ABSTRACT
Title of dissertation: QUANTUM IMPURITY REGIME OF
CIRCUIT QUANTUM ELECTRODYNAMICS
Nitish Jitendrakumar Mehta, Doctor of Philosophy, 2022
Dissertation directed by: Professor Vladimir E Manucharyan
Department of Physics
In this thesis we describe a novel regime of cavity quantum electrodynamics, where a single
atom is coupled to a multi-mode Fabry-Perot cavity with a strength much larger than its free spectral
range. In this regime, the atom acting as a quantum impurity mediates interactions between many-
body states of radiation in the multi-mode cavity. This novel regime of cavity QED is experimentally
realized by coupling superconducting artificial atoms to a high impedance 1-D superconducting
transmission line cavity. We study the problem of single photon decay in these strongly non-linear
cavities with discrete energy levels. By engineering the properties of the artificial atoms, we alter
interaction and connectivity between many-body states of radiation, and we observe two distinct
effects. For the case of a multi-mode Fabry-Perot coupled to a fluxonium artificial atom, the inter-
actions mediated by the atom attempts to down convert a single photon into many low frequency
photons but fails because of limited connectivity in the many-body Fock space. This phenomenon of
many-body localization of radiation gives rise to striking spectral features where a single standing
wave resonance of the cavity is replaced by a fine structure of satellite peaks. On the other hand,
for the case of a transmon coupled galvanically to the cavity, the interaction splits a single photon
at high energy into a shower of odd number of lower energy photons. In this case the single stand-
ing wave resonance of the cavity acquires a shorter lifetime which can be calculated using Fermi?s
golden rule and matches our theoretical model without any adjustable parameters.
QUANTUM IMPURITY REGIME OF
CIRCUIT QUANTUM ELECTRODYNAMICS
by
Nitish Jitendrakumar Mehta
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
2022
Dissertation Committee:
Professor Vladimir E. Manucharyan, Advisor
Professor Alicia Kolla?r
Professor Alexey Gorshkov
Professor Victor Yakovenko
Professor Andrew Childs, Dean?s Representative
? Copyright by
Nitish Jitendrakumar Mehta
2022
Acknowledgments
During the course of my graduate career at UMD, I have had the pleasure to interact with and
learn from a wide range of people with diverse backgrounds and I have greatly benefited from
their support, wisdom and guidance. So, it is my pleasure to start this thesis by acknowledging my
gratitude to all the incredible people who have made this thesis possible.
I would like to begin by thanking my advisor, Professor Vladimir Manucharyan for giving me the
invaluable opportunity to work in the superconducting circuits lab at UMD. I will forever be grateful
for the time, energy and patience with which Vlad taught me everything there is to know about
becoming a good scientist. From fundamental concepts in physics to the right choice of colors in
a paper plot. Some of my most fond memories of graduate school have been the times when we
discussed physics in his office (and later during COVID on Zoom). It is impossible to understate his
impact on the work described in this thesis and the impact, he has had on me as a scientist.
I am also very grateful to Professor Cristiano Ciuti, who has gradually become my second (the-
ory) advisor during our two years of collaboration. Cristiano was a constant companion in the long
and often frustrating road to come up with the theory for the fluxonium impurity experiments de-
scribed in chapter 5. Cristiano taught me the nuances of condensed matter theory and offered me
tremendous help throughout the project. He also rescued me numerous times when I got stuck in
calculations.
Thanks are due to Professor Alicia Kollar, Professor Alexey Gorshkov, Professor Victor Yakovenko,
ii
iii
and Professor Andrew Childs for agreeing to serve on my thesis committee and for sparing their
invaluable time reviewing the manuscript. I would especially like to thank Professor Yakovenko and
Professor Gorshkov on joining the committee on a short notice.
With a separate paragraph I would like to thank Dr. Roman Kuzmin who was my direct mentor
in the lab. All the projects described in this thesis have been done in either direct guidance or
collaboration with Roman. I will forever be grateful to Roman for teaching me everything from
theoretical calculations about quantum impurity models to operating the dilution refrigerator and
being patient even when I made mistakes repeatedly. I hope in the future I can adopt the methodical
and thorough approach towards science that Roman epitomizes.
My colleagues at the Superconducting Circuits laboratory (both past and present) have con-
tributed to this work in many ways, they deserve a special mention. I would like to thank Nick
Grabon and Ray Mencia for fabricating a subset of the devices described in this thesis and for train-
ing me in the dark arts of nanofabrication. On the fabrication side, I am also thankful of Natalia
Pankratova who devised the original recipe to make Josephson junction arrays in our lab. I would
like to thank postdoc Dr. Ivan Pechenezhskiy for our numerous discussions on my projects and also
writing the superconducting circuits simulation python package. It was only because of your initial
code base that I was able to simulate the spectra for more complicated circuits later in my PhD. I
thank Dr. Yen-Hsiang Lin and Haonan Xiong for designing the 3-D waveguide sample holder used
in this work. It was an absolute pleasure to work closely with Aaron on the Spin chain project in
the last half of my PhD. Although I did not work directly with Nathaneal Cottet, Long Nuygen, and
Hanho Lee, it was always fun to learn about their experiments in the lab, that helped me broaden
my vision.
I would also like to acknowledge help and support from the administrative staff of the Physics
department especially Josiland Chambers. She was incredibly kind and patient and helped me
iv
navigate all the administrative tasks right from starting my PhD to defense. I would be lost without
her help.
Outside of UMD, I would like to thank Professor Vijayraghavan at TIFR, who gave me, a confused
electrical engineering and physics undergrad a chance to work in his lab without prior research
experience. Vijay was also the person who introduced me to the fascinating world of circuit QED
and connected me to Vlad. In retrospect his advice of coming to Maryland and joining Vlad?s group
was easily one of the best pieces of advice I have received.
I owe my deepest thanks to my family - my brother, my mother and my father who have always
stood by me and guided me through my career. Words cannot express the gratitude I owe them. To
my loving wife, Aggie, thank you for your support and being by my side, it was a pleasure to go on
this graduate school journey with you. I cannot imagine finishing this PhD without my family and I
dedicate this thesis to you all.
Contents
Acknowledgments ii
Contents v
1 Introduction 1
1.1 Multi-mode cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Quantum impurity regime of cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Circuit Theory 9
2.1 Introduction to circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Linear superconducting circuits: Microwave photons . . . . . . . . . . . . . . . . . . 16
2.2.1 Lumped element L-C oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Standing wave modes on a transmission line . . . . . . . . . . . . . . . . . . . 18
2.2.3 Foster?s decomposition of transmission line . . . . . . . . . . . . . . . . . . . 21
2.3 Non-linear superconducting circuits: Artificial atoms . . . . . . . . . . . . . . . . . . 23
v
2.3.1 Charge qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Flux qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Coupling between linear and non-linear circuits . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Charge qubit coupled to multi-mode transmission line resonator . . . . . . . . 28
2.4.2 Flux qubit coupled to multi-mode transmission line resonator . . . . . . . . . 33
3 Experimental Techniques 43
3.1 Nanofabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Microwave reflectometry setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 3-D waveguide sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.3 Room temperature setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Multi-mode Fabry-Perot cavity 48
4.1 Josephson junction transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Transmission line Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.2 Multi-mode Fabry-Perot cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.3 Interactions between standing wave modes . . . . . . . . . . . . . . . . . . . 54
4.2 Control Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Characterization Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 Intrinsic lifetime of modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Two tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.3 Extracting Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Summary of measurements and conclusion . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Summary of device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
vi
5 Fluxonium impurity coupled to multi-mode cavity 65
5.1 Design of the device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Microscopic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Control experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.1 Single tone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4.2 Single tone Vs flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Characterization experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5.1 Bare Fabry-Perot cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5.2 Fluxonium parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6 Reflection spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.7 Interpretation of spectroscopy: Many-body localization of photons . . . . . . . . . . . 82
5.8 Towards strongly correlated photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.10 Summary of device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Transmon impurity galvanically coupled to multi-mode cavity 88
6.1 Design of the device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Microscopic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.1 Quartic non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.2 Phase-slip non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4 Control experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 Characterization experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.1 Large EJ/EC device (0a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
vii
6.5.2 Inelastic scattering of photon off phase-slip . . . . . . . . . . . . . . . . . . . 100
6.6 Comparison to material related losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.8 Summary of device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Summary and Outlook 110
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Tunable platform to study MBL physics in 1-D . . . . . . . . . . . . . . . . . . . . . . 111
7.3 Implementation of Q-RAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Appendices 113
A Benchmnarking 114
B Calculation of the reflection coefficient 118
Bibliography 120
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List of Figures
1.1 Cavity QED schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Multi-mode cavity QED schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Many-body Fock space near energy k? without disorder . . . . . . . . . . . . . . . . 6
1.4 Many-body Fock space near energy k? with disorder . . . . . . . . . . . . . . . . . . 7
2.1 Current phase relation of three basic two terminal components used in circuit QED . 11
2.2 Schematic and circuit model for Josephson junction . . . . . . . . . . . . . . . . . . . 13
2.3 Potential energy profile of a Josephson junction . . . . . . . . . . . . . . . . . . . . . 14
2.4 Schematic and circuit model of a 1-D transmission line. . . . . . . . . . . . . . . . . . 19
2.5 Foster?s decomposition of an arbitrary admittance . . . . . . . . . . . . . . . . . . . . 21
2.6 Circuit model and transition spectrum of charge qubits. . . . . . . . . . . . . . . . . . 23
2.7 Circuit model and transition spectrum of flux qubits. . . . . . . . . . . . . . . . . . . 27
2.8 Circuit model for a transmon coupled to a multi-mode transmission line resonator . . 29
2.9 Coupling constants for a transmon coupled to a multi-mode transmission line resonator 30
2.10 Circuit model for a fluxonium coupled to a multi-mode transmission line resonator . 34
2.11 Coupling constants for a fluxonium coupled to a multi-mode transmission line resonator 35
2.12 Renormalization of fluxonium inductive energy. . . . . . . . . . . . . . . . . . . . . . 39
ix
3.1 Crogenic and room temperature measurement setup . . . . . . . . . . . . . . . . . . 45
3.2 3-D waveguide sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Schematic and circuit model of high impedance Josephson junction transmission line. 49
4.2 Schematic and circuit model of multi-mode Fabry-Perot cavity made using a Josepshon
junction array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 One tone spectroscopy of multi-mode Fabry-Perot cavity. . . . . . . . . . . . . . . . . 56
4.4 Fit of the reflection coefficient to theory. . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Internal and external quality factors plotted versus normalized frequency. . . . . . . 59
4.6 Two tone spectroscopy of the multi-mode Fabry-Perot cavity. . . . . . . . . . . . . . . 61
4.7 Reconstructed dispersion relation of standing wave modes in the multi-mode Fabry-
Perot cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8 Reconstructed dispersion relation of all the multi-mode Fabry-Perot cavities . . . . . 63
5.1 Fluxonium impurity device schematic and optical image . . . . . . . . . . . . . . . . 66
5.2 Schematic representation of the three wave mixing effective Hamiltonian . . . . . . . 68
5.3 One tone spectroscopy of the fluxonium impurity device . . . . . . . . . . . . . . . . 72
5.4 One tone spectroscopy as a function of externally applied flux through the fluxonium
loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Mode spacing vs frequency for ?ext/2? = 0 and ?ext/2? = 0.5 . . . . . . . . . . . . . 75
5.6 Multi-particle excitation spectrum of multi-mode cavity coupled to a fluxonium . . . 77
5.7 Three wave mixing strength and qubit component in standing wave modes . . . . . . 78
5.8 Coarse and fine spectroscopy of multi-mode cavity coupled to fluxonium . . . . . . . 80
5.9 Spectroscopy of two-particle states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.10 Spectroscopy of beyond two-particle states . . . . . . . . . . . . . . . . . . . . . . . . 82
5.11 Towards strongly correlated photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
x
6.1 Transmon impurity device schematic and optical image . . . . . . . . . . . . . . . . . 89
6.2 Schematic representation of inelastic scattering of photons by a phase slip . . . . . . 92
6.3 One tone spectroscopy of transmon coupled to multi-mode Fabry-Perot cavity . . . . 95
6.4 Density of single particle states in a multi-mode cavity coupled to a transmon . . . . 96
6.5 Transmon induced photon-photon interactions between standing wave modes of the
cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.6 Photon loss probability in a single round trip across the multi-mode cavity . . . . . . 100
6.7 Elastic and Inelastic scattering response of photons off the transmon qubit . . . . . . 101
6.8 Qxternal quality factor measured as a function of the external flux . . . . . . . . . . . 102
6.9 Comparison of inelastic scattering rate with theory . . . . . . . . . . . . . . . . . . . 104
6.10 Example of coupling between single photon and multi-photon states induced by trans-
mon impurity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.11 Comparison to material related loss mechanisms . . . . . . . . . . . . . . . . . . . . 107
A.1 Simulated spectrum for a multi-mode cQED device using brute force numerics . . . . 116
xi
List of Tables
4.1 Bare Fabry-Perot cavity device parameters . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Fluxonium impurity device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1 Galvanically coupled transmon impurity device parameters . . . . . . . . . . . . . . . 109
xii
Nomenclatures
Abbreviations
1-D One dimensional
AC Alternating current
cQED Circuit quantum electrodynamics
dB decibels
DC Direct current
FSR Free spectral range
GHz gigahertz
HEMT High Electron Mobility Transistor
IPA Isopropyl Alcohol
MBL many-body localization
MMA Methyl methacrylate
PMMA Poly (methyl methacrylate)
xiii
SEM Scanning electron microscope
SQuID Superconducting quantum interference device
VNA Vector network analyzer
Physical constants
?0 Vacuum permittivity
? Reduced Planck constant
?0 Vacuum premeability
?0 Magnetic flux quantum
c Speed of light in vacuum
e Electron charge
h Planck constant
kB Boltzman constant
RQ Resistance quantum
Symbols
?ij Cross-Kerr interaction between cavity mode i, j
? Free spectral range of Fabry-Perot cavity
?(3) Spacing of 3-particle states in a Fabry-Perot cavity
?(5) Spacing of 5-particle states in a Fabry-Perot cavity
? Length of Fabry-Perot cavity
xiv
? Radiative linewidth of atom
? Decay rate
?in Intrinsic decay rate
a? annihilation operator
b? annihilation operator
c? annihilation operator
n? Cooper pair number operator
? Decay rate
L lagrangian
e Second atomic state index
f Third atomic state index
g First atomic state index; atom cavity coupling constant
h Forth atomic state index
?
i ?1
? Width of the first bloch band of a transmon qubit
? frequency in radians
?p Plasma frequency of a Josephson junction
?eg Transition frequency of an atom from ground to excited state
? Area of unit circle
xv
? Superconducting phase across a Josephson junction
?ext external phase bias of a superconducting loop
|?i? Wavefunction of ith eigenstate of a many-body Hamiltonian
a Unit cell size
c Capacitance per unit length
Cg Capacitance to ground
CJ Capacitance of Josephson junction
EC Charging energy, defined as EC = e2/2C
EJ Josephson energy
Ek Energy of kth Fabry-Perot mode
E 2L Inductive energy, defined as EL = (?0/2?) /L
feg ?eg/2?
H Hamiltonian
I Current
i Summation index
Ic Critical current of Josephson junction
j Summation index
k Summation index; Wave-vector
l Inductance per unit length
xvi
LJ Linearized inductance of a Josephson junction
N Number of junctions/modes in a transmission line
ng Gate charge
O Big-O notation
P Power
Q Charge operator
Qext Extrinsic quality factor
Qint Intrinsic quality factor
R Resistance
t Time
U Unitary transformation
V Voltage; Purturbation to Hamiltonian
Y admittance
Z impedance
Z? wave impedance of a transmission line
xvii
Citations to Published Work
Parts of chapters 2, 4, 5, and 6 have appeared in the following papers:
? N. Mehta, C. Ciuti, R. Kuzmin, and V. E. Manucharyan. Down-conversion of a single photon
as a probe of many-body localization. arXiv:2203.17186
? N. Mehta, C. Ciuti, R. Kuzmin, and V. E. Manucharyan. Superstrong coupling circuit quantum
electrodynamics revisited. in preparation
? R. Kuzmin, N. Grabon, N. Mehta, A. Burshtein, M. Goldstein, M. Houzet, L. I. Glazman and
V. E. Manucharyan. Inelastic scattering of a photon by a quantum phase slip. Phys. Rev. Lett.,
126:197701, May 2021
? R. Kuzmin, R. Mencia, N. Grabon, N. Mehta, Y. H. Lin, and V. E. Manucharyan. Quantum
electrodynamics of a superconductor-insulator phase transition. Nature Physics, 15(9):930-
934, 2019
? R. Kuzmin, N. Mehta, N. Grabon, R. Mencia, and V. E. Manucharyan. Superstrong coupling in
circuit quantum electrodynamics. npj Quantum Information, 5:20, 2019
xviii
Chapter 1
Introduction
Quantum electrodynamics (QED) is a branch of physics describing a remarkable list of fundamental
phenomena produced by the quantum nature of the electromagnetic field [1, 2, 3]. In addition to
this, Quantum electrodynamics either directly or indirectly has also led to the invention of many
important and ground-breaking practical applications including the laser, GPS, and more recently
quantum computers [4, 5]. One of the simplest models of Quantum electrodynamics is that of cavity
QED, where a single atom interacts with a single mode of quantized electromagnetic field. In this
system the electromagnetic modes can be spatially confined and the corresponding vacuum fields
can be dramatically enhanced [6]. A celebrated manifestation of cavity QED is the strong coupling
Figure 1.1: Atom with transition frequency ?eg/2? interacts coherently with radiation inside a Fabry-
Perot cavity of length ?. The linewidth of the standing wave modes in the cavity and atom are given
by ? and ? respectively. The free spectral range of the cavity is given by ? = 1/? = v/2?.
1
regime where the interaction strength between the atom and the electromagnetic fields g exceeds
the intrinsic loss rates of the atom and the cavity ?, ? respectively (see Fig. 1.1). In this regime
coherent exchange of single excitations can occur at the rate given by g, the vacuum Rabi frequency
[6]. This strong coupling regime has attracted extensive attention in the last two decades and
also forms the basis of most quantum computing architectures. More recently, new strong coupling
regimes of cavity QED have been explored, in particular the ultrastrong coupling regime, achieved
when the anti-resonant (non-rotating-wave) terms of light-matter interaction become relevant [7, 8,
9, 10] with just one cavity mode, such physics emerges when the vacuum Rabi frequency becomes
comparable to the photon and atom transition frequencies. In the ultrastrong coupling regime,
subspaces with different excitation numbers are coupled as a result of breaking of particle number
conservation. As comprehensively reviewed in Refs. [11, 12, 13], the ultrastrong coupling regime
has been investigated in many different platforms, including semiconductor systems [14, 15] and
superconducting quantum circuits [16, 17].
1.1 Multi-mode cavity QED
Another kind of strong coupling regime that instead has been relatively overlooked so far in both
theory and experiment, can be obtained in multi-mode resonators where the vacuum Rabi frequency
exceeds the free spectral range of the resonator, namely the frequency difference between consecu-
tive standing-wave modes. This regime of cavity QED was first pointed out theoretically in Ref. [18]
and was coined as superstrong coupling regime and is mathematically defined as ?? ? 1, where ?
is the round trip time of photons in the cavity and ? is the atom?s spontaneous emission linewidth
in the limit of infinite system size. The main predicted manifestation is the hybridization of the
atom transition with several single-photon states in different modes, thus modifying their spatial
profile [18]. In addition, this regime also enables coupling of states with different excitation num-
bers as states with multiple excitations become resonant with a single excitation state because of the
presence of low frequency modes (see Fig. 1.2). This regime has been demonstrated recently via
microwave spectroscopy on a superconducting platform consisting of a transmission line resonator
coupled to a single transmon-type atom [19, 20, 21]. The main observation in the experiments is
2
Figure 1.2: A schematic description of the energy levels of a multi-mode cavity coupled to a single
qubit. The modes of the cavity have frequencies ?j with j = 1, 2, 3, ..., while ?a is the qubit transition
frequency. On the left, the diagram depicts the energy levels of bare single-particle excitations (only
one photon in one mode or only a qubit excitation). The qubit excited state can hybridize with
several quasi-resonant single-photon states over a frequency window ?. On the right, example of
many-body states where multiple photons occupy different modes. Due to the existence of low-
frequency modes, single-particle excitations can be resonant to multi-particle states.
that the frequencies of a large number of resonator modes around the atom transition frequency
are shifted. However, the coupling was too low to expect quantum many-body effects with single
photons. Another important point to make here is that these experiments were done using a weakly
non-linear atom that can be modeled as a harmonic oscillator with a perturbative Kerr non-linearity.
The corresponding Hamiltonian [20] is equivalent to a multi-mode bosonic Hopfield model [22].
In other words, as the atom hybridizes with more modes of the cavity, its non-linearity gets diluted
in the harmonic modes of the cavity. As a result, counter intuitively, increasing the coupling would
actually results in a more linear system.
1.2 Quantum impurity regime of cavity QED
From many-body physics perspective, the most intriguing scenario occurs by replacing a weakly-
anharmonic atom [23, 19, 20, 21] with a strongly anharmonic one in a superstrongly coupled multi-
mode cavity QED platform. In fact, quantum electrodynamics in this case maps onto the strongly
3
correlated quantum impurity models of condensed matter physics [24, 25, 26, 27, 28]. Quantum
impurity models are among the simplest systems that display novel many-body effects, [29, 30, 31]
the models consist of a large number of otherwise uncoupled degrees of freedom interacting with a
local quantum impurity. In the cavity QED platform the role of the impurity is played by the atom
and the non-interacting modes are the standing wave modes of the Fabry-Perot cavity. The atom in
this case mediates interactions between a large subset of the cavity modes in direct analogy with
quantum impurity models like the Kondo model where a single spin half impurity mediates coupling
between a sea of non-interacting electrons[29, 30]. Theoretically, the computational complexity to
make predictions on these models scales exponentially in the number of modes coupled simultane-
ously to the impurity [32]. The advantage of simulating quantum impurity models in a cavity QED
setting is that, for sufficiently large coupling strengths, the many-body effects can be completely
developed, while only needing to control one quantum degree of freedom which is in stark contrast
to other experimental investigations of many-body phenomena that require the precise control of
many individual quantum degrees of freedom [33, 34, 35, 36, 37, 38, 39, 40, 41].
To understand the origin of many-body dynamics in our multi-mode cavity QED system, let us
ignore the effect of the atom, for the time being, and examine the Fock space of a cavity with
uniformly-spaced mode frequencies fk = k?, where k = 1, 2, ... (see Fig. 1.3). Consider exciting
a single high-frequency photon in the mode k ? 1. This single-particle state would be degenerate
with about O(k) two-particle states, as well as O(k2) three-particle states and so on so forth, as
illustrated with blue and red color states in Fig. 1.3. What would be the effect of an atom at fre-
quency feg ? fk? In the single-particle approximation, the atom merely adds an energy-dependent
phase-shift to reflected photons. This linear effect [23, 21, 42] amounts to weakly perturbing the
standing wave modes in the frequency window |fk ? feg| ? ?, where ? coincides with the atom?s
spontaneous emission linewidth in the limit of infinite system size and satisfies ? ? ? ? feg (the
superstrong coupling condition). At the multi-particle level, though, the atom can induce a multi-
particle interaction described in the most general way by the following effective Hamiltonian:
H? ?
= fka?
?
ka?k + V?3 + V?4 . . . , (1.1)h
k>0
4
?
V? = g A ? ? ?i i k,k1,k2,...ki a?k a?k . . . a?k a?k + h.c. (1.2)1 2 i
k,k2,...ki>0
Here a? (a??k k) is the operator that annihilates (creates) a photon at frequency fk in the standing-wave
mode k, which already includes a small atom admixture (due to hybridization with the atom). The
atom as a separate entity is completely eliminated from this picture. The interaction energy scales
gi are linked to the parameters of the atoms and the cavity. The matrix elements Ak,k are of1,...ki
order unity for ki = k but rapidly decay for |k ? ki| ? ?/?.
The?interaction term V?i in Eq. 1.2 coherently couples a single-particle state a??k|0? to multi-particle
states a??i k |0?. Therefore, the relevant many-body Fock space of Hamiltonian (Eq. 1.1) has ai
tree-like connectivity, similar to the case of many quantum impurity problems in condensed matter
physics [43]. The fate of a single excitation in such system can now be thought of as a hopping
problem in this fictional lattice of many-body Fock space, where the hopping amplitude is given by
the matrix elements of V?i. In the case of a disorder free ideal system the hopping leads to delocaliza-
tion of the single excitation in the many-body Fock spa?ce, where an init?ially excited photon in mode
?k irreversibly decays by down-conversion at a rate ? k ,k ,...k |?0|a? V? a?? a??k i k k . . . a?? 2k |0?| ?(fk ?1 2 i 1 2 i
fk ) given by Fermi?s Golden Rule [44]. However, for the case of system with finite disorder ini
the single particle spectrum, the massive degeneracy of the multi-particle states is lifted. Figure 1.4
shows the Hilbert space of such a system near mode k. In this case the fate of a single excitation
becomes a many-body problem and absence of diffusion in the Fock space is called many-body lo-
calization, by analogy with Anderson localization in the real space [45]. This general phenomenon
was subsequently predicted to occur in many other physical settings [46, 47, 48]. Being the only
known mechanism to prevent isolated interacting systems from reaching thermal equilibrium, MBL
has been attracting a vast attention from the theoretical physics community [49, 50].
1.3 Thesis overview
In this thesis we describe experiments and theory where we implement the quantum impurity regime
of multi-mode QED using the platform of cQED (circuit Quantum Electrodynamics) [51]. Experi-
mentally, this regime has been hard to achieve because it requires engineering strong coupling while
5
Figure 1.3: Energy levels of a disorder free multi-mode Fabry-Perot cavity with free spectral range
? near mode number k. The Hilbert space is partitioned based on the number of excitations in each
many-body state.
6
Figure 1.4: Energy levels of a disordered multi-mode Fabry-Perot cavity with free spectral range ?
near mode number k. The Hilbert space is partitioned based on the number of excitations in each
many-body state.
7
simultaneously also decreasing the free spectral range of the cavity. We can illustrate this diffi-
culty by an example for atom field coupling in 1-D, the coupling strength is typically proportional
? ?
to g ? 1/ ? and the free spectral range is ? ? 1/?, so the ratio g/? ? ?. The slow growing
nature of the function g/? is the reason why all of the recent experimental demonstrations of su-
perstrong coupling regime require cavities with extremely large spatial dimensions[23, 52]. The
first main technical innovation of this work is our ability to engineer Fabry-Perot cavity where the
speed of light is slowed by a factor of 100 compared to the speed of light in vacuum. This allows us
to implement a multi-mode cavity QED system in a compact form factor. Second innovation is the
galvanic connection between the Fabry-Perot cavity and the artificial atoms which gives us access to
practically infinite coupling and allows us to reach the quantum impurity regime. In addition we are
also able to engineer different connectivity in the many-body Fock space of the Fabry-Perot modes
by engineering the non-linearity of the artificial atoms. In two different experiments by engineering
sparse and dense connectivity in the many-body Hilbert space we observe both the localization and
delocalization of single excitation states in the many-body Fock space.
The thesis is organized as follows. Chapter 2 gives a brief overview of the physics of quantum
electrical circuits. Here we derive and explain the Hamiltonian of the multi-mode Fabry-Perot cavi-
ties and artificial atoms used in the thesis. Chapter 3 discusses the experimental tools used in the
work. Chapter 4 describes the experimental implementation of the multi-mode Fabry Perot cavity
using a long chain of Josephson junctions. The next two chapters 5, 6 describe in detail two specific
implementations of multi-mode cavity QED in the quantum impurity regime. In the first case we are
able to engineer the connectivity in the Fock space such that the single photon excitation gets local-
ized and we observe the violation of Fermi?s golden rule. In the second case we engineer the qubit
impurity in the exact opposite limit where a single excitation in a high frequency mode of the cavity
decays into a shower of odd number of lower frequency photons. We confirm this by measuring the
lifetime of the photon of the cavity modes and match the decreased lifetime to a Fermi?s golden rule
calculation. The last chapter summarizes the results and discusses some future work.
8
Chapter 2
Circuit Theory
In this chapter we provide a brief introduction to the quantum theory of superconducting circuits.
We start by introducing the circuit elements used to construct to the devices in the experiments
described in this work. We then proceed to describe the construction of the two main ingredients
in our work namely the linear multi-mode Fabry-Perot cavity and artificial atoms using the circuit
elements. And finally we conclude by deriving the couplings between the two class of circuits.
The last two sections of this chapter derive the microscopic many-body circuit Hamiltonians of the
multi-mode circuit QED systems described in this thesis.
2.1 Introduction to circuit QED
As introduced in the last chapter, circuit quantum electrodynamics (cQED) is a solid state implemen-
tation of the cavity QED setup. cQED studies the coherent interaction between artificial atoms made
of superconducting elements with discrete energy levels, and microwave photons that live inside
superconducting cavities. The ability to engineer properties of both the artificial atoms and photons
though clever circuit design and nano-fabrication enables this platform to study the physics of light
matter interactions in regimes experimentally inaccessible by natural atoms and light. These proper-
ties have made cQED as one of the leading platforms for both digital and analog quantum informa-
tion processing. Excellent state-of-the-art reviews can be found in these references [53, 54, 55, 56].
9
Before we begin to construct complicated quantum many-body systems using electrical circuits,
we need to understand under what conditions do the electric and magnetic fields inside these macro-
scopic circuit elements can be thought of in terms of quantum mechanical operators. The quantum
theory of circuits was pioneered by Michel Devoret and Steven Girvin in [57, 58], and this chapter
will closely follow their lecture notes. The first vital ingredient in constructing quantum electrical cir-
cuits is superconductivity! Superconductivity is described by the condensation of pairs of electrons
of opposite spin (Cooper pairs) into a ground state significantly gaped from the single-excitation
band of states where a single Cooper pair is broken. This excitation gap is effectively responsible
for reducing the number of degrees of freedom and allowing current to flow in the superconductor
without dissipation, thereby allowing these collective charge modes to be simply described by the
charge built up at a node or by the supercurrent flowing in a wire connecting two nodes.
Reducing the Hilbert space of superconducting circuits to just a single degree of freedom (charge
or flux) is not enough to justify a quantum mechanical treatment. Not only do we need to cool
down the circuit to introduce superconductivity, but we need to operate at a temperature where
the thermal fluctuations of the charge and flux are much smaller than the quantum ones associated
with the microwave resonance of the circuit kBT ? ??. This is the reason why all circuit QED
experiments involve anchoring the circuit to cold stage of a dilution refrigerator typically operating
in the 10 ? 15 mK. This temperature is much smaller than energy of a 5 GHz microwave photon
which corresponds to 240 mK. These circuits are described by many levels associated with different
quanta of microwave excitations. Therefore, it becomes equally critical to engineer modes with high
quality factor Q ? 1 [59] in order to distinguish these energy states and quantize the circuit. All of
the circuits described in this thesis can be decomposed into a network of three basic two terminal
components, namely capacitors, inductors and Josephson junctions. Each of these components is
defined by the total superconducting phase across it and the the total current flowing through it.
Each element is also defined by its unique current-phase relation which links the current flowing
through the element and the superconducting phase across it as a function of time
I(t) = f(t)V (t), (2.1)
and the total electrical energy stored in such an element is given by:
10
Figure 2.1: Current phase relation of three basic two terminal components used in circuit QED
? t
E(t) = V (t?)I(t?)dt?. (2.2)
??
For the purposes of describing the circuits in the Lagrange-Hamilton formalism, its instructive to
define two additional variables which will serve as the generalized coordinates for each component.
The coordinates can either be the magnetic flux across the component which is related to the voltage
as: ? t
?(t) = V (t?)dt?, (2.3)
??
or the charge flowing through the component which is related to the current as:
? t
Q(t) = I(t?)dt?. (2.4)
??
Since an arbitrary circuit can contain more branches than there are degrees of freedom, it becomes
more intuitive to define the network in terms of the flux and charge variables at each node. The
reader is encouraged to visit a more detailed description of this method of nodes in [59, 58]. The
choice of these node flux and charge variables in a network of two terminal components is akin to
choosing a set of generalized coordinates in a classical mechanics problem. This choice of variables,
as will be shown in the subsequent sections of this chapter merely defines the gauge for the problem
at hand and does not affect the dynamics of the problem.
11
2.1.1 Capacitors
Capacitors are formed by two conducting surfaces separated by an insulator or a gap. The voltage
across this element is then directly proportional to the charge accumulated on the conducting sur-
faces and the inverse of the proportionality constant is know as the capacitance. This leads to the
following expression for the energy stored in a capacitor:
1
V (t) = Q(t), (2.5)
C
1
E(t) = C(V (t))2, (2.6)
2
(Q(t)?Q(??))2
E(t) = . (2.7)
2C
2.1.2 Inductors
Similarly, an inductor is a piece of superconducting wire where the voltage across the wire is pro-
portional to the rate of change of current flowing through the wire and the proportionality constant
is known as the inductance of the wire. This leads to the following expression for the energy stored
in an inductor:
dI(t)
V (t) = L , (2.8)
dt
1
E(t) = L(I(t))2, (2.9)
2
(?(t)? ?(??))2
E(t) = . (2.10)
2L
2.1.3 Josephson junction
A Josephson junction is formed by a thin layer of an insulator sandwiched between two bulk su-
perconductors. This is a special element which introduces non-linearity and is responsible for all of
the interesting physics that can be studied using circuit QED. The origin of the non-linearity of this
element comes from the coherent tunneling of Cooper pairs across the thin insulting barrier which
12
Figure 2.2: (a) Josephson tunnel junction formed by sandwiching a weak link between two super-
conductors. (b) Effective circuit model of a Josephson junction as a non-linear element denoted by
the cross in parallel with its self capacitance. (c) Effective linearized circuit model of the Josephson
junction.
in turn is a result of the overlap of the ground state wavefunctions in the two superconducting elec-
trodes [60]. If we assume the number of Cooper pairs on the left electrode as NL and on the right
electrode as NR then the phenomenon of tunneling can be described by the following Hamiltonian:
1 ?
HJJ = ? EJ (|m??m+ 1|+ |m+ 1??m|) , (2.11)
2
m
here m denotes the Cooper pairs transferred across the tunnel junction and the parameter EJ de-
termines the rate of this tunneling. This Hamiltonian is identical to a one-dimensional tight binding
model and its eigenstates are plane waves given by:
?+?
|?? = eim?|m? , (2.12)
m=??
here ? = ka , a = 1 is the wavevector of the plane waves. Applying this ansatz to the tunneling
Hamiltonian we get the well known cosine dispersion of the eigenvalues:
HJJ |?? = ?EJ cos?|?? . (2.13)
We can now also defin?e the variable conjugate to the wavevector which is the total number of
Cooper pair tunneled n = m m|m??m|. In this representation ? can now be thought as the position
13
Figure 2.3: (a) Cosine potential of a large area Josephson junction where the dynamics of the phase
particle is confined to a single well. (b) Cosine potential of small area Josephson junction where the
tunneling of the phase particle across multiple wells is important.
14
of a fictional particle in a cosine potential and the number of Copper pairs as the momentum of this
particle.
We are now ready to derive the phase current relationship of the Josephson junctions. For this
we consider a voltage biased junction whose Hamiltonian is given by:
HJJ = ?EJ cos?? 2eV n. (2.14)
By writing the equations of motion for the variables ? and n we arrive at:
?n HJJ
= = (E /?) sin? , (2.15)
?t ? J??
??? HJJ
= = 2eV.
?t ?n
The first equation of motion is the DC Josephson relation which gives the current phase relation of
the Josephson junction as [61]:
I = Ic sin? , (2.16)
here I = 2ec ? EJ , is the maximum possible dissipationless current which can flow through the tunnel
junction. The second equation of motion is the AC Josephson relation which gives the AC response
of the junction to a DC bias voltage:
2e
?(t) = ?(0) + V t , (2.17)
?
I(t) = Ic sin(?(0) + ?t) , (2.18)
where, ? = 2eV/? is the frequency of the AC current.
This is not the complete picture of a Josephson junction since every physical Josephson junction
also comes with a corresponding self capacitance formed by the two superconducting electrodes sep-
arated by a dielectric. Including the effect of the capacitor, the Hamiltonian of an isolated Josephson
15
junction can then be written as:
HJJ = 4E n
2
C ? EJ cos?, (2.19)
here EC is the energy associated with the self capacitance. In the phase particle analogy the self
capacitance takes the role of mass of the fictional phase particle. Depending on the relative values of
the Josephson (EJ) and charging (EC) energies the dynamics of the phase particle can be broadly
divided into two categories (see Fig 2.2, 2.3). This first is when the ratio EJ/EC ? 1, in this case
the phase particle is confined to a single Josephson well and the cosine potential can be approxi-
mated to its lowest order. In this case the Josephson junction acts similar to an LC oscillator with an
effective linearized Josephson inductance given by:
( ? )2 1
LJ = (2.20)
2e EJ
?
=
2eIc
These types of junctions in the thesis will be refereed to as linear junctions. The second case is when
EC ? EJ then the dynamics of the phase particle involves tunneling across multiple cosine well and
in this case in general the full cosine term of the potential is important.
2.2 Linear superconducting circuits: Microwave photons
Now, that we have finished describing the basic building blocks, we can now start combining these
elements like Lego blocks to construct more complex composite circuits. In this section we will
describe the construction of linear superconducting circuits that play the role of light in the light
matter coupling studied by circuit Quantum Electrodynamics (cQED).
2.2.1 Lumped element L-C oscillator
The simplest composite circuit that can be constructed using the linear elements is that of a parallel
combination of inductor and a capacitor which forms a lumped element LC resonator. The dynamics
16
of this circuit can be described by a single variable the flux across the inductor or its conjugate
variable the charge on the capacitor. Using the current phase relations developed in the last section
we can get the energy stored in the inductor and capacitor which gives us the following Lagrangian
for the circuit:
C??2 ?2
L = ? , (2.21)
2 2L
and the corresponding Euler-Lagrange equations of motion:
? ??? = , (2.22)
LC
?
where the natural frequency of oscillation of the circuit is given by ? = 1/ LC. The charge across
the capacitor can then be represented as :
?L
Q = = C??. (2.23)
???
Applying the Legendre transform to the Lagrangian gives us the Hamiltonian of the circuit:
Q2 ?2
H = Q??? L = + . (2.24)
2C 2L
To write the quantum Hamiltonain of this circuit we promote the charge and flux variables to quan-
tum operators obeying the commutation relation [??, Q?] = i?. For convenience we introduce new
dimensionless variables of the flux and charge where the flux is normalized by the superconducting
flux quantum ? = ?/(?/2e) and the charge is normalized by the charge of a Cooper pair n = Q/2e.
Going forward in this thesis the circuit Hamiltonains will be written in terms of these new variables
and their corresponding quantum operators which satisfy the commutation relation [??, n?] = i.
The Hamiltonian of the lumped element L-C oscillator can now be written also in terms of anni-
hilation and creation operators defined as:
?
?? = ? ?ZPF (a?+ a? ),
2e (2.25)
n? = i2e nZPF (a?? a??),
17
? 1H? = ??(a? a?+ ) (2.26)
2
where ?ZPF and nZPF are the zero point fluctuations of the phase and Cooper pair number variables
given by: ?
Z
?ZPF = ? ,?RQ (2.27)
RQ
n
? ZPF
= .
4?Z
In these expressions Z = L/C is the impedance of the L-C circuit and the constant R 2Q = h/4e is
the superconducting resistance quantum.
2.2.2 Standing wave modes on a transmission line
Most circuit QED experiments only involve a single mode of a superconducting resonator where a
simple lumped element model can be used however for this work, we specifically study multi-mode
circuit QED where multi-mode structure of the device plays an important role. In our case this multi-
mode structure is due to the standing wave modes on a two-wire superconducting transmission line
resonator. In this section, we present the standard approach to finding the normal modes of such a
distributed resonator, first using a classical description of the circuit.
Here we consider the specific case of a simple one dimensional two-wire telegraph style trans-
mission line (see Fig. 2.4 (a)). Fig 2.4 (b) shows the equivalent lumped element circuit mode for
such a transmission line, where the distributed system is represented as a ladder of lumped element
inductances l ? dx in series and capacitances c ? dx to ground. Using the method prescribed in
section 2.2.1 we define the reduced node fluxes associated to each node as ?j and we can write the
Lagrangian of the system:
( ? )2 N??1 [1 2 ? (? 2 ]j+1 ? ?j)L = c?? dx , (2.28)
2e 2 j 2ldx
j=0
18
Figure 2.4: (a) Circuit schematic of a distributed 1-D transmission line resonator. (b) Lumped
element circuit mode showing the capacitance and inductace per unit length and the reduced node
fluxes at each node.
19
and the corresponding Hamiltonian as:
N??1 [ n2 ( )2
2 j ? (?j+1 ? ? 2
]
j)
H = 4e + . (2.29)
2cdx 2e 2ldx
j=0
By considering the continuum limit of this Hamiltonian when the unit cell size of the lumped model
is taken to zero such that Ndx = ? we get the following:
? ? [ n(x)2 ( ? )2 (? ?(x))2 ]x
H = dx 4e2 + , (2.30)
0 2c 2e 2l
where we have used the fact that ?x?(x) = limdx?0(?j+1 ? ?j) and we have defined the Cooper
pair density as n(x) = nj/dx. Writing the Hamilton?s equation of motion for Eq 2.30, we arrive at
the wave propagation equation for the transmission line as
2
2 ? ?(x, t) ? ?
2?(x, t)
v = 0, (2.31)
?x2 ?t2
?
where v = 1/ lc, is the speed of light in the transmission line. We can solve the wave equation by
using the method of separation of variables by using the ansatz:
?(x, t) = M(x)N(t), (2.32)
we arrive that the equations N?(t) = ??2N(t) and M?(x) = ?k2M(x), with ? being the frequency of
the standing wave modes and k being the wavevector. The spatial and temporal mode profile of the
standing wave modes can be written explicitly as:
N(t) = A cos(?t) +B sin(?t),
(2.33)
M(x) = C cos(kx) +D sin(kx),
and the constants A,B,C, and D are set by the boundary conditions of the transmission line at
x = 0 and x = ?. For the case of a open circuit boundary condition on both ends we can arrive at
the result where the wavevector is quantized as kj = j?/? and the corresponding frequencies for the
standing wave modes as ?j = j?0, where the ?0/2? is the frequency of the fundamental mode of
20
Figure 2.5: (a) Foster?s decomposition of an arbitrary admittance function Y (?) (b) Foster?s decom-
position of an open section of a 1-D transmission line resonator of length ?. The exact values of the
capcitances and the inductances in the decomposition is given by Eq. 2.37
the transmission line. By following the quantization procedure for a Harmonic oscillator discussed
in section 2.2.1 we arrive at the Hamiltonian of the transmission line written as:
??
H? = ?? (a??
1
j j a?j + ) (2.34)2
j=0
2.2.3 Foster?s decomposition of transmission line
A more convenient harmonic mode decomposition of the standing wave modes on a transmission
line can be found by using the Foster?s theorem. Foster?s theorem [62] states that any physical
admittance function Y (?) can be represented by an infinite number of series LjCj-circuits connected
in parallel, as shown in Fig. 2.5 (a). The admittance Y F (?) of such lumped element circuit is given
by: ??
F i?CjY (?) = . (2.35)
(1? ?jLjCj)j=1
To find the values of Lj and Cj , it is sufficient to match the poles and the residues of the two
analytical functions Y (?) and Y F (?). In this section we will first derive the Foster?s decomposition
for the simplest possible case of an open section of a 1-D transmission line of length ?, characterized
21
by the speed of light v and the wave impedance Z?. The admittance of such a transmission line of
length ? is given by [63]:
i
Y (?) = tan(??/v) . (2.36)
Z?
Matching the poles and residues of functions given by Eq. (2.36) and Eq. (2.35) results in the
equally spaced poles at frequencies ?j = 2??(j ? 1/2), where the free spectral range is ? = v/2?.
The inductances and capacitances read:
Z?
Lj = L1 = , (2.37)
4?
1 ? 1Cj = .
Z??2? (j ? 1/2)2
We can generalize the Foster?s decomposition method for a line with an arbitrary wave dispersion
k(?), frequency-dependent wave impedance Z?(?) and an arbitrary reactive termination, charac-
terized by the phase shift ?(?). The admittance of such a line is given by:
1 (e?i?(?) ? e?2ik(?)?)
Y (?) = ? ? , (2.38)Z?(?) e i?(?) + e 2ik(?)?
whose poles ?j are implicitly defined by:
k(?j)? = ?(j ? 1/2) + ?(?j) . (2.39)
In general, Eq. (2.39) produces non-uniformly spaced modes. Nevertheless, the parameters Lj and
Cj can still be expressed in a compact form:
?
Z?(?j) ??k
L = ?j ?? , (2.40)2 ?? ?=?j
2 ?? ??
Cj =
?2
? . (2.41)
jZ?(?j) ??k ?=?j
22
Figure 2.6: (a) Circuit schematic in the most general form of a charge qubit (b) Wavefunction and
energy of the first four eigenstates of a transmon qubit. (c, d) Energy levels of a charge qubit as a
function of the the ng for two different ratios of EJ/EC .
2.3 Non-linear superconducting circuits: Artificial atoms
In this section we will describe the construction of non-linear superconducting circuits that play the
role of artificial atoms. There is an entire periodic table [57] of superconducting artificial atoms
but for this brief introduction we will focus on two widely used circuits with distinct topology. In
the first example a single Josephson junction is connected to two isolated superconducting islands
and in the second one we consider a Josephson junction shunted by a very large kinetic inductance
forming a loop.
23
2.3.1 Charge qubits
Charge qubits are the simplest circuits (in terms of circuit complexity) that can be operated as a
qubit. They essentially only consist of a single superconducting island connected to ground by a
Josephson junction and the charge on the island is biased by a voltage source. This voltage source
induces an offset charge on the island given by 2eng = VgCg. We will work with the dimensionless
variable which converts the induced charge to the number of Cooper pairs. The junction has a self
capacitance CJ and can also be shunted by an additional capacitance CS . The Hamiltonian of such
a circuit is governed by the superconducting phase across the junction ??J and its conjugate variable
the Cooper pair number operator n?J . The Hamiltonian can be written as:
H? = 4EC(n?? n )2g ? EJ cos(??J), (2.42)
here EC is the total charging energy of the island given by EC = e2/2(CS + CJ + Cg). Note
that this Hamiltonain is exactly the same as that of a Josephson junction but now with the Coulomb
interaction associated with the capacitance taken into account. The Hamiltonian describes a fictional
phase particle in a periodic cosine potential where the Josephson energy defines the amplitude of
the cosine potential and the charging energy defines the inverse of the mass of this particle. This
Hamiltonian can be exactly solved and the solutions are given by the well known Mathieu functions
[64]. The charge qubits can be broadly classified into two categories based on the relative value of
the ratio between the Josephson tunneling energy and the charging energy EJ/EC . We will first
discuss the simple case when EJ/EC ? 1 and then proceed to describe what happens when the
EJ/EC is decreased.
2.3.1.1 Transmon
When the Josephson junction in Fig 2.6 (a) is shunted by a large capacitance, we can think of this as
a scenario where the tunneling potential barrier for the fictional phase particle is enhanced and the
mass is increased. In this scenario the tunneling between the adjacent well is strongly suppressed
and the phase particle wave function is strongly localized in one cosine well (Fig. 2.6 (b)) with
small phase excursions around the minimum. We can also estimate using the WKB approximation
24
the charge dispersion of the energy levels of this Hamiltonian. The band structure for the mth energy
level of the transmon can be written as Em(ng) = Em(ng = 0.25)+ 12?m cos(2?ng), where the charge
dispersion ?m is given by [64]:
?
24m+5 2( E )m/2+3/4 ?
?m ? J(?1)mEC exp(? 8EJ/EC). (2.43)
m! ? 2EC
This expression shows that the energy levels of the transmon become exponentially insensitive to
the charge offests as the ratio EJ/EC is increased. This justifies the Taylor series expansion of the
cosine potential which leads to the following transmon Hamiltonian:
( 1 1 )
H?transmon = 4E
2 2 4 6
C n?J ? EJ ? ??J + ??J +O(??J) . (2.44)2 24
In this approximation the transmon can be thought of as a quantum harmonic oscillator with a weak
Kerr type non-linearity.
EC
H?transmon = ?? ? ?ega? a?? a? a?a??a?, (2.45)
2
?
where the transition energy between the lowest two states is given by ?eg = 8EJEC ? EC . For
the experiments described in chapter 6, we incorporate flux tunability in the transmon by replacing
the Josephson junction with a SQuID loop. The SQuID loop can be thought as a single Josephson
junction whose Josephson energy depends on the flux threading the loop given by:
?
EJ(?ext) = (EJ + E 2
2
1 J ) cos(?ext/2) 1 + d tan (?ext/2), (2.46)2
here EJ , EJ are the Josephson energies of the two junctions in the SQuID loop and the parameter1 2
d = |EJ1 ? EJ2 |/(EJ1 + EJ2) is dependent on the asymmetry of the two junctions.
2.3.2 Flux qubits
Flux qubits consist of a different topology of circuits where the Josephson junction is shunted by
an inductor forming a superconducting loop. The change in topology of the circuit has profound
consequences on the mathematical description. The difference originates from the fact that now
25
the potential landscape for the phase across the Josephson junction is no longer a periodic function
and the phase is not defined on a circle ?j ? (0 ? 2?) but on a real line. Instead of periodic
boundary conditions imposed on the wavefunction we can now impose the boundary condition of
the wavefunction going to zero as ?J ? ?. Including the energy of an inductor to the Hamiltonian
of a Josephson junction we arrive at the Hamiltonian for a flux qubit:
1
H? = 4E (n? ? n )2 + E ??2C J g L J ? EJ cos(??J ? ?ext). (2.47)2
Because the charge is now a continuous variable we expect on physical grounds that a static offset
charge ng can be completely screened and should not affect the energy. Mathematically this can be
seen as performing a unitary gauge transformation and physically this can be thought of as a DC
short that can screen all the offset charge to the ground. Before we proceed to describe the spectrum
of flux qubit, we should note that there is an additional control knob in the circuit which comes from
the superconducting flux quantization condition and this can be thought of as the flux qubit analog
of the offset charge variable ng. As with the case of offset charge, we see that the resulting spectrum
must be periodic in the offset flux.
Given our ability to engineer EJ , EL, EC and the external flux ?ext threading the superconduct-
ing loop we can create a wide variety of potential landscapes and as a by-product a wide variety of
level structures in these qubits. Some of the most famous incarnations of the flux qubits are three
junction flux qubit [65], C-shunt flux qubit [66], fluxonium [67]. Transition energy and wavefunc-
tions for flux qubit in the three junction flux qubit regime and fluxonium regime are plotted in Fig.
2.7. In this work we work in the fluxonium regime of the flux qubit which is described next.
2.3.2.1 Fluxonium qubit
The fluxonium consists of a large inductance (and correspondingly small EL ? 1 GHz) that shunts
a Josephson junction with a EJ/EC ? 1. The large inductance of the fluxonium is supplied by the
kinetic inductance of a Josephson junction array. The Hamiltonian is given by:
H? = 4E n?2
1
C J + EL??
2
J ? EJ cos(??J ? ?ext). (2.48)2
26
Figure 2.7: (a) Circuit schematic in the most general form of a flux qubit (b,c) Circuit schematic for
three junction flux qubit (b) and fluxonium qubit (c). (d-f) Transition energy spectrum, Wavefunc-
tion and energy of the first four eigenstates of a three junction flux qubit. (g - i) Same but for flux-
onium qubit. Flxuonium parameters used in generating the plots - (EJ = 8.12 GHz, EC = 5.6 GHz,
EL = 1.42GHz ). FLux qubit parameters used in generating the plots - (EJ = 120GHz, EC = 6GHz,
EL = 60 GHz )
27
The fluxonium circuit displays rich physics and has a lot of appealing properties which make it
a leading qubit for the purposes of quantum computing but for the purposes of the experiments
described in chapter 5, its three most important properties are its large anharmonicity, symmetry
breaking of its potential and high coherence. All three of these properties are due to the inductive
energy term in the Hamiltonian. First, the inductive energy term creates an effective double well
potential which can be made asymmetric in ? by the application of a external magnetic flux though
the qubit loop (see Fig. 2.7 (g -i)) which is a crucial element for experiments described in chapter
5. This also makes the fluxonium transition spectrum highly anharmonic and allows us to truncate it
into an effective two level system. Lastly, the two orders of magnitude higher inductance compared
to a three junction flux qubit enables fluxonium to retain a reasonably high coherence time of 5 ?s
even when biased away from the half flux sweet spot [68, 69], where the potential displays its
asymmetry in ?. The above mentioned reasons make fluxonium the ideal choice of to study the
quantum impurity effects in multi-mode cQED as described in chapter 5.
2.4 Coupling between linear and non-linear circuits
In this section we derive the interaction Hamiltonian for circuits where we couple the artificial
atoms to the microwave photons. We study in depth two specific multi-mode circuit QED Hamilto-
nians where an open section of a 1-D transmission line is coupled galvanically to a transmon and a
fluxonium atom. Our goal is to construct the interacting circuit Hamiltonian in the form:
H? = H?atom + H?modes + H?int, (2.49)
where the first two terms describe respectively an atom and a set of non-interacting bosonic modes
hosted by the transmission line, while the third term describes their interaction.
2.4.1 Charge qubit coupled to multi-mode transmission line resonator
Here we derive the circuit Hamiltonian of a transmon circuit galvanically coupled to an open section
of a 1-D transmission line. The equivalent circuit schematic and circuit model is shown in Fig. 2.8.
28
Figure 2.8: (a) Circuit Schematic of a transmon galvanically cooupled to an open section of a 1-D
transmission line resonator (b, c) Equivalent circuit model in the flux (charge) gauge where the
transmission line resonator is replaced by the Foster?s decomposition.
29
Figure 2.9: Couplings constants calculated in the flux gauge (a) and charge gauge (b) for transmon
parameters EC = 0.35 GHz and EJ = 35 GHz coupled to a transmission line with Z? = 10 K? and
speed of light 3? 108 m/s.
Note that we have two possible choices for the reduced node fluxes for this circuit, we can either use
the reduced flux across the capacitors Ci as the node fluxes or we can use the reduced flux across
the inductors Li as the node fluxes. The two choices result in different Hamiltonian but in the end
they both describe the same physical system. We call the first choice of variables as the flux gauge
since in this description of the circuit the interaction between the transmon and the standing wave
modes of the transmission line is mediated by the flux variable and the latter choice as charge gauge
since in this cause the interaction is mediated by the charge variable.
2.4.1.1 Flux gauge
For the choice of the node fluxes shown in Fig. 2.8 (b), we can write the classical circuit Lagrangian
L = T ? V with T and V given by:
?
1 ?2 1T = C 2J ??J + Ci??i , (2.50)2
i?2=1?
? (?i ? ?J)
2
V = EJ cos(?J) + .
2L
i=1 i
30
In order to get the Hamiltonian of this circuit, we define the conjugate momenta (charges) to the
generalized fluxes shown in Fig. 2.8 (b):
?L
qi = = Ci??i, (2.51)
???i
?L
qJ = = Ci??J ,
???J
We can arrive at the Hamiltonian by applying Legendre transform and quantizing the classical oper-
ators using previously described methods as:
1
H? 2 2atom = 4EC n?J + EL??J ? EJ cos(??J), (2.52)2
??
H? = ?? a??modes i i a?i, (2.53)
i=1
??
(f)
H?int = hgi (a?
?
i + a?i ), (2.54)
i=?1 ?
where EC = e2/2C 2J and EL = (?/2e) ? i 1/Li and ?i = 1/ LiCi. Unfortunately, in the
flux gauge the separation of Hamiltonian (2.49) into the three terms has a number of disturbing
properties. To start, the value of the coupling constant (f)g grows with i approximately as i1/2i (see
Fig. 2.9 (a)). Therefore, truncating the model to a finite number of low-energy modes cannot be
readily justified. In addition, the transmon Hamiltonian has an additional renormalized inductive
energy EL that also depends on the total number of modes used in the Foster?s decomposition of
the transmission line (N). In fact, the parameter EL diverges as EL ? ? for N ? +?. Such a
behavior of EL is problematic for interpreting the dynamics of the renormalized atom?s Hamiltonian.
We show in the next section that all these problems in the interpretation of the Hamiltonian can be
solved by making a gauge transformation.
31
2.4.1.2 Charge gauge
We can now follow the same procedure as in the last section but for the choice of variables shown
in Fig. 2.8 (c). The circuit lagrangian L = T ? V is then given by:
??1 1
T = C 2J ??J + Ci(??i ? ?? 2J) (2.55)2 2
i=1
?? ?2
V = ?EJ cos(?J) + i (2.56)
2L
i=1 i
In order to get the Hamiltonian of this circuit, we define the conjugate momenta (charges) to the
generalized node fluxes in Fig. 2.8 (c):
?L ??
qJ = = CJ ??J + Ci(??J ? ??i),
???J i=1 (2.57)
?L
qi = = Ci(??i ? ??J)
???i
By applying the Legendre transform to the Hamiltonian we can get the following Hamiltonian:
?? ??
H? = 4E (n? ? n? )2C J i ?
1
EJ cos(??J) + 4E n?
2 + E ??2C i L i . (2.58)i 2 i
i=1 i=1
Note ?that we can also arrive at exactly the same Hamiltonian by applying the unitary trasformation
U = ei??J n?ii to the Hamiltonian in the flux gauge. We can write the Hamiltonian in the form of
Eq. 2.49 to get:
H?atom = 4E n?
2
C J ? EJ cos(??J) (2.59)
??
? (c)H?modes = ?i a?
?
i a?i (2.60)
i=1
??
H?int = ? (c)in?J hgi (a?i ? a?
?
i ) (2.61)
i
32
Note that in this gauge, all of the disturbing properties of the flux gauge vanish. First, the atom part
of the Hamiltonian exactly matches the disconnected transmon Hamiltonian. Second, the coupling
?
between the transmon and the standing wave modes of the transmission line scales as (c)gi ? 1/ i,
such that (c)gi ? 0 as i ? ?, this property allows us to truncate the high frequency modes for
numerical calculations.
2.4.2 Flux qubit coupled to multi-mode transmission line resonator
In this section we derive the circuit Hamiltonian of a flux qubit galvanically coupled to an open
section of a 1-D transmission line (see Fig. 2.10 (a)). The coupling between the flux qubit and the
transmission line is quantitatively characterized by x the ratio of the qubit loop inductance shared
with the transmission line. Here we derive the circuit Hamiltonian for our multi-mode cavity QED
system in the mixed gauge, whose variable choice is shown in Fig. 2.10 (d). Note that the choice
of the parameter i0 interpolates between the pure charge gauge (i0 = 0) and the pure flux gauge
(i0 = N). The variable choices for these specific cases are shown in Fig 2.10 (b, c).
2.4.2.1 Derivation of the circuit Hamiltonian
We link the dependent variable ?M to the independent variables ?i with i = 1, 2, . . . N . Using the
current conservation at node M , marked by the gray circle in Fig. 2.10 (d), we get:
?i0 ?i? ? 0 ?M M ?M ?J ?i ?i
+ + = + ? . (2.62)
xL (1? x)L Li (1? x)L L Li=1 i=1 i i>i i0
We can now write the circuit?s classical Lagrangian L = T ? V in terms of the generalized fluxes
?i = ?i(?/2e) and ?J = ?J(?/2e). The kinetic and potential energy contributions to the Lagrangian
are given by:
1 T
T = ?? C?? , (2.63)
2
1
V = ?TL?1?? EJ cos(?J ? ?ext) , (2.64)
2
33
Figure 2.10: (a) Circuit Schematic of a fluxonium galvanically cooupled to an open section of a
1-D transmission line resonator (b, c) Equivalent circuit model in the flux (charge) gauge where the
transmission line resonator is replaced by the Foster?s decomposition. (d) Equivalent circuit model
in the hybrid gauge where we interpolated continuously between the pure flux and charge gauges.
34
Figure 2.11: Couplings constants calculated in the flux gauge (a), charge gauge (b) and mixed gauge
(c, d) for fluxonium (EC = 5.6 GHz, EJ = 8.12 GHz and EL = 1.42 GHz) coupled to a transmission
line cavity of length 6 mm with Z? = 10 k? and speed of light 3 ? 108 m/s with varying geometric
factors.
35
where we have defined the vector
? ? (?0 = ?J , ?1, ?2, . . . ?N ) . (2.65)
The symmetric capacitance matrix C and the symmetric inductance matrix L have the following
matrix elements:
?
C 20,0 = CJ + x? Cj , (2.66)
j>i0
L? ?
C0,0<i?i0 = ?x? Cj , (2.67)Li j>i0
L? ?
C0,i>i = ?x?Ci ? x? Cj , (2.68)0 Li j>i0
L?
Ci?=0,j=? 0 = Ci?i,j + Ci?[i??i0 ? 1] (2.69)LjL? L? L?
+ Cj?[j ? i0 ? 1] + Ck ,
Li Li Lj
k>i0
(1? x?)
(L?1)0,0 = , (2.70)
(1? x)L
(L?1 x?)0,0<i?i = , (2.71)0 Li
(L?1)0,i>i = 0 , (2.72)0
36
(L?1 1)i?=0,j ?=0 = ?i,j (2.73)
L? Li
+ ?[i? i0 ? 1]?[j ? i0 ? 1] ,
LiLj
where the variables L? and x? are given by:
( )
? i ?11 1 ?0 1L = + + , (2.74)
xL (1? x)L L
i=1 i
L?
x? = . (2.75)
(1? x)L
Note that ? is the discrete Heaviside step function. Note that for the case of pure charge gauge,
L? = x(1? x)L and x? ? x.
In order to derive the Hamiltonian of this circuit, we define the conjugate momenta (charges) to
the generalized coordinates shown in Fig. 2.10 (c):
?L
q = = C?? . (2.76)
???
The Hamiltonian can be obtained by applying a Legendre transform to the Lagrangian:
H = qT ??? 1L = qTC?1 1q + ?TL?1?? EJ cos(?J ? ?ext) . (2.77)
2 2
The circuit Hamiltonian can be written as a sum of three parts. The atomic part of the Hamiltonian
reads:
2 1Hatom = 4ECnJ + E?L?
2
J ? EJ cos(?J ? ?ext) , (2.78)2
where EC = e2(C?1)0,0/2 and E?L = (?/2e)2(L?1)0,0, which formally depends on the (0, 0) element
of the inverse capacitance and inductance matrices. Remarkably, it can be proved analytically that
(C?1)0,0 = 1/CJ , irrespective of choice of i0 and (L?1)0,0 = 1/L for i0 = 0. This is relevant, because
it shows that the atom Hamiltonian defined in Eq. (2.78) is not renormalized by the transmission
37
line parameters and depends only on the bare fluxonium parameters for the pure charge gauge.
Whereas in the mixed gauge the atom inductive energy is renormalized and is a function of both x
and i0. The worst case scenario occurs for the flux gauge where the inductive energy of the atom
now depends on the number of modes included in the lumped element Foster?s decomposition of
the transmission line. The renormalization of the inductive energy as a function of the total number
of modes included in the flux gauge is shown in Fig 2.12.
The photonic part of the Hamiltonian, which is quadratic in the charge and flux variables, reads:
?N N1 1 ?
H = (C?1) ?1modes i,j qiqj + (L )i,j?i?j . (2.79)
2 2
i,j=1 i,j=1
Finally, the interaction between the fluxonium degrees of freedom and the transmission line
modes is given by: ?N ?N
H = q (C?1int J ) ?10,iqi ? ?J (L )0,i?i . (2.80)
i=1 i=1
2.4.2.2 Renormalization due to the diamagnetic terms
In order to quantize the classical Hamiltonian we introduce photon annihilation (creation) operators
a?i(a?
?
i ) as: ?
?
q?i = i (a?i ? a??i ) , (2.81)2RQ
?
?RQ
??i = (a?
?
i + a?i ) , (2.82)2
where R 2Q = h/4e is the superconducting resistance quantum. Note that any other resistance value
could be used as the dummy operators a? (a??i i ) merely define a starting basis for the cavity modes.
The operator H?modes can be diagonalized via a Bogoliubov transformation, because it is a quadratic
function of the photon operators. To proceed, we introduce a vector of annihilation operators a?:
a?T ? (a?1, a?2 . . . a?N ) . (2.83)
38
Figure 2.12: Relative renormalization of the inductive energy of the fluxonium atom in the flux
gauge as a function of the total number N of modes included in the Foster?s decomposition of the
transmission line. Same parameters as in Fig. 2.11. The legend indicates the values of the geometric
fraction x.
39
The Hamiltonian Hmodes can be rewritten in the following matrix form:? ? ? ?
1 a? b?
H? = (a??,T T
? ? 1 ?,T T ? ?
modes , a? )?M? ? = (b? , b? )???
2 ? 2 ?
? , (2.84)
a? b?
where ? is the diagonal 2N ? 2N matrix such that ?ii = 1 for i ? N and ?ii = ?1 for i > N . The
other 2N ? 2N matrix to deal with is:
? ?
X Y
M ?= ? ?? . (2.85)
?Y ?X
The symmetric N ?N matrices X and Y are defined as:
X = R (L?1i,j Q ) ?1i,j +RQ (C
?1)i,j , (2.86)
Y = R (L?1i,j Q ) ?1 ?1i,j ?RQ (C )i,j , (2.87)
where i, j ? {1, 2, ..., N}. Finally the diagonal matrix ? = diag(?1, ?2, .., ?N ,??1,??2, ...,??N )
contains the frequencies of the normal Bogoliubov modes. It is defined as M = Q?Q?1, so that the
Bogoliubov mode operators are:
?? ?? ?? ?? b? ? a?1 ??? = Q ? ? , (2.88)
b? a??
where the matrix Q can be expressed in the form :
Q ?
? ?
? V U= ?? . (2.89)
?U ?V
The Hamiltonian H? ?modes can be diagonalized in terms of bosonic normal-mode operators b?i (b?i ):
?N
(m)
H?modes = ?? b??i i b?i , (2.90)
i=1
40
where (m)?i are the corresponding normal-mode frequencies.
The coupling of the atom to the transmission line is represented by the interaction Hamiltonian:
?N4
H?int = in?J EC (C?1CJ)0,i(a? ? a??i i ) (2.91)?1/2? i=1N
? EL?? ?1J (L L)0,i(a? + a??i i ) .4?3/2
i=1
In order to express Hint in terms of the normal-mode transmission line operators b?? , we need to
invert the Bogoliubov transformation as follows:
?N ?N
a?i = V
?
i,? b?? + Ui,? b?? . (2.92)
?=1 ?=1
Note that the conservation of the bosonic commutation rules, i.e., [b? , b
?
?? ] = ??,?? , imposes the
following normalization condition for the Bogoliubov transformation coefficients:
?N
U2i,? ? V 2i,? = 1 . (2.93)
i=1
2.4.2.3 The mixed gauge QED Hamiltonian
We have now all the ingredients to formulate the multi-mode cavity QED Hamiltonian in terms of
the fluxonium operators ??J , n?J , and the transmission line normal mode operators b?i (b?
?
i ). The total
Hamiltonian is H = Hatom + Hmodes + Hint, where Hatom is given by Eq. (2.78), Hmodes is given by
Eq. (2.90) and Hint can be expressed as:
?N
(m,c)
H?int = in? hg (b? ? b??J i i i ) (2.94)
? i=1N
? (m,f)? ?J hgi (b?i ? b?i ) ,
i=1
41
with the mixed gauge coupling constants given by:
N
(m,c) 4 ?
hg ?1i = EC (C CJ)0,j(V1/2 j,i ? Uj,i) , (2.95)?
j=1
?N
(m,f) 1
hgi = E (L
?1
L L)3/2 0,j(Vj,i + Uj,i) . (2.96)4?
j=1
Note that for the special case of i0 = 0 (charge gauge) the flux coupling constants are identically
zero and for the case of i0 = N (flux gauge) the charge coupling constants are identically zero.
Moreover, the flux coupling constants in the mixed gauge (m,f)gi for i < i0 can be written analytically
as:
?
(m,f) ? (m)
RQ
hgi = ?i x . (2.97)4?Z?(i+ 1/2)
42
Chapter 3
Experimental Techniques
In this chapter we summarize the main experimental methods used throughout this thesis.
3.1 Nanofabrication
Fabrication of all the devices described in this work was done following well established techniques
evolved from more than two decades of extensive research on superconducting circuits [70]. All
our devices are fabricated on high resistivity Si of thickness 380 ?m. The Josephson junctions in
our devices are Al-Alox-Al tunnel junctions. These use aluminium oxide as the insulating layer in
the tunnel junction. We choose this because the growth of aluminum oxide is unique in that it is
self-terminating and thus has reliable thickness. This is important as the critical current and thus
the Josephson energy (EJ) is exponentially sensitive to the width of the insulating layer. In addition
there has been much study in use of these junction and methods have been found to grow low loss
junctions [71].
To fabricate the Josephson junctions in the device we used the Dolan bridge technique. Also
known as the shadow evaporation technique, this fabrication method allows small, overlapping
structures to be fabricated with very high accuracy and reproducibility. [72]. We use a chip size of
9 ?m ? 16 ?m to pattern our devices. A double layer resist of MMA (500 nm) and PMMA (100 nm)
43
is applied before using a Elionix electron bean lithography to pattern the device features. After
development of the patterns, Al deposition and growth of the Alox was performed using a Plassys
e-beam evaporation system. We used the double angle method where we deposited 20 nm of Al at
a rate of 1 nm/sec. The Alox growth was done at 100 mBar pressure for 20 mins. The second layer
of 40 nm Al deposition was done before a final oxidation at 10 mBar to give a capping layer for a
consistent top surface of the junctions.
3.2 Microwave reflectometry setup
3.2.1 Cryogenic setup
Our experiments were performed in a Blue Fors dilution refrigerator with a base temperature of 7?
15mK. We used a bottom loader dilution refregirator which allowed us to exchange samples without
warming up the whole unit in a relatively fast time scale of 7Hours. This was crucial in being able to
cycle through a lot of devices. The input line was attenuated by 60dBm to suppress the thermal noise
coming in from higher temperature stages of the refrigerator. The placement of the attenuators was
guided by the cooling power at different stages (see Fig. 3.1). Since the measurements were done
in a reflection geometry, we used a Pasternack directional coupler (2?20GHz) to physically separate
the input and output signals. Note that the input signal was attenuated by an additional 16 dBm
becacuse of the directional coupler. The reflected signal was directed to a Caltech HEMT ( High
Electron Mobility Transistor) cryogenic amplifier with a 35 dB gain and 5 K noise temperature in a
broad frequency range of 4 ? 12 GHz. Two Quinstar isolators (4 ? 12 GHz) were connected before
the amplifier to guard the sample from any spurious microwaves travelling back on the output
lines especially from the HEMT amplifier. All the components were thermally anchored to their
respective stages. An in house built 3.6 cm radius superconducting solenoid coil was attached to the
cold finger of the refrigerator to provide uniform perpendicular magnetic field used to flux bias the
superconducting loops of the SQuID transmon and fluxonium qubits. The coil was connected to a
44
45
Figure 3.1: The full microwave reflectometery setup used in the experiments and described in detail
in section 3.2.
Figure 3.2: (a) Optical image of the 3-D copper waveguide sample holder. The image shows the
placement of a sample silicon chip in the box and the launcher used to couple microwaves from
coaxial cable to the chip. (b) Transmission signal measured when the waveduide box is empty at
room temperature.
Yokogawa GS 200 DC current source at room temperature using a 30 ? twisted pair wire. The 3-D
waveguide sample holder was bolted to the cold finger of the dilution refrigerator and a Cryoperm
cylinder was used to shield the flux sensitive devices from stray magnetic fields.
3.2.2 3-D waveguide sample holder
The devices were placed in a 3-D waveguide sample holder for measurements. The waveguide
consists of a 9 mm ? 27 mm ? 25 mm copper box with two SMA ports. One port was permanently
connected to a cryogenic 50 ? termination. The other port was connected to a coax to waveguide
launcher used to guide microwaves travelling on the coaxial lines into the copper waveguide. The
launcher and cooper box dimension were chosen such that we have a relatively uniform coupling
in the frequency range of 5 ? 10 GHz. We would like to stress at this point that we did not use
any wire-bonding to couple microwaves into our devices. This setup minimizes the typical parasitic
capacitances due to the measurement setup seen by the devices and helps to get a clean spectroscopy
signal.
46
3.2.3 Room temperature setup
All measurements described in this work are done with continuous wave RF signals, we were able
to get by with a relatively simple room temperature electronics setup. The main equipment in our
setup was the 4 ? Port ZNB20 Rohde and Schwarz VNA (Vector Network Analyzer). The ability of
the VNA to perform fast measurement and demodulation of reflection signals was vital in obtaining
the spectroscopy results presented in chapters 4 ? 6. For two-tone experiments the additional tone
was combined using a directional coupler identical to the one used in the cryogenic setup.
47
Chapter 4
Multi-mode Fabry-Perot cavity
In this chapter we describe the experimental implementation and characterization of multi-mode
Fabry-Perot cavities made using a long array of Josephson junctions. We present reflection spec-
troscopy measurements of such Fabry-Perot cavities. Using these measurements we extract the dis-
persion of the standing wave modes and their intrinsic lifetime.
4.1 Josephson junction transmission line
Our transmission line consists of two parallel chains of large area Josephson junctions separated by
2 ? 10 ?m fabricated on an insulating silicon chip. The junctions are Al/Alox Josephson junctions
fabricated using the Dolan bridge method described in detail in section 3.1. Typical junction dimen-
sions used are 1? 3 ?m? 0.4 ?m which translates to EJ/EC ? 100. This safely places the junctions
in the linear regime where the Josephson junction can be considered an oscillator with a linearized
inductance given by Eq. 2.20. The inductance per junction in the transmission lines is of the order
of a few nH, which makes it approximately 104 times larger than a wire of similar dimensions. An
optical micrograph of a typical transmission line fabricated along with the SEM micrograph of the
Josephson junctions is shown in Fig. 4.1 (a), (b) respectively. The minimal circuit model of our
transmission line (see 4.1 (d)) includes two linear chains of Josephson junctions with energy EJ
48
Figure 4.1: (a) Optical image of a section of high-impedance transmission line made of array
of Josephson junctions. (b) Schematic of an array of Josephson junctions. (c) Scanning electron
micrograph of an array of Josephson junctions. (d) Exact Circuit model of the transmission line. (e)
Circuit model of the transmission line in the linear approximation.
49
in parallel with the Josephson capacitance CJ coupled with the inter-chain capacitance Cg and con-
nected to ground by the capacitance C?g . The unit cell size of the array of Josephson junctions is
a = 0.6 ?m.
4.1.1 Transmission line Lagrangian
We follow the procedure outlined in section 2.2.2 to arrive at the circuit Lagrangian of the trans-
mission line for the choice of reduced node flux variables shown in Fig 4.1 (d).
L = T ? V, (4.1)
where the kinetic energy is given by:
? (?? 2? ?,i ? ???,i?1) (?? 2? ?,i ? ???,i?1) (???,i ? ? 2 2 2?? ?,i) ??? ?,i ???,i
T = + + + + , (4.2)
2EC 2E ? ?i i C 2Eg 2E 2Ei i g i g
and the potential energy is given by:
?( )
V = ? EJ cos (??,i ? ??,i?1) + EJ cos (??,i ? ??,i?1) . (4.3)
i
It is convenient to work in the continuum limit and write the Lagrangian of the system as:
? [ (? ?? (x, t))2x ? a (?x???(x, t))2a (???(x, t)? ?? 2?(x, t))
L = dx + + + (4.4)
2EC 2EC 2Ega
??2?(x, t) ??
2
?(x, t) E
]
J
? + ? + [cos(?x??a) + cos(?x??a)] .2Ega 2Ega a
We define new variables ?+(x, t) = ??(x, t) + ??(x, t) and ??(x, t) = ??(x, t) ? ??(x, t) to get the
following Lagrangian:
? [ (?x???(x, t))2a (?x??+(x, t))2a (???(x, t))2( 1 1 ) (? ?? (x, t))2x +
L = dx + + + + (4.5)
4EC 4EC 2a E ?g 2Eg] 4E?ga2EJ 1 1
+ [cos( ?x??(x, t)a)? cos( ?x?+(x, t)a)] .
a 2 2
50
Since we work with large area Josephson junctions in this chapter such that the EJ/EC of each
junction is ? 1. Each individual junction in our transmission line can be thought of as a harmonic
oscillator. As was discussed in section 2.1.3 it is safe to approximate the cosine potential to its
lowest order to give us a quadratic Lagrangian. The Lagrangian can be separated into two parts as:
? [ ( L = L?)+ L+, ] (4.6)(???(x, t))2 1 1 (?x???(x, t))2a EJa
L? = dx? [ + ? + ? (? ?
2
x ?) ,
2a Eg 2Eg 4EC 4
(?x??+(x, t))
2 (?x??+(x, t))
2a E a ]? JL+ = dx ? + (?x?+)2 .4Ega 4EC 4
We now write the Euler Lagrange equations of motion for the two Lagrangians to get the wave
equation satisfied by the waves propagating in this transmission line:
a
?2
EJa 1
x?
2
t ??(x, t) + ?
2
x??(x, t)? ?2t ??(x, t) = 0, (4.7)2EC 2 E?ga
a 2 2 EJa 1?x?t ?+(x, t) + ?
2? 2x +(x, t)? ? ?t ?+(x, t) = 0,2EC 2 Ega
here we have defined the effective ground charging energy for the differential mode as E??1g =
E?1g + (2E
?)?1g . The equations in Eq. 4.7 define two separate modes of propagation of the plane
waves in the transmission line each corresponding to the oscillations of the voltage and current
defined by the superconducting phase density ?+(x, t) (common mode) and ??(x, t) (differential
mode). The common mode corresponds to the voltage oscillating in phase in the upper and lower
chains while the differential mode corresponds to the voltage oscillating out of phase in the upper
and lower chains.
4.1.2 Multi-mode Fabry-Perot cavity
We construct our Fabry-Perot cavity by simply taking a finite section (here 10mm long) of the two
parallel chains of Josephson junctions and short circuiting it on the right end and connecting a
dipole antenna on the left end. This dipole antenna is also used to couple radiation in and out of this
Fabry-Perot cavity (see Fig. 4.2). This finite length Fabry-Perot now quantizes the phase oscillations
51
Figure 4.2: (a) Optical image of a multi-mode Fabry-Perot cavity made by terminating a high-
impedance transmission line on the right by a short circuit and on the left by a weakly coupled
antenna port. (b,c) Circuit schematic and lumped element circuit model of the same.
52
defined by Eq. 4.7 into standing wave modes and we can arrive at the dispersion relation of these
collective modes of the phase field in the transmission line by imposing the boundary conditions
corresponding to the two ends. To solve this boundary value problem for the wave equations we
use the method of separation of variables discussed in section 2.2.2 and write the phase field in
the transmission line as ??(+)(x, t) = M?(+)(x)N?(+)(t). For the differential mode the boundary
conditions define a zero voltage at the right end at x = ? and zero current at x = 0 and for the
common mode the boundary condition corresponds to zero current at both ends. This directly leads
us to the differential equation of the space and time parts of the phase oscillations:
d2N?(+)(t)
+ ?2
dt2 ?(+)
N?(+)(t) = 0, (4.8)
d2M?(+)(x)
+ k2?(+)M?(+)(x) = 0,dx2
with boundary conditions:
dM?(x = 0)
= 0, (4.9)
dx
M?(x = ?) = 0,
dM+(x = 0)
= 0,
dx
dM+(x = ?)
= 0,
dx
This leads to the quantization of the wave vector for the two sets of modes in the Fabry-Perot cavity:
? i?
k?,i = + , (4.10)
2? ?
i?
k+,i = .
?
This finally leads us to the dispersion relations for the differential and common standing wave modes
in the Fabry-Perot cavity.
?
?a 12EJ E?gk???(k?) = , (4.11)
1 + a2k2
E?g
? 2EC
53
??a EJE?gk+?+(k+) = . (4.12)
E?
1 + a2k2 g+EC
These dispersion relations can be further simplified into a two parameter form that reads:
? v?k???(k?) = , (4.13)
1 + (v?k?/?p)2
? v+k+?+(k+) = . (4.14)
1 + (v k 2+ +/?p)
These equations define two sets of modes with acoustic dispersion in the low frequency with a speed
of propagation given by:
v? = ? a , (4.15)
2LJCg
a
v+ = ? ? , (4.16)LJCg
?
and a ultraviolet propagation cutoff given by ?p = 1/ LJCJ , the self resonance frequency of the
junctions f?orming the Fabry-Perot cavity. The wave impedance of the transmission line is given
by Z? = 2LJ/Cg. By changing the area of the Josephson junctions, the distance between the
Josephson junction chains and length we can design Fabry-Perot cavities with a wide ranges of
impedance, speed of light and FSR (free spectral range).
4.1.3 Interactions between standing wave modes
The interactions between the standing wave modes of our cavity are as a result of the non-linearity
of the Josephson junctions forming the cavity. In the last section we ignored all the non-quadratic
terms in the cosine potential of the junctions to arrive at the dispersion relation of the cavity. To
derive the interactions between the modes of the cavity we add the 4th order term in the cosine
potential as a perturbation given by:
54
?2E
(
J 1 1
)
V = (? ? (x, t))2(? ? (x, t))2a4 + (? ? (x, t))4 4
1
x + x ? x ? a + (?
4 4
x?+(x, t)) a . (4.17)
a 64 96 96
This perturbation introduces a negative self-Kerr non-linearity in all the standing wave modes
of the cavity. In addition, the perturbation is also responsible for an all to all dispersive cross-Kerr
interaction among the standing wave modes of the cavity. By quantizing the phase density and
promoting the phase and the conjugate charge to quantum operators we arrive at the quantum
Hamiltonian of the cavity:
?
H? = ?? a??
?
i i a? + ??
? ?
i i,j a?i a?ia?j a?j . (4.18)
i>0 i,j>0
The second term in the Hamiltonian gives the interaction between modes. For i = j it is the self Kerr
interaction and for i ?= j it is the cross Kerr interaction. The exact matrix elements can be calculated
from the lumped element model of the cavity and the detailed procedure is given in ref [73].
4.2 Control Experiments
In this section we describe the first experiment that we perform to confirm the functioning of the
Fabry-Perot cavity. Before we start the measurements we install the chip containing the Fabry-Perot
cavity in a 3-D copper waveguide box with a single broadband microwave input/output port see
(Fig 3.2 (a). The details of the 3-D waveguide design are presented in section 3.2.1. The simplest
experiment we perform on the device is to measure the reflection coefficient of continuous tone
microwaves generated by a Rhode and Swartz VNA. To launch microwaves we use a coaxial-to-
waveguide launcher with good matching in the range 5 ? 10 GHz (see section 3.2 (b)). A typical
trace reveals an ordered set of discrete resonances, which we associate with the standing wave
modes of the Fabry-Perot (see Fig. 4.3). One important point to note here is that the geometry
of the coaxial cable to waveguide launcher in our experiment is chosen such that the microwave
signals can only couple to the differential modes in our device and not the common modes. For
this reason, moving forward we will ignore the effect of the common modes in the system and
55
Figure 4.3: Reflection magnitude and phase response of a typical multi-mode Fabry-Perot made
using a array of Josephson junctions. The insets shows the change in the mode spacing as the
mode frequency increases because of the ultra-violet cutoff at the self resonance frequency of the
Josephson junctions.
56
concentrate only on the differential modes. By measuring the spectral distance between adjacent
resonances and knowing the length ? of the Fabry-Perot cavity we estimate the speed of light in the
cavity v = (?i+1 ??i)/(2??) ? 1.75? 106 m/s ? c/172. This measurement confirms that indeed our
transmission line acts like a 1-D Fabry-Perot cavity where the light is slowed down by more than a
factor of 100. Another important thing to note in this measurement is the vanishing of the magnitude
and phase response of the cavity resonances as the frequency goes below 4 GHz and above 9 GHz.
This is primarily due to the cutoff of the 3-D waveguide used to house the chip which is at ? 7GHz.
4.3 Characterization Experiments
In this section we describe more extensive and detailed experiments that are performed to accurately
characterize our multi-mode Fabry-Perot cavity.
4.3.1 Intrinsic lifetime of modes
In addition to revealing the positions of the standing wave modes in our Fabry-Perot cavity we use
the frequency dependent reflection coefficient measurement in the vicinity of each resonance to
characterize the lifetime of the modes. We fit the data to the well known expression:
?N 2i(? ? ?i)/? ? (Qexti i )?1 + (Qint)?1S (?) = i11 , (4.19)
2i(? ? ?i)/? + (Qext)?1 + (Qint ?1i=1 i i i )
where ?i is the resonance frequency of the ith Fabry-Perot standing wave mode,
,int(ext)
Qi is the
internal (external) quality factor of the resonance. Qint parameterizes the loss of energy in the
standing wave mode inside the transmission line device. Whereas, Qext can be viewed as a measure
of opacity of the mirror at the antenna end of the chain. We found that, at frequencies below 7GHz,
the opacity grows on reducing the frequency, which is consistent with the propagation cutoff of
our copper waveguide. The opacity also has a tendency to increase as the frequency grows above
10GHz, which may be related to a partial Anderson localization of microwaves in the chain and their
decoupling from the antenna end [74]. As a result, one-tone spectroscopy becomes inefficient far
outside the 5?10GHz pass band of our coaxial-to-waveguide launching system. For this reason, the
57
Figure 4.4: One tone spectroscopy overlayed with the fit to Eq. 4.19 used to extract the internal
and external quality factor of the standing wave modes of the Fabry-Perot cavity.
58
Figure 4.5: Inernal (left panel) and external (right panel) quality factor of the standing wave modes
of all the Fabry-Perot cavities plotted against the dimensionless parameter ?/?p
59
frequency range in Fig. 4.3 is limited. Finally, we note that our reflection data fit exceptionally well
to the above expression for S11(?), as shown in the Fig 4.4, which allows very accurate extraction
of both and int(ext)?i Qi .
We are also able to explain the intrinsic loss of energy in our standing wave modes. The limiting
mechanism of loss in our device is due to the absorption of photons in the dielectric of the Josephson
junctions. We model this loss by adding a small complex part to the junction capacitance CJ ?
CJ(1 + i tan ?). Using this we are able to arrive at the expression for the internal quality factor of
the standing wave modes:
2 (? )( (int p ? ? )2)pQ (?) = 1 , (4.20)
tan ? ? ?
here tan ? is the dielectric loss tangent. The internal quality factors data acquired across multiple
devices fit very well to this model with a dielectric loss tangent of tan ? = 10?3 (see Fig. 4.5).
This value is consistent with other experiments done using Al/Alox Josepshon junctions [75]. This
knowledge of the intrinsic loss mechanism in isolated Fabry-Perot modes will be crucial in latter
chapters when we introduce impurity in the Fabry-Perot cavity.
4.3.2 Two tone spectroscopy
To further characterize the cavity and to extract the cavity mode frequencies beyond 7 ? 12 GHz
range, we use a two tone spectroscopy technique where we exploit the weak cross-Kerr interaction
between the standing wave modes of the cavity. First, the readout mode is selected in the pass band.
Reflection amplitude and phase at a properly chosen frequency near the resonance are measured as
a function of the frequency of the second tone, which is scanned to look for other modes. The cross-
Kerr effect results in the shifting of the frequency of the readout mode due to the photon occupation
of every other mode in the system (see Fig. 4.6). Using this technique it is possible reconstruct
the full dispersion relation of the standing wave modes in the cavity and index the modes (see Fig.
4.7). This dispersion is in excellent agreement with the simple two parameter expression derived
in section 4.1.1. We fit the extracted mode frequencies to obtain the speed of light in the Fabry-
Perot cavity and the ultra-violet cutoff which corresponds to the self resonance frequency of the
junctions forming the transmission line. Fig. 4.7 (a) clearly shows the i = 1 mode of the cavity
60
Figure 4.6: (a) The read-out mode while the probe tone is in (red) and out off resonance (blue)
with the standing wave mode at 215 MHz. (b) The read-out tone reflection magnitude measured at
the left shoulder of read-out mode line as a function of the probe tone frequency.
at 40 MHz, at approximately half the mode spacing, which correctly reflects the open and short
circuit boundary conditions of the cavity mathematically expressed in Eq. 4.1.2. This observation
confirms that wave propagation occurs along the entire length of the system and the spectrum is
gapless. Fluctuations of the mode spacing as a function of the mode index are found to be within
a few percent, independently of the wavenumber (Fig. 4.7 (c)), thus showing minimal disorder of
junction parameters along the entire length of the system.
4.3.3 Extracting Impedance
By fitting the measured dispersion relation of standing wave modes in the cavity we get an accurate
estimate of the speed of light and the plasma frequency cutoff. However, our Fabry-Perot cavity is
characterized completely by 3 independent circuit parameters CJ , LJ , and Cg.?From the measure-?
ment of the speed of light and plasma?cutoff we have access to 1/ LJCJ and 1/ LJCg. In order to
estimate the wave impedance Z? = LJ/Cg we use two separate methods. The first one exploits
the fact that the Al oxide growth is typically self-terminating. It implies that the Josephson junction
capacitance is fully defined by the junction area. We use SEM to measure the junction areas along
the chain. The CJ is found then as an average junction area multiplied by an empirical constant
61
Figure 4.7: (a) The magnitude of reflection coefficient as a function of the probe frequency. Discrete
standing wave resonances are indexed one by one starting from the very first mode at about 40MHz.
The third resonance is a spurious mode and is discarded. (b) The reconstructed dispersion relation
(blue markers) and theoretical fit to Eq. 4.13 (solid line). (c) The mode spacing as a function
of the mode index. The sharp periodic outliers originate from the stitching error of the e-beam
lithographer, otherwise invisible in device images.
62
Figure 4.8: Reconstructed dispersion relation of all the multi-mode Fabry-Perot cavities. Parameters
are listen in Table. 4.1
63
45fF/m2. The second method of finding the wave impedance takes the known geometrical dimen-
sions of the chains and uses the well-known result for the capacitance of the coplanar stripes in order
to calculate Cg. These methods give the impedance values which are different from each other by
no more than 20 % (Table. 4.1). This number can be considered as our accuracy for an absolute
impedance measurement. We take the average of these two values as our estimate for Z?. Their
?
difference divided by 2 gives the impedance uncertainty.
4.4 Summary of measurements and conclusion
In summary, the reflection spectroscopy experiments on our devices confirm that we are able to
engineer on chip multi-mode Fabry-Perot cavities. Exploiting the kinetic inductance of the Josephson
junctions we are able to achieve two orders of magnitude reduction in the speed of light and an
almost two orders of magnitude increase in the wave impedance of the cavity. This allows us to have
a compact massively multi-mode cavity which can host ? 100 standing wave modes in the frequency
range of 0 ? 10 GHz. Furthermore, by measuring the quality factor of the modes we confirm that
the photons in the standing wave modes are well isolated and have line widths ?/? ? 100. This is
an important ingredient in exploring many-body effects in the system, since we want the intrinsic
dissipation of the modes to be much smaller than the strength of the many-body effects.
4.5 Summary of device parameters
Device v/106(m/s) ?p/2?(GHz) Z?(k?) EJ/EC Length (mm)
a 1.87? 0.05 24.9? 0.02 12.1? 0.6 87? 9 10
b 22.6? 0.06 26.7? 0.05 0.7? 0.05 729? 23 10
d 8.2? 0.01 21.5? 0.03 2.2? 0.1 224? 18 10
e 2.11? 0.03 20.9? 0.02 7.3? 0.4 448? 52 10
Table 4.1: Device parameters obtained from fitting the measured dispersion relation to theory.
64
Chapter 5
Fluxonium impurity coupled to
multi-mode cavity
In this chapter we describe the experimental implementation and characterization of multi-mode
cavity QED system where we couple a fluxonium artificial atom to the multi-mode Fabry-Perot cavity
introduced in the last chapter. We characterize the effect of the fluxonium impurity on the Fabry-
Perot cavity by measuring the frequency of the standing wave modes of the cavity as a function of the
external flux threading the fluxonium atom. We derive an effective three wave mixing Hamiltonian
which can split a single excitation in our cavity into many low frequency excitations but fails due to
many-body localization. Through spectroscopy measurements we show signatures of the many-body
localization of photons which manifests itself as the splitting of standing wave resonance into a fine
structure of multi-particle resonances.
5.1 Design of the device
The system we study in this chapter consists of a two-wire transmission line of length ? = 6 mm,
characterized by its wave impedance Z? and speed of light v, as shown in Fig. 5.1 (a). This trans-
mission line is terminated on one end by a superconducting loop made up of an array of Josephson
junctions including one weak junction that forms the fluxonium artificial atom. The fluxonium is
65
Figure 5.1: Optical micrograph (a) and circuit schematic (b) of the system showing a two-wire
transmission line terminated by a superconducting fluxonium atom. The transmission line is char-
acterized by its wave impedance Z?, speed of light v, and length ?. The fluxonium is defined by the
Josephson energy EJ , the charging energy EC of the small junction and the loop inductance L. A
fraction x (shown as region 2 on optical image) of the fluxonium loop inductance is shared with the
transmission line, coupling galvanically the atom to the transmission line?s standing wave modes.
The atom transition frequency can be tuned by changing the external flux bias (?/2e)?ext threading
the superconducting loop.
66
galvanically connected to the transmission line such that the fluxonium loop junction array shares a
fraction x (0 < x ? 1) of the junctions with the transmission line. These shared junctions are respon-
sible for the coupling of the standing wave modes of the transmission line to the fluxonium, thus
creating an interacting multi-mode circuit QED system. For given fluxonium and transmission line
parameters, the coupling depends on the geometric factor x. The energy levels of the fluxonium can
be controlled by varying an external magnetic flux (?/2e)?ext pierced through the fluxonium loop.
In this chapter we primarily discuss the measurement results from a device which shares x = 0.5
of its inductance with the transmission line. In addition to this a number of samples with varying
geometric factor x were characterized and we summarize the results from those measurements in
the last section.
5.2 Microscopic Hamiltonian
The system described in this work can be modeled as a fluxonium coupled inductively to a high-
impedance transmission line cavity. With careful selection of the gauge we can arrive at the exact
microscopic Hamiltonian of our system (see section 2.4.2). The Hamiltonian consists of three terms,
the first two terms define the bare fluxonium qubit and the bare non-interacting bosonic modes of
the transmission line cavity. The third term defines the coupling between the two
H? = H?qubit + H?modes + H?int. (5.1)
Explicitly the three terms are defined as:
H? 2
1 2
qubit/h = 4EC n?J + EL??J ? EJ cos(??J ? ?ext) , (5.2)2
??
(m)
H?modes/h = fi b?
?
i b?i , (5.3)
i=1
?? ??
(m,f) (m,c)
H?int/h = ???J gi (b?i + b?
?
i ) + in?J gi (b?
?
i ? b?i) , (5.4)
i=1 i=1
67
Figure 5.2: (a) Sketch depicting the frequency and wavefunctions of the cavity modes with free
spectral range ?. The columns show a subset of multi-particle states nearly resonant and coupled
with the single-particle state occupying the mode k. (b) Schematic representation of connectivity in
the many-body Fock space with a tree diagram whose nodes depict the many-body states and the
edges connect the states that have a direct matrix element mediated by the three wave mixing term
V? .
here the Hamiltonian parameters are related to an exact circuit model. The corresponding circuit
model and detailed procedure to arrive at the specific Hamiltonian parameters can be found in
section 2.4.2.
5.3 Effective Hamiltonian
In this section, we derive an effective Hamiltonian treatment that can be applied to our system when
the coupling between the standing wave modes of the cavity and the fluxonium is such that ? ? feg,
where feg is the frequency of the fluxonium transition from the ground state to the first excited state
and ? is its radiative width into a semi-infinite transmission line (where the standing-wave modes
form a continuum). In doing so, we only consider the transition from the ground state |g? to the
first excited state |e?. Within such a two-level approximation, the fluxonium atom Hamiltonian can
be approximated as:
Hqubit/h = feg(?ext)|e??e|. (5.5)
The flux-dependent transition frequency feg in Eq. (5.5) is calculated by a numerical diagonal-
ization of the full fluxonium Hamiltonian given in Eq. (5.2). We also consider the Hamiltonian of
our system in the mixed gauge, where we recall that the first i0 modes of the cavity are coupled to
68
the atom predominantly via the flux variable and the higher i > i0 cavity modes are coupled pre-
dominantly via the charge variable. This is convenient not only for the behavior of the qubit-cavity
coupling constants at low and high frequencies, but also because for a strongly anharmonic qubit
the two-level approximation is excellent in such a gauge, as verified with brute force numerics in the
Appendix A.
We start by dividing the Fabry-Perot standing wave modes into two sets. The first set includes
modes with frequencies much smaller than the fluxonium qubit transition frequency at ?ext = ?
(mode index i such that 1 ? i ? i0). The second set of modes includes those that quasi-resonantly
hybridize with the qubit transition. We will denote k the index of such modes where i0 < k ? N .
Note here that N is the high frequency cutoff chosen such that fN ? feg which ensures that modes
above mode number N also do not hybridize with the qubit transition. In the first step, we ignore for
the time being the coupling of the fluxonium to the low-frequency bosonic modes of the transmission
line and diagonalize the following Hamiltonian
?N ? ?
(m) (m,f) (m,c)
H?0 = ?? ? ? ?eg|e??e|+ ??i b?i b?i ? ??J hgi (b?i + b?i ) + in?J hgi (b?i ? b?i ) , (5.6)
i=1 i>i0 i>i0
in the single-excitation subspace spanned by the basis states:
{ }
B0 = |e?|0?, |g?b??k|0? . (5.7)
1?k?N
Note that the single-excitation eigenstates of the Hamiltonian (5.6) consist of bare transmission line
modes for k ? i0 with excitation frequency (m)?k /2?. For k > i0 they have the form:
a??
?
k|G? = W
?
k,0|e?|0?+ Wk,k? |g?b?k? |0? , (5.8)
k?
where fk = ?k/2? is the corresponding single-polariton excitation frequency. The term ?polariton?
state in used here in analogy to similar states appearing in semiconductor systems. The Hamiltonian
in Eq. (5.6) can now be rewritten as
69
?
H?0 = ??ia??i a?i . (5.9)
i>0
As a second step of our procedure, we now include the effect of the coupling of low-frequency
modes to the fluxonium in Eq. (5.9). The coupling term is given by
?i0 ?i0
(m,f) (m,c)
V? = ???J hgi (b?i + b?
?
i ) + in?J hgi (b?
?
i ? b?i ) . (5.10)
i=1 i=1
As it can be directly inspected, this term gives rise to an effective coupling between the bare transmis-
sion line modes and the polaritonic modes via a three-wave mixing process. Indeed, this elementary
process down converts one particle states a?? |0? into two-particle states a?? ?k k? a?i |0?, with excitations in
another polaritonic mode k? and one excitation in a low-frequency photon mode i. The correspond-
ing coupling is quantified by the following matrix element:
? | ? ? (m,f)G a? ?kV? a?k? a?i |G? = hgi Wk,0Wk?,0(?e|??J |e? ? ?g|??J |g?) . (5.11)
Likewise, each of these two-particle states can couple to a subset of three-particle states, which
retain the low-frequency photon in mode i from the previous generation, i.e. the low-frequency
photons are stable, and so forth. This finally leads to the effective Hamiltonian of our multi-mode
cQED system:
? i? ??i0 ?H? ? ?eff = ??ka?ka?k + hg iAk,k? a?ka?k? a?i + h.c. , (5.12)
k>0 k,k?>i0
where
A ?k,k? = Wk,0Wk?,0 , (5.13)
(m,f)
g = g1 (?e|??J |e? ? ?g|??J |g?) . (5.14)
Note that the matrix element Ak,k? is proportional to the qubit components in the polaritonic modes
participating in the three-wave-mixing process. via the factor W ?k,0Wk?,0. Moreover, the three-wave-
mixing interaction strength g is finite only when there is a breaking of the flux symmetry of the
70
Josephson atom potential energy (necessary to have ?e|??J |e? ? ?g|??J |g? ?= 0), which is achieved
away from ?ext = n?. In addition, the three-wave-mixing strength g is proportional to the coupling
?
of the fluxonium to the low frequency mode i. In the flux gauge, it is given by (m,f) (m,f)gi = g1 i, as
it can be seen from Eq. (2.97).
The effective Hamiltonian can now be used to construct the full spectrum of our system by an
iterative procedure that consists of enlarging the Hilbert space to include multi-particle states. Two-
excitation states can be accounted for by considering the subspace spanned by the following basis of
states: ?{ }? ? i0<k?NB1 = B0 a?ka?i |0? . (5.15)
1?i?i0
In this expanded Hilbert space, the single-polariton states are no longer eigenstates of the Hamil-
tonian. This is due to the fact that the single-polariton states now can hybridize with two-photon
states due to the photon-photon interactions mediated by the fluxonium. Given the large number of
frequency degeneracies between single-polariton states and two-photon states, determined by the
relation (m)?k = ?k? + ?i , the diagonalization of the effective Hamiltonian produces a myriad of
energy anticrossings in the spectrum. If we want to include the effect of three-photon states, we
need to enlarge the basis of the Hilbert space by considering the states:
? ?{ }i0<k,k??N
B2 = B0 B1 a?? a??k k? a?
?
i |0? . (5.16)
1?i?i0
This process of expanding the Hilbert space can be iterated to include n-photon states, where we
have one excitation in the polaritonic mode and n? 1 excitations in low-frequency photon modes.
5.4 Control experiments
In this section we describe two simple control experiments that are preformed to ensure that the two
main components namely the transmission line and the fluxonium atom are functioning as intended.
71
Figure 5.3: Reflection magnitude and phase as a function of CW microwave probe tone when there
is zero external flux threading the fluxonium loop.
72
5.4.1 Single tone
Here we repeat the most basic experiment that we performed on a Fabry-Perot cavity described in
the last chapter. We measure the reflection coefficient at the single port of the 3-D copper waveguide
as a function of the CW microwave tone when there is no magnetic flux threading the fluxonium
loop. The response reveals an almost equally spaced forest of resonances indistinguishable from
the response of the bare Fabry-Perot cavity shown in Fig 4.3 in chapter 4. By measuring the
distance between the consecutive resonances (?(7GHz) = 178MHz) and knowing the length of the
transmission line, we can estimate the speed of light as v ? 2? 106 m/s. The measurement of speed
of light and mode spacing confirms that the junctions of the transmission line are superconducting
and there is no break in the line and the microwave propagation happens along the full length of
the device.
5.4.2 Single tone Vs flux
In this next experiment we repeat the same reflection coefficient measurement but this time scanning
a smaller frequency range of 200 MHz around 7 GHz for different values of the DC current flowing
through the superconducting coil. The result of this measurement is displayed in Fig. 5.4 which
shows that the standing wave resonance of the cavity is modulated periodically as a function of
DC current flowing through the coil and the flux threading the fluxonium loop. This measurement
shows that the fluxonium loop is connected and is superconducting. In addition, we observe no
vacuum Rabi splittings in the spectroscopy data as the fluxonium frequency is tuned through 7GHz.
Instead we observe that multiple modes of the cavity shift simultaneously which proves we have
achieved the superstrong coupling regime of cavity QED.
5.5 Characterization experiments
Together these two experiments show that the transmission line and fluxonium are working as ex-
pected and now we can proceed to characterize the device in more detail.
73
Figure 5.4: Colorplot of reflection magnitude as a function of externally applied magnetic flux
threading the fluxonium loop. The signal is clearly flux periodic which shows that the fluxonium
loop is closed and superconducting. This periodicity of the reflection signal with coil current is used
to calibrate the magnetic flux threading the fluxonium loop.
74
Figure 5.5: Mode spectral spacing (difference between the frequencies of two consecutive modes)
versus frequency when the flux bias is ?ext/2? = 0 (red circles) and ?ext/2? = 0.5 (black circles).
The decrease of mode spacing at low frequency is due to the antenna capacitance, while at high
frequencies it occurs owing to the plasma cut-off frequency at 26.94 GHz. Note also the Lorentzian
dip in the mode spacing around the qubit transition frequency for the case of ?ext/2? = 0.5.
5.5.1 Bare Fabry-Perot cavity
In the experiment we can directly measure the bare transmission line frequencies (m)(fk ) by perform-
ing spectroscopy of the single particle excitation frequencies when the flux through the qubit loop
is set to an integer multiple of 2?. In this configuration the three wave mixing term induced by the
qubit goes to zero. Furthermore, the qubit transition frequency is tuned to its highest (feg > 15GHz)
and the transmission line modes below 10GHz are effectively decoupled from the qubit. The single-
particle energy spacing in our transmission line resonator in this case are fitted with high accuracy
by a simple two-parameter model, which has an ultra-violet cut-off given by the plasma frequency
75
of the Josephson junctions [42] given by:
v
?(f) = (1? (f/f )2)3/2p (5.17)
2?
where ? is the length of the transmission line and fp is the plasma frequency of the Josephson
junction making the transmission line. In addition to high frequency, the mode spacing is also
distorted in the very low frequency limit due to the antenna capacitance. The amplitude of the
random disorder fluctuations in the mode spacing is of the order of a few MHz, which is about 1%
of the average mode spacing. This is probably due to the area fluctuations in the array of Josephson
junctions making the slow-light Fabry-Perot resonator.
5.5.2 Fluxonium parameters
In the presence of the qubit, the energy spacing of the Fabry-Perot modes is distorted and displays a
dip, as shown in Fig. 5.5. In this case the mode spacing as a function of frequency is given by :
v 2 2 2 2
?(f) = (1? (f/f )2)3/2 4? (f ? feg) + ? ?p + (5.18)
2? ?
Here feg is the ground to excited state transition frequency of the qubit. When the qubit is
tuned to its minimum energy configuration corresponding to ?ext = ?, we fit the dip in the mode
spacing with a Lorentzian of width ? = 1.43 GHz [42]. This puts the device deep in the superstrong
coupling regime with ?/? ? 8. We also checked that the qubit width remains constant for the
frequency range of 5 GHz to 10 GHz. This distortion in the mode spacing is as a result of linear
hybridization of the qubit |e? ? |g? transition with the bare Fabry-Perot modes. This hybridization
of the flux dependent qubit transition with the cavity modes is responsible for the flux modulation
of the single-particle spectrum of our effective Fabry-Perot cavity. Since, the flux modulation of
the single-particle energy spectrum encodes information of the qubit, we use the experimentally
measured single-particle spectrum as a function of the flux threading the superconducting loop of
the qubit to extract the bare qubit parameters.
To accomplish this, we fit the single-particle mode frequencies with the energy eigenvalues of
the truncated Hamiltonian of Eq 5.1, restricting the Hilbert space to the single-excitation subspace
76
Figure 5.6: First column from the left: excitation frequency versus the applied flux bias for the
single-particle eigenstates (black solid lines). The filled black circles are the experimental data used
for the fitting procedure described in the text. Second column: energy of the uncoupled two-particle
states (red lines). Third column: same but for the three-particle states (green lines).
77
78
Figure 5.7: (a) Photon-photon interaction scale g as a function of the external flux. (b) Color
plot of the matrix elements Ak,k? versus the mode indices k, k? when the qubit frequency is tuned
feg = 7.2GHz (around mode index 39) at ?ext/2? = 0.36. Note that the width of the local maximum
in Ak,k? is given by ?/?.
(see Eq. 5.7). We use a standard least mean squares minimization algorithm to find the optimal
values (EC , EL, EJ). The data points used in the fitting algorithm are shown in Fig 5.6 (a) as filled
black circles and the corresponding calculated frequencies are represented by the soli?d black lines.
The Hamiltonian in the single-excitation subspace can then be written as H?0/h = k>0 fka?
?
ka?k,
where a??(a?k) denotes the normal mode creation (destruction) operator and fk is the corresponding
normal mode frequency, which is now a function of the flux threading the qubit loop. The hybrid
single-particle states can now be written as a linear superposition of the qubit excited state and
single photon state in one of the transmission line modes.
a?? ?k|G? = b?k|G? , k ? i0 (5.19)
a??
?
k|G? = Wk,0|e?|0?+ W
?
k,k? |g?b?k? |0? , k > i0 (5.20)
k?>i0
Note that the hybridization coefficient Wk,0 goes to zero rapidly as the detuning between the normal
mode frequency fk and the qubit transition frequency feg exceeds ? (see Fig 5.7(b)). Having the
single-particle energies, we can also generate the bare two and three-particle energy spectrum, as
shown in Fig 5.6 (b) and (c) respectively. Obtaining the bare Fabry-Perot and fluxonium parameters
also allows us to completely reconstruct the effective Hamiltonian parameters of Eq. 5.12. The
reconstructed three wave mixing strength g and the matrix elements Ak,k? are plotted in Fig. 5.7
(a), (b) respectively.
5.6 Reflection spectroscopy
In this section we report detailed single and two tone spectroscopy of the system by measuring the
reflection coefficient |S11(?)| of microwaves on the single port of the 3-D waveguide (see Chapter 3)
in which the chip containing the device is housed. Our experiment consists of measuring the reflec-
tion magnitude |S11| at the port end of the cavity, following previously established procedures [42].
The incident power excites the cavity with less than one photon, on average, and we checked that
the resulting spectra are power-independent.
79
Figure 5.8: (a) Linear (low-power limit) reflection spectroscopy signal |S11|(dB) of the multi-
mode circuit QED system as a function the external flux bias. The black-dashed lines show the fitted
|g? ? |e? transition frequency of the fluxonium atom, while the red-dotted lines depict the calculated
energy spectrum in the single-excitation manifold. (b) Closer look at the spectra that, depending on
the frequency, can display an additional fine structure with satellite resonances.
80
Figure 5.9: Reflection magnitude S11 (dB) near modes k = 35 and k = 37, data for ?ext/2? < 0.5,
theory in the two-particle approximation without adjustable parameters for ?ext/2? > 0.5. The red
lines show uncoupled two-particle energy levels taken directly from the lower-frequency data. The
insets show a line cut for an illustrative flux bias ?ext/2? = 0.356.
81
Figure 5.10: (a) Reflection magnitude |S11|(dB) near modes 49, 43 and 41. Insets (left to right):
(left) zoom on the spectroscopy data in the 100 MHz wide windows; (right) calculated spectrum,
no adjustable parameters. (b) Energies of all possible (left), with non-zero matrix element (right)
uncoupled single-particle (black), two-particle (red), three-particle (green), and four-particle (blue)
states in the 100 MHz window. (c) Examples of many-body Fock space trees describing the localiza-
tion dynamics at selected flux values, indicated by the vertical dashed line in the insets of (a). Note,
density of interconnected states is is much smaller than that of all multi-particle states.
5.7 Interpretation of spectroscopy: Many-body localization of
photons
Let us first discuss the coarse frequency resolution data (Fig. 5.8 (a)). It shows the expected sequence
of standing wave resonances for modes k = 25, 26, ..., 50, whose frequencies fk are modulated by the
magnetic flux ?ext due to the qubit admixture. The flux-modulation rapidly vanishes for modes with
k < 25. The measured values fk agree with a single-particle (linear) hybridization model detailed in
82
section 5.5.2 (Fig. 5.8 (a), dotted lines). Fitting this model provides reliable estimates of all circuit
parameters (see Table 5.1 in particular row with x = 0.5). For example, we obtain ? ? 1.4 GHz
(?/? ? 8) as well as the uncoupled qubit frequency (Fig. 5.8 (a), dashed line). The flux knob can
continuously modify our system, including switching off the interaction V? at ?ext = 0, ?. In addition
to adjusting the single-particle spectrum, we can maximize the interaction effects for a given mode k
by setting feg(?ext) ? fk ??. This condition graphically corresponds to the sloped region of fk(?ext)
in Fig. 5.8 (a).
A higher resolution data reveals dramatic deviations from the single-particle picture (Fig. 5.8
(b)). Namely, the cavity resonances with k > 30 anticross many new flux-dependent lines. The
splitting size varies from line to line, but remains much smaller than ?. The most unusual property
of these new spectral lines is that their density is much larger than 1/?, especially towards higher
frequencies. Indeed, where would so many states possibly come from in a cavity excited by at most
one photon? In fact, these are precisely the multi-particle states introduced in chapter 1. In fact,
one can use the fk-data to duly check that each new resonance around the modes k = 31 ? 39 in
Fig. 5.8 (b) satisfies the two-particle matching condition fk = fk? + fj , for selected integer pairs
k?, j < k. For k ? 30 there are no matching two-particle states in our system irrespective of the flux
bias.
Zooming in to several modes (k = 35, 37) we find a remarkably good agreement between data
and the two-particle spectrum of Hamiltonian (Eq. 5.12), produced without adjustable parameters
(Fig. 3). The matrix elements of V? were calculated using the circuit parameters obtained from
fitting the single-particle model( see section 5.5.2). The theoretical |S11|-signal in Fig. 5.9 involves
the two-particle wavefunctions as well as the values of the intrinsic and extrinsic quality factors of
the cavity resonances, which were measured at the non-interacting flux bias ?ext = 0. We stress that
linear spectroscopy only lights up states with a substantial single-particle component, which is why
the two-particle resonances appear as a ?fine structure? around the cavity mode frequencies.
The example spectra shown in Fig. 5.9 are inconsistent with the Fermi?s Golden rule decay rate
theory, but they have a straightforward interpretation in terms of localization in the many-body
Fock space. We focus on a specific realization of our system at ?ext/2? = 0.356 (Fig. 5.9 - insets).
Thus, mode k = 35 appears as a single resonance (as well as all modes k < 35) and it is accurately
83
described by a single-particle wavefunction, |??? ? a??35|0?. A photon excited in this mode does not
make any attempt to split because the nearest available two-particle states are far detuned compared
to the interaction scale. By contrast, a slightly higher frequency mode k = 37 fragments into at least
eight resolved resonances. Each of them is described by a many-body localized wavefunction, for
instance (as marked in Fig. 5.9 - insets), |??? ? 0.58 a?? ? ? ? ?37|0? + 0.102 a?25a?12|0? + 0.184 a?24a?13|0? +
0.512 a?? a??23 14|0? ? 0.53 a?
? ?
22a?15|0? ? 0.223 a?
?
21a?
?
16|0? ? 0.125 a?
? a??20 17|0?. This wavefunction consists of a
superposition of the original single-particle state, with a probability 0.582 ? 34%, and several nearly-
resonant two-particle states. One may say that a photon excited in mode k = 37 does make attempts
to split, e.g. into a pair of photons in modes 22, 15 or 23, 14. However, since these secondary photons
are stable (k < 35), the down-conversion dynamics localizes to a few two-particle sites in the Fock
space and hence remains coherent.
The extent of localized states rapidly grows for higher-frequency modes (Fig. 5.10 (a)). In partic-
ular, the data around modes k = 41, 43 contains extra-fine spectral features that cannot be explained
within the two-particle approximation but agree with the three-particle spectrum of Hamiltonian
(Eq. 5.12). We illustrate the composition of many-body resonances using the Fock-space hopping
trees. For clarity, we retain only the strongest branches, for which the hopping amplitude (the ma-
trix element of V? ) exceeds the energy mismatch. In the example of mode k = 41 (at ?ext = 0.39),
a single-particle state a??41|0? hybridizes predominantly with one two-particle state a?
?
35a?
?
6|0?, which
itself hybridizes with four three-particle states a?? a?? a??21 14 6|0?, a?
? ? ? ? ? ? ? ? ?
22a?13a?6|0?, a?23a?12a?6|0?, a?24a?11a?6|0?.
The tree terminates at those four three-particle states because the corresponding third-generation
photons are stable (see Fig. 5.9 (b)). Same analysis applies to the spectrum around mode k = 43
(?ext/2? = 0.367), which reveals not only three-particle states but also an increasing density of
two-particle states.
As the density of multi-particle states grows with energy, one may wonder if the Fermi?s Golden
rule recovers for higher-frequency modes. We note, though, that our system has a finite coupling
to an external bath, both due to leaking of photons into the measurement apparatus and materials
absorption. Therefore, as the localization extends to larger and larger number of multi-particle
states, their frequency spacing would eventually become comparable to their external linewidth,
which in our system is of order 1 MHz. In this case, the fine spectral structure would merge into
84
Figure 5.11: (a - c) Reflection spectroscopy signal |S11| (expressed in dB) near 7.5GHz mode of the
Fabry-Perot cavity for devices with geometric coupling ratio x = 0.31 (a), x = 0.5 (b) and x = 1 (c)
. (d - f) Intrinsic quality factor measured for the standing wave modes in the same three devices as
(a - c) for ?ext/2? = 0 and ?ext/2? = 0.5.
a single broad resonance, effectively recovering the applicability of the Fermi?s Golden rule in the
otherwise localized system. This fundamental effect limits our present resolution to two- and three-
particle states, as can be seen by the example of mode k = 49, where one expects four-particle states
to appear (Fig. 5.10).
5.8 Towards strongly correlated photons
In this section we describe the spectroscopy data for devices with geometric ratio x < 0.5 and
x > 0.5. We observe that as we decrease x, the width of the qubit transition decreases and the qubit
hybridizes with less number of modes ?/?. The direct consequence of this on the spectrum is the
decrease in the density of two-particle anticrossings. This is due to the fact that the matrix element
decreases more rapidly as the factor Ak,k? decreases faster as |k ? k?| grows. The exact opposite
happens for the case of x > 0.5, we observe an increased density of two-particle anticrossings. As
we continue increasing x, we observe that for x = 1, the spectrum changes qualitatively. Instead of
an increased density of multi-particle anti-crossings we observe that the standing wave resonance in
85
this case is completely washed away into what looks like a continuum. This is the regime where the
qubit width becomes comparable to ? ? feg and the effective Hamiltonian described in section 5.3
is no longer applicable. In this regime we can no longer make a two-band distinction between low
and high-frequency modes which brakes the strict selection rule that limit the connectivity in the
many-body fock space. Here the single particle state of the cavity becomes dressed in all possible
multi-photon states resonant with the single particle state creating a truly strongly correlated photon
state.
Additional evidence of this direct coupling of single-particle state to multi-particle states can be
seen by measuring the quality factor of the cavity standing wave resonances at the external flux
bias points of ?ext/2? = 0 and ?ext/2? = 0.5. At these specific flux bias points the three-wave
mixing non-linearity goes to zero. The main difference between the two flux bias points is the qubit
frequency. For ?ext/2? = 0 the qubit is tuned to its maximum frequency whereas for ?ext/2? = 0.5
it is tuned to its minimum frequency. We observe that when ? ? feg there is no difference in the
intrinsic quality factors for the two flux bias points however as the ? increases and approaches feg,
we observe a more than an order of magnitude drop in the intrinsic quality factor of resonances
(see Fig. 5.11). This drop in the quality factor is due to the de-localization of the single-particle
excitation into the many-body Fock space. A precise theoretical model of this decay of single particle
excitation is currently not available at the time of the writing of this work and is the subject of
ongoing theoretical and experimental work.
5.9 Conclusion
To summarize, we showed that spontaneous down-conversion of a single photon in the quantum im-
purity regime of cavity QED is a many-body phenomenon and it is subject to many-body localization
(MBL). Localization leads to a breakdown of Fermi?s Golden rule description of the down-conversion
rate, by analogy with the original prediction for a confined Fermi-liquid quasiparticle [43]. As a re-
sult, the standing wave modes fragment into a spectacular fine structure of multi-particle resonances,
which unambiguously distinguishes MBL from either the delocalization regime (single interaction-
broadened resonance) or the non-interacting photons (single sharp resonance).
86
5.10 Summary of device parameters
Device (x) v/106(m/s) Z?(k?) ?(MHz) ?(GHz) EJ(GHz) EC(GHz) EL(GHz)
0.07 2.18 9.33? 0.5 182 < 0.2 8.12? 0.05 5.69? 0.04 1.42? 0.01
0.3 1.82 10.05? 1.04 151 0.61? 0.03 2.62? 0.10 6.97? 0.2 1.12? 0.02
0.5 2.36 8.97? 0.67 197 1.43? 0.02 8.9? 0.6 8.0? 0.4 1.39? 0.05
0.65 2.03 9.62? 0.01 169 2.10? 0.1 3.62? 0.3 6.72? 0.2 1.04? 0.01
1 1.93 10.4? 0.4 161 - - - -
Table 5.1: Device parameters extracted by performing characterization experiments detailed in sec-
tion 6.5. Note that the length of all the devices presented in this table was 6 mm.
87
Chapter 6
Transmon impurity galvanically
coupled to multi-mode cavity
In this chapter we describe the experimental implementation and characterization of multi-mode
cavity QED system where we galvanically couple a flux tunable transmon qubit to the multi-mode
Fabry-Perot cavity introduced in the chapter 4. We study the elastic and inelastic scattering of
microwaves off the impurity. We characterize the elastic scattering by measuring the frequency shift
induced by the tranmson on the standing wave modes of the Fabry-Perot cavity. We characterize the
inelastic scattering by measuring the line width of the standing wave modes. Furthermore, we derive
an effective photon-photon coupling Hamiltonian for our system and explain the measurements
across multiple devices using the theory.
6.1 Design of the device
The system we study in this chapter consists of a two-wire transmission line of length ? = 6 mm
(except for device 0a, where ? = 12 mm), characterized by its wave impedance Z? and speed
of light v, identical to the one shown in Fig. 4.2 (a) but with a small modification at the right
end. Instead of a short circuit now we terminate this transmission line by a two junction SQuID
(Superconducting Qunatum Interference Device) which forms our flux tunable transmon artificial
88
Figure 6.1: Optical micrograph (a) and circuit schematic (b) of the system showing a two-wire
transmission line terminated by a superconducting transmon qubit. The transmission line is char-
acterized by its wave impedance Z?, speed of light v, and length ?. The transmon is defined by
the Josephson energy EJ , the charging energy EC of the small junction. The two wires of the
transmission line are terminated with the SQuID loop, coupling galvanically the qubit to the cavity?s
standing wave modes. The qubit transition frequency can be tuned by changing the external flux
bias (?/2e)?ext threading the SQuID loop.
89
atom. The SQuID geometry allows us to tune the effective Josephson energy EJ and through it the
transmon resonance frequency.
6.2 Microscopic Hamiltonian
The system described in this work can be modeled as a transmon qubit coupled galvanically to
a high-impedance transmission line cavity. By decomposing the extended transmission line into
its lumped element foster form and using the charge gauge we can arrive at the exact microscopic
Hamiltonian of our system (see section 2.4.2). The Hamiltonian consists of three terms, the first two
terms define the bare transmon qubit and the bare non-interacting bosonic modes of the Fabry-Perot
cavity. The third term defines the coupling between the two.
H? = H?qubit + H?modes + H?int, (6.1)
Explicitly the three terms are defined as:
H?qubit/h = 4EC n?
2
J ? EJ cos(??J), (6.2)
??
? (c)H?modes/ = ?i a?
?
i a?i, (6.3)
i=1
??
H?int/h = ? (c)in?J gi (a?i ? a?
?
i ), (6.4)
i=1
here the Hamiltonian parameters are related to an exact circuit model. The corresponding circuit
model and detailed procedure to arrive at the specific Hamiltonian parameters can be found in
section 2.4.1.2.
90
6.3 Effective Hamiltonian
In this section, we derive the effective Hamiltonian treatment that can be applied to the system in
two different parameter regimes. First we will consider the parameter regime where the EJ/EC ? 1
of the transmon qubit. In this case the transmon qubit can be modelled as an harmonic oscillator
with a perturbative Kerr type non-linearity. In the second case we will consider the regime where the
EJ/EC is decreased and a new type of non-linearity arises that of phase-slip across the transmon?s
Josephson junction.
6.3.1 Quartic non-linearity
As the transmon consists of an effective Josephson junction with EJ/EC ? 1, it can be simply
replaced by an harmonic oscillator with a resonance freque?ncy equal to the frequency of the lowest?
two energy levels ?eg = 8EJEC ?EC and impedance RQ 4EC/?2EJ with an effective Kerr non-
linearity equal to EC . In this case our system reduces to a system of coupled harmonic oscillators.
We can then write the effective Hamiltonian of the system as:
H? ? ?
?
2 ?
eff/? = ?ega?0a?0 ? EC(a?0a?0) + ?j a?( j
a?j (6.5)
? ? j>0g )2? j(a?0 ? a?? ? ?0) gj(a?j ? a?j ) + (a? + a? ) .1/2 j j
j>0 j>0 ?eg
This Hamiltonian is a textbook Caldeira-Leggett model of a quantum degree of freedom interacting
with an Ohmic bath [76]. The last term is the rigorously derived ?A2?-term of quantum optics. Note
that in our model g2j ? 1/j, which regularizes perturbative series automatically, without the need for
sophisticated analysis recently proposed for a capacitively connected transmon [19, 77]. We treat
the weak non-linearity of the transmon as a perturbation and diagonalize the remaining quadratic
terms of the effective Hamiltonian by a Bogoliubov transformation and arrive at the following final
result:
? ? ?
H?eff/? = ?j c??j c?j ? K (c?
?
j j c?j)
2 ? ? ? ?i,j c?i c?ic?j c?j , (6.6)
j>0 j>0 i,j
91
Figure 6.2: (a) Illustration showing inelastic scattering process that splits an incident single photon
at x = 0 in one resonant photon and even number of low-frequency photons. (b) Schematic showing
the coupling between many-body states induced by the transmon.
where
?? ?? ? ?? ?? a?? ?? u v???? c??= ? , (6.7)
a?? ?v u c??
and
Kj = EC(v
2 2 2
1,j ? u1,j) /?, (6.8)
? 2 2i,j = 4EC(v1,j ? u1,j)(v2 21,j ? u1,j)/?.
This effective Hamiltonian describes a new multi-mode Fabry-Perot cavity with flux dependent stand-
ing wave modes ?j , j = 1, 2, . . . with an additional photon-photon Kerr type interaction. The flux
dependence and non-linearity of the cavity are both acquired as a result of hybridization of the
transmon oscillator with the bare Fabry-Perot cavity modes.
92
6.3.2 Phase-slip non-linearity
As we decrease the EJ/EC of the transmon coupled to the cavity a new type of non-linearity appears
from the quantum phase-slip fluctuations across the transmon junction [78, 79]. Namely, tunneling
of the phase between the equivalent minima of the Josephson potential.
To derive the effective Hamiltonian we start with the microscopic Hamiltonian in the form given
in Eq. 2.58. We begin by dividing the transmission line modes into two sets following a procedure
similar to the one followed in section 5.3. The first set of modes consists of high-frequency modes
quasi resonant with the transmon and the low frequency modes far detuned from the transmon. This
two-band distinction is always possible as long as the ?eg ? ?. Here ? is the spontaneous emission
linewidth of the transmon into the bare cavity modes in the infinite length limit of the cavity. The
microscopic Hamiltonian can be re-written as:
? ?? ( 1 ) ?
H? = 4EC(n?J ? n?i)2 ? EJ cos(??J) + 4EC n?2i + E 2Li ??i ? 8EC n?J n?i. (6.9)i 2
i?i0 i=1 i>i0
In the first step, we ignore for the time being the effect of the low-frequency bosonic modes of
the transmission line on the transmon and diagonalize the following Hamiltonian
? ?
H?0/? = ?eg| (c) (c)e??e|+ ?i b?
?
i b?i ? in?J gi (b?i ? b?
?
i ). (6.10)
i>0 i>i0
This Hamiltonian represents a model where the transmon resonance is hydridized with the standing
wave resonances of the Fabry-Perot cavity. This Hamiltonian can be easily diagonalized to get:
?
H?0/? = ? a??i i a?i, (6.11)
i>0
where now ?i/2? represents the excitation frequency of the normal modes of the device which are
the result of hybridization of the transmon excited state |e? with the single photon states in the bare
93
Fabry-Perot cavity modes defined as
a??
?
k|g?|0? = Wk,0|e?|0?+ Wk,k? |g?b?
?
k? |0? , (6.12)
k?>0
Note that for k ? i0 the hybridization of the qubit with the modes is completely neglected in our
model, i.e., ? ? and (c)a?k = b?k ?k = ?k .
We now include the effect of the low frequency modes. A quick comparison of the qubit part
of the Hamiltonian with the charge qubit Hamiltonian reveals that the charge operators for the low
frequency modes take the place of the gate charge ng described in Eq 6.9. It is possible to calculate
the sensitivity of the transmon excited state energy level (see Eq. 2.43) to this gate charge and
arrive at the final effective Hamiltonian which describes the interaction between the low and high
frequency modes:
?
H? ?eff/? = ?ka?ka?k + V? , (6.13)
k>0 ?
V? = ?|e??e| cos(2? n?i),
? i?i0? ?V? = ? W ?k,0Wk?,0a?ka?k? cos(2? n?i).
k,k?>i0 i?i0
This interaction describes an effective non-linear Fabry-Perot cavity which allows for scattering of
a photon in a high-frequency mode k to a nearly resonant high frequency mode k? while emitting
or absorbing an even number of low frequency modes. The scattering rate ? is proportional to the
amplitude of phase slips across the transmon small junction when the transmon is in the first excited
state. The exact expression for ? is given by Eq. 2.43.
6.4 Control experiments
We perform the same control experiments as described in section 5.4 to confirm that the microwaves
propagate in the full length of the transmission line and the SQuID loop is not disconnected. In the
interest of space and time we skip its details here.
94
Figure 6.3: (a) Reflection magnitude as a function of probe frequency and external flux through the
transmon SQuID loop. The color scale (not shown) is optimized to make the position of resonances
maximally visible. The three insets have the same width along the frequency axis and correspond to
the three boxed regions on the main plot. (b) An example slice of spectroscopy signal at the external
flux 0.525 ? h/2e, shown by the horizontal dashed line, where the transmon resonance is expected
at around 7 GHz.
6.5 Characterization experiments
To characterize the devices we follow previously described single and two tone reflection spec-
troscopy techniques (see sections 4.2, 4.3.2) and use the spectroscopy data to extract the parame-
ters for the bare Fabry-Perot cavity and the transmon qubit. In the following subsections we describe
the main results of the experiments and compare them to the theory developed in the section 6.3.
First we describe in detail experimental results from a device where the EJ/EC of the SQuID junc-
tions is large such that the phase slip amplitude is negligible. Next we will show the effect of
progressively decreasing the EJ/EC of the transmon when the phase slip amplitude becomes large
and induces inelastic scattering of photons.
6.5.1 Large EJ/EC device (0a)
6.5.1.1 Device parameters
In order to extract the parameters of the bare Fabry-Perot cavity we tune the flux threading the
loop of the transmon?s SQuID loop to an integer multiple of ?0. In this configuration the transmon
resonance is tuned to the plasma cuttoff ?p/2? and the modes of the bare Fabry-Perot cavity in the
95
Figure 6.4: (a) Single-particle density of photon states (DOS) vs. frequency (see text). The fluctu-
ations are reproducible and reflect a small disorder in the junction parameters. The fit (solid red
line) is a combination of a Lorentzian resonance shape combined with the background transmission
line DOS growing due to the van Hove singularity at the plasma frequency ?p/2? = 22.6 GHz. (b)
DOS extracted for a device with Z? = 9.8 k?. (c) DOS as function of external flux threading the
transmon SQuID loop. The two dashed lines indicate the fit to a transmon?s transition and a fixed
width of the DOS peak.
96
frequency range 0 ? 15 GHz are effectively decoupled from the transmon impurity. We use the one
tone and two tone reflection spectroscopy data measured at ?ext = 0 to reconstruct the dispersion
relation of the standing wave modes following procedure of chapter 4. We fit the dispersion of the
modes to obtain the speed of light and the plasma cutoff of the cavity. The wave impedance of the
cavity is estimated using the procedure detailed in section 4.3.3.
In order to characterize the properties of the impurity we perform spectroscopy as a function of
flux through the transmon?s SQuID loop as shown in Fig. 6.3. We observe a dense set of regularly
spaced resonances, with a mode spacing of about ? ? 50?60 MHz, which are disturbed periodically
in flux (only a fraction of the period is shown). This flux-dependent disturbance is associated with
the tuning of the transmon?s resonance across the spectrum of the bare Fabry-Perot cavity. The
unique feature of these data compared to previous multi-mode circuit experiments [23, 80] is that
there are no individual vacuum Rabi splittings. In fact, at a fixed flux, the raw spectroscopy trace
versus frequency contains no obvious sign of the transmon resonance (Fig. 6.3 (b)). However, upon
tuning the flux, we observe that multiple resonances shift by nearly the same amount, indicating
that many modes are equally sensitive to the tuning of the transmon (see three insets in Fig. 6.3).
A clear picture of an transmon simultaneously coupled to many Fabry-Perot comes from ex-
tracting the density of states (DOS) conventionally defined as (?/?)/(?j+1 ? ?j), where ?j is the
measured frequency of jth mode. This quantity is readily available since every individual mode is
frequency-resolved in our reflection spectroscopy. The obtained DOS has a broad peak, involving
about 20 consecutive modes, and slowly increasing with frequency background (Fig. 6.4(a)). The
peak line-shape fits well to a simple Lorentzian function with unit area, which correctly reflects one
additional state coming from the transmon. The DOS peak position shifts with flux and perfectly fol-
lows the disconnected transmon?s resonance, assuming a reasonable junction asymmetry and that
the maximal plasma frequency is that of the other chain junctions (Fig. 6.4 (c)). Importantly,
the fitted peak width is frequency-independent and matches the theoretical radiative linewidth
? = 1/2?Z?CJ ? 600 MHz, within ?50 MHz. In a control experiment, we have checked that
a device with a twice higher wave impedance, Z? = 9.8 k?, fabricated using a skinnier junction
chain, narrows the DOS peak precisely by a factor of two (Fig. 6.4).
97
Figure 6.5: (a) Power-dependence of the spectroscopy signal for the DOS peak at 7.12 GHz. The
modes inside the DOS peak shift more strongly with power than the ones outside. (b) Dispersive
shifts ?i,j as a function of i-modes frequencies for a fixed second-tone driving of the mode j = 108
at 7.014GHz, near the top of the DOS resonance (marked with an arrow). Blue markers indicate the
same measurement for a largely detuned atom. Theory (solid and dashed lines) has no adjustable
parameters except for the Y-axis scale (see section 6.5.1.2). The error bars are the standard de-
viations of the measured dispersive shifts. (c) Power-dependence of the shifts of selected modes
indicated by matching markers
98
6.5.1.2 Photon-photon interactions
To test the Hamiltonian (Eq. 6.6) further, we measure the transmon induced photon-photon inter-
actions. We verified the presence of the induced self-Kerr effect in the normal modes of the device
using high power spectroscopy. The modes inside the DOS peak clearly shift much more in response
to a high-power driving compared to the modes outside of it (Fig. 6.5(a)). At some threshold power,
we observe that the resonances within the DOS peak snap onto their uncoupled values. This effect,
standard in circuit QED [81], can be explained as the SQuID split-junction effectively acquiring an
infinite inductance because the swing of its over-driven phase exceeds 2?.
The cross-Kerr interaction can be characterized more accurately. For a chosen flux bias when
the transmon resonance is tuned to 7.2 GHz, we drive a mode j = 108, corresponding to the peak
in the DOS at a convenient atomic frequency around 7 GHz. The power is chosen such that the
mode is populated by an approximately one photon on average. Reflection measurements are then
performed on the neighboring modes i = j ? 1, j ? 2, . . . with a second low-power drive being
consecutively on or off. The mode frequency in every measurement was found from the fit of the
reflection coefficient to Eq 4.19. The shift is then defined as an average mode frequency change
between the on and off state of the second tone. The procedure was repeated for various second-
tone powers (see Fig. 6.5 (c)). To eliminate the detrimental effects of the flux jitter, we chose
the shortest possible time between two consecutive measurements. Moreover, the frequency of the
second tone was modulated with a time scale much smaller than the measurement time and with
an amplitude larger than the jitter amplitude. This allowed us to create a constant population of
the driving mode disregarding its frequency fluctuation. This procedure allowed us to measure the
dependence of ?i,j on i , j and compare it with our perturbative calculation. The ?i,j is maximal
for the DOS peak maximum, and drops rapidly as the i-modes leave the ?-neighborhood of the
transmon resonance. Although the measured shifts have some fluctuations of presently unknown
nature, the overall data match the theory well using the photon number of the second drive tone as
the only adjustable parameter.
99
Figure 6.6: The measured positions of standing wave resonances as a function of flux through the
split-junction loop in device 3a . The color shows probability to lose a single photon in one round-
trip time.
6.5.2 Inelastic scattering of photon off phase-slip
In this section we describe measurements on devices where we progressively decrease the size of
the transmon?s SQuID junctions. This decreases the ratio of EJ/EC of the transmon and increases
the amplitude of the phase slips across its Josephson junction. To extract the parameters of the bare
transmission line we follow the same procedure as the last section where we measure the standing
wave resonances of the device when the reduced flux (?ext) threading the SQuID loop is set to an
integer multiple of 2?. In order to extract the parameters of the transmon we measure the standing
wave resonances of the cavity as a function of external flux threading the SQuID loop. We observe
that multiple standing wave resonances of the cavity shift as a result of tuning the frequency of the
transmon as expected. In addition, we also observe that the standing wave resonances acquire a
larger linewidth when the transmon is in resonance with a given standing wave mode. This is in
stark contrast to the splitting of the standing wave resonance observed in the case of a fluxonium
impurity (see chapter 5).
The effect of the transmon impurity on the photons in the transmission line can be separated
100
Figure 6.7: Measured elastic phase shift (top) and intristic loss rate (bottom) for devices with
progressively larger charging energy EC indicated on the plots. In each device, the external flux
bias ?ext is tuned such that ?eg/2? ? 6.5?7.5 GHz. The blue markers show data at ?ext = 0, where
the impurity is effectively switched off. The dashed line represents the background dielectric loss
inside the transmission line. Device parameters are given in Table 6.1.
into two parts. The first part is the elastic scattering effect where all the photons in the transmission
line reflect from the transmon boundary with a phase shift ?(?) depending on the frequency of
the incoming photons. This effect results in the shift in the standing wave mode frequencies of
the Fabry-Perot cavity described in the last section. This shift can also be thought of as the result
of hybridization of the transmon resonance with the bare resonances of the transmission line as
described in Eq. 6.10. The shift ?(?) can be completely characterized by comparing the frequencies
of the hybridized standing wave modes of the cavity with the bare standing wave modes of the cavity
when the transmon is maximally detuned (?ext = 0, 2?) by the following formula:
?k(?ext = 0)? ?k(?ext)
?(?k(?ext)) = , (6.14)
?k+1(?ext = 0)? ?k(?ext = 0)
where ?k(?ext) are the normal modes of the Fabry-Perot cavity at a given flux bias and ?k(?ext = 0)
are the bare Fabry-Perot cavity modes.
The second part is the inelastic scattering effect where a photon in mode k upon reflection
from the transmon boundary down-converts from a given standing wave mode into odd number of
photons in lower frequency modes. This effect is due to the interaction induced by the transmon
described by the interaction Hamiltonian in Eq. 6.3.2. This interaction can be thought of as the
101
Figure 6.8: External quality factors as a function of frequency and external flux for device 3a.
The strong frequency dependence of the Qext towards the lower frequencies is the result of the
propagation cut-off of our copper waveguide at 7 GHz.
interaction between a single-particle state in standing wave mode k with many-particle states with
one excitation in mode close to k and an even number of excitations in low frequency modes. As
long as the many-body level spacing of all possible final states that a single photon can down convert
into is smaller than the scattering rate into them, then this process looks like there is an intrinsic
absorption mechanism for this mode. In this limit we can calculate the decay rate of each standing
wave mode using the Fermi?s golden rule [82]. The end result of this decay is the broadening of the
standing wave modes of the Fabry-Perot cavity.
To characterize the effect of the transmon we perform reflection spectroscopy of the standing
wave modes of our effective Fabry-Perot cavity as a function of the magnetic flux threading the
SQuID loop of the transmon. As noted in the previous chapters these spectroscopy experiments are
performed while populating the modes with less than one photon on average, and we checked that
the spectroscopy response remained power independent. By fitting the reflection spectroscopy data
near each resonance with Eq. 4.19, we were able to extract precisely the excitation frequency and
the quality factors of each of the standing wave modes of the effective Fabry-Perot cavity. This allows
102
us to extract the elastic phase shift and the intrinsic loss rate of each of the standing wave modes.
The elastic phase shift ?(?) expectedly winds by ? across the transmon?s resonance. Fitting this shift
with the standard formula for an oscillator we also extract ?, the spontaneous emission linewidth
of the transmon. We check that this quantity remains independent of the transmon frequency. The
loss rate in each standing wave mode is flux independent in device 0a with EC = 0.39 GHz, and it
can be explained by the background dielectric absorption in our bare Fabry-Perot cavity (see section
4.3.1). However, already for EC = 0.66 GHz in device 1a, there is a noticeable deviation of ?(?)
from the background at ?ext = 0, and this deviation rapidly grows with EC (Fig. 6.7, lower panels).
The anomalous dissipation is maximal for modes located in the ?-vicinity of the transmon resonance
at ?eg, defined in Fig. 6.7 as ?(?) = ?/2.
Subtracting the background loss of each device from ?(?), we interpret the remaining rate ?in(?)
as the rate of photon decay due to inelastic scattering at the impurity. Several properties of ?in
support our interpretation. The maximal decay rate ?in(? = ?eg) grows by an order of magnitude
on reducing ?eg by only a few GHz. Such a strong frequency dependence of ?in(? = ?eg) eliminates
the possibility of mundane absorption due to either a lossy dielectric or quasiparticle tunneling in the
impurity junction (see section 6.6). In fact, the growth of ?in(?eg) at lower frequencies is atypical
to materials loss (see section 6.6). Furthermore, the rate ?in(?eg) vanishes in device 0a, which
features the fastest variation of mode frequency with ?eg [the sharpest function ?(?) near ? = ?eg
in Fig. 6.7]. Such an observation eliminates inhomogeneous broadening mechanism due to slow
fluctuations of ?eg in time. We have also explicitly measured the external quality and checked that
the measured port coupling is insensitive to flux bias, and ?in is insensitive to increasing the port
coupling (see Fig. 6.8).
Within the experimental uncertainty on model parameters, Fermi?s golden rule decay calculations
matches the data from all four devices with Z? > RQ without adjustable parameters (see Fig. 6.9)
[82, 83]. Either increasing EC or reducing ?eg with the flux knob exponentially increases ?, which
in turn causes a rapid growth of ?in(?eg). The effect of impedance is weaker but more complex. In
particular, our analysis breaks down for Z? ? RQ, in which case photons are likely produced in the
entire frequency range. Devices with Z? < RQ exhibit similar, by order of magnitude, decay rates
?eg, compared to those by devices with Z? > RQ with similar values of EC [Fig. 6.9 (left panel)
103
Figure 6.9: Inelastic scattering rate ?in/? (colored markers) for devices with Z? > RQ (left panel)
and Z? < RQ (right panel). The width of theory lines (colored bands) comes from uncertainty in
the device parameters. The error bars are the standard errors of ?in/? at the resonance. The color
code represents nominally identical values of EC .
104
Figure 6.10: An example of the coupled many-body states satisfying energy conservation condition
for the decay of the mode k = 47 in device 2a (?eg = ?47). The many-body spectrum (right) is
obtained by summing all possible combinations of three (blue) and five (green) one-photon fre-
quencies, measured experimentally (left). The frequency range in the right panel is equal to the
measured half-linewidth of the k = 47 mode. The visualization in the central panel illustrates the
composition of the multi-photon states from the measured one-photon spectrum. Note the most
likely decay channels ?47 ? ?46 +2?1 and ?47 ? ?45 +4?1 involving the lowest frequency mode at
?1/2? = 63MHz are highlighted with dotted lines.
105
vs (right panel)]. However, a quantitative comparison in case Z? < RQ requires more advanced
theoretical models than those presently available at the time of writing of this thesis.
The reason for the observation of the increased linewidth of the standing wave modes instead of
discrete many-body resonances as observed in chapter 5 is the delocalization of a single excitation
state in the many-body Fock space. Let us illustrate the large number of decay channels available for
a single photon, using an example of mode 47 in device 2a. The flux ?ext is tuned such that ?eg/2? ?
?47/2? ? 6.476GHz, and the measured mode linewidth ?47 = 11MHz. Using extended spectroscopy
data (Fig. 6.10, left panel), we identified those three-photon and five-photon combinations, whose
energy matches ??47/2? within the half-linewidth h ? 5.5 MHz. This construction reveals a large
number of states with a relatively uniform three-photon [?(3)1MHz, Fig. 6.10, blue states] and five-
photon [?(5)50 kHz, Fig. 6.10, green states] level spacing. Final states involving higher number
of photons are also available and they would form even denser spectrum. We checked that most
three-photon states with energies (?i + ?j + ?k) couple relatively uniformly, and same applies to
five-photon states. The energy uniformity property comes from a small amount of disorder and
dispersion in the single-particle spectrum (see chapter 4 section 4.3.2), which breaks the otherwise
massive degeneracy of multi-photon states. These observations justify the treatment of our finite-
size transmission line as an infinite one and apply Fermi?s golden rule to calculate the lifetime of the
standing wave modes.
6.6 Comparison to material related losses
As was already mentioned earlier, the observed inelastic rate and its behavior are inconsistent with
the material-related loss mechanisms. To make this point more clear, Fig. 6.11 compares the
measured ?in/? to the predictions for the dielectric loss and quasiparticles induced loss. To make
such a comparison, we modeled the dissipation as a resistor Rs connected to the end of the trans-
mission line, in parallel with the transmon. The value of Rs at a frequency ?s can be found by
matching the model predictions to the observed inelastic rate at this frequency. By assuming that
the resistance comes either from the dielectric Rs = 1/(?sCJ tan ?), or from quasiparticle tunneling
106
Figure 6.11: ?in/? at resonance with the transmon as a fuction of the transmon frequency (star
markers) and predictions for the dielectric (solid line) and quasiparticle induced loss (dashed line)
in the transmon junction. Numbers on the lines indicate the values of loss tangent tan ? and quasi-
particle density xqp.
107
?
Rs = RQ/(2xqp)EC/ 2?sc?s, we can estimate the dielectric loss tangent tan ? and the quasipar-
ticle density xqp. ?sc is the superconducting gap of aluminum. We see in Fig. 6.11, that both
mechanisms demonstrate a frequency dependence opposite to the observed one. Moreover, the in-
elastic rate around 7.4GHz translates to loss tangent tan? ? 0.1 or quasiparticle density xqp ? 0.01.
Matching the data to the material-related loss at lower frequencies will result in even higher tan ?
and xqp. Such numbers are unrealistically high, taking into account the fact that the same junctions
and a similar RF setup are used in fabrication and measurements of high-coherence qubits [68] [69].
6.7 Conclusion
In summary, we engineered a multi-mode circuit QED setup in the quantum impurity regime where
the transmon impurity induces interaction between all odd-order multi-photon states. This interac-
tion can split a single incident photon into a large number of lower-energy photons with probability
near unity. Inserting such an efficient inelastic scattering center inside a closed Fabry Perot resonator
makes the photon lifetime comparable to the round-trip time, in which case the standing-wave reso-
nances are damped by the photon-photon interaction to the degree prohibiting the use of free-photon
description of the quantum electromagnetic field in the resonator.
6.8 Summary of device parameters
108
Device v/106(m/s) ?p/2?(GHz) Z?(k?) ?(MHz) ?(GHz) EC(GHz)
0a 1.65 22.6 5.36? 0.67 137 0.6? 0.02 0.39? 0.05
1a 1.93 23.0 5.26? 0.4 161 1.03? 0.03 0.66? 0.05
2a 1.78 22.5 5.36? 0.05 149 1.70? 0.02 1.11? 0.01
3a 2.25 27.3 4.44? 0.24 188 2.50? 0.04 1.35? 0.08
4a 1.89 23.2 5.22? 0.21 158 3.10? 0.09 1.96? 0.1
0b 1.99 24.5 9.82? 0.04 166 0.32? 0.03 0.38? 0.03
1b 2.32 26.2 9.25? 1.16 194 0.66? 0.02 0.74? 0.10
2b 1.37 17.5 13.61? 0.92 114 0.63? 0.02 1.05? 0.08
3b 1.75 22.0 10.89? 0.42 146 1.18? 0.04 1.56? 0.08
4b 1.72 21.0 11.38? 0.01 143 1.32? 0.07 1.83? 0.1
Table 6.1: Device parameters extracted by performing characterization experiments detailed in sec-
tion 6.5. Note that the length of all the devices presented in this table was 6mm, except for Devices
0a and 0b which were twice longer.
109
Chapter 7
Summary and Outlook
7.1 Summary of Results
In this thesis, we presented a series of experiments where we engineered the quantum regime of
cavity quantum electrodynamics. We achieved this by coupling superconducting Josephson atoms to
a multi-mode Fabry-Perot cavity. By leveraging the high kinetic inductance of electrons and galvanic
coupling we were able to enter a regime of light-matter interaction where a single atom interacts
with multiple modes of the cavity simultaneously. In this regime we found that the decay of a
single photon becomes a problem of many-body physics as the atom acting as an impurity mediates
interactions between many-body states of radiation. In particular we experimentally implemented
this regime for the case of two different impurity atoms.
In the first experiment we coupled a fluxonium artificial atom to the cavity. In this case our
experiments revealed that the atom mediates a three-wave mixing interactions between the modes
of the multi-mode cavity. This interaction attempts to down convert a single excitation in a high
frequency mode of the cavity into lower frequency excitations but fails due to lack of connectivity
in the many-body Fock space. Many-body localization of photons manifests itself as the violation
of Fermi?s golden rule and the splitting of the standing wave resonance into discrete well resolved
peaks due to the coherent hybridization of many-body states of radiation with a single particle state.
We directly measure this spectrum of many-body states of radiation and identify each peak with its
110
wavefunction composed of a specific many-body content.
In the second experiment we coupled a transmon artificial atom to the cavity. In this case we
found that for large EJ/EC the main effect of the transmon on the standing wave modes of the cavity
was a frequency dependent phase shift. The photon-photon interactions were suppressed because
of the transmon?s weak anharmonicity. However, we observed that by decreasing the EJ/EC of
the transmon turns on another kind of non-linearity that of phase slips across the small junction of
the transmon. In addition, we also observed that this non-linearity induces interactions between
all many-body states of radiation of odd parity which splits a single photon excitation into a large
number of lower frequency photons. In stark contrast with the case of the fluxonium impurity here,
the single excitation diffuses into the many-body Fock space and the only measurable consequence
of this effect is an increased linewidth of the standing wave modes of the Fabry-Perot cavity.
7.2 Tunable platform to study MBL physics in 1-D
Our experiments puts forward multi-mode circuit quantum electrodynamics as a possibly useful ex-
perimental platform to further explore fundamentals of MBL. In comparison to spin chains or lattice
fermions, here the frequency space takes the role of the real space, the number of participating
degrees of freedom is set by the excitation frequency, and the connectivity of the many-body Fock
space can be tailored by varying the properties of the qubit circuit, serving as one of the cavity mir-
rors. In the experiments presented in this thesis, we implemented a local tree-like connectivity by
harnessing the inversion symmetry breaking and the strong anharmonicity properties of fluxoniums.
At the opposite extreme we also engineered a quantum phase-slip impurity, which can maximize the
Fock-space connectivity, in which case the down-conversion rate indeed accurately obeys the Fermi?s
Golden Rule [84]. In future experiments, the disorder can be tailored via nanofabrication, the in-
teraction can be quenched by rapidly tuning the flux knob, and multi-mode tomography techniques,
developed for bosonic qubits [85], can be applied here to monitor the entanglement dynamics in
the time-domain.
In addition, future experiments can probe the cross-over of our system from many-body localized
to de-localized regime. One proposal to implement this could be by adding a tunable boundary
111
condition to the left end of the device discussed in chapter 5. To understand this lets consider the
spectrum of many-body states near a single particle state k ? 1. For the case of an open boundary
condition on the other end of the impurity will result in a spectrum where the n and n + 1 particle
subspaces will be maximally detuned which would result in the localization of a single particle
excitation state similar to the case described in detail in chapter 5. However, for the case of a
short boundary at the other end of the impurity will result in all n and n+ 1 particle states to be in
resonance, this will result in de-localization. The ability to tune this boundary condition in situ via
a large area SQuID can implement this transition in a single device.
7.3 Implementation of Q-RAM
The devices presented in chapters 5 and 6 also have a direct application for the purposes of quan-
tum information processing. In particular, the small mode volumes and high quality factors make
the standing wave modes in the Josephson junction transmission line ideal as quantum memories.
In addition, by coupling the transmon and fluxonium qubits in the strong dispersive regime, all
to all interactions can be induced between the memory modes by applying off-resonant drives on
the qubits [86]. In comparison to existing proposals that involve directly exciting the qubit, this
scheme can offer a substantial improvement in gate fidelity. The ability to engineer mode frequen-
cies via nano-fabrication to obtain non-uniform mode spacing also makes this platform attractive for
practical proposes.
The experimental realization of quantum information processing using multi-mode cQED has the
potential to solve two of the most pressing challenges in continuous variable quantum computation
namely scaling and slow gate speeds. The all to all connectivity in this architecture also paves the
way to realize quantum random access memories which is a prerequisite in important quantum
algorithms like the Grover?s search [87].
112
Appendices
113
Appendix A
Benchmnarking
In this appendix we benchmark the effective Hamiltonian and the procedure to calculate the eigen-
spectrum for a multi-mode cavity QED system described in chapter 5 in Eq. (5.12) in the mixed
gauge, against brute-force diagonalization for finite-size systems.A full numerical diagonalization of
the Hamiltonian in Eq. (2.49) rapidly becomes a daunting task, because the Hilbert space size grows
exponentially with the number N of bosonic modes and the number of photons involved. Here we
present results for a system with N = 6 bosonic modes. Moreover, we also report results in the other
two gauges (charge and flux gauge) including multiple fluxonium levels and show that the same
results can be obtained by approximating the fluxonium as a two-level system in the mixed gauge.
In our numerical calculations, we consider a fluxonium atom described by the parameters EJ =
8.12 GHz, EC = 5.69 GHz, EL = 1.42 GHz. For the cavity resonator, we consider a two-wire trans-
mission line with parameters Z? = 9865k?, v = 2.18?106m/s, ? = 300?m. In this example the line
shares a fraction x = 0.2 of the inductance with the fluxonium. The unequal spacing of the modes
in the transmission line is achieved simply by introducing the ultra-violet cutoff due to the plasma
frequency ?p for wave propagation in the transmission line. The unequal mode spacing breaks the
degeneracy of different multi-photon states and isolate anti-crossings between any two given states
in the many-body spectrum of the system.
Note that we have also derived the circuit Hamiltonian of the system in both the flux and the
114
charge gauges following the procedure described in section 2.4.2. We have diagonalized the cir-
cuit Hamiltonian in both gauges using the lowest 40 levels of the uncoupled photonic modes of the
transmission line and the 7 lowest-energy levels of the fluxonium. In Fig. A.1 (a), we report the en-
ergy eigenvalues in such two gauges with the prediction of the effective Hamiltonian (mixed gauge,
two-level approximation for the fluxonium). The excellent agreement between the three gauges
in Fig A.1 (a)) not only illustrates the gauge invariance of our multi-mode cQED model, but also
serves as the additional proof that we have used a large enough Hilbert space in diagonalizing the
Hamiltonian. In particular, it also validates the two-level approximation for the fluxonium in the
mixed gauge. A zoom in of the eigenspectrum plot near the vacuum Rabi splitting of the third pho-
tonic mode reveals an additional anti-crossing. This anti-crossing can be explained as the spectral
manifestation of the hybridization between the single-particle state a??4|G? and the two-photon state
with one excitation in polariton mode k = 3 and one excitation in mode k = 1, namely the state
a?? a??3 1|G?. To reproduce these spectral features with the effective Hamiltonian derived in section 5.3,
first we diagonalize the Hamiltonian only in the single-excitation subspace, where we ignore the
coupling of the fluxonium to the lowest transmission line mode, and obtain the polaritonic frequen-
cies ?k for k > 1. Then we include the effect of the coupling of the lowest mode by expanding the
Hilbert space to include the two-photon and three-photon states. We finally calculate the spectrum
by diagonalizing the following 7? 7 matrix:
? Heff = (A.1)? ?? (m,f) (m,f)? ?3/2? 0 0 g1 A3,3 g1 A3,4 0 0? ?? (m,f) (m,f) ?? 0 ?4/2? 0 g ?1 A4,3 g1 A4,4 0 0? ??? (m,f) (m,f)? 0 0 ?5/2? g1 A5,3 g1 A5,4 0 0 ???? (m,f) (m,f) (m,f) ? ?? (m,f)
? (m,f) ?
?g1 A3,3 g1 A3,4 g1 A3,5 ?3/2? + ?1/2? 0 2g1 A3,2 2g1 A3,3? ? ? ?
?
? (m,f) (m,f) (m,f) (m,f) (m,f) ??g1 A4,3 g1 A4,4 g1 A ?4,5 ?4/2? + ?1/2? 0 2g1 A4,2 2g1 A4,3?? ? ? ?
?
(m,f) (m,f)
? 0 0 0 2g ?? 1
A2,3 2g1 A2,4 ?2/2? + 2? ?1/2? 0 ??? ? (m,f) ? (m,f) ?0 0 0 2g1 A3,3 2g1 A3,4 0 ?3/2? + 2? ?1/2???
. (A.2)
115
Figure A.1: (a) Excitation spectrum of the multi-mode circuit QED system (parameters of cavity and
fluxonium in the text) for N = 6 unequally spaced modes obtained by numerical diagonalization
in the charge (red dotted lines) and flux (solid blue lines) gauge. These results have been obtained
by including the 7 lowest-energy fluxonium eigenstates. The spectrum obtained with the effective
Hamiltonian 5.12 in the mixed gauge is displayed by the solid black lines. Note that the curves in
the three gauges excellently overlap. (b) Zoom in of the spectrum near the anticrossing of mode
k = 3 with the bare |e? ? |g? transition, showing hybridization between the two-photon state with
frequency (?3 +?1) and a single-polariton state with frequency ?4. The red dotted line is the result
in the charge gauge with 7 levels, while the black solid line shows the energy levels calculated using
the effective Hamiltonian described in Eq. (5.12) (mixed gauge, two-level approximation for the
fluxonium). (c) Same as in (b), but this time showing the anti-crossing between the two-photon
state with frequency (?4 + ?1) and the three-photon state with frequency (?3 + 2?1).
The red dashed line in Fig. A.1 (b,c) shows the corresponding results, which are in excellent
agreement with the full numerical diagonalization result. The figure shows the anticrossing due
to the hybridization between the single-polariton state a??4|G? and the two-photon state consisting
116
of one excitation in polariton mode k = 3 and one excitation in the low-frequency mode k = 1,
namely a?? ?3a?1|G?. The results display also an anticrossing between the two-photon state a?
?
4a?
?
1|G? and
three-photon state a??3(a?
?
1)
2|G?.
117
Appendix B
Calculation of the reflection
coefficient
In this appendix we summarize the procedure to theoretically generate the refection spectra of the
multi-mode cavity QED device presented in chapter 5. In order to theoretically calculate the linear
microwave reflection spectra, we first consider the expression for the bare transmission line modes:
? ( ? (bare))f fN 2i i 1 1(bare) ? ext + intf Qi Qi
S11(f) = ( i ) , (B.1)
? (bare)f f
i=1 2i i 1 1(bare) + +
f Q
ext Qint
i i i
where Qint and Qexti i are respectively the internal and external quality factors of the bare transmission
line modes. To calculate the reflection coefficient in the presence of the qubit, the expression is
replaced by the product over the many-body eigenstates of the effective Hamiltonian in Eq. 5.12 of
the main text: ? ( ? (many?body))N(s) f f2i( i 1 1(many?body) ?f ) Q?ext + Q?intS i i i11(f) = ? , (B.2)? (many body)f f 1 1
i=1 2i i(many?body) +
f Q?ext
+
i Q?
int
i i
where (many?body)fi are the excitation frequencies (with respect to the ground state) with corre-
sponding eigenfunctions |? ? and N (s)i are the total number of eigenstates of the Hamiltonian in the
s-particle Hilbert subspace. Note that the external (Q?exti ) and internal (Q?
int
i ) quality factors of the
118
the many-body states depend on the single-particle weight in the many-body eigenstate. This can be
calculated by weighing the eigenstate quality factors with their single-particle fractions, namely:
?
int(ext)
(Q? )?1 |? | ? | ?|2 ?int(ext)i = ?i a?k 0 (Q k )
?1 . (B.3)
k>i0
Here the single-particle quality factors depend on the bare transmission line mode quality factors
given by: ?
?int(ext) ?1 | |2 int(ext)(Q k ) = Wk,i (Q )?1i . (B.4)
i>i0
119
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