 to be equivalent to the one given by Eq. B.1,
the phase definition is ? = ?1 ? ?s/2 + ?.
For a SQUID loop shunted by an inductor (Fig. 4.6), we can follow the same procedure,
noting that ?1 = ? ? ?ext, where ? is the phase across the inductor. We can rewrite the
Josephson terms in the Hamiltonian given by Eq. 4.1 as
( ) ( ){ ( ) ( )}
?1 + ?2 ?1 ? ?2 ? ?1 + ?2 ?1 ? ?2VJ = EJ cos( ) cos? 2 ( ){ 1 d(tan2 )2 (tan)} 2 (B.12)
?s ?s ?s ?s
= EJ cos ?1 ? cos 1? d tan ?1 ? tan .? 2 2 2 2
APPENDIX B. EFFECTIVE JOSEPHSON ENERGY IN SQUID 228
Define a variable ? as before, with tan ? = d tan(?s/2), we have
cos (?s/2)
VJ = EJ ?cos (?? 1 ? ?s/2cos ? (+ ?))
?s
= EJ cos ((?s/2) 1 + d)2 tan
2 cos (?
? 1 ? ?s/2 + ?) (B.13)2
?s
= EJ cos ?1 ? + ? .eff 2
The flux through SQUID loop simply tunes the effective Josephson energy of the circuit.
The split-junction fluxonium Hamiltonian can thus be rewritten as
( )
1 ?s
H = 4E n2C + EL?
2 ? EJ eff cos ?? ?ext ? + ? . (B.14)
2 2
Appendix C
Coupled Circuits Hamiltonians
Call me Ishmael.
Herman Melville
C.1 Capacitive Coupling
?
Consider two coupled LC oscillators with resonant frequencies ?1 = 1/ L1C1 and ?2 =
?
1/ L2C2 as shown in Fig. C.1(a). If we wish to turn one of them into a qubit, we can
conveniently add the Josephson energy term in the final step. The Lagrangian for two
capacitively coupled LC circuits with coupling capacitance Cc is
L 1 ?
2 1 ?2 1
= C ??2 ? 1 + C 2 21 1 2??2 ? + Cc(??1 ? ?? 22) . (C.1)2 2L1 2 2L2 2
Using the transformation given by Eq. 3.7, we get
Q21 ?
2 Q2 ?2 Cc
H = ? +
1 + 2? +
2 + Q1Q2, (C.2)
2C1 2L1 2C2 2L2 C1C2
where C ? = (C C +C C +C C )/(C +C ) and C ?1 1 2 1 c 2 c 2 c 2 = (C1C2+C1Cc+C2Cc)/(C1+Cc). The
229
APPENDIX C. COUPLED CIRCUITS HAMILTONIANS 230
coupling capacitance Cc thus slightly renormalizes the shunting capacitances [70]. Writing
the Q operators in terms of ladder operators (see section 3.1), we get
( )
h? C C C 1/4c 1 2
Hc = ? (a? ? a)(b? ? b) = ?h?gRR(a? ? a)(b? ? b). (C.3)
2 C1C2 L1L2 ?
For qubit-resonator coupling, the convention is to writeQ1 = 2en for qubit,Q = i h? (a?2 2Z ?R
a), so the coupling Hamiltonian takes the form
? ?
Cc h? CR
Hc = i2e n(a
? ? a) = h?g ?qRn(a ? a). (C.4)
CqCR 2 LR
For qubit-qubit coupling, we often write
2 CcHc = 4e n1n2 = JCn1n2. (C.5)
C1C2
The coupling constant gqR is used to simulate the dispersive shift [127], and JC is used to
compute qubit spectrum. The coupling constant gRR can be simulated in Ansys Desktop
(section 5.2) by assigning a lump circuit boundary condition to the qubit mode, making it
behave like an LC oscillator. We can extract the vacuum coupling coefficient 2gRR, which
equals to the anti-crossing amplitude. This can then be converted to gqR and JC . A spec-
troscopy fit in the qubit-cavity anti-crossing region is used to experimentally extract the
coupling constant gqR, as shown in Fig. 6.23(b). Two-qubit spectrum can be used to fit and
extract JC , as shown in Fig. 6.15.
C.2 Inductive Coupling
Here we consider two harmonic oscillators coupled via a mutual inductance L3 as shown
in Fig. C.1(b). Defining the flux ?1, ?2, and ?3 as shown in Fig. C.1(b), we can write the
APPENDIX C. COUPLED CIRCUITS HAMILTONIANS 231
Figure C.1: Coupled circuits diagrams (a) Capacitively coupled LC circuits. (b) Induc-
tively coupled LC circuit.
Lagrangian of the circuit as
L 1 2 1 2 ? (?1 ? ?3)
2 2 2
? (?2 ? ?3) ?= C1??1 + C 32??2 ? . (C.6)2 2 2L1 2L2 2L3
We can write the potential part as a function of only ?1 and ?2 by using the condition on
the current going through the mutual inductor L3,
?3 ?1 ? ?3 ?2 ? ?3
I3 = = + . (C.7)
L3 L1 L2
Let L3/L1 = r1 and L3/L2 = r2, we get
r1?1 + r2?2
?3 = . (C.8)
1 + r1 + r2
Replacing ?3 in Eq. C.6 gives us
{ }
L 1 1 1 1P = [(1 + r 2 2 22 2)?1 ? r2?2] + [(1 + r1)?2 ? r1?1] + [r1?1 + r2?2] .(1 + r1 + r2) 2L1 2L2 2L3
(C.9)
APPENDIX C. COUPLED CIRCUITS HAMILTONIANS 232
The Hamiltonian of the coupled systems is
H = H1 +H2 +Hc, (C.10)
where the individual uncoupled terms H1, H2 each has a normalized inductance, and the
coupling term Hc involves ? 2 21?2. By grouping terms with ?1, ?2, and ?1?2 in Eq. C.9, we
can write
[ ( ) ]
Q21 1 1 2 1 1H1 = + [ (1 + r 22 2) + + r2C1 2(1 + r 11 + r2) L1 (L2 L3) ]?21,
Q2
H = 2
1 1 1 1
2 + [ (1 + r1)2 + + r2 ?2, (C.11)2C2 2(1 + r1 + r2)2 L2 L1 L3] 2 2
? 1 r2(1 + r2) r1(1 + r1) ? r1r2Hc = + ?1?2.
(1 + r1 + r2)2 L1 L2 L3
These are the general expressions for the Hamiltonians. In the weak coupling limit,
L1, L2  L3, H1 and H2 are modified by L3 by a negligible amount, while the coupling
term Hc becomes [128]
( )
L 23 h? L3
Hc = ? ?1?2 = ? ?1?2. (C.12)
L1L2 2e L1L2
For inductively coupled qubit-cavity system [173], we can add a Josephson energy term
to H1. For fluxonium, we can take the limit L1  L2, L3, so r1 = L3/L1 ? 0. The
Hamiltonians in Eqs. C.11 become
Q2 ?2
H 11 = +
1 ? EJ cos(?1 ? ?ext),
2C1 2L1
Q2 ?2
H = 2 + 22 , (C.13)
2C2 2(L2 + L3)
L3
H ?c = ? ?1?2 = ?gqR?1(a+ a ).
L1(L2 + L3)
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