Waveguide Quantum Electrodynamics in Sup

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Waveguide Quantum Electrodynamics in Superconducting Circuits
Citation
Kim, Eun Jong
(2022)
Waveguide Quantum Electrodynamics in Superconducting Circuits.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/bscv-b073.
Abstract
Achieving an efficient interface of light and matter has been a principal goal in the field of quantum optics. A burgeoning paradigm in the study of light-matter interface is waveguide quantum electrodynamics (QED), where quantum emitters are coupled to a common one-dimensional waveguide channel. In this scenario, cooperative effects among quantum emitters emerge as a result of real and virtual exchange of photons, giving rise to new ways of controlling matter.
Superconducting quantum circuits offer an exciting platform to study quantum optics in the microwave domain with artificial quantum emitters interfaced to engineered photonic structures on chip. Beyond revisiting the experiments performed in atom-based platforms, superconducting circuits enable exploration of novel regimes in quantum optics that are otherwise prohibitively challenging to achieve. Moreover, the unprecedented level of control over individual quantum degrees of freedom and good scalability of the system provided by state-of-the-art circuit QED toolbox set a promising direction towards the study of quantum many-body phenomena.
In this thesis, I discuss waveguide QED experiments performed in superconducting quantum circuits where transmon qubits are coupled to engineered microwave waveguides. Employing the high flexibility and controllability of superconducting quantum circuits, we realize and explore various schemes for generating waveguide-mediated interactions between superconducting qubits. We also demonstrate an intermediate-scale quantum processor based on a dispersive waveguide QED system involving ten superconducting qubits, exploring quantum many-body dynamics in a highly controllable fashion. The work described in the thesis marks an important step towards the construction of scalable architectures for quantum simulation of many-body models and realization of efficient coupling schemes for quantum computation.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Waveguide QED; Quantum Information; Quantum Many-Body Physics; Circuit QED
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Painter, Oskar J.
Group:
Institute for Quantum Information and Matter, Kavli Nanoscience Institute
Thesis Committee:
Faraon, Andrei (chair)
Brandao, Fernando
Painter, Oskar J.
Preskill, John P.
Defense Date:
11 February 2022
Non-Caltech Author Email:
ekim7206 (AT) gmail.com
Record Number:
CaltechTHESIS:02122022-205429202
Persistent URL:
DOI:
10.7907/bscv-b073
Related URLs:
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Description
DOI
Article adapted for Chapter 3.
DOI
Article adapted for Chapter 4.
DOI
Article adapted for Chapter 5.
ORCID:
Author
ORCID
Kim, Eun Jong
0000-0003-4879-8819
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
14495
Collection:
CaltechTHESIS
Deposited By:
Eun Jong Kim
Deposited On:
20 Apr 2022 19:42
Last Modified:
21 May 2025 22:27
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Waveguide Quantum Electrodynamics in
Superconducting Circuits

Thesis by

Eun Jong Kim

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2022
Defended February 11th, 2022

Eun Jong Kim
ORCID: 0000-0003-4879-8819

iii

To my wife Eunjin Hong and my daughter Hayden
without whom I could not have put a successful end to this journey.
And to my parents Meewon Park and Sunsuk Kim
who made me who I am today.

ACKNOWLEDGEMENTS
Now has been the moment long awaited since the first day I came to Caltech, a hot
summer day of Southern California in 2016 when my long journey towards PhD
began. I am feeling grateful for many of whom I had the privilege to interact with
along this adventure, which I would like to acknowledge here.
I want to first thank my advisor Prof. Oskar Painter for the endless support, encouragement, trust, and guidance you constantly showed me during the course of
my PhD. No words can express how much I feel indebted to you. In particular, I
appreciate for being open to trying out new things in the lab and supporting all of
my research from the heart. From you, I learned to enjoy scientific research, not
being overwhelmed by technical difficulties, and to be confident of my skills and
knowledge when dealing with projects.
Also, I would like to express special gratitude to my colleague Xueyue (Sherry)
Zhang who I spent the most time with in the lab and the office. I enjoyed numerous
hours we spent in the lab and the enlightening discussions we had, which were
crucial to moving the experiments forward and coming up with new ideas for the
projects. I believe your sharpness and patience in approaching research will pay off,
and I wish you the best of luck in completing your PhD.
Painter group class of 2022, Vinicius Ferreira and Jash Banker, I want to thank
you guys for being nice friends and colleagues along the long journey to PhD.
After coming through numerous ups and downs (usually more downs than ups),
including the first publication of papers, struggles and successes in projects, and the
outbreak of pandemic, I am truly happy that we finally made it to the end. I want
to congratulate you all and hope you have a great start of your future careers after
graduation.
I cannot help mentioning how I fortunate I was to learn from the most brilliant
mentors, the former postdocs of Painter group Mohammad Mirhosseini (now faculty
at Caltech EE), Andrew Keller (now at AWS Center for Quantum Computing), and
Alp Sipahigil (now faculty at UC Berkeley), from an early stage of my PhD. I owe you
countless cups of coffee for the instruction of various theoretical and experimental
techniques, skills to overcome technical challenges, and the academic mindsets.
I also want to acknowledge our technical staff Barry Baker and former senior graduate
students Mahmoud Kalaee, Gregory MacCabe, Hengjiang (Jared) Ren, Jie (Roger)

Luo, and Michael Fang for their help in keeping up the lab in shape and training
various tools used for my research. I would like to thank the postdocs Srujan
Meesala, Clai Owens, Mo Chen, David Lake and graduate students Steven Wood,
Gihwan Kim, Andreas Butler, Piero Sameer Sonar, and Utku Hatipoglu who joined
the group during the latter part of my PhD for helpful discussions and everyday
chitchats. Best of luck in your future endeavors!
Outside of Caltech, I want to thank wonderful collaborators I worked with during
my PhD. I learned a lot from discussion with theory collaborators Prof. Ana AsenjoGarcia (Columbia University) and Prof. Darrick Chang (ICFO) who helped us in the
magic cavity project, Dr. Miguel Bello and Prof. Alejandro González-Tudela (CSIC)
who helped out with quickly ramping up the topological waveguide QED project.
Also, I am feeling grateful to Daniel Mark and Prof. Soonwon Choi at MIT for
the discussions we had about the on-going project on studying quantum many-body
physics in our quantum processor. I believe that our collaboration will come to
fruition very soon. Also, special thanks to my Israeli friends Niv Drucker, Yonatan
Cohen, and colleagues from Quantum Machines for endless technical support on
the OPX.
Last but not least, I want to thank my family members for their unconditional
encouragement and support. To my parents Meewon Park and Sunsuk Kim, who
should’ve been waiting for this moment more than I did from 6000 miles away full
of heart. You’re not only great parents but also the greatest people I’ve ever known.
Thank you for making me believe that I can do anything and everything in life. To
my wonderful wife Eunjin Hong, who bravely ventured into becoming my lifelong
companion and always stayed with through all the goods and hardships. I am so
lucky to have a wife like you, and I love you with all of myself. To my soon-to-beborn daughter Hayden, who responded to my calling from Eunjin’s tummy with a
lot of fetal movement. I’m so excited to meet you soon!

ABSTRACT
Achieving an efficient interface of light and matter has been a principal goal in
the field of quantum optics. A burgeoning paradigm in the study of light-matter
interface is waveguide quantum electrodynamics (QED), where quantum emitters
are coupled to a common one-dimensional waveguide channel. In this scenario,
cooperative effects among quantum emitters emerge as a result of real and virtual
exchange of photons, giving rise to new ways of controlling matter.
Superconducting quantum circuits offer an exciting platform to study quantum optics
in the microwave domain with artificial quantum emitters interfaced to engineered
photonic structures on chip. Beyond revisiting the experiments performed in atombased platforms, superconducting circuits enable exploration of novel regimes in
quantum optics that are otherwise prohibitively challenging to achieve. Moreover,
the unprecedented level of control over individual quantum degrees of freedom and
good scalability of the system provided by state-of-the-art circuit QED toolbox set
a promising direction towards the study of quantum many-body phenomena.
In this thesis, I discuss waveguide QED experiments performed in superconducting quantum circuits where transmon qubits are coupled to engineered microwave
waveguides. Employing the high flexibility and controllability of superconducting
quantum circuits, we realize and explore various schemes for generating waveguidemediated interactions between superconducting qubits. We also demonstrate an
intermediate-scale quantum processor based on a dispersive waveguide QED system involving ten superconducting qubits, exploring quantum many-body dynamics
in a highly controllable fashion. The work described in the thesis marks an important step towards the construction of scalable architectures for quantum simulation
of many-body models and realization of efficient coupling schemes for quantum
computation.

vii

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] V. S. Ferreira, J. Banker, A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim,
M. Mirhosseini, and O. Painter, “Collapse and revival of an artificial atom
coupled to a structured photonic reservoir,” Phys. Rev. X 11, 041043 (2021)
10.1103/PhysRevX.11.041043,
E.K. participated in the conception of the project and contributed to the
writing of the manuscript.
[2] E. Kim, X. Zhang, V. S. Ferreira, J. Banker, J. K. Iverson, A. Sipahigil, M.
Bello, A. González-Tudela, M. Mirhosseini, and O. Painter, “Quantum electrodynamics in a topological waveguide,” Phys. Rev. X 11, 011015 (2021)
10.1103/PhysRevX.11.011015,
E.K. came up with the concept and planned the experiment, performed the
device design and fabrication, performed the measurements, analyzed the
data, and participated in the writing of the manuscript. E.K. and X.Z. contributed equally to this work.
[3] M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller,
A. Asenjo-Garcia, D. E. Chang, and O. Painter, “Cavity quantum electrodynamics with atom-like mirrors,” Nature 569, 692–697 (2019) 10.1038/
s41586-019-1196-1,
E.K. came up with the concept and planned the experiment, performed the
device design and fabrication, performed the measurements, analyzed the
data, and participated in the writing of the manuscript. E.K. and M.M. contributed equally to this work.
[4] M. Mirhosseini, E. Kim, V. S. Ferreira, M. Kalaee, A. Sipahigil, A. J. Keller,
and O. Painter, “Superconducting metamaterials for waveguide quantum
electrodynamics,” Nat. Commun. 9, 3706 (2018) 10.1038/s41467-01806142-z,
E.K. carried out the device design and fabrication, performed the measurements, analyzed the data, and contributed to the writing of the manuscript.

viii

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . .
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Quantum light-matter interface . . . . . . . . . . . . . . . . . . . .
1.2 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Quantum many-body physics . . . . . . . . . . . . . . . . . . . . .
1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter II: Waveguide Quantum Electrodynamics . . . . . . . . . . . . . . .
2.1 From cavity QED to waveguide QED . . . . . . . . . . . . . . . . .
2.2 Quantum emitters coupled to a waveguide with linear dispersion . . .
2.3 Waveguide QED in a dispersive photonic channel . . . . . . . . . . .
2.4 Experimental realizations of waveguide QED . . . . . . . . . . . . .
Chapter III: Device Fabrication and Experimental Techniques . . . . . . . . .
3.1 Fabrication of superconducting quantum circuits . . . . . . . . . . .
3.2 Microwave packaging of quantum devices . . . . . . . . . . . . . . .
3.3 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Room-temperature electronics and signal processing . . . . . . . . .
Chapter IV: Waveguide-mediated Cooperative Interactions of Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Strong-coupling regime of waveguide QED: cavity QED with atomlike mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Spectroscopic characterizations . . . . . . . . . . . . . . . . . . . .
4.4 Time-domain characterizations . . . . . . . . . . . . . . . . . . . .
4.5 Compound atomic mirrors . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter V: Development of Superconducting Metamaterials for Waveguide
QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Band-structure analysis and spectroscopy . . . . . . . . . . . . . . .
5.3 Physical realization using lumped-element resonators. . . . . . . . .
5.4 Disorder and Anderson localization . . . . . . . . . . . . . . . . . .
5.5 Anomalous Lamb shift near the band-edge . . . . . . . . . . . . . .
5.6 Enhancement and suppression of spontaneous emission . . . . . . .

iv
vi
vii
vii
xii
10
12
12
13
21
27
31
31
32
35
38
46
46
47
50
52
56
58
58
65
65
69
70
71
71
73

ix
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter VI: Quantum Electrodynamics in a Topological Waveguide . . . . . 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Description of the topological waveguide . . . . . . . . . . . . . . . 78
6.3 Properties of quantum emitters coupled to the topological waveguide 81
6.4 Quantum state transfer via topological edge states . . . . . . . . . . 88
6.5 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . 90
Chapter VII: An Intermediate-Scale Quantum Processor Based on Dispersive
Waveguide QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Metamaterial quantum processor . . . . . . . . . . . . . . . . . . . 94
7.3 Single-qubit characterization . . . . . . . . . . . . . . . . . . . . . 97
7.4 Interaction between qubit-photon bound states . . . . . . . . . . . . 98
7.5 High-fidelity single-shot readout . . . . . . . . . . . . . . . . . . . . 100
7.6 Quantum many-body dynamics . . . . . . . . . . . . . . . . . . . . 102
7.7 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . 104
Chapter VIII: Outlook and Future Directions . . . . . . . . . . . . . . . . . . 105
8.1 Opportunities for studying many-body physics . . . . . . . . . . . . 105
8.2 New directions for scaling up quantum processors . . . . . . . . . . 107
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Appendix A: Fabrication Details . . . . . . . . . . . . . . . . . . . . . . . . 140
Appendix B: Designing Josephson Junctions for Transmon Qubits . . . . . . 147
Appendix C: Supplementary Information for Chapter 4 . . . . . . . . . . . . 151
Appendix D: Supplementary Information for Chapter 5 . . . . . . . . . . . . 165
Appendix E: Supplementary Information for Chapter 6 . . . . . . . . . . . . 183

LIST OF FIGURES

Number
Page
1.1 Quantum light-matter interface . . . . . . . . . . . . . . . . . . . . 2
1.2 Cavity quantum electrodynamics . . . . . . . . . . . . . . . . . . . . 3
1.3 Stages of quantum information processing . . . . . . . . . . . . . . . 8
1.4 Overview of our work . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Transition from cavity QED to waveguide QED . . . . . . . . . . . . 12
2.2 Single quantum emitter coupled to a waveguide . . . . . . . . . . . . 14
2.3 Multiple quantum emitters coupled to a waveguide . . . . . . . . . . 16
2.4 Superradiance and subradiance of two quantum emitters . . . . . . . 17
2.5 Coherent waveguide-mediated exchange interaction . . . . . . . . . . 19
2.6 Exchange interaction between two quantum emitters . . . . . . . . . 21
2.7 Emitter-photon bound state . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Waveguide-mediated interactions between emitter-photon bound states 25
2.9 Examples of experimental platforms for waveguide QED . . . . . . . 28
3.1 Microwave packaging of quantum devices . . . . . . . . . . . . . . . 32
3.2 Cryogenic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Microwave synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Downconversion and demodulation . . . . . . . . . . . . . . . . . . 42
4.1 Waveguide-QED setup . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Single-qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Vacuum Rabi splitting . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Vacuum Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Characterization of the dark-state cavity . . . . . . . . . . . . . . . . 55
4.6 Compound atomic mirrors, 𝑁 = 4 . . . . . . . . . . . . . . . . . . . 57
4.7 Scanning electron microscope image of the fabricated device . . . . . 59
4.8 Schematic of the measurement chain inside the dilution refrigerator . 62
5.1 Microwave metamaterial waveguide . . . . . . . . . . . . . . . . . . 67
5.2 Disorder effects and qubit-waveguide coupling . . . . . . . . . . . . 68
5.3 Measured dispersive and dissipative qubit dynamics . . . . . . . . . 72
5.4 State-selective enhancement and inhibition of radiative decay . . . . 74
6.1 Topological waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Directionality of qubit-photon bound states . . . . . . . . . . . . . . 82

xi
6.3
6.4
7.1
7.2
7.3
7.4
7.5
7.6
B.1
C.1
C.2
D.1
D.2
D.3
D.4
E.1
E.2
E.3
E.4
E.5
E.6
E.7
E.8
E.9
E.10
E.11
E.12
E.13

Probing band topology with qubits . . . . . . . . . . . . . . . . . . . 85
Qubit interaction with topological edge modes . . . . . . . . . . . . 87
Metamaterial waveguide . . . . . . . . . . . . . . . . . . . . . . . . 94
Description of the metamaterial quantum processor . . . . . . . . . . 96
Single-qubit characterization . . . . . . . . . . . . . . . . . . . . . . 97
Qubit-qubit interactions . . . . . . . . . . . . . . . . . . . . . . . . 99
Multi-qubit readout characterization . . . . . . . . . . . . . . . . . . 101
Quantum walk of two photons along qubit-photon bound states . . . 103
Scanning electron micrograph of a Josephson junction . . . . . . . . 150
Effect of thermal occupancy on extinction . . . . . . . . . . . . . . . 153
Level structure of the atomic cavity and linear cavity . . . . . . . . . 154
Circuit diagram of metamaterial waveguide . . . . . . . . . . . . . . 165
Characterization of lumped element resonators . . . . . . . . . . . . 174
Circuit diagram for a transmon qubit coupled to a metamaterial
waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Qubit lifetime as a function of resonance frequency . . . . . . . . . . 178
Modeling of the topological waveguide . . . . . . . . . . . . . . . . 183
Band structure of the realized topological waveguide under various
assumptions discussed in App. E.2 . . . . . . . . . . . . . . . . . . . 191
Eigenspectrum of the finite-sized topological circuit . . . . . . . . . 192
Eigenfrequencies of the system under 100 disorder realizations in
coupling elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Optical micrograph of Device I (false-colored) . . . . . . . . . . . . 197
Schematic of the measurement setup inside the dilution refrigerator
for Device I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Tapering section of Device I . . . . . . . . . . . . . . . . . . . . . . 200
Understanding the directionality of qubit-photon bound states . . . . 202
External coupling rate of qubit-photon bound states away from the
reference frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Understanding the swirl pattern . . . . . . . . . . . . . . . . . . . . 205
Topology-dependent photon scattering on various qubit pairs . . . . . 207
Optical micrograph of Device II (false-colored) . . . . . . . . . . . . 208
Schematic of the measurement setup inside the dilution refrigerator
for Device II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

xii

LIST OF TABLES
Number
Page
2.1 Experimental platforms for studying waveguide QED . . . . . . . . . 28
4.1 Qubit characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 59
C.1 Parameters used for fitting Rabi oscillation curves . . . . . . . . . . . 159
C.2 Decay rate and decoherence rate of dark states . . . . . . . . . . . . 159
D.1 Measured resonance parameters for metamaterial waveguide . . . . . 172
E.1 Qubit coherence in the middle bandgap. . . . . . . . . . . . . . . . . 196
E.2 Qubit parameters on Device II . . . . . . . . . . . . . . . . . . . . . 208

Chapter 1

INTRODUCTION
Since its inception in the 20th century, quantum mechanics had a profound impact on humans, revolutionizing the physical understanding of nature. Starting
from Planck’s quantum hypothesis to explain the black-body radiation [1]—a mere
heuristic correction to the classical mechanics at that point—the quantum mechanics
became a major theory in fundamental science, permeating through a broad range of
disciplines such as atomic, molecular, and optical (AMO) physics, condensed matter
physics, particles physics, and even quantum chemistry. Not only of fundamental
importance, the quantum theory also spawned major technological developments
such as lasers [2, 3] and transistors which are integral to modern telecommunication
and digital electronics we use in everyday life.
While the counter-intuitive aspects of quantum theory, often exemplified by the
famous thought experiments such as Schrödinger’s cat [4] and Bell’s inequality
[5], were considered purely theoretical in the early days, the unprecedented ability
to control and measure individual quantum degrees of freedom, such as atoms,
photons, and even macroscopic objects, developed in modern era of physics has
placed such exotic quantum nature within the experimental reach. This has sparked
new directions in quantum physics where information theoretic tools are utilized
for understanding complex quantum many-body phenomena and strongly correlated
states of matter. In particular, such approach can open the door to novel and powerful
technologies that can benefit from truly quantum mechanical effects such as quantum
superposition and quantum entanglement beyond the semi-classical description.
The burgeoning interest in quantum science we see in recent years, highlighted by
industry-fueled efforts to build a practical quantum computer [6–10] and a variety of
quantum simulators [11–16], is expected to again transform the world in many ways.
Numerous ways to systematically control, measure, and understand quantum systems
await further explorations and are anticipated to give birth to practical quantum
applications. We present a step towards this goal by demonstrating quantum control
experiments in a new platform. In the following, I will provide a gentle introduction
to the thesis by explaining the historical context on relevant scientific disciplines
and providing an outline of the remaining chapters.

Γ0
Γ1D

Figure 1.1: Quantum light-matter interface. A one-dimensional photonic channel (light
blue) is coupled to an atom (orange) at a rate Γ1D . The atom has a spontaneous emission
rate Γ0 to free space.

1.1

Quantum light-matter interface

Interaction between electromagnetic field and matter lies at the heart of various phenomena we observe in nature. The basic optical processes such as reflection of light
and dispersion of materials originate from the interplay between oscillating fields
and charged particles (e.g., electrons) governed by laws of classical electrodynamics
[17]. In the quantum regime, the same problem is mapped to the interaction between
a single photon and a single atom [18] which results in non-classical phenomena
such as anti-bunching of photons emitted by an atom [19] and quantum beat of Rabi
oscillation [20]. In this respect, engineering an efficient interface of light and matter
in a controllable fashion has been a central goal in atomic physics [21] and quantum
optics [22], and is of fundamental significance.
Coupling of an atom to a 1D photonic channel
The general picture of quantum light-matter interface can be understood in terms
of the simplest scenario of coupling a 1D photonic channel with a single atom, as
described in Fig. 1.1. In this case, the useful rate that couples the atom to the desired
channel is represented by the decay rate Γ1D , which is often compared with the rate
Γ0 of spontaneous emission. The ratio Γ1D /Γ0 of the two rates is a key figure of
merit quantifying the efficiency of atom-photon coupling and is known to depend
on the following relation [23]:
𝜎 𝑐
Γ1D
(1.1)
Γ0
𝑣𝑔
Here, the first factor is the ratio of the scattering cross-section 𝜎0 of the atom to
the effective mode area 𝐴. The second factor is the group index of the introduced
photonic channel given by the ratio of speed of light 𝑐 in vacuum to the group
velocity 𝑣 𝑔 of the desired mode.
Achieving a strong atom-photon coupling is naturally a challenging task. An atom
in free space has the resonant scattering cross-section of 𝜎0 = 3𝜆20 /(2𝜋), where

Figure 1.2: Cavity quantum electrodynamics. a, A cavity QED setup, consisting of an
atom (orange, decay rate 𝛾) coupled to photons inside a cavity (green, leakage rate 𝜅) with
a coupling rate 𝑔. b, A microwave cavity made of two superconducting niobium mirrors
used in Prof. Serge Haroche’s group. c, An optical cavity used in Prof. Jeff Kimble’s
group. d, A circuit QED setup consisting of a coplanar waveguide resonator coupled to
a superconducting qubit (blue inset) on chip. The panels b, c, and d are adapted from
Refs. [25], [26], and [27], respectively.

𝜆0 is the linewidth of the atomic transition [21]. This means that the area 𝐴 of
the coupled mode has to be very small (on the order of 𝜆20 ) in order to achieve an
appreciable atom-photon coupling according to Eq. (1.1). One could imagine using
tightly focused beam to reduce the mode area [24], but the diffraction limit makes it
challenging to accomplish 𝜎0 /𝐴 on the order of unity.
Cavity quantum electrodynamics
One successful method to overcome this challenge is by utilizing an electromagnetic
mode confined in a resonator such as a Fabry-Pérot cavity. In this case, the efficiency
of coupling is enhanced by the number of passes that a photon makes across the
atom before leaking out of the cavity. Also, the placement of a resonator alters the
photonic vacuum of the environment for the atom, greatly modifying the rate of
spontaneous emission. This field where an atom is interfaced with photons inside a
cavity is known as the cavity quantum electrodynamics (QED) [28, 29], illustrated in
Fig. 1.2. A cavity QED system is described by the Jaynes-Cummings Hamiltonian
[30]
𝐻ˆ = ℏ𝑔( 𝑎ˆ 𝜎
ˆ + + 𝑎ˆ † 𝜎
ˆ −)
(1.2)
where 𝑔 is the coupling between an atom and a cavity photon, 𝑎ˆ (𝑎ˆ † ) is the annihilation (creation) operator of cavity photons, and 𝜎
ˆ − (𝜎
ˆ + ) is the lowering (raising)
operator of atomic pseudospin-1/2. In addition to the Hamiltonian, the cavity QED

system is further described by the the decay rate 𝛾 of the atom and the photon
leakage rate 𝜅 of the cavity. In this case, the figure of merit Γ1D /Γ0 can be mapped
to the cooperativity
(2𝑔) 2
C=
(1.3)
𝜅𝛾
Achieving cooperativity exceeding the unity (i.e., C > 1) where the atom-field
interaction overwhelms the dissipative processes, known as the strong-coupling
regime of cavity QED, is the key to observing exotic quantum phenomena, which
has been successfully demonstrated in a variety of platforms.
The field of cavity QED was pioneered by Prof. Serge Haroche at Ecole Normale
Supérieure in the microwave domain by using Rydberg atoms with superconducting
cavities [31] (see Fig. 1.2b). A parallel endeavor in the group of Prof. Jeff Kimble at
Caltech [32] was pursued in the optical domain by employing high-finesse optical
Fabry-Pérot cavities [33] (see Fig. 1.2c) and microtoroidal resonators [34]. The
implementation of cavity QED is not restricted to atomic systems but more generally achievable with artificially engineered quantum emitters such as quantum dots,
color centers in diamond, rare-earth ions, or superconducting qubits. In particular,
the cavity QED framework revisited in systems of superconducting qubits and microwave resonators was specifically named the circuit QED [35–38] (see Fig. 1.2d),
laying the groundwork for numerous efforts to build a quantum computer.
Alternative approaches to realize a strong atom-photon interaction
While cavity QED has been the predominant approach for achieving strong atomphoton interactions at the single-photon level, there are several drawbacks to this
approach as the system size grows. First, cavities are intrinsically finite-sized
objects as they are physically confined the electromagnetic structure. Represented
as a zero-dimensional object (node) in a graph, scaling up such systems will require
envisioning a network of quantum nodes [26, 39]. Another consideration is related
to applications where a high bandwidth is necessary for inducing photon-photon
interactions [40], which will inevitably degrade the cooperativity defined in Eq. (1.3).
Various 1D-scalable approaches to build strongly coupled quantum light-matter
interfaces ensued for these reasons.
One alternative method is to utilize the cooperative enhancement of coupling of an
ensemble of atoms [41]. From the early days in quantum physics, it has been shown
that the radiation of a collection of emitters much faster and stronger than that of
a single atom, known as the Dicke superradiance [42, 43]. Utilizing the collective

excitation of an atomic ensemble it is possible to realize the cooperative Γ1D scaling
linearly with the number of atoms.
Another promising approach is based on utilizing guided modes of an engineered
waveguide, called the waveguide QED [44, 45], which is being investigated in a variety of physical platforms including atoms coupled to optical nanofiber or nanophotonic structures. Our approach to waveguide QED experiments in superconducting
quantum circuits is realized by employing superconducting qubits coupled to onchip microwave transmission lines or engineered superconducting metamaterial
waveguides. This topic will be discussed in a greater depth in Chapter 2.
Novel types of light-matter interface
In addition to the directions mentioned above, there has been various efforts to
realize exotic types of quantum light-matter interface employing novel properties
of photonic baths [46, 47]. Here, I enumerate a few interesting examples related to
this approach.
First, the spin-momentum locking of strongly confined guided modes in nanophotonic structures are being studied as chiral light-matter interfaces [46]. This was
experimentally investigated in cold atoms coupled to a nanofiber [48] or coupled via
whispering-gallery mode of a microsphere resonator [49] and quantum dots coupled
to a nanophotonic waveguide [50]. In such settings, the unidirectional coupling of
quantum emitters to a 1D photonic mode realizes a cascaded quantum system [51]
which can be used to perform steady-state generation of entangled states of emitters
with driven-dissipative methods [52].
Second, there has been novel efforts to synthesize strongly-correlated quantum
matter in topological photonic structures that breaks the time-reversal symmetry.
For example, a twisted optical cavity where photons experience artificial gauge field
together with Rydberg atom-mediated photon-photon interactions were employed
to create fractional quantum Hall states of light [53]. In line with this, a Chern
insulator made of a two-dimensional array of 3D superconducting cavities with
superconducting qubits are being investigated [54].
Finally, the topological property of photonic bath can alter the properties of quantum
emitters, which can be used to imprint novel types of interaction leading to an exotic
quantum many-body state [55, 56]. Our work on the topological waveguide QED
will be discussed in this context in Chapter 6.

1.2

Quantum computation

The emergence of the field of quantum computation [57] dates back to 1982 when
Richard Feynman discussed the problem of simulating quantum systems [58]. He
explained the infeasibility of simulating quantum systems with a classical computer
and postulated a computer based on quantum systems to achieve this. It is an
interdisciplinary field of science associated with the convergence of three major
scientific areas that appeared in the 20th century—quantum theory, computer science,
and information theory.
Power of quantum computation
The fact that quantum systems are hard to simulate with a classical computer implies
that quantum computers capable of solving certain kinds of classically intractable
problems (NP-hard), the most trivial example of which is the simulation of quantum
many-body systems discussed in Sec. 1.3. However, the power of quantum computation was not generally appreciated until 1994 when Peter Shor discovered an
efficient algorithm for prime factorization on a quantum computer [59], which is
otherwise prohibitively hard to tackle to date. This was soon followed by the discovery of another important quantum algorithm for search of unstructured database,
known as the Grover’s algorithm [60], which drew a wider attention to the field.
Even today, many applications of quantum computing are being envisioned, further
strengthening the potentials of quantum computers.
Quantum error correction
Contrary to the optimistic perspectives on quantum computing from its potential
computational power, there has been many skeptical views associated with its feasibility [61]. Practically, quantum computation is experimentally a daunting task due
to the conflicting requirement that qubits strongly interact with each other while the
interaction of qubits to the environment has to be greatly suppressed except when
we interrogate qubits for control and measurement. The inevitable build-up of coherent and incoherent errors during quantum computation will make the approach
unscalable, making the sophisticated quantum algorithms ineffectual.
In order to resolve this issue, ideas on quantum error correction were conceived from
the early days in the field of quantum computation [62, 63]. The basic principles of
quantum error correction is to encode logical quantum information in a subspace of
Hilbert space of a larger number of physical qubits, thereby using quantum entanglement to introduce sufficient redundancy. Measurement of error syndromes (few

combinations of multi-qubit Pauli operators) allows one to detect and correct bitflips (𝑋-type errors) and phase-flips (𝑍-type errors) without destroying the original
quantum information. However, one needs to consider the errors associated with the
error correction process itself. Fault-tolerant protocols are proven to allow reliable
quantum computation even with errors provided the error rate per physical gate or
time step is below some constant threshold value, known as the threshold theorem
[64]. Therefore, quantum error correction serves as an important foundation for a
scalable quantum computation.
A widely accepted viable approach for quantum error correction is the surface code
[65, 66], which is based on topological error correcting codes named toric code
invented by Sergey Bravyi and Alexei Kitaev [67]. This scheme utilizes physical
qubits laid out in a 2D checkerboard pattern alternating between data qubits and
measure qubits. The measure qubits themselves alternates between 𝑋 syndrome and
𝑍 syndrome qubits for measurement of 𝑋 𝑋 𝑋 𝑋 and 𝑍 𝑍 𝑍 𝑍 operators of adjacent
data qubits, respectively, and the +1 eigenspace of all syndrome operators define
the logical code space. The surface code is considered the most feasible direction
in quantum error correction, requiring only nearest-neighbor connectivity between
large number of physical qubits and tolerating high error rate on the order of ∼ 1 %
[66] compared with other error-correcting codes.
Another interesting direction is to encode a logical quantum bit of information in
levels of a harmonic oscillator, known as the bosonic error correction [68, 69]. A
harmonic oscillator has an infinite number of evenly spaced levels and in principle
can provide hardware-efficient redundancy for encoding in contrast to approaches
involving multiple physical qubits described above. In addition, the dominant error
in harmonic oscillators is the photon loss error 𝑎, further simplifying the kind of
error that needs to be corrected. Examples of well-known bosonic codes include
the Gottesman-Kitaev-Preskill (GKP) code [70] (analog of quadrature-amplitude
modulation), the cat code [71] (analog of phase-shift keying), and the binomial
code [72] where a logical quantum bit of information is encoded in superposition
of position/momentum eigenstates, coherent states, and Fock states, respectively. In
particular, assisted by the low error rates of 3D circuit QED architectures, bosonic
error correction with cat code has reached a milestone of break-even [73] where the
lifetime of logical qubit was comparable to that of the physical constituent (cavity
photon).

Figure 1.3: Stages of quantum information processing. Seven stages of quantum information processing described by Devoret and Schoelkopf. The figure is taken from Ref. [74].

Experimental approaches to quantum computation
Experimental efforts to build a quantum computer started with technical development of building blocks of quantum computation and proof-of-principle experiments
to demonstrate quantum algorithms in a variety of platforms including nuclear magnetic resonance (NMR), trapped ions, superconducting qubits, neutral atoms, and
photons, to name a few. In 2013, pioneers in superconducting qubits Michel Devoret and Robert Schoelkopf at Yale University presented their perspectives on the
development stages of quantum information processing [74], illustrated in Fig. 1.3.
They argued that the three first stages associated with single- and multi-qubit operations as well as the quantum non-demolition measurement has been achieved but the
next step—demonstrating a logical qubit with a lifetime greater than the underlying
physical qubits—poses significant technical challenges.
Systematic efforts to accomplish an error-corrected logical qubit are under way in
both the academic and industrial fronts, but realizing a logical qubit outperforming
physical qubits persists to be a difficult task due to technical challenges associated
with various error processes. In this respect, Prof. John Preskill at Caltech coined the
term noisy intermediate-scale quantum (NISQ) to describe [75] the current progress
of the field with quantum processors of ∼ 100 qubits without error correction. The
NISQ devices will be complicated enough to be simulated with the most powerful
classical computers in the world, best represented by the quantum supremacy ex-

periment [76], but nevertheless limited by noise to harness the full computational
power of quantum computers.
In parallel with the frontier to push the boundary of quantum error correction, new
types of building blocks for quantum computation are continually being developed.
Examples of this include multi-mode circuit QED [77, 78] and schemes for scaling up
error-corrected bosonic systems [79]. In line with this, we will describe a waveguidebased high-bandwidth approach to realize a scalable long-range interactions between
qubits in Chapter 7, which remains relatively unexplored.
1.3

Quantum many-body physics

Understanding strongly correlated phases of matter has been a long-standing goal
in the field of condensed matter physics, with the famous examples of fractional
quantum Hall effect [80] and high-𝑇𝑐 superconductivity [81], first discovered in
the 1980s, still lacking established microscopic theories. Coherent control over
engineered quantum systems provides a novel alternative approach for tackling such
formidable tasks by enabling the quantum simulation of many-body phenomena
[82, 83]. In addition, quantum control experiments are central to the study of
fundamental topics ranging from quantum thermalization [84, 85] and dynamics of
quantum entanglement [86] to novel quantum phases of matter.
Pioneering experiments in quantum simulation utilized laser-cooled and trapped
cold neutral atoms to study Bose- and Fermi-Hubbard models [87]. Quantum gas
microscopes with in-situ imaging at single-atom and single-site resolution [88, 89]
enabled explorations of quantum correlations of many-body systems [90–92], followed by a bottom-up approach to assemble arrays of atoms trapped in optical
tweezers [12, 15]. In parallel, various quantum simulation experiments were performed using trapped ions [93, 94], resulting in numerous impressive studies in
quantum information propagation [95], dynamical quantum phase transition [96],
and discrete time crystals [97, 98], to name a few.
Superconducting quantum circuits have recently emerged as a promising platform
to study quantum many-body physics. Compared to the traditional AMO systems, superconducting qubits offer new possibilities to study higher-order quantum
many-body effects at a high repetition rate, with fully arbitrary local qubit control,
quantum non-demolition readout, and real-time feedback control without technical
difficulties associated with cooling and trapping atoms. Investigations of quantum
many-body physics in superconducting circuits took place in architectures with cou-

Light-matter interface

Quantum Computation

- Cavity QED

- Single-qubit operation

- Waveguide QED

- Multi-qubit operation

- Topological photonics
- Metamaterial

Our
Work

- QND measurement
- Real-time feedback

Many-body Physics
- Dynamics of quantum entanglement
- Novel phases of matter
- Quantum thermalization

Figure 1.4: Overview of our work. The work in the thesis is viewed from a broad context
of quantum light-matter interface, quantum computation, and quantum many-body physics,
described in Secs. 1.1-1.3.

pling limited to nearest neighbor in 1D [99–101] and 2D [102, 103] lattice or with
resonator-mediated all-to-all coupling [104–106] which is not scalable. Our quantum processor based on superconducting metamaterials discussed in Chapter 7 goes
beyond this by realizing a scalable approach to create tunable long-range interaction
between qubits.
1.4

Outline of the thesis

My doctoral research at Caltech lies at the intersection of broad disciplines of science
discussed above, as illustrated in Fig. 1.4. In particular, we perform experimental
studies of the burgeoning field of waveguide QED by utilizing micro-fabricated
superconducting quantum devices. Below, I provide a brief description on the
organization of the thesis.

11
First, I will begin the thesis by introducing the field of waveguide QED. Specifically,
I will motivate the transition from cavity QED to waveguide QED and discuss the
basic concepts and various physical realizations of waveguide QED in Chapter 2.
Next, I will describe the experimental techniques which have been employed
throughout the work in the thesis in Chapter 3. This includes the device fabrication
procedure, microwave packaging of device, cryogenic setup, and room-temperature
electronics that are standard in the field of superconducting quantum circuits [107]
or newly developed in the lab.
In the remaining parts of the thesis, I will provide detailed descriptions on the
experiments performed during my PhD:
• In Chapter 4, I discuss the interaction between superconducting qubits mediated by a common waveguide channel in the strong-coupling regime, which
is published in Ref. [108].
• In Chapter 5, a new approach to scaling up quantum processors with a compact waveguide QED architecture based on superconducting metamaterials is
explored, which is published in Ref. [109].
• In Chapter 6, we utilize our ability to engineer a novel topological photonic
structure and demonstrate an exotic type of qubit-qubit interaction mediated
by this structure, which is published in Ref. [56].
• In Chapter 7, I will describe our current progress on building a large-scale
quantum processor based on superconducting metamaterials.
Finally, I will conclude the thesis by discussing the implications of our work to the
field and by providing an outlook and future directions in Chapter 8.

12
Chapter 2

WAVEGUIDE QUANTUM ELECTRODYNAMICS
We have discussed the historical context and overview of quantum-light matter
interfaces in Sec. 1.1 with a gentle introduction to the field of waveguide QED. In
this chapter, we provide a more detailed description of waveguide QED systems by
explaining the theory and the experimental realizations.
2.1

From cavity QED to waveguide QED

Cavity quantum electrodynamics (QED) studies the interaction of a quantum emitter
with a single electromagnetic mode of a high-finesse cavity with a discrete spectrum
[28, 29]. In this canonical setting, a large emitter–photon coupling is achieved
by repeated interaction of the emitter with a single photon bouncing many times
between the cavity mirrors. Coupling multiple quantum emitters to a common cavity
is also shown to induce exchange-type photon-mediated interactions between the
emitters [110–112], realizing a quantum information processing architecture with
all-to-all connectivity. The cavity is formed by confining the electromagnetic field

···

DOS

FSR

···

Frequency

···

DOS

···

Frequency
Figure 2.1: Transition from cavity QED to waveguide QED. a, Left: a cavity QED
system where quantum emitters are coupled to a common cavity. Right: the density of
states (DOS) inside a cavity plotted against the frequency shows a set of peaks associated
with discrete modes spaced by free spectral range (FSR). The FSR is inversely proportional
to the physical length scale of the cavity and hence the size of the cavity cannot grow
indefinitely while maintaining the single-mode picture. b, Left: a waveguide QED system
where quantum emitters are coupled to a common 1D waveguide channel. Right: the DOS
of a waveguide plotted against the frequency shows a continuum of modes.

13
into a small volume and therefore has a finite length scale 𝐿 that limits the number
of quantum emitters that can be simultaneously coupled to the cavity. Increasing
the size 𝐿 of the cavity inevitably comes at the expense of breakdown of the singlemode picture due to the decrease in free spectral range (FSR), which is inversely
proportional to the system size (Fig. 2.1a). Therefore, one needs to consider an
alternative strategy of connecting multiple cavity QED systems in order to scale up,
outlined in Prof. Jeff Kimble’s description of quantum internet [26].
Recently, there has been much interest in achieving strong light–matter interaction
in a cavity-free system such as a waveguide [40, 44, 45]. Waveguide QED refers to
a system where a chain of quantum emitters are coupled to a common 1D photonic
channel with a continuum of electromagnetic modes over a large bandwidth, visualized in Fig. 2.1b. While it is more challenging to achieve strong emitter-photon
interaction than the cavity QED scenario, utilizing waveguide-based architectures
provides a natural solution for scaling up the system in 1D. Also, the high bandwidth of the system enables using propagating photons as the basis for quantum
information processing [40, 113].
2.2

Quantum emitters coupled to a waveguide with linear dispersion

A canonical waveguide QED system consists of an array of quantum emitters which
are interfaced with a common 1D waveguide channel with linear dispersion. In
this scenario, the quantum emitters are characterized by the decay rate Γ1D to the
desired waveguide channel and the parasitic decay rate Γ0, as illustrated in Fig. 2.2a.
The most important figure of merit in such systems representing the strength of
emitter-photon coupling is the Purcell factor [114, 115], given by
𝑃1D ≡

Γ1D
Γ0

(2.1)

which characterizes the collection efficiency of the emitter’s radiation to the desired
waveguide channel. In the following, I provide a heuristic overview of the basic
processes in this setting. A more general description of the system in the form of
input-output theory is discussed in Refs. [116, 117].
Single quantum emitter coupled to a waveguide
The simplest scenario in waveguide QED is when a single quantum emitter is
coupled to the waveguide as illustrated in Fig. 2.2a. Even in this simple setting,
the interaction between the emitter and photons in the waveguide can give rise to
various output fields depending on photon statistics of the incident field.

â→
in

â→
out

Γ1D

b 1.0

c 1.0

R(∆)

â←
out

0.5

0.0
1.0
P1D

T (∆)

0.0
1.0

0.5

0.1
10
100

0.5
0.0
−2

Ω/Γ
0.01
0.3
0.5
10

0.5
0.0

−1

∆/Γ

−2

−1

∆/Γ

Figure 2.2: Single quantum emitter coupled to the waveguide. a, A quantum emitter is
coupled to a 1D waveguide channel with decay rates Γ1D to the waveguide and Γ0 to other
spurious channels. b, The reflectance 𝑅(Δ) and transmittance 𝑇 (Δ) associated with singlephoton scattering are plotted as a function of detuning Δ normalized to the total decay rate
Γ = Γ1D + Γ0, for a few different values of Purcell factor 𝑃1D = {0.1, 1, 3, 10, 100}. c, The
reflectance 𝑅(Δ) and transmittance 𝑇 (Δ) of coherent field are plotted against normalized
detuning Δ/Γ with Purcell factor of 𝑃1D = 100, for a few different values of drive amplitude
Ω/Γ = {0.01, 0.3, 0.5, 1, 10}.

Single-photon scattering
We first consider the effects in the linear regime where the saturation of quantum
emitters is negligible, which can be realized by using a single-photon wavepacket
or a weak coherent field as input to the system. The reflectivity and transmittivity
in this case can be written as [115, 118, 119]
𝑟 (Δ) =

Γ1D /2
𝑖Δ − (Γ1D + Γ0)/2

𝑡 (Δ) = 1 + 𝑟 (Δ),

(2.2)

where Δ = 𝜔−𝜔0 is the detuning of the probe photon from the resonant frequency 𝜔0
of the emitter. The transmission and reflection spectrum, represented by reflectance
𝑅(Δ) ≡ |𝑟 (Δ)| 2 and transmittance 𝑇 (Δ) ≡ |𝑡 (Δ)| 2 , have the Lorentzian lineshapes
centered at the emitter’s resonance as illustrated in Fig. 2.2b. On resonance Δ = 0
(𝜔 = 𝜔0 ), the reflectance and the transmittance can be simplified into
2
𝑃1D
𝑅(Δ = 0) =
, 𝑇 (Δ = 0) =
(2.3)
1 + 𝑃1D
(1 + 𝑃1D ) 2

15
indicating that the emitter becomes reflective 𝑅 ≈ 1 (𝑇 ≈ 0) in the regime of high
Purcell factor 𝑃1D
1. In other words, a single quantum emitter with high Purcell
factor can act as a near-perfect mirror for a single photon resonant to the emitter
propagating along the waveguide. This can be attributed to the destructive interference of the incident photonic wavepacket with the emitter’s radiation in the forward
direction. The on-resonance extinction measured from low-power spectroscopy of
a single quantum emitter offers a simple method to extract the Purcell factor from
experiments. It is interesting to note that the mirror-like property of quantum emitters at the single-photon level can be leveraged to create a cavity-like confinement
of photonic excitation in a finite segment of the waveguide, allowing us to revisit
cavity QED-like effects in a waveguide QED setting [109, 120] (see also Chapter 4).
Multi-photon scattering
Using a multi-photon input to the system can result in various non-trivial effects
due to the interference of emitter’s radiation with the incident field from which the
emitter can absorb only one photon at a time. For example, it has been shown
that a degenerate two-photon state input results in three possible cases of strongly
correlated output photon pairs consisting of (i) a pair of anti-bunched photons that
are reflected, (ii) a pair of bunched photons that are transmitted (two-photon bound
state), and (iii) a pair of transmitted and reflected photons propagating in the opposite
directions [121, 122].
Inputting a coherent state to the system, the reflection and transmission spectrum in
Eq. (2.2) are also modified, collecting a factor associated with the saturation of the
emitter [115]:

Γ1D
𝑟 (Δ) = −

𝑖Δ + Γ1D2+Γ
2
Γ1D +Γ 0
Ω2
Δ +
+ 2

𝑡 (Δ) = 1 + 𝑟 (Δ).

(2.4)

Here, Ω denotes the Rabi frequency, the rate at which the emitter is driven by the
input field. The effect of saturation is shown in Fig. 2.2c. Setting Ω = 0 reduces
Eq. (2.4) into the earlier case of single-photon scattering in Eq. (2.2), while larger
spectroscopy power increases the effective linewidth of Lorentzian curve due to
saturation.
Cooperative effects in waveguide QED
When multiple quantum emitters are coupled to a common waveguide channel,
the exchange of photons in the waveguide between emitters result in cooperative

Γ1D,3

Γ1D,2

Γ1D,1

Γ3

Γ2

Γ1

···
x1

ΓN

···

Γ1D,N

···

Figure 2.3: Multiple quantum emitters coupled to a waveguide. Quantum emitters are
coupled to a waveguide channel at positions 𝑥 = 𝑥 𝑗 along the waveguide, with waveguide
decay rate Γ1D, 𝑗 and spurious decay rate Γ0𝑗 (1 ≤ 𝑗 ≤ 𝑁).

effects such as correlated decay and exchange interaction [116, 120]. For simplicity,
we consider the case where quantum emitters are placed along the waveguide at
position 𝑥 = 𝑥 𝑗 , with identical transition frequency 𝜔0 , waveguide decay rates
Γ1D, 𝑗 , and spurious decay rate Γ0𝑗 , as illustrated in Fig. 2.3 ( 𝑗 = 1, 2, · · · , 𝑁). The
photons propagating along the waveguide is assumed to have group velocity 𝑣 (i.e.,
wavelength of 𝜆 = 2𝜋𝑣/𝜔0 or wavevector 𝑘 = 𝜔0 /𝑣). In cases where propagation
delay is negligible compared to the timescale of radiation of emitters, i.e., 𝑣
Γ1D 𝐿
where 𝐿 is the length scale associated with the largest distance between emitters,
the Born-Markov approximation holds and we can describe these processes with a
Lindblad master equation after tracing out the photonic degrees of freedom [116,
120]:
𝑖 ˆ
𝜌 = − [ 𝐻,
𝜌] + L [𝜌].
(2.5)
d𝑡
Here, 𝜌 is the density operator in the subspace of emitters, 𝐻ˆ is the Hamiltonian
describing the coherent processes and L is the Liouville superoperator specifying
the incoherent processes.
Correlated decay
The correlated decay is caused by the cooperative emission of real photons to the
waveguide channel from an ensemble of emitters. For example, a photon emitted
from a quantum emitter can propagate to the other emitter at a distance 𝑑 along
the waveguide by collecting a phase factor 𝑒𝑖𝑘 𝑑 before interacting with the second
emitter, and vice versa. Since the emitters share the same waveguide bath of
photons, the emission from multiple emitters can be coherently added, resulting
in interference of emission processes. A prime example of the correlated decay
is the Dicke superradiance [42, 43] where the emission of an ensemble of excited
emitters distributed within distances much smaller than the resonant wavelength
(𝑑
𝜆) radiates at a rate much faster than that of a single emitter. Such processes

Γ1D

(i) Γ

(ii) Γ

Γ1D

···

λ/2

···

1.0

R(∆)

0.5
0.0
1.0

T (∆)

0.5
0.0
−2

−1

∆/Γ

Figure 2.4: Superradiance and subradiance of two quantum emitters. a, Two quantum
emitters are coupled to a waveguide with a separation of (i) 𝑑 = 𝜆/2 or (ii) 𝑑 = 𝜆. The
electric field of a resonant photon propagating along the waveguide is illustrated, with
regions of positive (negative) value shaded in red (blue). Due to the phase associated
with photon propagation, the anti-symmetric (symmetric) superposition of single excitation
states of emitters is superradiant to the waveguide when 𝑑 = 𝜆/2 (𝑑 = 𝜆). The other
superposition state with opposite symmetry is subradiant to the waveguide, having zero
waveguide decay rate. b, The reflectance 𝑅(Δ) and transmittance 𝑇 (Δ) of such two-emitter
waveguide QED system plotted against the detuning Δ normalized to the total decay rate
Γ = Γ1D + Γ0 of a single emitter (red solid lines). The black dashed lines represent the
spectra for a single-emitter waveguide QED system. Here, it is assumed that the Purcell
factor is 𝑃1D = 100.

in waveguide QED can be described by a Liouville superoperator Lc given by
1 + −
Lc [𝜌] =
Γ𝑖 𝑗 𝜎
ˆ 𝑖 𝜌𝜎
ˆ 𝑗 − {𝜎
ˆ 𝜎
ˆ , 𝜌}
(2.6)
2 𝑗 𝑖
𝑖, 𝑗
where
Γ𝑖 𝑗 =

Γ1D,𝑖 Γ1D, 𝑗 cos (𝑘 |𝑥𝑖 − 𝑥 𝑗 |).

(2.7)

The diagonal terms (𝑖 = 𝑗) in Eq. (2.7) correspond to the self-decay of an emitter to
the waveguide that was present in the case of single emitter, i.e., Γ𝑖𝑖 = Γ1D,𝑖 . The
off-diagonal terms (𝑖 ≠ 𝑗) in Eq. (2.7) represent the rate of correlated decay which
is proportional to the geometric mean of waveguide decay rates Γ1D of a pair of
emitters and a sinusoidal function dependent on the phase 𝜙𝑖 𝑗 = 𝑘 |𝑥𝑖 − 𝑥 𝑗 | associated
with propagation of a photon along the waveguide between the emitters.
Restricting our analysis to the case of two emitters with identical waveguide decay
rate Γ1D, 𝑗 = Γ1D and spurious decay rate Γ0𝑗 = Γ0 for 𝑗 = 1, 2, it can be shown that
the magnitude of correlated decay rate is maximized if the emitters are separated
by a integer multiple of half-wavelength, i.e., 𝑑 = |𝑥 1 − 𝑥 2 | = 𝑛𝜆/2 where 𝑛 is an
integer (equivalently, 𝑘 𝑑 = 𝑛𝜋). In this case, Equation (2.6) can be rewritten as
1 + −
Lc [𝜌] =
Γ1D,𝜇 𝜎
ˆ 𝜇 𝜌𝜎
ˆ 𝜇 − {𝜎
ˆ 𝜎
ˆ , 𝜌}
(2.8)
2 𝜇 𝜇
𝜇=𝐵,𝐷

18
in terms of new operators
ˆ 1± + (−1) 𝑛 𝜎
ˆ 2±
ˆ 1± − (−1) 𝑛 𝜎
ˆ 2±
, 𝜎
ˆ𝐷 =
(2.9)
and modified waveguide decay rates Γ1D,𝐵 = 2Γ1D and Γ1D,𝐷 = 0. This means that
an entangled single-excitation state
ˆ 𝐵± =

|𝑒i1 |𝑔i2 + (−1) 𝑛 |𝑔i1 |𝑒i2
(2.10a)
has a superradiantly enhanced decay rate equal to the sum of decay rates of emitters,
hence the name bright state (also referred to as superradiant state). It can be shown
that the effective Purcell factor of the two-emitter bright state |𝐵i is 𝑃1D,𝐵 = 2Γ1D /Γ0,
two times the single-emitter Purcell factor1. On the other hand, the other state with
the opposite symmetry
|𝐵i ≡ 𝜎
ˆ 𝐵+ |𝑔i1 |𝑔i2 =

|𝑒i1 |𝑔i2 − (−1) 𝑛 |𝑔i1 |𝑒i2
(2.10b)
named dark state (also referred to as subradiant state), is effectively decoupled
from the waveguide channel and therefore has zero waveguide decay rate. The
subspace of Hilbert space composed of such non-radiative states to the waveguide
channel is referred to as the decoherence-free subspace, consisting of the only
long-lived states in a waveguide QED system which could be used as resources for
quantum information processing [123]. Note that the bright (dark) state has the
same (opposite) symmetry as a photon propagating along the waveguide, collecting
a phase factor 𝑒𝑖𝑘·𝑛𝜆/2 = (−1) 𝑛 , as illustrated in Fig. 2.4a. It can be also shown
that the coherent exchange interaction between the emitters is zero at such distances
𝑑 = 𝑛𝜆/2 (see below), so the self and the correlated decay alone are sufficient to
describe the full picture.
|𝐷i ≡ 𝜎
ˆ 𝐷+ |𝑔i1 |𝑔i2 =

The signature of superradiance can be measured with spectroscopic methods at low
enough probe power. It can be shown that the transmission and reflection spectrum of
such two-qubit system shows a Lorentzian lineshape with full-width half-maximum
linewidth of 2Γ1D corresponding to the bright state |𝐵i, as shown in Fig. 2.4b. The
dark state |𝐷i, decoupled from the waveguide channel, cannot be probed by using
the waveguide channel and therefore cannot be easily accessed. A novel scheme to
utilize waveguide-mediated coherent interaction to perform coherent control over
the inaccessible dark state |𝐷i, proposed by Darrick Chang and colleagues [120],
will be experimentally investigated in Chapter 4 [108].
1 More generally, the Purcell factor of the most superradiant state of an array of quantum emitters

is proportional to the number of emitters 𝑁.

19
|g 1 |g 2 â†k |{0}

(2)

(1)

∆k

|e 1 |g 2 |{0}

J12,k

|g 1 |e 2 |{0}

Figure 2.5: Coherent waveguide-mediated exchange interaction. Energy level diagram
of single-excitation subspace of a system involving two quantum emitters (state labeled by
subscript indices 1, 2) and a waveguide (state labeled by mode index 𝑘). The single-emitter
excited state |𝑒i1 |𝑔i2 |{0}i (|𝑔i1 |𝑒i2 |{0}i) of emitter 1 (2) is coupled to a single-photon state
|𝑔i1 |𝑔i2 𝑎ˆ †𝑘 |{0}i of mode 𝑘 at detuning Δ 𝑘 from the frequency of the emitters with coupling
rate 𝑔 𝑘(1) (𝑔 𝑘(2) ) (blue, solid). This induces virtual exchange between the two single-emitter
excited states at a rate 𝐽12,𝑘 = −𝑔 𝑘(1) 𝑔 𝑘(2) /Δ 𝑘 (red, dashed) for each off-resonant mode 𝑘,
without transfer of real photons.

Exchange interaction
There also exists a mechanism to induce coherent interactions between emitters
coupled to a common waveguide channel, known as the exchange interaction. This
process arises from virtual exchange of photons mediated by off-resonant modes
of the waveguide which form a continuum without transferring real photons to the
dissipative waveguide, as illustrated in Fig. 2.5.
The coherent exchange interaction can be described by a Hamiltonian 𝐻ˆ c written as
𝐻ˆ c =

𝐽𝑖 𝑗 𝜎
ˆ 𝑖+ 𝜎
ˆ 𝑗− ,

(2.11)

𝑖, 𝑗

where

1p
Γ1D,𝑖 Γ1D, 𝑗 sin (𝑘 |𝑥𝑖 − 𝑥 𝑗 |).
(2.12)
The diagonal terms (𝑖 = 𝑗) in Eq. (2.12) correspond to the self-interaction of an
emitter with itself which is zero 𝐽𝑖𝑖 = 0. The off-diagonal terms (𝑖 ≠ 𝑗) in Eq. (2.12)
represent the rate of exchange interaction which is proportional to the geometric
𝐽𝑖 𝑗 =

2 While the Eq. (2.12) predicts that the waveguide-mediated exchange interaction is infinite-

ranged, this form is obtained under the Born-Markov approximation in the limit where the propagation
delay is small compared to the inverse of relevant bandwidth for waveguide photons. At long
distances, the non-negligible propagation delay will significantly invalidate the assumption, resulting
in non-Markovian phenomena that cannot be described by a simple effective master equation in
Eq. (2.5).

20
mean of waveguide decay rates Γ1D of a pair of emitters and a sinusoidal function
dependent on the phase 𝜙𝑖 𝑗 = 𝑘 |𝑥𝑖 − 𝑥 𝑗 | associated with propagation of a photon
along the waveguide between the emitters. It is interesting to note that the form of
𝐽𝑖 𝑗 is purely sinusoidal without reduction at large inter-emitter distances, providing
potential for realizing long-range coherent interactions2.
Restricting our analysis to the case of two emitters with identical waveguide decay
rate Γ1D, 𝑗 = Γ1D for 𝑗 = 1, 2, it can be shown that the magnitude of exchange
interaction rate is maximized if the emitters are separated by a half-integer multiple of
half-wavelength, i.e., 𝑑 = |𝑥 1 −𝑥 2 | = (𝑛+1/2)𝜆/2 where 𝑛 is an integer (equivalently,
𝑘 𝑑 = 𝑛𝜋 + 𝜋/2) as illustrated in Fig. 2.6a. In this case, Equation (2.11) can be
rewritten as
Γ1D + −
(𝜎
ˆ1𝜎
ˆ2 + 𝜎
ˆ 2+ 𝜎
ˆ 1− ),
(2.13)
𝐻ˆ c = (−1) 𝑛
giving the maximal achievable interaction rate of 𝐽 = (−1) 𝑛 Γ1D /2 between the
two emitters. Note that the correlated decay rate Γ12 vanishes to zero in this
configuration, while the self decay of each emitter to the waveguide at the rate Γ1D
remains. Since the magnitude of coherent coupling rate cannot exceed the total decay
rate of individual emitter, i.e., 2|𝐽 | ≤ Γ = Γ1D + Γ0, it is intrinsically forbidden to
realize a strong coherent coupling of two emitters in waveguide QED, which would
have resulted in vacuum Rabi splitting in spectroscopic characterization. Instead,
the transmission and reflection spectrum of such a system shows the interference
of coherent coupling and strong decay, giving rise to a non-Lorentzian lineshape
illustrated in Fig. 2.6b.
Such intrinsic limitations forbid us from efficiently utilizing the long-range interactions, which could be useful for performing non-local gates in quantum computation
and also for the study of many-body physics. There has been two major ways to overcome this challenge. The first approach is to only use states that are non-radiative
to the waveguide channel, the decoherence-free subspace, with precise positioning of emitters along the waveguide. The second method is to make the emitters
themselves non-dissipative, which makes the protocol for controlling emitters much
simpler. One example of this is to utilize an emitter coupled to a waveguide at
multiple locations to allow the emitter’s radiation from multiple coupling points
to destructively interfere, known as the giant atom [124]. The other example is
to utilize emitter-photon bound states inside the bandgap regime of a dispersive
waveguide [125], discussed in Sec. 2.3

Γ1D

(ii) Γ

Γ1D

···

λ/4

R(∆)

(i) Γ

···

0.5
0.0
1.0

T (∆)

b 1.0

3λ/4

0.5
0.0

−2

−1

∆/Γ

Figure 2.6: Exchange interaction between two quantum emitters. a, Two quantum
emitters are coupled to a waveguide with a separation of (i) 𝑑 = 𝜆/4 or (ii) 𝑑 = 𝜆/3.
The electric field of a resonant photon propagating along the waveguide is illustrated, with
regions of positive (negative) value shaded in red (blue). Due to the phase associated with
photon propagation, the coupling of second emitter with a mode at a positive detuning from
the emitters’ frequency, having a shorter wavelength, will have a sign opposite (identical) to
that of the first emitter when 𝑑 = 𝜆/4 (𝑑 = 3𝜆/4), resulting in a positive (negative) exchange
interaction. b, The reflectance 𝑅(Δ) and transmittance 𝑇 (Δ) of such two-emitter waveguide
QED system plotted against the detuning Δ normalized to the total decay rate Γ = Γ1D + Γ0
of a single emitter (red solid lines). The spectra for a single-emitter waveguide QED system
is plotted with black dashed lines as a reference. Here, it is assumed that the Purcell factor
is 𝑃1D = 100.

2.3

Waveguide QED in a dispersive photonic channel

While the original concept of waveguide QED revolved around coupling quantum
emitters to a waveguide inside a transmission band, a new paradigm to induce
long-range photon-mediated interactions between quantum emitters inside a photonic bandgap was proposed in Ref. [125]. In this scheme, photonic structures are
engineered to host a bandgap where the propagation of photons are prohibited. Tuning the emitters’ frequencies inside the bandgap, the emitters cannot radiate to the
waveguide channel while the coherent interaction between distant emitters are still
allowed by means of exchange of virtual photons. Compared to the case when the
emitters are tuned inside the transmission band discussed in Sec. 2.2, the bandgap
regime offers a practical direction to achieve strong and long-range coupling between emitters without suffering from significant dissipation to the photonic band.
In this section, I introduce the basic concepts of this direction to achieve a scalable
quantum many-body system with long-range connectivity.
Emitter-photon bound states
When the transition frequency of a quantum emitter is tuned inside a photonic
bandgap of an electromagnetic structure, the spontaneous emission of the emitter
is forbidden due to the absence of resonant photonic modes to absorb the emitted

ωc
ω0
ωb

−π/d

π/d

Figure 2.7: Emitter-photon bound state. a, Illustration of a quantum emitter (green
circle) coupled to a 1D dispersive waveguide channel (gray) represented as a periodic
electromagnetic structure with lattice constant 𝑑. The emitter’s frequency is tuned inside
the bandgap of the waveguide, inducing a emitter-photon bound state exponentially localized
at a length scale 𝜉, shaded in blue. b, An example of photonic bandstructure of a dispersive
waveguide. The transmission band (bandgap) is shaded in red (gray). Tuning the bare
transition frequency 𝜔0 of the emitter (green arrow) below the frequency 𝜔 𝑐 of band-edge,
the emitter-photon bound state exists at a frequency 𝜔 𝑏 lower than that of the emitter (blue
arrow).

photons. Instead, the emitted photon is scattered back to the original emitter,
resulting in a coupled eigenstate of photonic modes of the electromagnetic structure
and the emitter. This is known as the emitter-photon bound state3, first predicted by
Sajeev John and Jian Wang in 1990 [126] in the context of Anderson localization of
light [127]. In the emitter-photon bound state, the emitter is dressed with a photonic
tail exponentially localized with respect to the emitter, gaining a spatial extent. This
length scale of the emitter-photon bound state can be adjusted by controlling the
detuning of the emitter from the edge of the photonic band.
The Hamiltonian of a quantum emitter coupled to such 1D photonic structure, a
dispersive waveguide (an example illustrated in Fig. 2.7), can be written as
Õ
𝐻 = ℏ𝜔0 |𝑒ih𝑒| +
ℏ𝜔 𝑘 𝑎ˆ 𝑘 𝑎ˆ 𝑘 +
ℏ 𝑔 𝑘 𝑎ˆ 𝑘 |𝑔ih𝑒| + 𝑔 𝑘 𝑎ˆ 𝑘 |𝑒ih𝑔| ,
(2.14)

where 𝜔0 is the transition frequency of the emitter, 𝑎ˆ 𝑘 (𝑎ˆ †𝑘 ) is the annihilation
(creation) operator of photonic mode at wavevector 𝑘 satisfying the canonical commutation relation [ 𝑎ˆ 𝑘 , 𝑎ˆ †𝑘 0 ] = 𝛿 𝑘,𝑘 0 , and 𝑔 𝑘 is the momentum-space coupling between
the emitter and the waveguide photons. Here, |𝑔i and |𝑒i denote the ground state
and the excited state of the emitter, respectively. Limiting our analysis to the single3 Also known as the atom (qubit)-photon bound state if an atom (a qubit) plays the role of a two-

level emitter.

23
excitation manifold, we look for a emitter-photon bound state of the following form
|𝜙 𝑏 i = cos 𝜃|{0}i|𝑒i + sin 𝜃

𝑐 𝑘 𝑎ˆ †𝑘 |{0}i|𝑔i

(2.15)

ˆ 𝑏 i = ℏ𝜔 𝑏 |𝜙 𝑏 i. Here,
that satisfies the time-independent Schrödinger equation 𝐻|𝜙
|{0}i represents the vacuum state of photon modes and the coefficients 𝑐 𝑘 are
normalized by 𝑘 |𝑐 𝑘 | 2 = 1. Writing out the algebraically independent terms in the
equation we get
𝜔0 cos 𝜃 +
𝑔 ∗𝑘 𝑐 𝑘 sin 𝜃 = 𝜔 𝑏 cos 𝜃
(2.16a)

𝜔 𝑘 𝑐 𝑘 sin 𝜃 + 𝑔 𝑘 cos 𝜃 = 𝜔 𝑏 𝑐 𝑘 sin 𝜃

(2.16b)

From equating the coefficients in Eq. (2.16b), we obtain the probability amplitudes
at wavevector 𝑘 to be
𝑔𝑘
(2.17)
𝑐𝑘 =
(𝜔 𝑏 − 𝜔 𝑘 ) tan 𝜃
Substituting the Eq. (2.17) into Eq. (2.16a), we get a transcendental equation for
evaluating the energy of the emitter-photon bound state
𝜔 𝑏 = 𝜔0 +

|𝑔 𝑘 | 2
𝜔𝑏 − 𝜔 𝑘

(2.18a)

|𝑔 𝑘 | 2
(𝜔 𝑏 − 𝜔 𝑘 ) 2

(2.18b)

subjected to the normalization condition
tan2 𝜃 =

For a generic 1D waveguide with a quadratic dispersion relation, a simple analytical
relation to describe the emitter-photon bound state can be derived. We specifically
consider the dispersion relation of the form
𝜔 𝑘 = 𝜔𝑐 + 𝛼(𝑘 − 𝑘 0 ) 2 ,

(2.19)

where 𝜔𝑐 is the frequency of the band-edge, 𝛼 > 0 is the curvature of the photonic
band, and 𝑘 0 is the wavevector at which the band-edge occurs.
Assuming that the coupling of emitter and the waveguide locally occurs at position
𝑥 = 𝑥0 with the strength 𝑔, the emitter-photon interaction Hamiltonian takes the
form 𝐻ˆ int = 𝑔( 𝑎ˆ †x0 |𝑔ih𝑒| + 𝑎ˆ x0 |𝑒ih𝑔|) in terms of real-space annihilation operator
𝑎ˆ 𝑥0 = √1 𝑘 𝑒𝑖𝑘𝑥0 𝑎ˆ 𝑘 where 𝑁 is the number of modes inside the band. This is

24
translated into the last term in Eq. (2.14) with momentum-space coupling given by
𝑔 𝑘 = 𝑔𝑒 −𝑖𝑘𝑥0 / 𝑁. In this case, the Eqs. (2.18a)-(2.18b) are simplified into4

and

𝑔2 𝑑
𝜔0 − 𝜔 𝑏 = p
2 𝛼(𝜔𝑐 − 𝜔 𝑏 )

(2.20a)

𝑔2 𝑑
𝜔0 − 𝜔 𝑏
tan2 𝜃 = p
4 𝛼(𝜔𝑐 − 𝜔 𝑏 ) 3 2(𝜔𝑐 − 𝜔 𝑏 )

(2.20b)

Here, 𝑑 is the shortest length scale (lattice constant) of the waveguide that determines
the first Brillouin zone. It can be seen from Eq. (2.20a) that the emitter-photon
bound state inside the bandgap (𝜔 𝑏 < 𝜔𝑐 ) has a frequency lower than the bare
emitter frequency (𝜔 𝑏 < 𝜔0 ) due to the negative Lamb shift from hybridization
with photonic modes at higher frequencies. Also, combining Eqs. (2.20a)-(2.20b),
it can be shown that the photonic component of the bound state becomes
−1
4 𝛼
3/2
sin 𝜃 = 1 + 2 (𝜔𝑐 − 𝜔 𝑏 )
𝑔 𝑑

(2.21)

This means that the emitter-photon bound state becomes more photon (emitter)-like
as the frequency get closer to (farther from) the band-edge frequency 𝜔𝑐 .
One can also evaluate the real-space coefficients 𝑐 𝑥 = √1 𝑘 𝑒𝑖𝑘𝑥 𝑐 𝑘 of photonic part
of the wavefunction |𝜙 𝑏 i using Eq. (2.17)5:
𝑔𝑑
𝑐𝑥 = − p
𝑒𝑖𝑘 0 (𝑥−𝑥0 ) 𝑒 −|𝑥−𝑥0 |/𝜉
2𝛼(𝜔0 − 𝜔 𝑏 )
where

𝜉=

𝜔𝑐 − 𝜔 𝑏

4 In the derivation, the summation Í

(2.22)

(2.23)

𝑘 over the first Brillouin zone was replaced with the integral
𝑑𝑘
in
the
thermodynamic
limit
(𝑁
→ ∞) whose upper and lower limits are extended from ±𝜋
2𝜋
to ±∞ assuming that the integration performed at high |𝑘 | values are negligible. Also, the integral
identities

𝑥
d𝑥
d𝑥 2
𝑎𝑥
−1 𝑥
tan
= 3 2
+ 3 tan−1
𝑎 +𝑥
𝑎 +𝑥
2𝑎 𝑎 + 𝑥
2𝑎

are employed.
5 Here, the integral identity
∫ ∞

𝑒 𝑖𝑎𝑥
d𝑥 2
= 𝜋𝑒 −|𝑎 |
𝑥 +1
−∞

is used.

(i)

ωc

(ii)
−π/d

π/d

Figure 2.8: Waveguide-mediated interactions between emitter-photon bound states.
When two quantum emitters are tuned inside the bandgap, the spatial overlap of corresponding emitter-photon bound states induces emitter-emitter interaction. The range of interaction
depends on the localization length 𝜉 which is long when the emitters are tuned close to the
band-edge frequency 𝜔 𝑐 (i, green arrow), while the interaction is short-ranged deep inside
the bandgap (ii, blue arrow).

is the localization length. Equations (2.22)-(2.23) directly shows a few important
properties of the photonic component of the bound state. First, the magnitude of
𝑐 𝑥 exponentially decays at a length scale 𝜉 with respect to the location 𝑥 = 𝑥 0 of
the emitter. The localization length 𝜉 characterizes the effective spatial extent of
the emitter-photon bound state. Second, the probability amplitude 𝑐 𝑥 collects an
additional factor 𝑒𝑖𝑘 0 (𝑥−𝑥0 ) associated with propagation with wavevector 𝑘 = 𝑘 0 of
the band-edge. Also, the localization length is inversely proportional to the detuning
of bound state from the band-edge, i.e., 𝜉 ∝ (𝜔𝑐 − 𝜔 𝑏 ) −1/2 , diverging as 𝜔 𝑏 → 𝜔𝑐 .
This means that the spatial extent of the emitter-photon bound state can take a wide
range of values depending on the frequency tuning of the emitter inside the bandgap.

Waveguide-mediated interactions between emitter-photon bound states
The spatially extended nature of the emitter-photon bound state has been viewed
as a method to induce effective interactions between emitters not long since the
first theoretical investigation of emitter-photon bound states [128, 129]. When two
resonant emitters are tuned inside the bandgap of a common photonic structure,
it was shown that the photonic band off-resonantly mediates exchange interaction
between the emitters in a way similar to how each emitter is dressed by photonic
modes to form the emitter-bound state (self-interaction). Such interaction takes
a special form that falls off exponentially with the distance between the emitters,
whose length scale 𝜉 can be tuned by adjusting the frequency of the emitters inside

𝑗,𝑘

(2.24)
( 𝑗)
where 𝜔0 is the frequency of the emitter 𝑗, 𝑎ˆ 𝑘 (𝑎ˆ †𝑘 ) is the annihilation (creation)

operator of photonic mode at wavevector 𝑘 satisfying the canonical commutation
( 𝑗)
relation [ 𝑎ˆ 𝑘 , 𝑎ˆ †𝑘 0 ] = 𝛿 𝑘,𝑘 0 , and 𝑔 𝑘 is the momentum-space coupling between the
emitter 𝑗 and the waveguide. Here, |𝑔i 𝑗 and |𝑒i 𝑗 denote the ground state and the
excited state of the emitter 𝑗, respectively.
Again, we look for a emitter-photon bound state in the single-excitation manifold of
the following form
|𝜙 𝑏 i = cos 𝜃|{0}i 𝑐 𝑞(1) |𝑒i1 |𝑔i2 + 𝑐 𝑞(2) |𝑔i1 |𝑒i2 +sin 𝜃
𝑐 𝑘 𝑎ˆ †𝑘 |{0}i|𝑔i1 |𝑔i2 (2.25)

ˆ 𝑏 i = ℏ𝜔 𝑏 |𝜙 𝑏 i. Here,
that satisfies the time-independent Schrödinger equation 𝐻|𝜙
|{0}i represents the vacuum state of photon modes and the coefficients 𝑐 𝑘 are
Í ( 𝑗)
normalized by 𝑘 |𝑐 𝑘 | 2 = 1 and 𝑐 𝑞 by 𝑗 |𝑐 𝑞 | 2 = 1. Writing out the algebraically
independent terms in the equation we get
( 𝑗)
( 𝑗) ( 𝑗)
( 𝑗) ∗
(2.26a)
𝜔0 𝑐 𝑞 cos 𝜃 +
𝑔 𝑘 𝑐 𝑘 sin 𝜃 = 𝜔 𝑏 𝑐 𝑞 cos 𝜃,

( 𝑗) ( 𝑗)

𝑔𝑘 𝑐𝑞

cos 𝜃 + 𝜔 𝑘 𝑐 𝑘 sin 𝜃 = 𝜔 𝑏 𝑐 𝑘 sin 𝜃.

(2.26b)

Equation (2.26b) can be simplified into an expression for coefficients 𝑐 𝑘 of photonic
modes:
Í ( 𝑗) ( 𝑗)
𝑗 𝑔𝑘 𝑐𝑞
𝑐𝑘 =
(2.27)
(𝜔 𝑏 − 𝜔 𝑘 ) tan 𝜃
Substituting Eq. (2.27) into Eq. (2.26a), we obtain an eigenequation for probability
( 𝑗)
amplitudes 𝑐 𝑞 of quantum emitters given by
𝑐 𝑞(1)
𝜔0(1) + 𝐽11
𝐽12
𝑐 𝑞(1)
𝜔 𝑏 (2) =
(2.28)
𝑐𝑞
𝐽21
𝜔0(2) + 𝐽22 𝑐 𝑞(2)
where

𝐽𝑖 𝑗 =

Õ 𝑔 (𝑖) 𝑔 ( 𝑗)

𝜔𝑏 − 𝜔 𝑘

(2.29)

27
The matrix 𝐽𝑖 𝑗 is Hermitian and represents the effective interaction between the
emitters mediated by photons of the transmission band. The diagonal terms (𝑖 = 𝑗)
correspond to self-interaction of a quantum emitter with itself, also known as the
Lamb shift identical to Eq. (2.18a).
Considering again a 1D waveguide with a quadratic dispersion relation described
in Eq. (2.19) where each emitter 𝑗 is coupled at position 𝑥 = 𝑥 𝑗 of the waveguide
( 𝑗)
(resulting in momentum-space coupling 𝑔 𝑘 = 𝑔𝑒 −𝑖𝑘𝑥 𝑗 / 𝑁), we can readily evaluate
the sum over 𝑘 in the thermodynamic limit. Following procedures similar to the
derivation of Eq. (2.22), Equation (2.29) reduces to
𝑔2 𝑑
𝑒𝑖𝑘 0 (𝑥𝑖 −𝑥 𝑗 ) 𝑒 −|𝑥𝑖 −𝑥 𝑗 |/𝜉 ,
𝐽𝑖 𝑗 = − p
2 𝛼(𝜔𝑐 − 𝜔 𝑏 )

(2.30)

where 𝜉 is the localization length defined in (2.23). It can be easily seen that the
interaction between emitters mediated by the dispersive waveguide in Eq. (2.30)
follows the spatial shape of emitter-photon bound state in Eq. (2.22), collecting a
phase factor 𝑒𝑖𝑘 0 Δ𝑥 and an attenuation factor 𝑒 −|Δ𝑥|/𝜉 associated with suppressed
propagation of a photon inside the bandgap along the displacement Δ𝑥 = 𝑥𝑖 − 𝑥 𝑗
between the emitters.
2.4

Experimental realizations of waveguide QED

The physical processes discussed in Secs. 2.2-2.3 are applicable to a generic system
consisting of two-level quantum emitters coupled to a 1D photonic waveguide
channel. This paradigm of waveguide QED has been experimentally realized in
various combinations of emitters and waveguides such as atoms coupled to a tapered
nanofiber [130, 131] or a nanophotonic waveguide [44], artificial atoms such as
quantum dots coupled to a nanophotonic waveguide [132], and superconducting
qubits coupled to a transmission line in the microwave domain, as illustrated in
Fig. 2.9. In this section, I will provide a brief introduction to various platforms
for waveguide QED experiments and compare the characteristics of the platforms,
summarized in Table 2.1.
Cold atoms
Experiments in waveguide QED were first performed in cold atomic systems by
tightly focusing a probe field toward a single trapped atom [24], where it was shown
that a single atom could cause an extinction of ∼ 10 % in transmission spectrum
of the field. However, as noted in Sec. 1.1 and in Eq. (1.1), achieving strong
atom-photon interactions with this approach is challenging due to a small atomic

1 mm

200 μm

Figure 2.9: Examples of experimental platforms for waveguide QED. a, Atoms are
interfaced with tapered optical nanofiber (top) or an alligator photonic crystal waveguide
(bottom). b, A quantum dot is interfaced with a photonic crystal waveguide. c, Superconducting qubits are interfaced with a coplanar waveguide transmission line (top) or a
machined 3D waveguide (bottom). The panel a is adapted from Refs. [133, 134]; b is
adapted from Ref. [132]; c is adapted from Refs. [108, 135].

Quantum Emitter
Atom

Waveguide
Nanofiber
PhC waveguide

Quantum dot
SC qubit

Plasmonic waveguide
On-chip TL
3D waveguide

𝑃1D

∼ 10−2 –10−1

∼ 103

∼ 10−1 –100
∼ 101 –102
∼ 100 –101
∼ 102
∼ 102

<5
∼ 10
<5

Ref.
[136]
[137]
[132]
[138]
[108]
[135]

Table 2.1: Experimental platforms for studying waveguide QED. State-of-the-art experiments in waveguide QED, performed in various combinations of quantum emitters and
waveguides, are summarized. Here, typical values of the highest achieved Purcell factor
𝑃1D and the number of resonant quantum emitters 𝑁 are compared between platforms. PhC:
photonic crystal, SC: superconducting, TL: transmission line.

scattering cross section 𝜎0 = 3𝜆20 /(2𝜋) at best comparable to the focused mode area
𝐴 ∼ 𝜆20 (diffraction limit). Also, this technique cannot be extended to the case of
many atoms since the beam waist diverges rapidly under tight focusing.
For these reasons, alternative approaches to achieve strong atom-photon interactions
were investigated using guided modes of 1D photonic structures which realizes small
mode area. Optical nanofibers with radius of few hundred nanometers were first
used to couple atoms to the evanescent tail of guided nanofiber mode [139]. Trapped
atoms near the nanofiber, it was shown that 𝑁 ∼ 103 atoms could be coupled to
the photonic mode [140]. However, suboptimal mode overlap between atoms and
photons at the evanescent tail result in small single-atom Purcell factor. To overcome
this challenge, a novel direction to employ engineered nanophotonic waveguides was

29
pursued. In particular, a Purcell factor on the order of unity was achieved for the
first time with atoms by interfacing with a photonic crystal waveguide engineered
to have the highest mode intensity at the atoms’ position [137, 141]. Also, a similar
structure was utilized to study atom-atom interactions inside the bandgap regime
discussed in Sec. 2.3, especially near the band-edge [142]. An outstanding challenge
in this approach is to trap the atoms near the nanophotonic structures [134, 143],
which has been difficult due to strong attractive forces at the dielectric surface [144].
Quantum dots
There has been numerous investigations to interface quantum dots with nanophotonic
structures [132] such as a photonic crystal waveguide or a plasmonic waveguide
[138], realizing single-emitter Purcell factors on the order of 10–100. However,
most of the studies to date were limited to the case of a single quantum dot with
photons being conceived as resources for quantum applications [145].
Superconducting qubits
Superconducting circuits offer a promising platform to study waveguide QED in the
microwave domain due to strong emitter-photon coupling and a wide variety of microwave photonic structures that could be fabricated on chip with high flexibility. In
superconducting circuits, the strong sub-wavelength confinement of electromagnetic
modes in transverse direction and large transition dipole moment of superconducting
qubits themselves result in high Purcell factors. The first notable waveguide QED
experiment in superconducting circuits was performed by Astafiev and colleagues
[146] where they have used a superconducting flux qubit coupled to a transmission
line on chip. The resonance fluorescence showed strong extinction in transmission
spectrum of 94 %, indicating a Purcell factor beyond unity 𝑃1D > 1. Experiments
involving multiple qubits [147] were also investigated, demonstrating strong cooperative interactions between qubits mediated by the waveguide channel [108]. There
has also been new efforts to utilize machined 3D microwave waveguides together
with multiple qubit chips placed along the waveguide [135, 148].
The current limitations of superconducting qubit platform is related to the scalability.
The wavelength at microwave frequencies (few GHz) is on the order of a few
centimeters and therefore channeling qubits with a microwave waveguide requires
a large footprint on chip or a large 3D enclosure. Therefore, a compact microwave
structure for waveguide QED must be envisioned [109] in order to increase the
effective size of the system. Also, additional wiring for individual addressing of

30
qubits is necessary to harness the full power of superconducting qubits, which
makes the scale-up more challenging. Nevertheless, with the technology developed
for state-of-the-art quantum processors [6, 76], realizing a waveguide QED system
with 𝑁 ∼ 102 superconducting qubits is expected to be feasible.

31
Chapter 3

DEVICE FABRICATION AND EXPERIMENTAL TECHNIQUES
Performing experiments with superconducting quantum circuits requires a broad
technical knowledge ranging from fabrication of device, designing packaging of the
device, engineering of cryogenic setup and microwave electronics, and experimental
methods to control and read out qubits. In this chapter, I will summarize the
techniques developed and utilized in the experiments described in the remaining
chapters.
3.1

Fabrication of superconducting quantum circuits

The devices used in the thesis are fabricated on 500 𝜇m-thick high-resistivity (>
10 kΩ · cm) silicon (Si) wafers diced into 1 cm × 1 cm or 2 cm × 1 cm chips1. As
the first step, alignment markers (positive 20 𝜇m square) are created by performing
electron-beam lithography on Zeon ZEP520A resist, development in Zeon ZED-N50
developer, and electron-beam evaporation of 150 nm-thick niobium (Nb). Next,
the ground plane, waveguides, resonators and qubit capacitors are patterned by
electron-beam lithography with the same resist and development procedure as above,
followed by electron-beam evaporation of 120 nm aluminum (Al). The Josephson
junctions for qubits are then patterned on MMA(8.5)MAA EL11 and 950 PMMA
A42 bilayer resist stack and fabricated using double-angle electron-beam evaporation
on suspended Dolan bridges [149], where 60 nm and 120 nm Al evaporation are
intervened by controlled exposure to oxygen. We then perform 5 min argon (Ar)
ion mill and 140 nm Al evaporation to form a bandage layer [150] that electrically
contacts the Al layers defined in previous steps. Finally, the airbridges are patterned
using grayscale electron-beam lithography on a trilayer resist stack consisting of
950 PMMA A93, MMA(8.5)MAA EL114, and 950 PMMA A9 and developed in
a mixture of isopropyl alcohol (IPA) and deionized (DI) water [151]. After 2 hr of
resist reflow at 105 ◦ C, electron-beam evaporation of 140 nm Al is performed at a
rate of 1 nm/s following 5 min of Ar ion milling. The bandage and the airbridge
steps are often combined into a single-layer process when only a small number of
1 The wafers are diced from Silicon Valley Microelectronics, Inc.
2 Polymethyl methacrylate with molecular weight of 950k, 4 % in Anisole.
3 Polymethyl methacrylate with molecular weight of 950k, 9 % in Anisole.
4 Mixture of PMMA and 8.5 % methacrylic acid, 11 % in Etyl Lactate.

Figure 3.1: Microwave packaging of quantum devices. a, A device is packaged in 16-port
packaging for 1 cm × 1 cm chips developed in 2017, consisting of a printed circuit board
(PCB) soldered to MMPX connectors and a machined aluminum enclosure. b, A device
is packaged in 26-port packaging for 2 cm × 1 cm chips developed in 2021, consisting of a
printed circuit board (PCB) soldered to SMPM connectors and a machined copper enclosure.

airbridges are necessary. Liftoff processes are performed at the end of each layer
in N-methyl-2-pyrrolidone at 160◦ C for at least 1.5 hr. For detailed descriptions on
each fabrication step, refer to App. A. We also implement additional calibrations
involving few rounds of test fabrication and imaging prior to fabrication of the
junction layer for accurate fabrication of Josephson junctions, which is outlined in
App. B.
3.2

Microwave packaging of quantum devices

The fabricated devices are wire-bonded to a printed circuit board (PCB) for electrical
connection to external control wiring and packaged inside a metal enclosure for
electromagnetic shielding and good thermalization. For high-fidelity control of
multiple qubits, it is crucial for microwave packaging to have low crosstalk and
to accommodate high wiring density without spurious modes [152]. Also, the
packaging must be mechanically robust and thermally stable to be able to undergo
multiple rounds of experiments with reconfiguration and thermal cycling. In this
section, I will introduce the microwave packages used for experiments described in
the thesis.
16-port packaging for 1 cm × 1 cm chips
The 16-port device packaging system illustrated in Fig. 3.1a was developed by
former undergraduate student Paul Dieterle in 2017 and has enabled a multitude of
successful experiments in Refs. [56, 108, 109, 153].

33
The core of the PCB in this packaging is Arlon AD1000 (dielectric constant 𝜖𝑟 = 10.2
and loss tangent tan 𝛿 = 0.0023 at 10 GHz) with the thickness of 20 mils (508 𝜇m),
0.5 oz (18 𝜇m) copper electrodeposited on the top surface, 2 oz (72 𝜇m) copper
electrodeposited on the bottom surface. The top and bottom copper films of the
PCB are connected by a large number of copper vias of diameter 10 mil (254 𝜇m)
which are periodically arranged in a two-dimensional grid spaced by 1 mm, in
order to minimize stray resonances. Planar microwave transmission lines in the
form of conductor-backed coplanar waveguide (CBCPW) [154] are patterned on
the PCB to interface the chip’s input and output ports to end-launch surface mount
MMPX connectors (Huber+Suhner 92_MMPX-S50-0-1/111_NM) with operating
frequencies up to 65 GHz. The PCB is electroplated with 50 𝜇in (1.27 𝜇m) of wirebondable soft gold without nickel (i.e., non-magnetic) to prevent oxidation and to
improve thermal contact with the enclosure. The designed PCB was manufactured
by Hughes Circuits, Inc. The MMPX connectors are soldered to the PCB by using
low-temperature Sn42/Bi57.6/Ag0.4 solder paste5 and a heat gun.
The enclosure (commonly referred to as “box”) for the chip and the PCB is manufactured by machining either oxygen-free high-conductivity (OFHC) copper (C101)
for best thermalization or aluminum alloy (6061-T651) for shielding against external magnetic field6. It consists of a top part (cover) and a bottom part (mount
plate) joined with UNC #2-56 brass screws. The cover has a 11.4 mm × 11.4 mm
× 1.5 mm rectangular opening above the chip surface. The mount plate was initially designed without opening but a pocket of 8 mm × 8 mm × 3 mm was milled
out in later revisions to suppress cavity resonances formed by the mount plate and
the ground plane of the chip due to high dielectric constant 𝜖𝑟 = 11.65 of the Si
substrate.
While many experiments were conducted in this packaging, there has been several
issues found over time. First, there is no mechanism to tightly mount the PCB to the
enclosure. This made the wirebonding of the chip to PCB challenging and is also
expected to make the thermalization of PCB inefficient. Second, there were many
cases when the MMPX connectors soldered to PCB “pop out” due to mechanical
force unintendedly acting on the connectors when closing the lid of the box, making
the assembly process very challenging. This is due to the over-constrained design of
5 Chip Quik SMDLTLFP, melting point 138 ◦ C.
6 Aluminum becomes a superconductor at temperatures below ∼1 K and can repel magnetic field

due to the Meissner effect.

34
the assembly associated with mating of connectors, PCB, and the enclosure. Finally,
MMPX connectors are costly relative to other kinds of RF connectors to be used
as consumables. These issues are addressed in the new packaging design which is
described below.
26-port and 16-port packaging for 2 cm × 1 cm chips
In 2021, we have developed a new packaging standard illustrated in Fig. 3.1b to
enable larger-scale experiments and to accommodate larger number of control lines.
In particular, packaging for 2 cm × 1 cm chips was conceived to enable fabrication of
longer superconducting metamaterial waveguides for constructing 10-qubit system
to study quantum many-body physics.
The PCB has a large footprint of 45 mm × 45 mm to allow for up to 26 RF connectors
along its perimeter. We use the same PCB core (Arlon AD1000 with 2 oz/0.5 oz
electrodeposited copper) as the old packaging version and pattern similar CBCPW
transmission lines to connect on-chip launchers to RF connectors. We place vias
of diameter 10 mil (254 𝜇m) as “fences” surrounding each waveguide trace and the
footprint of each RF connector to suppress spurious resonances. Also, the vias are
placed along the inner edges of the PCB and in a two-dimensional grid spaced by
1 mm automatically positioned according to the PCB fabrication rules. We decided
to use the SMPM connector7, a miniature version of the well-known SMP connector
considered the availability and the cost. We use full-detent through-hole connectors
(Amphenol RF 925-138J-51P) soldered to the PCB rather than surface mount ones
to achieve mechanically robust connections. Stub resonances caused by ∼ 2 mm
extruded pins of this connector are expected to lie above ∼ 38 GHz which will not
cause problems in the experiments. The PCB is gold-plated in a similar manner as
described above. The SMPM connectors are soldered to the PCB from the bottom
side by using eutectic Sn63/Pb37 solder paste8 and a heat gun.
The machined OFHC copper enclosure consists of the cover and the mount plate.
The cover has an opening of 21.4 mm × 11.4mm × 1.6mm above the top surface of
the chip. The mount plate has a 20 mm × 10 mm wide and 3 mm deep pocket to be
placed under the chip to suppress the resonances above 10 GHz. Also, pedestals at
the corners of the pocket provide mechanical support of the chip. The top cover and
the mount plate are assembled using four UNC #2-56 brass screws.
7 SMPM is also known as mini-SMP or Corning Gilbert GPPO, with maximum operating

frequency of 65 GHz.
8 Chip Quik SMD291AX10, melting point 183 ◦ C

35
During the test of this packaging, we have observed degradation of mating between
the PCB connectors and right-angled cable connectors with thermal cycling. To
prevent this, we have later added another copper piece to the assembly for cable
connectors to be tightly clamped to the PCB connectors, which solved the problems.
Assembly procedure
The PCB is soldered to RF connectors and then sonicated in acetone and IPA in
order to remove residues of solder paste. The enclosure parts made of aluminum
(copper) are cleaned with Transene Aluminum Etchant A (1.0M citric acid) followed
by rinse in DI water and IPA prior to the assembly.
After cleaning, the PCB and the fabricated chip is placed on top of the mount plate
and wire-bonded. In case of the new packaging, the fabricated chip is bonded
to the pedestals by using GE Varnish (VGE-7031) to provide strong mechanical
support prior to wire-bonding and cryogenic heatsink [155]. Also, the PCB is
tightly attached to the mount plate by using four UNC #1-64 brass screws with a thin
layer of Apiezon N grease [156] applied at the interface to maintain good thermal
conductance at cryogenic temperatures. The box is then closed by joining the cover
and the mount plate with brass screws.
The packaging assembly is then installed vertically to the sample mount machined
with OFHC copper and mounted to the mixing chamber plate of the dilution refrigerator. We use non-magnetic semi-flexible coaxial cables (Micro-Coax UT-085CFORM) to route input/output signals between the PCB connectors and the cryogenic
semi-rigid coaxial cables discussed in Sec. 3.3. After this, we enclose the packaging
assembly with two 1.5 mm-thick cylindrical Cryophy magnetic shields of outer diameters 70 mm and 90 mm and heights 185 mm and 200 mm, respectively. Finally,
there exists a large cylindrical mu-metal shield (thickness 1 mm, inner diameter
395 mm, height 750 mm) placed inside the vacuum can of the dilution refrigerator
for additional protection from external magnetic field.
3.3

Cryogenic setup

The packaged devices are cooled down to low temperatures for deposited metals to
become superconductors [𝑇𝑐 = 1.2 K (9.26 K) for Al (Nb)] and more importantly for
quantum effects to appear. The devices in the thesis are characterized in Bluefors
LD-250 dilution refrigerator (named “DF2”) with a base temperature of 𝑇 . 7 mK.
Proper engineering and placement of cryogenic microwave components [157–160]
cannot be overlooked as thermal photons due to bad thermalization at milli-Kelvin

300 K

Twisted-Pair Wire
BeCu-SS Coax
NbTi-NbTi Coax
SS-SS Coax
Semi-flexible Coax

50 K

HEMT Amplifier

XY / RO
Input

Fast Flux
Input

DC Flux
Input

JTWPA
Pump

RO
Output

JTWPA

Circulator
50 Ω Load
Dir. Coupler (20 dB)

2 × 2 RF Switch

MXC

K&L

400M

K&L

20 dB Cryo. Att. (XMA)

K&L

20 dB Cryo. Att. (QMC)
10 dB Cryo. Att. (QMC)
Eccosorb Filter

Device

400M

Mini-Circuits VLFX-400+

K&L

K & L Low-pass filter
RCR Low-pass filter
DC-coupled Bias Tee

Figure 3.2: Cryogenic Setup. A typical wiring diagram of our dilution refrigerator, where
the meaning of symbols are enumerated on the right. Orange dashed boxes indicate magnetic
shields. The capacitor symbol represents inner/outer DC blocks for breaking ground loops.

temperatures [161, 162] can result in thermal excitation of qubits [163, 164] or
dephasing associated with residual thermal photons in readout resonators [165,
166]. Here, I will provide detailed description of the cryogenic setup in DF2,
illustrated in Fig. 3.2. For a general introduction to cryogenic engineering and
operating principles of dilution refrigerator, I refer the readers to Refs. [167, 168].
Input lines
The input coaxial cables are thermalized at each stage of the refrigerator with a series
of cryogenic attenuators9 to reduce the thermal noise from the room-temperature
environment. To be specific, for input drive lines for XY control and readout
resonators, we place 1 dB at 50 K plate, 20 dB at 4 K plate, 1 dB at still plate, 10 dB
9 We use cryogenic attenuators 2082-6418--CRYO with stainless steel (SS) enclosure from

XMA Corporation at stages with temperature above 100 mK (50 K, 4 K, Still). For stages with
temperature below 100 mK (CP, MXC), we try to use attenuators made with OFHC copper enclosure
for best thermalization. Examples of such cold attenuators include ones from Prof. B. S. Palmer’s
lab [157] and QMC-CRYOATT- from Quantum Microwave. Here, is the value of attenuation
in units of dB.

37
at cold plate (CP), and 30 dB at mixing chamber (MXC) plate with stages connected
by semi-rigid SS-SS coaxial cables (Micro-Coax UT-085-SS-SS). Along the input
coaxial lines for fast flux control, we only place 20 dB at 4 K plate and 0 dB at
other stages. Each fast flux line is additionally filtered with a reflective low-pass
filter with passband width of 400 MHz (Mini-Circuits VLFX-400+). in order to
prevent high-frequency noise from entering the device while keeping full analog
bandwidth offered by arbitrary waveform generators with sample rate of 1 GSa/s.
For DC flux inputs, we use low-Ohmic twisted-pair wiring consisting of copper
from room temperature to 4 K and NbTi/CuNi (superconducting) from 4K to MXC
manufactured by Bluefors. The DC wiring goes through a cryogenic RCR low-pass
filter10 thermalized to the 4 K plate via a copper braid. The DC and the fast flux
signals are combined with a DC-coupled bias tee11 anchored to the MXC plate.
Output lines
The output signal is directed through a pair of cryogenic isolators12 thermally
anchored to the MXC plate of the dilution refrigerator and is subsequently sent into an
amplifier chain consisting of a high electron mobility transistor (HEMT) amplifier13
at the 4 K stage via a series of semi-rigid superconducting NbTi–NbTi coaxial cables
(KEYCOM NbTiNbTi085A). Note that good thermalization of HEMT amplifiers
by a direct contact with machined OFHC copper parts (rather than braided copper
straps) is crucial to achieving the smallest added noise. After the HEMT amplifier we
use semi-rigid beryllium copper (BeCu)–SS coaxial cables (Micro-Coax UT-085BSS) to estabish a low-loss connection to higher-temperature stages at the expense of
a slightly higher thermal conductivity than the SS–SS ones used for the input lines.
For rapid single-shot readout of qubits, it is necessary to employ near quantumlimited parametric amplifiers such as Josephson parametric amplifier (JPA) [169]
or Josephson traveling-wave parametric amplifier (JTWPA) [170, 171] at the first
amplification stage before the HEMT amplifier. To that end, we install JTWPA pro10 The RCR filter is manufactured by Aivon and consists of two series resistors (SMD 499 Ω ± 1 %,

giving total series resistance of 1 kΩ) and a shunt capacitance of 10.2 nF to the ground. This gives
the cutoff frequency of 𝑓𝑐 ≈ 33 kHz assuming source resistance 𝑅𝑆 = 10 kΩ and zero load resistance.
11 Mini-Circuits ZFBT-4R2GW+ is modified by replacing the series capacitor between the RF
and RF+DC ports with a short.
12 Low-Noise Factory 4–12 GHz cryogenic circulator LNF-CICIC4_12A terminated with Quantum Microwave QMC-CRYOTERM-0412.
13 Low-Noise Factory LNF-LNC4_8C for lowest-noise amplification in typical readout frequencies 4–8 GHz or LNF-LNC0.3_14A for wideband amplification across 0.3–14 GHz.

38
vided by Prof. William Oliver’s group at MIT Lincoln Laboratory on the MXC plate
with a cryogenic directional coupler (Quantum Microwave QMC-CRYOCOUPLER20) and a pair of wideband dual-junction cryogenic isolators12 for 50 Ω impedance
matching at the signal, idler, and pump frequencies. While the addition of cryogenic
isolators between the sample and the JTWPA results in insertion loss reducing the
quantum efficiency of readout, at least 30 dB of isolation was necessary in order to
prevent pump leakage and spurious JTWPA tones from reaching the device which
caused unintended driving of readout resonators and ac Stark shift of qubits. In addition, well-thermalized cryogenic isolators are necessary in order to attenuate the
residual room-temperature thermal photons propagating backwards from the output
lines without need for attenuators. The JTWPA is enclosed inside a Cryoperm shield
to prevent critical current suppression arising from external magnetic field.
Infrared filtering
Infrared (IR) radiation is known to generate excess quasiparticles in superconductors
by breaking Cooper pairs [172, 173], acting as a loss mechanism in superconducting
circuits. In order to prevent IR light from entering the device we take several
measures in the cryogenic setup. First, we place low-pass clean-up filters (K&L
Microwave 6L250-12000/T26000-OP/O) with the cutoff frequency of 12 GHz and
guaranteed 50 dB rejection up to 26 GHz along all the input/output lines to reflect
off any high-frequency radiation transmitted to the device via coaxial cables. For the
same purpose, coaxial absorptive Eccosorb filters [174–176] were manufactured in
lab or acquired from AWS Center for Quantum Computing and additionally installed
along the input lines. Finally, the interior of the radiation shield for the MXC stage
was painted with IR-blocking absorptive materials (a mixture of silica powder, fine
carbon powder, and 1 mm silicon carbide grains in Stycast 1266 epoxy) described
in Refs. [172, 177] to shield against stray IR radiation.
3.4

Room-temperature electronics and signal processing

The quantum device cooled down to cryogenic temperatures is interrogated with
various electronic circuitry and instruments. In this section, we introduce the
main electronic components at the room-temperature for controlling and measuring
superconducting qubits.

ωIF , φ
AI (t)

AQ (t)

cos
− sin
sin
cos

ωLO , φLO

T = 10 mK

SI (t)
LO RF

SQ (t)

V (t)

φd

Vd (t)

T = 300 K

Figure 3.3: Microwave synthesis. a, The baseband waveforms 𝐴I (𝑡) and 𝐴Q (𝑡) are
multiplied to phase-shifted copies of oscillating signals (cos, − sin, sin, cos) at intermediate
frequency (IF) 𝜔IF with phase 𝜙 and summed, resulting in upconverted IF waveforms 𝑆I (𝑡)
and 𝑆Q (𝑡). The IF waveforms are output from DAC channels of a controller instrument
(represented as box). b, The IF waveforms are multiplied to a local oscillator (LO) signal
with frequency 𝜔LO and phase 𝜙LO at a IQ mixer, resulting in RF signal 𝑉 (𝑡). c, The
RF signal 𝑉 (𝑡) synthesized at room temperature propagates to the device via a series of
cryogenic coaxial cables and components, acquiring a phase 𝜙 𝑑 associated with the delay
and response. The final voltage arriving at the device is denoted as 𝑉d (𝑡). Throughout the
figure, the signals corresponding to the real (imaginary) part of the corresponding complex
phasor representation is colored red (blue).

Synthesis of microwave pulses for qubit control and readout
We synthesize microwave pulses for driving superconducting qubits and readout
resonators with electronics at the room temperature and the resulting voltage signal
propagates to the device inside the dilution refrigerator.
Generation of IF signals from baseband waveforms
The first stage of synthesis of microwave signals happens inside the controller instruments as illustrated in Fig. 3.3a. We specify the real-valued baseband waveforms
𝐴I (𝑡) and 𝐴Q (𝑡), which act as envelope functions for real-valued intermediatefrequency (IF) voltage signals 𝑆I (𝑡) and 𝑆Q (𝑡) that are analog outputs from the
instrument. The relation between the specified baseband waveforms and the outputs
from digital-to-analog converter (DAC) channels are given by
𝑆I (𝑡)
cos(𝜔IF 𝑡 + 𝜙) − sin(𝜔IF 𝑡 + 𝜙) 𝐴I (𝑡)
(3.1)
𝑆Q (𝑡)
sin(𝜔IF 𝑡 + 𝜙) cos(𝜔IF 𝑡 + 𝜙)
𝐴Q (𝑡)
where 𝜔IF is the angular frequency of the IF signal and 𝜙 is the angle of frame
rotation. An equivalent representation of Eq. (3.1) using complex variables 𝑺(𝑡) ≡
𝑆I (𝑡) + 𝑖𝑆Q (𝑡) and 𝑨(𝑡) ≡ 𝐴I (𝑡) + 𝑖 𝐴Q (𝑡) is given by
𝑺(𝑡) = 𝑒𝑖(𝜔IF 𝑡+𝜙) 𝑨(𝑡).

(3.2)

This process can be done completely in the software by uploading the computed
IF waveform 𝑆I/Q (𝑡) of a full sequence of pulses to arbitrary waveform generators

40
(AWGs). In architectures based on field-programmable gate array (FPGA), we only
specify the baseband envelope 𝐴I/Q (𝑡) at the pulse level and this computation is done
at the hardware in real time. In our experiments, we have used Keysight M3202A
PXIe AWG in the former case and Quantum Machines Operator-X (OPX) in the
latter case. Both systems have the DAC sample rate of 𝑓s = 1 GSa/s and has the
analog bandwidth of 400 MHz.
Upconversion of signals from IF to RF
The IF signals from the DAC channels are multiplied with local oscillator (LO)
signal at the IQ mixer and are upconverted into radio frequency (RF) signal as
shown in Fig. 3.3b. The upconverted real-valued voltage signal 𝑉 (𝑡) from the IQ
mixer can be written as
𝑉 (𝑡) = 𝑆I (𝑡) cos(𝜔LO 𝑡 + 𝜙LO ) − 𝑆Q (𝑡) sin(𝜔LO 𝑡 + 𝜙LO )
= 𝐴I (𝑡) cos(𝜔𝑡 + 𝜙 + 𝜙LO ) − 𝐴Q (𝑡) sin(𝜔𝑡 + 𝜙 + 𝜙LO )

(3.3)

Here, 𝜔 = 𝜔LO + 𝜔IF is the upconverted frequency. An equivalent complex variable
representation with 𝑉 (𝑡) = Re [𝑽 (𝑡)] is given by
𝑽 (𝑡) = 𝑒𝑖(𝜔LO 𝑡+𝜙LO ) 𝑺(𝑡) = 𝑒𝑖(𝜔𝑡+𝜙+𝜙LO ) 𝑨(𝑡).

(3.4)

In our setup, the pair of IF outputs 𝑆I/Q (𝑡) from DAC channels are filtered by a
low-pass filter (Mini-Circuits VLFX-400+) to reject any possible spurious signals
above the Nyquist frequency ( 𝑓𝑠 /2 = 500 MHz) and are plugged into I/Q ports
of an IQ mixer14 mounted to the microwave chassis for temperature stability. We
place microwave attenuators on the I/Q ports (i) to limit the power into the IQ mixer
∼ 6 dB to below the 1 dB compression point in order to avoid non-linearity and
(ii) to provide good matching. We use Rohde & Schwarz SMB100A microwave
signal generators to synthesize LO signals for IQ mixers. The RF output 𝑉 (𝑡) from
the IQ mixers are amplified by a low-noise linear amplifier (Mini-Circuits ZX6083LN-S+) and sent into the microwave input lines of the dilution refrigerator. In
practice, this upconversion process with IQ mixers generates spurious tones due to
various imperfections in the mixing process, which needs to be calibrated [178].
For a general introduction to microwave mixers, refer to Refs. [179–182]. While the
14 We use IQ mixers manufactured from Marki Microwave. Specifically, the legacy models IQ-

0307 (3–7 GHz) and IQ-4509 (4.5–9 GHz), and more recently monolithic microwave integrated
circuit (MMIC) IQ mixers MMIQ-0218LXPC (2–18 GHz) were used in the experiments described
in the thesis.

41
analog bandwidth of each IF signal is 400 MHz, the IQ-combined single-sideband
upconversion to RF allows for frequencies in the range [−400, +400] MHz with
respect to the LO frequency (i.e., bandwidth of 800 MHz).
Propagation to the device
The RF signal 𝑉 (𝑡) synthesized at room temperature propagates to a XY drive line
of qubits or a feedline of readout resonators inside the dilution refrigerator after
passing through multiple coaxial cables and various microwave components such as
amplifiers, attenuators, and filters as illustrated in Fig. 3.3c. The ideal net effect of
propagation is frequency-dependent group delay 𝑡 𝑑 resulting in a phase shift. Then,
the qubit drive voltage 𝑉d (𝑡) can be written in the form
𝑉d (𝑡) = 𝑉 (𝑡 − 𝑡 𝑑 ) = 𝐴I (𝑡) cos(𝜔𝑡 + 𝜙 + 𝜙 𝑑 ) − 𝐴Q (𝑡) sin(𝜔𝑡 + 𝜙 + 𝜙 𝑑 ).
where 𝜙 𝑑 is the sum of phases that we don’t have control over.
Equivalent complex variable representation with 𝑉d (𝑡) = Re[𝑽 d (𝑡)] is given by
𝑽 d (𝑡) = 𝑒𝑖𝜙 𝑑 𝑽 (𝑡) = 𝑒𝑖(𝜔𝑡+𝜙+𝜙LO +𝜙 𝑑 ) 𝑨(𝑡).
In practice, the propagation of room-temperature RF signal to the device can be
affected by impedance mismatch of various microwave components inside the cryostat, adding undesirable response to the pulse. Proper engineering of cryogenic
microwave components is thus necessary in order to achieve high-fidelity qubit
control [183].
Example: amplitude- and phase-modulated baseband waveforms
Let us consider the most general case of baseband waveforms 𝐴I (𝑡) = 𝑎(𝑡) cos [𝜃 (𝑡)]
and 𝐴Q (𝑡) = 𝑎(𝑡) sin [𝜃 (𝑡)] [complex representation 𝑨(𝑡) = 𝑎(𝑡)𝑒𝑖𝜃 (𝑡) ], which
represents amplitude modulation with 𝑎(𝑡) and phase modulation with 𝜃 (𝑡). The
resulting IF outputs from DAC channels are given by
𝑆I (𝑡) = 𝑎(𝑡) cos [𝜔IF 𝑡 + 𝜙 + 𝜃 (𝑡)]
𝑆Q (𝑡) = 𝑎(𝑡) sin [𝜔IF 𝑡 + 𝜙 + 𝜃 (𝑡)]

𝑺(𝑡) = 𝑎(𝑡)𝑒𝑖[𝜔IF 𝑡+𝜙+𝜃 (𝑡)] .

After upconversion to RF, the signal becomes
𝑉 (𝑡) = 𝑎(𝑡) cos [𝜔𝑡 + 𝜙 + 𝜃 (𝑡) + 𝜙LO ]

or 𝑽 (𝑡) = 𝑎(𝑡)𝑒𝑖[𝜔IF 𝑡+𝜙+𝜃 (𝑡)+𝜙LO ] .

After propagation to the device, the drive voltage is given by
𝑉d (𝑡) = 𝑎(𝑡) cos [𝜔𝑡 + 𝜑(𝑡)],
where 𝜑(𝑡) = 𝜙 + 𝜃 (𝑡) + 𝜙LO + 𝜙 𝑑 is the sum of all phases.

(3.5)

V (t)

RF LO

SI (t)

ADC

cos(ωIF tm )

SI [m]

sin(ωIF tm )

+ AI [m]

ωLO , φLO
SQ (t)

ADC

cos(ωIF tm ) +

wQ [m] +

−wQ [m]

− sin(ωIF tm )

SQ [m]

wI [m]

AQ [m]

wI [m] +

Figure 3.4: Downconversion and demodulation. a, The RF output signal 𝑉 (𝑡) is mixed
with LO signal and downconverted to a pair of IF signals 𝑆I (𝑡) and 𝑆Q (𝑡) at a IQ mixer.
A low-pass filter is placed after the downconversion to reject spurious high-frequency
components above the LO frequency. b, The IF signals are digitized at ADC channels
and transformed into discrete-time signals 𝑆I [𝑚] and 𝑆Q [𝑚]. The discrete-time signals
are multiplied to phases-shifted copy of IF signals and summed, resulting in discretetime baseband waveforms 𝐴I [𝑚] and 𝐴Q [𝑚]. c, The baseband waveforms are summed with
integration weights specified by 𝑤 I [𝑚] and 𝑤 Q [𝑚], resulting in scalars 𝐼 and 𝑄. Throughout
the figure, the signals corresponding to the real (imaginary) part of the corresponding
complex phasor representation is colored red (blue).

Downconversion of readout signal and state discrimination
The output readout signals from the cryostat pass through a variety of microwave
components and are digitized to be further processed for qubit state discrimination.
Prior to the first room-temperature amplification, it is necessary to remove the
JTWPA pump leakage tone (dominant spurious tone from JTWPA) in order to avoid
saturation of subsequent amplifiers and mixers. To that end, the output readout
signals are passed through a room-temperature isolator15 followed by a Yttrium
iron garnet (YIG)-tuned band-reject filter (Micro Lambda Wireless MLBFR-0212)
whose rejection frequency set to that of the JTWPA pump. Then, we place a highgain room-temperature amplifier16 and a tunable microwave attenuator (Vaunix
LabBrick LDA-133) to adjust the power to sufficiently below 1 dB compression
points of subsequent microwave components while utilizing the full dynamic range.
Downconversion of signals from RF to IF
The RF output 𝑉 (𝑡) = 𝑏(𝑡) cos [𝜔𝑡 + 𝜑(𝑡)] is multiplied to two copies of the LO
signal with relative phase of 90◦ at the IQ mixer and downconverted into IF signal
as shown in Fig. 3.4a. The downconverted real-valued voltage signals 𝑆I/Q (𝑡) at I/Q
15 Fairview

Microwave dual-junction circulator FMCR1019 terminated with Mini-Circuits
ANNE-50L+ matched loads.
16 Narda-MITEQ AFS42-00101200-22-10P-42

43
ports of an IQ mixer are given by
𝑆I (𝑡) ∝ 𝑉 (𝑡) cos (𝜔LO 𝑡)
𝑏(𝑡)
{cos [(2𝜔LO + 𝜔IF )𝑡 + 𝜑(𝑡)] + cos [𝜔IF 𝑡 + 𝜑(𝑡)]} ,
𝑆Q (𝑡) ∝ −𝑉 (𝑡) sin (𝜔LO 𝑡)
𝑏(𝑡)
{− sin [(2𝜔LO + 𝜔IF )𝑡 + 𝜑(𝑡)] + sin [𝜔IF 𝑡 + 𝜑(𝑡)]} .
After low-pass filtering (Mini-Circuits VLFX-400+), the fast-oscillating terms at
frequency 2𝜔LO + 𝜔IF are discarded, and the resulting IF signals at I and Q ports
are written as
𝑆I (𝑡) = 𝑏(𝑡) cos [𝜔IF 𝑡 + 𝜑(𝑡)]
𝑆Q (𝑡) = 𝑏(𝑡) sin [𝜔IF 𝑡 + 𝜑(𝑡)]

𝑺(𝑡) = 𝑏(𝑡)𝑒𝑖[𝜔IF 𝑡+𝜑(𝑡)] ,

(3.6)

where 𝑺(𝑡) = 𝑆I (𝑡) + 𝑖𝑆Q (𝑡) is the complex representation of IF signals.
Digitization and Demodulation of IF signals
The downconverted IF waveforms 𝑆I/Q (𝑡) are rescaled with attenuators and IF
amplifiers (Mini-Circuits ZFL-500LNB+) to fit in the range of analog-to-digital
converter (ADC) channels. In the experiments described in the thesis, we use
ADC channels of AlazarTech ATS9870 PCIe digitizer or Quantum Machines OPX
with the sample rate of 1 GSa/s for digitization of IF waveforms. In order to avoid
aliasing, we additionally place a low-pass filter (Mini-Circuits VLFX-400+) to reject
spurious signals above the Nyquist frequency of the ADC channels.
Upon digitization, the IF signals are converted to discrete-time signals 𝑆I/Q (𝑡 𝑚 ) →
𝑆I/Q [𝑚] sampled at times 𝑡 = 𝑡 𝑚 (𝑚 = 0, 1, 2, · · · , 𝑁 − 1). Each discrete-time
sample recorded to ADC channels is multiplied to cosine and sine functions as
𝐴I [𝑚]
cos(𝜔IF 𝑡 𝑚 ) sin(𝜔IF 𝑡 𝑚 ) 𝑆I [𝑚]
(3.7)
𝐴Q [𝑚]
− sin(𝜔IF 𝑡 𝑚 ) cos(𝜔IF 𝑡 𝑚 ) 𝑆Q [𝑚]
and downconverted to the baseband discrete-time samples 𝐴I/Q [𝑚] (Fig. 3.4b),
which are summed with integration weights 𝑤 I/Q [𝑚] to reduce to scalars (Fig. 3.4c)
! 𝑁−1
Õ 𝑤 I [𝑚] 𝑤 Q [𝑚] 𝐴I [𝑚]
(3.8)
−𝑤 Q [𝑚] 𝑤 I [𝑚] 𝐴Q [𝑚]
𝑚=0
Using the complex notation for the baseband waveforms 𝑨[𝑚] ≡ 𝐴I [𝑚] + 𝑖 𝐴Q [𝑚],
the integration weights 𝒘 [𝑚] ≡ 𝑤 I [𝑚] + 𝑖𝑤 Q [𝑚], and the demodulated scalar

44
variables 𝒁 ≡ 𝐼 + 𝑖𝑄, equations (3.7)-(3.8) can be simplified into
𝒁=

𝑁−1

𝒘 [𝑚] 𝑨[𝑚] =

𝑚=0

𝑁−1

𝑒 −𝑖𝜔IF 𝑡 𝑚 𝒘 ∗ [𝑚]𝑺[𝑚].

(3.9)

𝑚=0

Given the IF signals in Eq. (3.6), the demodulated IQ variables are given by
𝐼=

𝑄=

𝑁−1
𝑚=0
𝑁−1

𝑏(𝑡 𝑚 ) 𝑤 I [𝑚] cos 𝜑(𝑡 𝑚 ) + 𝑤 Q [𝑚] sin 𝜑(𝑡 𝑚 ) ,

(3.10a)

𝑏(𝑡 𝑚 ) −𝑤 Q [𝑚] cos 𝜑(𝑡 𝑚 ) + 𝑤 I [𝑚] sin 𝜑(𝑡 𝑚 ) ,

(3.10b)

𝑚=0

or equivalently,
𝒁=

𝑁−1

𝒘 ∗ [𝑚]𝑏(𝑡 𝑚 )𝑒𝑖𝜑(𝑡 𝑚 ) .

(3.10c)

𝑚=0

In the simplest case of readout with uniform integration weights (𝑤 I [𝑚], 𝑤 Q [𝑚]) =
(cos 𝜙0 , sin 𝜙0 ), we obtain
𝐼=

𝑁−1

𝑏(𝑡 𝑚 ) cos [𝜑(𝑡 𝑚 ) − 𝜙0 ],

𝑄=

𝑚=0

𝑁−1

𝑏(𝑡 𝑚 ) sin [𝜑(𝑡 𝑚 ) − 𝜙0 ],

(3.11a)

𝑚=0

or equivalently,
𝒁=

𝑁−1

𝑏(𝑡 𝑚 )𝑒𝑖[𝜑(𝑡 𝑚 )−𝜙0 ] .

(3.11b)

𝑚=0

If the readout duration is long enough compared to the ringdown time 1/𝜅 of the
readout resonator, the readout resonator stays in the steady state most of the time
during the on-time of the readout pulse, giving near-constant readout signal with
magnitude 𝑏 ss ≈ 𝑏(𝑡 𝑚 ) and phase 𝜑ss ≈ 𝜑(𝑡 𝑚 ). This further simplifies Eq. (3.11a)(3.11b) into
𝐼 ≈ 𝑁 𝑏 ss cos (𝜙ss − 𝜙0 ),

𝑄 ≈ 𝑁 𝑏 ss sin (𝜙ss − 𝜙0 ).

(3.12a)

or equivalently,
𝒁 ≈ 𝑁 𝑏 ss 𝑒𝑖(𝜙ss −𝜙0 ) .

(3.12b)

Qubit state discrimination
An ensemble of demodulated readout signals {𝒁 |𝑖i } corresponding to each qubit
state |𝑖i (𝑖 = 𝑔, 𝑒, 𝑓 , . . .) form a blob of points in the 𝐼𝑄-plane following a Gaussian
distribution centered at 𝝁 |𝑖i and standard deviation 𝜎|𝑖i in the ideal case. It is
common to perform qubit state discrimination by drawing boundaries on the IQ

45
plane and classifying according to the region each point fall into. Misclassified
points count towards readout infidelity and the boundaries for state discrimination
should be chosen to minimize the infidelity.
In the simplest case, a line passing through the midpoint of means 𝒁 |𝑖i and 𝒁 | 𝑗i that
is perpendicular to the difference vector 𝒁 |𝑖i − 𝒁 | 𝑗i for classification between states
|𝑖i and | 𝑗i. Another well-known method to find the classification boundary is to fit
the histogram with a multi-mode Gaussian distribution [184, 185]. This assumes
that the ensemble of readout signals with preparation of each qubit state follows
a superposition of multiple Gaussian distributions corresponding to qubit states to
take into account the non-ideal effects during state preparation and measurement.
Machine learning-based classification methods [186] are also being widely used due
to the high-performance and simplicity. We use linear discriminant analysis (LDA)
trained with the collection of readout signals and corresponding state preparation
labels, linear map This only requires allows for real-time state discrimination in
Quantum Machines OPX.

46
Chapter 4

WAVEGUIDE-MEDIATED COOPERATIVE INTERACTIONS OF
SUPERCONDUCTING QUBITS
The simplest scheme to engineer interactions between qubits using a waveguide is
to utilize modes inside the transmission band. In Sec. 2.2, it was shown that strong
coherent interaction between two quantum emitters are intrinsically forbidden due
to the fact that the maximum waveguide-mediated exchange interaction rate cannot
exceed the rate of decay of emitters. A novel approach to circumvent this problem
was proposed in Ref. [120], by creating a cavity-like configuration of emitters where
subradiant states can coherently interact with another precisely positioned emitter
without significant decay to the waveguide. In this chapter, I will introduce our
experimental work on this topic, which is published in Ref. [108].
4.1

Introduction

It has long been recognized that atomic emission of radiation is not an immutable
property of an atom, but is instead dependent on the electromagnetic environment [187], and in the case of ensembles, also on the collective interactions between
the atoms [42, 43, 111, 112, 188]. In an open radiative environment, the hallmark of
collective interactions is enhanced spontaneous emission—super-radiance [42]—
with non-dissipative dynamics largely obscured by rapid atomic decay [189]. Here,
we observe the dynamical exchange of excitations between a single artificial atom
and an entangled collective state of an atomic array [120] through the precise
positioning of artificial atoms realized as superconducting qubits [190] along a
one-dimensional waveguide. This collective state is dark, trapping radiation and
creating a cavity-like system with artificial atoms acting as resonant mirrors in the
otherwise open waveguide. The emergent atom-cavity system is shown to achieve
a large interaction-to-dissipation ratio (cooperativity exceeding 100), reaching the
regime of strong coupling, in which coherent interactions dominate dissipative and
decoherence effects. Achieving strong coupling with interacting qubits in an open
waveguide provides a means of synthesizing multi-photon dark states with high efficiency and paves the way for exploiting correlated dissipation and decoherence-free
subspaces of quantum emitter arrays at the many-body level [191–194].
The collective interaction of atoms in the presence of a radiation field has been

47
studied since the early days of quantum physics. As first studied by Dicke [42], the
interaction of resonant atoms in such systems results in the formation of super- and
sub-radiant states in the spontaneous emission. While Dicke’s central insight—that
atoms interact coherently even through an open environment—was used to understand the radiation properties of an idealized, point-like atomic gas, the dynamical
properties of ordered, distant atoms coupled to open environments also exhibit
novel physics. In their most essential form, such systems can be studied within the
canonical waveguide quantum electrodynamics (QED) model [40]: atoms coupled
to a one-dimensional (1D) continuum realized by an optical fiber or a microwave
waveguide [132, 195]. Within this model, a diverse and rich set of phenomena
await experimental study. For instance, one can synthesize an artificial cavity QED
system [120], distill exotic many-excitation dark states with fermionic spatial correlations [191], and use classical light sources to generate entangled and quantum
many-body states of light [192–194].
4.2

Strong-coupling regime of waveguide QED: cavity QED with atom-like
mirrors

A central technical hurdle common to these research avenues—reaching the socalled strong coupling regime, in which atom-atom interactions dominate decay—is
experimentally difficult, especially in waveguide QED, because while the waveguide
facilitates infinite-range interactions between the atoms [196, 197], it also provides a
dissipative channel [198]. Decoherence through this and other sources destroys the
fragile many-body states of the system, which has limited the experimental stateof-the-art to spectroscopic probes of waveguide-mediated interactions [142, 147,
199]. However, by utilizing collective dark states, where the precise positioning of
atoms protects them from substantial waveguide-emission decoherence, the strong
coupling limit is predicted to be within reach [120]. Additionally, if the timescale
of single-atom emission into the waveguide is long enough to permit measurement
and manipulation of the system, the coherent dynamics can be driven and probed
at the single-atom level. Here, we overcome these hurdles with a waveguide QED
system consisting of transmon qubits coupled to a common microwave waveguide,
strengthening opportunities for a range of waveguide QED physics.
As a demonstration of this tool, we construct such an emergent cavity QED system
and probe its linear and non-linear dynamics. The system features an ancillary
probe qubit and a cavity-like mode formed by the dark state of two single-qubit
mirrors. Using waveguide transmission and individual addressing of the probe qubit,

Γ1D

Γ1D,p

κ = Γ′D

Γ1D

λ/4 m = 1

γ = Γ1D,p + Γ′p

b Probe qubit
···

100 μm

|g〉p|D〉

Γ′p Γ′ Γ′B

Γ1D,p

|g〉p|G〉

|g〉p|B〉
2Γ1D

1 mm

Mirror qubits

Q2 Q3

Q5 Q6

|e〉p|G〉

XY4

g=J

m = -1 λ/4

Γ′

···

Γ′p

Γ′

Type I
Type II

200 μm

Input

Output

Figure 4.1: Waveguide-QED setup. a, The top schematic shows the cavity configuration
of waveguide QED system consisting of an array of 𝑁 mirror qubits (𝑁 = 2 shown; green)
coupled to the waveguide with an inter-qubit separation of 𝜆 0 /2, with a probe qubit (red)
at the center of the mirror array. The middle schematic shows the analogous cavity QED
system with correspondence to waveguide parameters. The bottom panel shows the energylevel diagram of the system of three qubits (two mirror, one probe). The mirror dark state
|𝐷i is coupled
to the excited state of the probe qubit |𝑒ip at a cooperatively enhanced rate
of 2𝐽 = 2Γ1D Γ1D,p . The bright state |𝐵i is decoupled from the probe qubit. b, Optical
image of the fabricated waveguide QED circuit. Tunable transmon qubits interact via
microwave photons in a superconducting CPW (false-color orange trace). The CPW is used
for externally exciting the system and is terminated in a 50-Ω load. The insets show scanning
electron microscope images of the different qubit designs used in our experiment. The probe
qubit, designed to have Γ1D,p /2𝜋 = 1 MHz, is accessible via a separate CPW (XY4 ; falsecolor blue trace) for state preparation, and is also coupled to a compact microwave resonator
(R4 ; false-color cyan) for dispersive readout. The mirror qubits come in two types: type I,
with Γ1D /2𝜋 = 20 MHz and type II, with Γ1D /2𝜋 = 100 MHz. The figure is adapted from
Ref. [108].

we observe spectroscopic and time-domain signatures of the collective dynamics
of the qubit array, including vacuum Rabi oscillations between the probe qubit
and the cavity-like mode. These oscillations provide direct evidence of strong
coupling between these modes as well as a natural method of efficiently creating
and measuring dark states that are inaccessible through the waveguide. Unlike
traditional cavity QED, our cavity-like mode is itself quantum nonlinear, as we
show by characterizing the two-excitation dynamics of the array.
The collective evolution of an array of resonant qubits coupled to a 1D waveguide
can be formally described by a master equation [116, 120] of the form
𝜌¤̂ = − [ 𝐻ˆ eff , 𝜌]
ˆ +
Γ𝑚,𝑛 𝜎
ˆ ge
𝜌ˆ 𝜎
ˆ eg
𝑚,𝑛

(4.1)

49
𝑚 = |𝑔 ih𝑒 |, |𝑔i and h𝑒| are a qubit’s ground and excited states, respecwhere 𝜎
ˆ ge
tively, and 𝑚 and 𝑛 represent the indices of the qubit array. Within the Born-Markov
approximation, the effective Hamiltonian can be written in the interaction picture as
Õ
Γ𝑚,𝑛
𝑚 𝑛
𝐻ˆ eff = ℏ
𝐽𝑚,𝑛 − 𝑖
ˆ eg
ˆ ge ,
(4.2)
𝑚,𝑛

where ℏ = ℎ/2𝜋 is the reduced Planck constant. Figure 4.1a depicts the waveguide
QED system considered in this work. The system consists of an array of 𝑁 qubits
separated by distance 𝑑 = 𝜆 0 /2 and a separate probe qubit centered in the middle of
the array with one-dimensional waveguide decay rate Γ1D,p , and where 𝜆 0 = 𝑐/ 𝑓0
is the wavelength of the field in the waveguide at the transition frequency of the
qubits 𝑓0 . In this configuration, the effective Hamiltonian can be simplified in the
single-excitation manifold to
𝑖ℏΓ1D,p (p)
𝑖𝑁ℏΓ1D ˆ† ˆ
(p) ˆ†
𝑆B 𝑆B −
ˆ ee + ℏ𝐽 𝜎
ˆ ge 𝑆D + H.c. ,
(4.3)
𝐻eff = −
√ Í
𝑚 ∓𝜎
−𝑚 )(−1) 𝑚 are the lowering operators of the
ˆ ge
ˆ ge
where 𝑆ˆB , 𝑆ˆD = 1/ 𝑁 𝑚>0 ( 𝜎
bright collective state |𝐵i and the fully-symmetric dark collective state |𝐷i of the
qubit array, as shown in Fig. 4.1a, and where 𝑚 > 0 and 𝑚 < 0 denote qubits to
the right and left of the probe qubit, respectively. As shown by the last term in
the Hamiltonian, the probe qubit is coupled to this dark state at a cooperatively
√ p
enhanced rate 2𝐽 = 𝑁 Γ1D Γ1D,p . (H.c. is the Hermitian conjugate.) The bright
state super-radiantly emits into the waveguide at a rate of 𝑁Γ1D . The collective dark
state has no coupling to the waveguide, and a decoherence rate Γ0D which is set by
parasitic damping and dephasing not captured in the simple waveguide QED model
(see Sec. C.2 and Sec. C.3). In addition to the bright and dark collective states
described above, there exist an additional 𝑁 − 2 collective states of the qubit array
with no coupling to either the probe qubit or the waveguide [120].
The subsystem consisting of a coupled probe qubit and symmetric dark state of the
mirror qubit array can be described as an analog to a cavity QED system [120].
In this depiction, the probe qubit plays the role of a two-level atom and the dark
state mimics a high-finesse cavity, with the qubits in the 𝜆 0 /2-spaced array acting as
atomic mirrors (see Fig. 4.1a). In general, provided that the fraction of excited array
qubits remains small as 𝑁 increases, 𝑆ˆD stays nearly bosonic and the analogy to the
Jaynes-Cummings model remains valid. By mapping the waveguide parameters to
those of a cavity QED system, the cooperativity between probe qubit and atomic
cavity can be written as C = (2𝐽) 2 /(Γ1D,p + Γ0p )Γ0D ≈ 𝑁 𝑃1D . Here 𝑃1D = Γ1D /Γ0D is

50
the single qubit Purcell factor, which quantifies the ratio of the waveguide emission
rate to the parasitic damping and dephasing rates. Attaining C > 1 is a prerequisite
for observing coherent quantum effects. Referring to the energy level diagram of
Fig. 4.1a, by sufficiently reducing the waveguide coupling rate of the probe qubit
one can also realize a situation in which 𝐽 > (Γ1D,p + Γ0p ), Γ0D , corresponding to
the strong coupling regime of cavity QED between excited state of the probe qubit
(|𝑒ip |𝐺i) and a single photon in the atomic cavity (|𝑔ip |𝐷i) (see Sec. C.2). This
mapping of a waveguide QED system onto a cavity QED analog therefore allows us
to use cavity QED techniques to efficiently probe the dark states of the qubit array
with single-photon precision.
The fabricated superconducting circuit used to realize the waveguide-QED system
is shown in Fig. 4.1b. The circuit consists of seven transmon qubits (Q 𝑗 , where
𝑗 = 1-7), all of which are side-coupled to the same coplanar waveguide (CPW).
Each qubit’s transition frequency is tunable via an external flux bias port (Z1 -Z7 ).
We use the top-center qubit in the circuit (Q4 ) as a probe qubit. This qubit can be
independently excited via a weakly coupled CPW drive line (XY4 ), and is coupled to
a lumped-element microwave cavity (R4 ) for dispersive readout of its state. The other
six qubits are mirror qubits. The mirror qubits come in two different types (I and II),
which are designed with different waveguide coupling rates (Γ1D,I /2𝜋 = 20 MHz
and Γ1D,II /2𝜋 = 100 MHz) in order to provide access to a range of Purcell factors.
Type I mirror qubits also lie in pairs across the CPW waveguide and have rather
large (∼ 50 MHz) direct coupling.
4.3

Spectroscopic characterizations

We characterize the waveguide and parasitic coupling rates of each individual qubit
by measuring the phase and amplitude of microwave transmission through the
waveguide (see Fig. 4.2) [146]. Measurements are performed in a dilution refrigerator at a base temperature of 8 mK (see Sec. 4.7). For a sufficiently weak coherent
drive the effects of qubit saturation can be neglected and the on-resonance extinction
of the coherent waveguide tone relates to a lower bound on the individual qubit Purcell factor. Any residual waveguide thermal photons, however, can result in weak
saturation of the qubit and a reduction of the on-resonance extinction. We find an
on-resonance intensity transmittance as low as 2 × 10−5 for the type II mirror qubits,
corresponding to an upper bound on the CPW mode temperature of 43 mK and a
lower bound on the Purcell factor of 200. Further details of the design and measured
parameters of probe and each mirror qubit are provided in Sec. 4.7.

51
100

Transmittance, |t|2

100

10-2
10-2

10-1
10-2

10-4

Probe Q4

-2

Detuning (MHz)

10-3
Type I Q6
Type II Q1

-100

10-4
10-5-1 0 1
100

Detuning (MHz)

Figure 4.2: Single-qubit spectroscopy. Waveguide transmission spectrum across individual qubit resonances. The left panel shows the probe qubit (Q4 ); the right panel shows type I
(Q6 , green curve) and type II (Q1 , blue curve) mirror qubits). The inset shows a zoomed-in
view of the center of the curves with the same axes. From a Lorentzian lineshape fit of the
measured waveguide transmission spectra we infer Purcell factors of 𝑃1D = 11 for the probe
qubit and 𝑃1D = 98 (219) for the type I (type II) mirror qubit. The figure is adapted from
Ref. [108].

The transmission through the waveguide, in the presence of the probe qubit, can also
be used to measure spectroscopic signatures of the collective dark state of the qubit
array. As an example of this we utilize a single pair of type I mirror qubits (Q2 and
Q6 ), which we tune to a frequency where their separation along the waveguide axis
is 𝑑 = 𝜆 0 /2. The remaining qubits on the CPW are decoupled from the waveguide
input by tuning their frequency away from the measurement point. Figure 4.3a shows
the waveguide transmission spectrum for a weak coherent tone in which a broad
resonance dip is evident, corresponding to the bright state of the mirror qubit pair.
We find a bright-state waveguide coupling rate of Γ1D,B ≈ 2Γ1D = 2𝜋 × 26.8 MHz
by fitting a Lorentzian lineshape to the spectrum. The dark state of the mirror qubits
is not observable in this waveguide spectrum, but it becomes observable when
measuring the waveguide transmission with the probe qubit tuned into resonance
with the mirror qubits (see Fig. 4.3b). In addition to the broad response from the
bright state, in this case two spectral peaks appear near the center of the brightstate resonance (Fig. 4.3c). This pair of highly non-Lorentzian spectral features
result from the Fano interference between the broad bright state and the hybridized
polariton resonances formed between the dark state of the mirror qubits (the atomic
cavity photon) and the probe qubit. The hybridized probe qubit and the atomic
cavity eigenstates can be more clearly observed by measuring the transmission
between the probe qubit drive line (XY4 ) and the output port of the waveguide
(see Fig. 4.3d). As the XY4 line does not couple to the bright state owing to the
symmetry of its positioning along the waveguide, we observe two well-resolved

In M1

M1 Out

100
50

c 1

In M1 P M1 Out

0.5

-50

|t|2

-100

100

3×10-3

Z4 voltage (mV)

Transmittance, |t|2

0.5

-50

M1 P M1 Out

-100

6.58 6.59

6.6

6.61

Frequency (GHz)

6.58

6.59

6.6

6.61

Frequency (GHz)

Figure 4.3: Vacuum Rabi splitting. a, Transmission through the waveguide for two
type I mirror qubits (M1; Q2 and Q6 ) on resonance, with the remaining qubits on the
chip tuned away from the measurement frequency range. b, Transmission through the
waveguide as a function of the flux bias tuning voltage of the probe qubit (Q4 ). c, Waveguide
transmission spectrum for the probe qubit (P; Q4 ) and the mirror qubits tuned to resonance.
d. Transmission spectrum as measured between the probe qubit drive line XY4 and the
waveguide output as a function of flux bias tuning of the probe qubit. e, XY4 -to-waveguide
transmission spectrum for the three qubits tuned to resonance. The dashed red lines in d and
solid black line in e are the predictions of a numerical model with experimentally measured
qubit parameters. The prediction ine includes slight power broadening effects. The figure
is adapted from Ref. [108].

resonances (Fig. 4.3e) with mode splitting 2𝐽/2𝜋 ≈ 6 MHz, when the probe qubit
is nearly resonant with the dark state. Observation of vacuum Rabi splitting in the
hybridized atomic cavity-probe qubit polariton spectrum signifies operation in the
strong coupling regime.
4.4

Time-domain characterizations

To further investigate the signatures of strong coupling we perform time-domain
measurements in which we prepare the system in the initial state |𝑔ip |𝐺i → |𝑒ip |𝐺i
using a 10-ns microwave 𝜋 pulse applied at the XY4 drive line. Following excitation
of the probe qubit we use a fast (5 ns) flux bias pulse to tune the probe qubit into

53
|e〉p

|G〉

M1 P M1

Xπ

0.5

Population

|D〉

τ = TSWAP

Read

M2 P M2

|g〉p

fp
fM1
fp0
fM2

T1 = 134 ns

0.5

200

400

600

800

TSWAP = 83 ns

0.5

200

400

600

800

TSWAP = 36 ns

200

τ (ns)
400

600

800

Figure 4.4: Vacuum Rabi oscillations. Measured population of the excited state of the
probe qubit for three different scenarios. The top, red curve represents the probe qubit (P)
tuned to 𝑓p0 = 6.55 GHz, with all mirror qubits tuned away, corresponding to free population
decay. The middle, green curve represents the probe qubit tuned into resonance with a pair
of type I mirror qubits (M1; Q2 and Q6 ) at frequency 𝑓M1 = 6.6 GHz corresponding to
𝑑I = 𝜆0 /2. The bottom, blue curve represents the probe qubit tuned into resonance with
type II mirror qubits (M2; Q1 and Q7 ) at frequency 𝑓M2 = 5.826 GHz corresponding to
population decay time and half the oscillation time period for the spontaneous decay curve
and the vacuum Rabi oscillations, respectively. The figure is adapted from Ref. [108].

resonance with the collective dark state of the mirror qubits (the atomic cavity)
for a desired interaction time 𝜏. Upon returning to its initial frequency after the
flux bias pulse, the excited-state population of the probe qubit is measured via the
dispersively coupled readout resonator. In Fig. 4.4, we show a timing diagram
and plot three measured curves of the probe qubit’s population dynamics versus
𝜏. The top red curve corresponds to the measured probe qubit’s free decay, where
the probe qubit is shifted to a detuned frequency 𝑓p0 to eliminate mirror qubit
interactions. From an exponential fit to the decay curve we find a decay rate of
1/𝑇1 ≈ 2𝜋 × 1.19 MHz, in agreement with the result from waveguide spectroscopy
at 𝑓p0 . In the middle green and bottom blue curves we plot the measured probe
qubit’s population dynamics while interacting with an atomic cavity formed from

54
type I and type II mirror qubit pairs, respectively. In both cases the initially prepared
state |𝑒ip |𝐺i undergoes vacuum Rabi oscillations with the dark state of the mirror
qubits |𝑔ip |𝐷i. Along with the measured data we plot a theoretical model where the
waveguide coupling, parasitic damping, and dephasing rate parameters of the probe
qubit and dark state are taken from independent measurements, and the detuning
between probe qubit and dark state is left as a free parameter (see Sec. C.2). From
the excellent agreement between measurement and model we infer an interaction
rate of 2𝐽/2𝜋 = 5.64 MHz (13.0 MHz) and a cooperativity of C = 94 (172) for the
type I (type II) mirror system. For both mirror types we find that the system is well
within the strong coupling regime 𝐽
Γ1D,p + Γ0p , Γ0D , with the photon-mediated
interactions dominating the decay and dephasing rates by roughly two orders of
magnitude.
The tunable interaction time in our measurement sequence also permits state transfer
between the probe qubit and the dark state of the mirror qubits using an iSWAP gate.
We measure the dark state’s population decay in a protocol where we excite the
probe qubit and transfer the excitation into the dark state (see Fig. 4.5a). From an
exponential fit to the data we find a dark-state decay rate of 𝑇1,D = 757 ns (274 ns) for
type I (type II) mirror qubits, enhanced by approximately the cooperativity over the
bright-state lifetime. In addition to the lifetime, we can measure the coherence time
∗ = 435 ns
of the dark state with a Ramsey-like sequence (see Fig. 4.5b), yielding 𝑇2,D
(191 ns) for type I (type II) mirror qubits. The collective dark-state coherence time is
slightly shorter than its population decay time, hinting at correlated sources of noise
in the distantly entangled qubits forming the dark state (see discussion in Sec. C.3).
These experiments have so far probed the waveguide and the multi-qubit array with
a single excitation, where the cavity QED analog is helpful for understanding the
response. However, this analogy is not fully accurate for understanding multiexcitation dynamics, where the quantum nonlinear response of the qubits leads
to a number of interesting phenomena. To observe this, we populate the atomic
cavity with a single photon via an iSWAP gate and then measure the transmission
of weak coherent pulses through the waveguide. Figure 4.5c shows transmission
through the atomic cavity formed from type I mirror qubits before and after adding
a single photon. The sharp change in the transmissivity of the atomic cavity is
a result of trapping in the long-lived dark state of the mirror qubits. The dark
state has no transition dipole to the waveguide channel (see Fig. 4.5d), and thus
it cannot participate in absorption or emission of photons when probed via the

Population

T1 = 757 ns
T1 = 274 ns

0.5

Transmittance, |t|2

c 100
10-1

τ (μs)

|D〉

-2

10
6.51 6.55

6.59

Frequency (GHz)

|G〉

|E〉
|G〉

|B〉

2Γ1D

ℒ(τ)
0.5

1.5

T2 = 191 ns

τ (μs)

0.5

e 1
2Γ1D

Xπ/2

T2* = 435 ns

Population

ℒ(τ)

|G〉

b |g〉p Xπ/2
Population

|g〉p Xπ

0.8
0.6
0.4
0.2

τ (ns)

200 400 600

Figure 4.5: Characterization of the dark-state cavity a, Measurement of the population
decay time (𝑇1,D ) of the dark state of the type I (top, green) and type II (bottom, blue) mirror
∗ ) of the type I (top, green) and type
qubits. b, Corresponding Ramsey coherence time (𝑇2,D
II (bottom, blue) dark states. c, Waveguide transmission spectrum through the atomic cavity
without (brown data points) and with (orange data points) pre-population of the cavity. Here
the atomic cavity is initialized in a single photon state by performing an iSWAP gate acting
on the probe qubit followed by detuning of the probe qubit away from resonance. In both
cases the transmission measurement is performed using coherent rectangular pulses with a
duration of 260 ns and a peak power of 𝑃 ≈ 0.03(ℏ𝜔0 Γ1D ). Solid lines show theory fits
from numerical modeling of the system. d, Energy level diagram of the 0 (|𝐺i), 1 (|𝐷i,B),
and 2 (E) excitation manifolds of the atomic cavity indicating waveguide-induced decay
and excitation pathways. e, Rabi oscillation with two excitations in the system of the probe
qubit and atomic cavity. The shaded region shows the first iSWAP step in which an initial
probe qubit excitation is transferred to the atomic cavity. Populating the probe qubit with an
additional excitation at this point results in strong damping of subsequent Rabi oscillations
due to the rapid decay of state |𝐸i. The dashed brown curve is the predicted result for
interaction of the probe qubit with an equivalent linear cavity. In (c)-(e) the atomic cavity is
formed from type I mirror qubits Q2 and Q6 . The figure is adapted from Ref. [108].

56
waveguide. As a result, populating the atomic cavity with a single photon makes it
nearly transparent to incoming waveguide signals for the duration of the dark-state
lifetime. This is analogous to the electron shelving phenomenon, which leads to
suppression of resonance fluorescence in three-level atomic systems [200]. As a
further example, we use the probe qubit to prepare the cavity in the doubly excited
state via two consecutive iSWAP gates. In this case, with only two mirror qubits
and the rapid decay via the bright state of the two-excitation state |𝐸i of the mirror
qubits (refer to Fig. 4.5d), the resulting probe qubit population dynamics shown in
Fig. 4.5e have a strongly damped response (C < 1) with weak oscillations occurring
at the vacuum Rabi oscillation frequency. This is in contrast to the behavior of a
linear cavity (shown as the dashed brown curve in Fig. 4.5e), where driving the
second photon transition leads to persistent Rabi oscillations with a frequency that
is 2 times larger than vacuum Rabi oscillations. Further analysis of the nonlinear
behavior of the atomic cavity is provided in Sec. C.4.
4.5

Compound atomic mirrors

The waveguide QED chip of Fig. 4.1b can also be used to investigate the spectrum
of sub-radiant states that emerge when 𝑁 > 2 and there is direct interaction between
mirror qubits. This situation can be realized by taking advantage of the capacitive
coupling between co-localized pairs of type I qubits (Q2 and Q3 , or Q5 and Q6 ).
Although in an idealized 1D waveguide model there is no cooperative interaction
term between qubits with zero separation along the waveguide, we observe a strong
coupling (with the measured interaction rate, 𝑔/2𝜋 = 46 MHz) between the colocalized pair of mirror qubits Q2 and Q3 , as shown in Fig. 4.6a. This coupling
results from near-field components of the electromagnetic field that are excluded
in the simple waveguide model. The non-degenerate hybridized eigenstates of the
qubit pair effectively behave as a compound atomic mirror. The emission rate of
each compound mirror to the waveguide can be adjusted by setting the detuning,
Δ, between the pair. As illustrated in Fig. 4.6b, resonantly aligning the compound
atomic mirrors on both ends of the waveguide results in a hierarchy of bright and
dark states involving both near-field and waveguide-mediated cooperative coupling.
Probing the system with a weak continuous tone via the waveguide, we identify the
two super-radiant combinations of the compound atomic mirrors (Fig. 4.6c). Similar
to the case of a two-qubit cavity, we can identify the collective dark states via the
probe qubit. As evidenced by the measured Rabi oscillations shown in Fig. 4.6d,
the combination of direct and waveguide-mediated interactions of mirror qubits in

Z2 voltage (V)

0.3
|t|2
0 1

-0.3
6.1

Z4 voltage (V)

0.30

6.4

6.2

Population

6.4

6.45

|B1〉

6.5

0.5

Time (μs)

6.6

6.55

Probe qubit

|B2〉

|D2〉

|g〉p|D1〉 ↔ |e〉p|G〉
Decay of |D1〉

0.5

|S〉

6.6

6.5

|D1〉

6.3

Frequency (GHz)

|t|2
0.15

|A〉

|S〉

|g〉p|D2〉 ↔ |e〉p|G〉
Decay of |D2〉

0.5

Time (μs)

Figure 4.6: Compound atomic mirrors, 𝑁 = 4. a, Avoided mode crossing of a pair of type
I mirror qubits positioned on opposite sides of the CPW. Near the degeneracy point, the qubits
form a pair of compound eigenstates consisting of symmetric (|𝑆i) and anti-symmetric (| 𝐴i)
states with respect to the waveguide axis. b, Measured transmission through the waveguide
with the pair of compound atomic mirrors aligned in frequency. The two broad resonances
correspond to super-radiant states |𝐵1 i and |𝐵2 i as indicated. As we tune the frequency
of the probe qubit, we observe the (avoided-crossing-like) signatures of the interaction of
the probe qubit with each dark state. c, Illustration of the single-excitation manifold of
the collective states of 𝑁 = 4 mirror qubits forming a pair of compound atomic cavities.
The bright (super-radiant) and dark (sub-radiant) states can be identified by comparing the
symmetry of the compound qubit states with the resonant radiation field pattern in the
waveguide. d, Probe qubit measurements of the two dark states, |𝐷 1 i and |𝐷 2 i. In these
measurements the frequency of each dark state is shifted to ensure 𝜆0 /2 separation between
the two compound atomic mirrors. The figure is adapted from Ref. [108].

58
this geometry results in the emergence of a pair of collective entangled states of the
four qubits acting as strongly-coupled atomic cavities with a frequency separation
of 4𝑔 2 + Δ2 .
4.6

Conclusion

In conclusion, we have realized a synthetic cavity QED system in which to observe
and drive the coherent dynamics that emerge from correlated dissipation in an open
waveguide, paving the way for several exciting research avenues beyond the work
presented here. Our current work has reached single-qubit Purcell factors exceeding
200, which is an order of magnitude larger than the experimental state of the art
in planar superconducting quantum circuits and on par with the values achievable
in less scalable three-dimensional architectures [135], but further improvement is
theoretically possible. With better thermalization to the waveguide [157] and coherence times in line with the best planar superconducting qubits [201], Purcell
factors in excess of 104 should be achievable. In this regime, with an already realized system size of 𝑁 = 4, a universal set of quantum gates with fidelity above
0.99 could theoretically be realized by encoding information in decoherence-free
subspaces [123]. Even without improved Purcell factors, the control demonstrated
here over the sub-radiant states of an atomic chain enables studies of the formation
of fermionic correlations between excitations and the power-law decay dynamics
associated with a critical open system in a modestly-sized array (𝑁 = 10) [191].
Further, the demonstrated ability to measure the population decay time and coherence time for the entangled states of multiple distant qubits provides a valuable
experimental tool with which to examine the sources of correlated decoherence in
circuit QED. Finally, reducing the frequency disorder of transmon qubits beyond
the values measured in our system (𝛿 𝑓 ≈ 60 MHz) and using a slow-light metamaterial waveguide [109] would allow chip-scale waveguide QED experiments with
a much larger number of fixed-frequency qubits, in the range 𝑁 = 10–100, where
the full extent of the many-body dynamics of large quantum spin chains can be
studied [192–194].
4.7

Methods

Qubits
We design and fabricate transmon qubits in three different variants for the experiment (see Fig. 4.7a-b): type I mirror qubits (Q2 , Q3 , Q5 , Q6 ), type II mirror qubits
(Q1 , Q7 ), and the probe qubit (Q4 ). The qubit frequency tuning range, waveg-

id
gu

ve
Wa

Meander

250 μm

5 μm

id
gu

100 μm

5 μm

Figure 4.7: Scanning electron microscope image of the fabricated device. a, Type I (Q2 ,
Q3 ) and type II (Q1 ) mirror qubits coupled to the coplanar waveguide (CPW). b, The central
probe qubit (Q4 ) and lumped-element readout resonator (R4 ) coupled to CPW. The inset
shows an inductive meander of the lumped-element readout resonator. c, A superconducting
quantum interference device (SQUID) loop with asymmetric Josephson junctions used for
qubits. d, An airbridge placed across the waveguide to suppress slotline mode.

𝑓max (GHz)
𝑓min (GHz)
Γ1D /2𝜋
(MHz)
Γ /2𝜋 (kHz)

Q1
6.052
4.861

Q2
6.678
5.373

Q3
6.750
5.389

Q4
6.638
5.431

Q5
6.702
5.157

Q6
6.817
5.510

Q7
6.175
4.972

94.1

16.5

13.9a,b 0.91

18.4b

18.1

99.5

430

< 341

< 760a,b

375b

185

998

a Measured at 6.6 GHz
b Measured without the cold attenuator

Table 4.1: Qubit characteristics. 𝑓max ( 𝑓min ) is the maximum (minimum) frequency of
the qubit, corresponding to “sweet spots" with zero first-order flux sensitivity. Γ1D is the
qubit’s rate of decay into the waveguide channel and Γ0 is its parasitic decoherence rate
due to damping and dephasing from channels other than the waveguide at 0 temperature.
All reported values are measured at the maximum frequency of each qubit, save for Q3 in
which case the values were measured at 6.6 GHz (marked with superscript a ). With the
exception of Q3 and Q5 (marked with superscript b ), all the values are measured with the
cold attenuator placed in the input line of the waveguide (see Sec. C.1).

60
uide coupling rate (Γ1D ), and parasitic decoherence rate (Γ0) can be extracted from
waveguide spectroscopy measurements of the individual qubits. The values for all
the qubits inferred in this manner are listed in Table 4.1. Note that Γ0 is defined
as due to damping and dephasing from channels other than the waveguide at zero
temperature. The inferred value of Γ0 from waveguide spectroscopy measurements
is consistent with this definition in the zero temperature waveguide limit (effects of
finite waveguide temperature are considered in Sec. C.1). The standard deviation
in maximum frequencies of the four identically designed qubits (type I) is found as
61 MHz, equivalent to ∼ 1% qubit frequency disorder in our fabrication process.
Asymmetric Josephson junctions are used in all qubits’ superconducting quantum
interference device (SQUID) loops (Fig. 4.7c) to reduce dephasing from flux noise,
which limits the tuning range of qubits to around 1.3GHz. For Q4 , the Josephson energies of the junctions are extracted to be 𝐸 J1 /ℎ = 18.4 GHz and 𝐸 J2 /ℎ = 3.5 GHz,
𝐽2
giving a junction asymmetry of 𝑑 ≡ 𝐸𝐸 𝐽𝐽 11−𝐸
+𝐸 𝐽 2 = 0.68. The anharmonicity is measured
to be 𝜂/2𝜋 = −272 MHz and 𝐸 𝐽 /𝐸𝐶 = 81 at maximum frequency for Q4 .
Readout
We have fabricated a lumped-element resonator (shown in Fig. 4.7b) to perform
dispersive readout of the state of central probe qubit (Q4 ). The lumped-element
resonator consists of a capacitive claw and an inductive meander with a pitch of about
1 𝜇m, effectively acting as a quarter-wave resonator. The bare frequency of resonator
and coupling to probe qubit are determined to be 𝑓r = 5.156 GHz and 𝑔/2𝜋 =
116 MHz, respectively, giving dispersive frequency shift of 𝜒/2𝜋 = −2.05 MHz
for Q4 at maximum frequency. The resonator is loaded to the common waveguide
in the experiment, and its internal and external quality factors are measured to be
𝑄 i = 1.3 × 105 and 𝑄 e = 980 below single-photon level. It should be noted that the
resonator-induced Purcell decay rate of Q4 is ΓPurcell
/2𝜋 ∼ 70 kHz, small compared
to the decay rate into the waveguide Γ1D,p /2𝜋 ∼ 1 MHz. The compact footprint
of the lumped-element resonator is critical for minimizing the distributed coupling
effects that may arise from interference between direct qubit decay to the waveguide
and the the Purcell decay of the qubit via the resonator path.
Suppression of spurious modes
In our experiment we use a coplanar transmission line for realizing a microwave
waveguide. In addition to the fundamental propagating mode of the waveguide,
which has even symmetry with respect to the waveguide axis, these structures

61
also support a set of modes with odd symmetry, known as slotline modes. The
propagation of a slotline mode can be completely suppressed in a waveguide with
perfectly symmetric boundary conditions. However, in practice perfect symmetry
cannot be maintained over the full waveguide length, which unavoidably leads to
presence of the slotline mode as a spurious loss channel for the qubits. Crossovers
connecting ground planes across the waveguide are known to suppress propagation
of slotline modes, and to this effect Al airbridges have been used in superconducting
circuits with negligible impedance mismatch for the desired CPW mode [202].
In this experiment, we place airbridges (Fig. 4.7d) along the waveguide and control
lines at specific distances set with the following considerations. Airbridges create
reflecting boundary for slotline mode, and if placed by a distance 𝑑 a discrete resonance corresponding to wavelength of 2𝑑 is formed. Therefore, placing airbridges
over distances less than 𝜆/4 apart, where 𝜆 is the wavelength of the mode resonant
with the qubits, pushes the slotline resonances of the waveguide sections between the
airbridges to substantially higher frequencies. In this situation, the dissipation rate
of qubits via the spurious channel is substantially suppressed by the off-resonance
Purcell factor ΓPurcell
∼ (𝑔/Δ) 2 𝜅, where Δ denotes detuning between the qubit transi1
tion frequency and the frequency of the odd mode in the waveguide section between
the two airbridges, and where the parameters 𝑔 and 𝜅 are the interaction rate of the
qubit and the decay rate of the slot-line cavity modes, respectively. In addition, we
place the airbridges before and after bends in waveguide, to ensure the fundamental
waveguide mode is not converted to the slot-line mode upon propagation [203].
Crosstalk in flux biasing
We tune the frequency of each qubit by supplying a bias current to individual flux
control lines (Z lines), which control the magnetic flux in the qubit’s SQUID loop.
In our system, the Z lines are attached to external wires in two forms with different
configurations, which allows the qubit frequency to be tuned in ‘slow’ and ‘fast’
timescales (See Fig. 4.8). The bias currents were generated via independent bias
voltages generated by seven arbitrary waveform generator channels, allowing for
simultaneous tuning of all qubits. In practice, independent frequency tuning of each
qubit needs to be accompanied by small changes in the flux bias of the qubits in the
near physical vicinity of the qubit of interest, owing to cross-talk between adjacent
Z control lines.
In this experiment, we characterized the crosstalk between bias voltage channels of

62
Waveguide
Input

20 dB

40 dB

Thin-film
attenuator

XY4

Fast Z

20 dB
225
MHz

Slow Z

64
kHz

Waveguide
Output

HEMT

300 K
50 K plate
4 K plate
Mixing
chamber
plate

225
MHz

(20 dB)

7.5
GHz

2-8.7
GHz

Figure 4.8: Schematic of the measurement chain inside the dilution refrigerator. The
four types of input lines, the output line, and their connection to the device inside a magnetic
shield are illustrated. Attenuators are expressed as rectangles with labeled power attenuation
and capacitor symbols correspond to direct-current blocks. The thin-film attenuator and a
circulator (colored red) are added to the waveguide input line and output line, respectively,
in a second version of the setup and a second round of measurements to further protect the
sample from thermal noise in the waveguide line. HEMT, high-electron-mobility-transistor.

the qubits in the following way. First, we tune the qubits not in use to frequencies
more than 800 MHz away from the working frequency (which is set as either
5.83 GHz or 6.6 GHz). These qubits are controlled by fixed biases such that their
frequencies, even in the presence of crosstalk from other qubits, remained far enough
from the working frequency and hence were not considered for the rest of the analysis.
Second, we tune the remaining in-use qubits to relevant frequencies within 100MHz
of the working frequency and record the array of biases 𝒗 0 and frequencies 𝒇 0 of all
qubits. Third, we varied the bias on only a single ( 𝑗th) qubit and linearly interpolated
the change in frequency ( 𝑓𝑖 ) of the other (𝑖th) qubits with respect to bias voltage
𝑣 𝑗 on the 𝑗th qubit, finding the cross talk matrix component 𝑀𝑖 𝑗 = (𝜕 𝑓𝑖 /𝜕𝑣 𝑗 ) 𝒗=𝒗 0 .
Repeating this step, we obtained the following (approximately linearized) relation
between the frequencies 𝒇 and bias voltages 𝒗 of qubits:
𝒇 ≈ 𝒇 0 + 𝑀 (𝒗 − 𝒗 0 ).
Finally, we took the inverse of the above relation to find bias voltages 𝒗 that is
required for tuning qubits to frequencies 𝒇 :
𝒗 ≈ 𝒗 0 + 𝑀 −1 ( 𝒇 − 𝒇 0 ).

63
An example of such crosstalk matrix between Q2 , Q4 , and Q6 near 𝒇 0 = (6.6, 6.6, 6.6)GHz
used in the experiment is given by
0.2683 −0.0245 −0.0033ª
𝑀 = −0.0141 −0.5310 0.0170 ® GHz/V
« 0.0016 0.0245 0.4933 ¬
This indicates that the crosstalk level between Q4 and either Q2 or Q6 is about 5%,
while that between Q2 and Q6 is less than 1%. We have repeated similar steps for
other configurations in the experiment.
Measurement Setup
Figure 4.8 illustrates the outline of the measurement chain in our dilution refrigerator. The sample was enclosed in a magnetic shield that was mounted at the
mixing chamber. We have outlined four different types of input lines used in our
experiment. Input lines to the waveguide and XY4 go through a direct-current block
at room temperature and then were attenuated by 20 dB at the 4 K stage, followed
by additional 40 dB of attenuation at the mixing chamber. The fast flux tuning lines
(Z3 , Z4 ) are attenuated by 20 dB and were filtered with a low-pass filter with corner
frequency at 225 MHz to minimize thermal noise photons while maintaining short
rise and fall time of pulses for fast flux control. The slow flux tuning lines (Z1 , Z2 ,
Z5 , Z6 , Z7 ) are filtered by an additional low-pass filter with 64 kHz corner frequency
at the 4K stage to further suppress noise photons. In addition, the waveguide signal
output path contained a high electron mobility transistor (HEMT) amplifier at the
4K plate. Three circulators were placed between the HEMT and the sample to
ensure (> 70 dB) isolation of the sample from the amplifier noise. In addition, a
series of low-pass and band-pass filters on the output line suppressed noise sources
outside the measurement spectrum.
A thin-film ‘cold attenuator’, developed by Palmer and colleagues at the University
of Maryland [157], was added to the measurement path in order to achieve better
thermalization between the microwave coaxial line and its thermal environment.
Similarly, an additional circulator was added to the waveguide measurement chain in
later setups to further protect the device against thermal photons (both the attenuator
and circulator are highlighted in red in the schematic in Fig. 4.8). The effect of this
change is discussed in Sec. C.1.

64
Dark state characterization
We characterized the collective dark state of mirror qubits with population decay
∗ using the cooperative interaction with the
time 𝑇1,D and Ramsey coherence time 𝑇2,D
probe qubit. For each configuration of mirror qubits, we obtain the Rabi oscillation
curve (see Figs. 4.4a, 4.6d) using a fast flux-bias pulse on the probe qubit as explained
in the main text. The half-period 𝑇SWAP of Rabi oscillation results in a complete
transfer of probe qubit population to the collective dark state and vice versa, hence
defining an iSWAP gate.
To measure the population decay time 𝑇1,D of the dark state, we excited the probe
qubit with a resonant microwave 𝜋-pulse, followed by applying an iSWAP gate.
This prepared the collective dark state |𝑔ip |𝐷i off-resonantly decoupled from the
probe qubit. After free evolution of the dark state for a variable duration 𝜏, another
iSWAP gate was applied to transfer the remaining dark state population back to the
probe qubit. Finally, we measured the state of the probe qubit and performed an
exponential fitting to the resulting decay curve.
∗ of the dark state as follows.
Likewise, we measured Ramsey coherence time 𝑇2,D
First, we excited the probe qubit to a superposition (|𝑔i + |𝑒i)p |𝐺i of the ground and
excited states by applying a detuned microwave 𝜋/2-pulse. Next, application of an
iSWAP gate maps this superposition to that of the dark state |𝑔ip (|𝐺i + |𝐷i). After a
varying delay time 𝜏, another iSWAP gate was applied, followed by a detuned 𝜋/2pulse on the probe qubit. Measurement of the state of the probe qubit resulted in a
damped oscillation curve the decay envelope of which gives the Ramsey coherence
time of the dark state involved in the experiment. Note that the fast oscillation
frequency in this curve is determined by detuning of the dark state with respect to
the frequency of the microwave pulses applied to the probe qubit.

65
Chapter 5

DEVELOPMENT OF SUPERCONDUCTING METAMATERIALS
FOR WAVEGUIDE QED
In the previous chapter, we observed that a waveguide having a linear dispersion
relation close to transition frequencies of superconducting qubits can be harnessed
to induce coherent long-range interaction in the strong-coupling regime. As noted in
Secs. 2.2 and 2.3, there are several drawbacks of this approach associated with having
to work with specific arrangement of qubit array in order to access decoherence-free
subspace, the only states in a waveguide QED system protected from radiation to the
waveguide channel, complicating the protocol for quantum information processing.
We divert our attention to an alternative direction utilizing a dispersive waveguide
for realizing long-range qubit-qubit interaction inside a photonic bandgap, while simultaneously protecting qubits from radiation to the waveguide. In this chapter, we
introduce our work on the development of a compact and scalable microwave structure based on metamaterial concept and characterization of this structure utilizing a
superconducting qubit, which is published in Ref. [109].
5.1

Introduction

Recently, there has been much interest in achieving strong light-matter interaction
in a cavity-free system such as a waveguide [40, 195]. Slow-light photonic crystal
waveguides are of particular interest in waveguide QED because the reduced group
velocity near a bandgap preferentially amplifies the desired radiation of the atoms
into the waveguide modes [137, 204, 205]. Moreover, in this configuration an
interesting paradigm can be achieved by placing the resonance frequency of the
atom inside the bandgap of the waveguide [126, 142, 206, 207]. In this case, the
atom cannot radiate into the waveguide but the evanescent field surrounding it gives
rise to a photonic bound state [126]. The interaction of such localized bound states
has been proposed for realizing tunable spin-exchange interaction between atoms in
a chain [125, 208], and also for realizing effective non-local interactions between
photons [209, 210].
While achieving efficient waveguide coupling in the optical regime requires the challenging task of interfacing atoms or atomic-like systems with nanoscale dielectric
structures [132, 140, 211–213], superconducting circuits provide an entirely differ-

66
ent platform for studying the physics of light-matter interaction in the microwave
regime [35, 195]. Development of the field of circuit QED has enabled fabrication
of tunable qubits with long coherence times and fast qubit gate times [190, 214].
Moreover, strong coupling is readily achieved in coplanar platforms due to the deep
sub-wavelength transverse confinement of photons attainable in microwave waveguides and the large electrical dipole of superconducting qubits [27]. Microwave
waveguides with strong dispersion, even “bandgaps” in frequency, can also be simply realized by periodically modulating the geometry of a coplanar transmission
line [180]. Such an approach was recently demonstrated in a pioneering experiment
by Liu and Houck [215], whereby a qubit was coupled to the localized photonic
state within the bandgap of a modulated coplanar waveguide (CPW). Satisfying the
Bragg condition in a periodically modulated waveguide requires a lattice constant
on the order of the wavelength, however, which translates to a device size of approximately a few centimeters for complete confinement of the evanescent fields in
the frequency range suitable for microwave qubits. Such a restriction significantly
limits the scaling in this approach, both in qubit number and qubit connectivity.
An alternative approach for tailoring dispersion in the microwave domain is to
take advantage of the metamaterial concept. Metamaterials are composite structures with sub-wavelength components which are designed to provide an effective
electromagnetic response [216, 217]. Since the early microwave work, the electromagnetic metamaterial concept has been expanded and extensively studied across a
broad range of classical optical sciences [218–220]; however, their role in quantum
optics has remained relatively unexplored, at least in part due to the lossy nature
of many sub-wavelength components. Improvements in design and fabrication of
low-loss superconducting circuit components in circuit QED offer a new prospect
for utilizing microwave metamaterials in quantum applications [221]. Indeed, high
quality-factor superconducting components such as resonators can be readily fabricated on a chip [222], and such elements have been used as a tool for achieving
phase-matching in near quantum-limited traveling wave amplifiers [170, 223] and
for tailoring qubit interactions in a multimode cavity QED architecture [224].
In this work, we utilize an array of coupled lumped-element microwave resonators
to form a compact bandgap waveguide with a deep sub-wavelength lattice constant
(𝜆/60) based on the metamaterial concept. In addition to a compact footprint,
these sort of structures can exhibit highly nonlinear band dispersion surrounding the
bandgap, leading to exceptionally strong confinement of localized intra-gap photon

a 10
bare waveguide
dispersion

Frequency (GHz)

bandgap region
(imaginary k-vector)

1/2 Ck

symmetry axis

ω+,k

ω-,k

1/2 Cr

kx (π/d)

20 μm

Transmission

0.8
0.6

Band Gap

0.4
0.2

4.5

5.5

6.5

7.5

Frequency (GHz)

Figure 5.1: Microwave metamaterial waveguide. a, Dispersion relation of a CPW loaded
with a periodic array of microwave resonators (red curve). The green line shows the
dispersion relation of the waveguide without the resonators. Inset: circuit diagram for a
unit cell of the periodic structure. b, Scanning electron microscope (SEM) image of a
fabricated capacitively coupled microwave resonator, here with a wire width of 500 nm.
The resonator region is false-colored in purple, the waveguide central conductor and the
ground plane are colored green, and the coupling capacitor is shown in orange. We have
used pairs of identical resonators symmetrically placed on the two sides of the transmission
line to preserve the symmetry of the structure. c, Transmission measurement for the realized
metamaterial waveguide made from 9 unit cells of resonator pairs with a wire width of 1 𝜇m,
repeated with a lattice constant of 𝑑 = 350 𝜇m. The blue curve depicts the experimental
data and the red curve shows the lumped-element model fit to the data.

68
l=d

CPW input

100 µm

LDOS (GHz -1μm-1)

10
100

-10

metamaterial
waveguide

10-5

SQUID loop
25 µm

x=d

Frequency (GHz)

100 µm

resonator
qubit

Figure 5.2: Disorder effects and qubit-waveguide coupling. a, Calculated localization
length for a loss-less metamaterial waveguide with structural disorder (blue circles). The
nominal waveguide parameters are determined from the fit to a lumped element model
(including resonator loss) to the transmission data in Fig. 5.1. Numerical simulation has
been performed for 𝑁 = 100 unit cells, averaged over 105 randomly realized values of the
resonance frequency 𝜔0 , with the standard deviation 𝛿𝜔0 /𝜔0 = 0.5%. The vertical green
lines represent the extent of the bandgap region. The red curve outside the gap is an analytic
model based on Ref. [225]. For comparison, the solid black curve shows the calculated
effective localization length without resonator frequency disorder but including resonator
loss. b, SEM image of the fabricated qubit-waveguide system. The metamaterial waveguide
(gray) consists of 9 periods of the resonator unit cell. The waveguide is capacitively coupled
to an external CPW (red) for reflective read-out. c, The transmon qubit is capacitively
coupled to the resonator at the end of the array. The Z drive is used to tune the qubit
resonance frequency by controlling the external flux bias in the superconducting quantum
interference device (SQUID) loop. The XY drive is used to coherently excite the qubit.
d, capacitively coupled microwave resonator. e, Calculated local density of states (LDOS)
at the qubit position for a metamaterial waveguide with a length of 9 unit cells and open
boundary conditions (experimental measurements of LDOS tabulated in Table D.1). The
band-edges for the corresponding infinite structure are marked with vertical green lines. f,
Normalized electromagnetic energy distribution along the waveguide versus qubit frequency
for the coupled qubit-waveguide system. The vertical axis marks the distance from the qubit
(𝑥/𝑑) in units of the lattice period 𝑑.

states. We present the design and fabrication of such a metamaterial waveguide, and
characterize the resulting waveguide dispersion and bandgap properties via interaction with a tunable superconducting transmon qubit. We measure the Lamb shift and
lifetime of the qubit in the bandgap and its vicinity, demonstrating the anomalous
Lamb shift of the fundamental qubit transition as well as selective inhibition and
enhancement of spontaneous emission for the first two excited states of the transmon
qubit.

69
5.2

Band-structure analysis and spectroscopy

We begin by considering the circuit model of a CPW that is periodically loaded
with microwave resonators as shown in the inset to Fig. 5.1a. The Lagrangian for
this system can be constructed as a function of the node fluxes of the resonator
and waveguide sections Φ𝑛𝑏 and Φ𝑛𝑎 [226]. Assuming periodic boundary conditions
and applying the rotating wave approximation, we derive the Hamiltonian for this
system and solve for the energies ℏ𝜔±,𝑘 along with the corresponding eigenstates
|±, 𝑘i = 𝛼ˆ ±,𝑘 |0i as (see Sec. D.1)
(Ω 𝑘 + 𝜔0 ) ± (Ω 𝑘 − 𝜔0 ) + 4𝑔 𝑘 ,
𝜔±,𝑘 =
(5.1)
(𝜔±,𝑘 − 𝜔0 )
𝑔𝑘
𝛼ˆ ±,𝑘 = q
𝑎ˆ 𝑘 + q
𝑏ˆ 𝑘 .
(5.2)
(𝜔±,𝑘 − 𝜔0 ) + 𝑔 𝑘
(𝜔±,𝑘 − 𝜔0 ) + 𝑔 𝑘
Here, 𝑎ˆ 𝑘 , and 𝑏ˆ 𝑘 describe the momentum-space annihilation operators for the bare
waveguide and bare resonator sections, the index 𝑘 denotes the wavevector, and the
parameters Ω 𝑘 , 𝜔0 , and 𝑔 𝑘 quantify the frequency of traveling modes of the bare
waveguide, the resonance frequency of the microwave resonators, and coupling rate
between resonator and waveguide modes, respectively. The operators 𝛼ˆ ±,𝑘 represent
quasi-particle solutions of the composite waveguide, where far from the bandgap
the quasi-particle is primarily composed of the bare waveguide mode, while in the
vicinity of 𝜔0 most of its energy is confined in the microwave resonators.
Figure 5.1a depicts the numerically calculated energy bands 𝜔±,𝑘 as a function of the
wavevector 𝑘. It is evident that the dispersion has the form of an avoided crossing
between the energy bands of the bare waveguide and the uncoupled resonators.
For small gap sizes, the midgap frequency is close to the resonance frequency of
uncoupled resonators 𝜔0 , and unlike the case of a periodically modulated waveguide,
there is no fundamental relation tying the midgap frequency to the lattice constant
in this case. The form of the band structure near the higher cut-off frequency 𝜔c+
can be approximated as a quadratic function (𝜔 − 𝜔c+ ) ∝ 𝑘 2 , whereas the band
structure near the lower band-edge 𝜔c− is inversely proportional to the square of
the wavenumber (𝜔 − 𝜔c− ) ∝ 1/𝑘 2 . The analysis above has been presented for
resonators which are capacitively coupled to a waveguide in a parallel geometry;
a similar band structure can also be achieved using series inductive coupling of
resonators (see Sec. D.1 and Fig. D.1).

70
5.3

Physical realization using lumped-element resonators.

A coplanar microwave resonator is often realized by terminating a short segment of
a coplanar transmission line with a length set to an integer multiple of 𝜆/4, where
𝜆 is the wavelength corresponding to the fundamental resonance frequency [180,
222]. However, it is possible to significantly reduce the footprint of a resonator
by using components that mimic the behavior of lumped elements. We have used
the design presented in Ref. [227] to realize resonators in the frequency range
of 6-10 GHz. This design provides compact resonators by placing interdigitated
capacitors at the anti-nodes of the charge waves and double spiral coils near the peak
of the current waves at the fundamental frequency (see Fig. 5.1b). The symmetry
of this geometry results in the suppression of the second harmonic frequency and
thus the elimination of an undesired bandgap at twice the fundamental resonance
frequency of the band-gap waveguide. A more subtle design criterion is that the
resonators be of high impedance. Use of high impedance resonators allows for a
larger photonic bandgap and greater waveguide-qubit coupling. For the waveguide
QED application of interest this enables denser qubit circuits, both spatially and
spectrally.
The impedance of the resonators scales roughly as the inverse square-root of the pitch
of the wires in the spiral coils. Complicating matters is that smaller wire widths
have been found to introduce larger resonator frequency disorder due to kinetic
inductance effects [228]. Here we have selected an aggressive resonator wire width
of 1 𝜇m and fabricated a periodic array of 𝑁 = 9 resonator pairs coupled to a CPW
with a lattice constant of 𝑑 = 350 𝜇m. The resonators are arranged in identical
pairs placed on opposite sides of the central waveguide conductor to preserve the
symmetry of the waveguide. In addition, the center conductor of each CPW section
is meandered over a length of 210 𝜇m so as to increase the overall inductance of the
waveguide section which also increases the bandgap. Further details of the design
criteria and lumped element parameters are given in Sec. D.2. The fabrication of
the waveguide is performed using electron-beam deposited Al film (See Methods).
Figure 5.1c shows the measured power transmission through such a finite-length
metamaterial waveguide. Here 50-Ω CPW segments, galvanically coupled to the
metamaterial waveguide, are used at the input and output ports. We find a midgap
frequency of 5.83 GHz and a bandgap extent of 1.82 GHz for the structure. Using
the simulated value of effective refractive index of 2.54, the midgap frequency gives
a lattice constant-to-wavelength ratio of 𝑑/𝜆 ≈ 1/60.

71
5.4

Disorder and Anderson localization

Fluctuations in the electromagnetic properties of the metamaterial waveguide along
its length, such as the aforementioned resonator disorder, results in random scattering
of traveling waves. Such random scattering can lead to an exponential extinction
of propagating photons in the presence of weak disorder and complete trapping of
photons for strong disorder, a phenomenon known as the Anderson localization of
light [229]. Similarly, absorption loss in the resonators results in attenuation of wave
propagation which adds a dissipative component to the effective localization of fields
in the metamaterial waveguide. Figure 5.2a shows numerical simulations of the
effective localization length as a function of frequency when considering separately
the effects of resonator frequency disorder and loss (see Sec. D.3 for details of
independent resonator measurements used to determine frequency variation (0.5%)
and loss parameters (intrinsic 𝑄-factor of 7.2 × 104 ) for this model). In addition
to the desired localization of photons within the bandgap, we see that the effects of
disorder and loss also limit the localization length outside the bandgap. In the lower
transmission band where the group index is largest, the localization length is seen
to rapidly approach zero near the band-edge, predominantly due to disorder. In the
upper transmission band where the group index is smaller, the localization length
maintains a large value of 6 × 103 periods all the way to the band-edge. Within
the bandgap the simulations show that the localization length is negligibly modified
by the levels of loss and disorder expected in the resonators of this work, and is
well approximated by the periodic loading of the waveguide alone which can be
simply related to the inverse of the curvature of the transmission bands of a lossless, disorder-free structure [125]. These results indicate that, even with practical
limitations on disorder and loss in such metamaterial waveguides, a range of photon
length scales of nearly four orders of magnitude should be accessible for frequencies
within a few hundred MHz of the band-edges of the gap (See Sec. D.4).
5.5

Anomalous Lamb shift near the band-edge

To further probe the electromagnetic properties of the metamaterial waveguide we
couple it to a superconducting qubit. In this work we use a transmon qubit [190] with
the fundamental resonance frequency 𝜔𝑔𝑒 /2𝜋 = 7.9 GHz and Josephson energy to
single electron charging energy ratio of 𝐸 J /𝐸 C ≈ 100 at zero flux bias (details of
our qubit fabrication methods can also be found in Ref. [230]). Figure 5.2b shows
the geometry of the device where the qubit is capacitively coupled to one end of the
waveguide and the other end is capacitively coupled to a 50-Ω CPW transmission

Lamb shift (MHz)

40
20
-20
-40

T1 lifetime (μs)

5.5

Frequency (GHz)

6.4

6.8

6.5

Figure 5.3: Measured dispersive and dissipative qubit dynamics. a, Lamb shift of
the qubit transition versus qubit frequency. b, Lifetime of the excited qubit state versus
qubit frequency. Open circles show measured data. The solid blue line (dashed red line)
is theoretical curve from the circuit model of a finite (infinite) waveguide structure. For
determining the Lamb shift frompmeasurement, the bare qubit frequency is calculated as a
function of flux bias Φ as ℏ𝜔𝑔𝑒 = 8𝐸 C 𝐸 J (Φ) − 𝐸 C using the extracted values of 𝐸 C , 𝐸 J , and
assuming the symmetrical SQUID flux bias relation 𝐸 J (Φ) = 𝐸 J,max cos(2𝜋Φ/Φ0 ) [190].
The lifetime characterization is performed in the time domain where the qubit is initially
excited with a 𝜋 pulse through the XY drive. The excited state population, determined from
the state-dependent dispersive shift of a close-by band-edge waveguide mode, is measured
subsequent to a delay time during which the qubit freely decays. Inset to b shows a zoomed
in region of the qubit lifetime near the upper band-edge. Solid blue (red) lines show the
circuit model contributions to output port radiation (structural waveguide loss), adjusted to
include a frequency independent intrinsic qubit life time of 10.86 𝜇s. The black dashed line
shows the cumulative theoretical lifetime.

73
line. This geometry allows for forming narrow individual modes in the transmission
band of the metamaterial, which can be used for dispersive qubit state read-out
[231] from reflection measurements at the 50-Ω CPW input port (see Sec. D.2 and
Table D.1). Figures 5.2e and 5.2f show the theoretical photonic LDOS and spatial
photon energy localization versus frequency for this finite length qubit-waveguide
system. Within the bandgap the qubit is self-dressed by virtual photons which
are emitted and re-absorbed due to the lack of escape channels for the radiation.
Near the band-edges surrounding the bandgap, where the LDOS is rapidly varying
with frequency, this results in a large anomalous Lamb shift of the dressed qubit
frequency [129, 207]. Figure 5.3a shows the measured qubit transition frequency
shift from the expected bare qubit tuning curve as a function of frequency. Shown
for comparison are the circuit theory model frequency shift of a finite structure with
𝑁 = 9 periods (blue solid curve) alongside that of an infinite length waveguide (red
dashed curve). It is evident that the qubit frequency is repelled from the band-edges
on the two sides of the bandgap, a result of the strongly asymmetric density of
states in these two regions. The measured frequency shift at the lower frequency
band-edge is 43 MHz, in good agreement with the circuit theory model. Note that
at the lower frequency band-edge where the localization length approaches zero due
to the anomalous dispersion (see Fig. 5.2a), boundary-effects in the finite structure
do not significantly alter the Lamb shift. Near the upper frequency band-edge,
where finite-structure effects are non-negligible due to the weaker dispersion and
corresponding finite localization length, we measure a qubit frequency shift as large
as −28 MHz. This again is in good correspondence with the finite structure model;
the upper band-edge of the infinite length waveguide occurs at a slightly lower
frequency with a slightly smaller Lamb shift.
5.6

Enhancement and suppression of spontaneous emission

Another signature of the qubit-waveguide interaction is the change in the rate of
spontaneous emission of the qubit. Tuning the qubit into the bandgap changes
the localization length of the waveguide photonic state that dresses the qubit (see
Fig. 5.2f). Since the finite waveguide is connected to an external port which acts
as a dissipative environment, the change in localization length ℓ(𝜔) is accompanied
by a change in the lifetime of the qubit 𝑇rad (𝜔) ∝ 𝑒 2𝑥/ℓ(𝜔) , where 𝑥 is the total
length of the waveguide (See Sec. D.5). In addition to radiative decay into the
output channel, losses in the resonators in the waveguide also contribute to the
qubit’s excited state decay. Using a low power probe in the single-photon regime we

Excited state population

100

τ = 1.1 μs
10-1

ωeg
ωfe

τ = 11 μs

LDOS

b 10

τ = 2.7 μs

10-1

ωeg
ωfe

τ = 5.7 μs

LDOS

Delay, τ (μs)

Figure 5.4: State-selective enhancement and inhibition of radiative decay. a, Measurement with the e-g transition tuned deep into the bandgap (𝜔eg /2𝜋 = 5.37GHz), with the
f-e transition near the lower transmission band (𝜔fe /2𝜋 = 5.01GHz). b, Measurement with
the e-g transition tuned near the upper transmission band (𝜔eg /2𝜋 = 6.51GHz), with the
f-e transition deep in the bandgap (𝜔fe /2𝜋 = 6.17GHz). For measuring the f-e lifetime, we
initially excite the third energy level |fi via a two-photon 𝜋 pulse at the frequency of 𝜔gf /2.
Following the population decay in a selected time interval, the population in |fi is mapped
to the ground state using a second 𝜋 pulse. Finally the ground state population is read using
the dispersive shift of a close-by band-edge resonance of the waveguide. g-e (f-e) transition
data shown as red squares (blue circles)

have measured intrinsic 𝑄-factors of 7.2 ± 0.4 × 104 for the individual waveguide
resonances between 4.6-7.4 GHz. Figure 5.3b shows the measured qubit lifetime
(𝑇1 ) as a function of its frequency in the bandgap. The solid blue curve in Fig. 5.3b
shows a fitted theoretical curve which takes into account the loss in the waveguide
along with a phenomenological intrinsic lifetime of the qubit (𝑇1,i = 10.8 𝜇s).
The dashed red curve shows the expected qubit lifetime for an infinite waveguide
length. Qualitatively, the measured lifetime of the qubit behaves as expected; the
qubit lifetime drastically increases inside the bandgap region and is reduced in the
transmission bands. More subtle features of the measured lifetime include multiple,
narrow Fano-like spectral features deep within the bandgap. These features arise
from what are believed to be interference between parasitic on-chip modes and low𝑄 modes of our external copper box chip packaging. In addition, while the measured
lifetime near the upper band-edge is in excellent agreement with the finite waveguide

75
theoretical model, the data near the lower band-edge shows significant deviation. We
attribute this discrepancy to the presence of low-𝑄 parasitic resonances, observable
in transmission measurements between the qubit XY drive line and the 50-Ω CPW
port. Possible candidates for such spurious modes include the asymmetric “slotline”
modes of the waveguide, which are weakly coupled to our symmetrically grounded
CPW line but may couple to the qubit. Further study of the spectrum of these modes
and possible methods for suppressing them will be a topic of future studies.
Focusing on the upper band-edge, we plot as an inset to Fig. 5.3b a zoom-in
of the measured qubit lifetime along with theoretical estimates of the different
components of qubit decay. Here the qubit decay results from two dominant effects:
detuning-dependent coupling to the lossy resonances in the transmission band of the
waveguide, and emission into the output port of the finite waveguide structure. The
former effect is an incoherent phenomena arising from a multi-mode cavity-QED
picture, whereas the latter effect arises from the coherent interference of band-edge
resonances which can be related to the photon bound state picture and resulting
localization length. Owing to the weaker dispersion at the upper band-edge, the
extent of the photon bound state has an appreciable impact on the qubit lifetime in
the 𝑁 = 9 finite length waveguide. This is most telling in the strongly asymmetric
qubit lifetime around the first waveguide resonance in the upper transmission band.
Quantitatively, the slope of the radiative component of the lifetime curve in the
bandgap near the band-edge can be shown to be proportional to the group delay (see
Sec. D.6), |𝜕𝑇rad /𝜕𝜔| = 𝑇rad 𝜏delay . The corresponding group index, 𝑛g ≡ 𝜏delay /𝑥, is
a property of the waveguide independent of its length 𝑥. Here we measure a slope
corresponding to a group index 𝑛𝑔 ≈ 450, in good correspondence with the circuit
model of the lossy metamaterial waveguide.
The sharp variation in the photonic LDOS near the metamaterial waveguide bandedges may also be used to engineer the multi-level dynamics of the qubit. A transmon
qubit, by construct, is a nonlinear quantum oscillator and thus has a multilevel
energy spectrum. In particular, a third energy level (|fi) exists at the frequency
𝜔gf = 2𝜔ge − 𝐸 C /ℏ. Although the transition g-f is forbidden by selection rules, the
f-e transition has a dipole moment that is 2 larger than the fundamental transition
[190]. This is consistent with the scaling of transition amplitudes in a harmonic
oscillator and results in a second transition lifetime that is half of the fundamental
transition lifetime for a uniform density of states in the electromagnetic environment
of the oscillator. The sharply varying density of states in the metamaterial, on the

76
other hand, can lead to strong suppression or enhancement of the spontaneous
emission for each transition. Figure 5.4 shows the measured lifetimes of the two
transitions for two different spectral configurations. In the first scenario, we enhance
the ratio of the lifetimes 𝑇eg /𝑇fe by situating the fundamental transition frequency
deep inside in the bandgap while having the second transition positioned near the
lower transmission band. The situation is reversed in the second configuration,
where the fundamental frequency is tuned to be near the upper energy band while
the second transition lies deep inside the gap. In our fabricated qubit, the second
transition is about 300 MHz lower than the fundamental transition frequency at zero
flux bias, which allows for achieving large lifetime contrast in both configurations.
5.7

Discussion

Looking forward, we anticipate that further refinement in the engineering and fabrication of the devices presented here should enable metamaterial waveguides approaching a lattice constant-to-wavelength ratio of 𝜆/1000, with limited disorder
and a bandgap-to-midgap ratio in excess of 50% (see Sec. D.7). Such compact, low
loss, low disorder superconducting metamaterials can help realize more scalable
superconducting quantum circuits with higher levels of complexity and functionality in several regards. They offer a method for densely packing qubits – both
in spatial and frequency dimensions – with isolation from the environment and
controllable connectivity achieved via bound qubit-waveguide polaritons [125, 199,
205]. Moreover, the ability to selectively modify the transition lifetimes provides simultaneous access to long-lived metastable qubit states as well as short-lived states
strongly coupled to waveguide modes. This approach realizes a transmon qubit
system with state-dependent bound state localization lengths, which can be used as
a quantum nonlinear media for propagating microwave photons [210, 232, 233], or
as recently demonstrated, to realize spin-photon entanglement and high-bandwidth
itinerant single microwave photon detection [234, 235]. Combined, these attributes
provide a unique platform for studying the many-body physics of quantum photonic
matter [236–239].

77
Chapter 6

QUANTUM ELECTRODYNAMICS IN A TOPOLOGICAL
WAVEGUIDE
We have introduced our first step to develop a scalable waveguide QED architecture based on engineered superconducting metamaterials in Chapter 5. By periodic
placement of sub-wavelength resonant microwave elements, we were able to realize
a dispersive and low-loss 1D waveguide channel which induces a huge anomalous
Lamb shift of a superconducting qubit close to edges of photonic bands, a prerequisite for inducing strong and long-range photon-mediated interactions between
qubit-photon bound states (see Sec. 2.3). This opens up opportunities to investigate
quantum electrodynamical properties of quantum emitters coupled to numerous
types of engineered photonic structures, which can be designed to induce exotic
characters on the emitters depending on the dispersion relation and the topology.
By utilizing the highly flexibility of waveguide QED tools we have developed in superconducting circuits, we demonstrate the physics of quantum emitters coupled to
a dispersive waveguide with topological properties, which is published in Ref. [56].
6.1

Introduction

Harnessing the topological properties of photonic bands [47, 240, 241] is a burgeoning paradigm in the study of periodic electromagnetic structures. Topological
concepts discovered in electronic systems [242, 243] have now been translated and
studied as photonic analogs in various microwave and optical systems [47, 241].
In particular, symmetry-protected topological phases [202] which do not require
time-reversal-symmetry breaking, have received significant attention in experimental studies of photonic topological phenomena, both in the linear and nonlinear
regime [244]. One of the simplest canonical models is the Su-Schrieffer-Heeger
(SSH) model [245, 246], which was initially used to describe electrons hopping
along a one-dimensional dimerized chain with a staggered set of hopping amplitudes between nearest-neighbor elements. The chiral symmetry of the SSH model,
corresponding to a symmetry of the electron amplitudes found on the two types
of sites in the dimer chain, gives rise to two topologically distinct phases of electron propagation. The SSH model, and its various extensions, have been used in
photonics to explore a variety of optical phenomena, from robust lasing in arrays

78
of microcavities [247, 248] and photonic crystals [249], to disorder-insensitive 3rd
harmonic generation in zigzag nanoparticle arrays [250].
Utilization of quantum emitters brings new opportunities in the study of topological physics with strongly interacting photons [251–253], where single-excitation
dynamics [254] and topological protection of quantum many-body states [255] in
the SSH model have recently been investigated. In a similar vein, a topological
photonic bath can also be used as an effective substrate for endowing special properties to quantum matter. For example, a photonic waveguide which localizes and
transports electromagnetic waves over large distances, can form a highly effective
quantum light-matter interface [29, 44, 132] for introducing non-trivial interactions
between quantum emitters. Several systems utilizing highly dispersive electromagnetic waveguide structures have been proposed for realizing quantum photonic
matter exhibiting tailorable, long-range interactions between quantum emitters [125,
199, 215, 256, 257]. With the addition of non-trivial topology to such a photonic
bath, exotic classes of quantum entanglement can be generated through photonmediated interactions of a chiral [46, 258] or directional nature [55, 259].
With this motivation, here we investigate the properties of quantum emitters coupled to a topological waveguide which is a photonic analog of the SSH model,
following the theoretical proposal in Ref. [55]. Our setup is realized by coupling
superconducting transmon qubits [190] to an engineered superconducting metamaterial waveguide [109, 153], consisting of an array of sub-wavelength microwave
resonators with SSH topology. Combining the notions from waveguide quantum
electrodynamics (QED) [40, 44, 132, 195] and topological photonics [47, 241],
we observe qubit-photon bound states with directional photonic envelopes inside a
bandgap and cooperative radiative emission from qubits inside a passband dependent on the topological configuration of the waveguide. Coupling of qubits to the
waveguide also allows for quantum control over topological edge states, enabling
quantum state transfer between distant qubits via a topological channel.
6.2

Description of the topological waveguide

The SSH model describing the topological waveguide studied here is illustrated
in Fig. 6.1a. Each unit cell of the waveguide consists of two photonic sites, A
and B, each containing a resonator with resonant frequency 𝜔0 . The intra-cell
coupling between A and B sites is 𝐽 (1 + 𝛿) and the inter-cell coupling between unit
cells is 𝐽 (1 − 𝛿). The discrete translational symmetry (lattice constant 𝑑) of this

J(1+δ) J(1−δ)

7.5

C0 L0

UBG

100 µm
P1

-25

6.5
–π

|S21| (dB)

Frequency (GHz)

-50

MBG

-25

LBG

-50

6.5

Frequency (GHz)

7.5

Figure 6.1: Topological waveguide. (a) Top: schematic of the SSH model. Each unit cell
contains two sites A and B (red and blue circles) with intra- and inter-cell coupling 𝐽 (1 ± 𝛿)
(orange and brown arrows). Bottom: an analog of this model in electrical circuits, with
corresponding components color-coded. The photonic sites are mapped to LC resonators
with inductance 𝐿 0 and capacitance 𝐶0 , with intra- and inter-cell coupling capacitance 𝐶 𝑣 ,
𝐶 𝑤 and mutual inductance 𝑀 𝑣 , 𝑀 𝑤 between neighboring resonators, respectively (arrows).
(b) Optical micrograph (false-colored) of a unit cell (lattice constant 𝑑 = 592 𝜇m) on a
fabricated device in the topological phase. The lumped-element resonator corresponding to
sublattice A (B) is colored in red (blue). The insets show zoomed-in view of the section
between resonators where planar wires of thickness (𝑡 𝑣 , 𝑡 𝑤 ) = (10, 2) 𝜇m (indicated with
black arrows) control the intra- and inter-cell distance between neighboring resonators,
respectively. (c) Dispersion relation of the realized waveguide according to the circuit
model in panel (a). Upper bandgap (UBG) and lower bandgap (LBG) are shaded in gray,
and middle bandgap (MBG) is shaded in green. (d) Waveguide transmission spectrum
|𝑆21 | across the test structure with 8 unit cells in the trivial (𝛿 > 0; top) and topological
(𝛿 < 0; bottom) phase. The cartoons illustrate the measurement configuration of systems
with external ports 1 and 2 (denoted P1 and P2), where distances between circles are used
to specify relative coupling strengths between sites and blue (green) outlines enclosing two
circles indicate unit cells in the trivial (topological) phase. Black solid curves are fits to the
measured data (see App. E.1 for details) with parameters 𝐿 0 = 1.9nH, 𝐶0 = 253fF, coupling
capacitance (𝐶 𝑣 , 𝐶 𝑤 ) = (33, 17) fF and mutual inductance (𝑀 𝑣 , 𝑀 𝑤 ) = (−38, −32) pH in
the trivial phase (the values are interchanged in the case of topological phase). The shaded
regions correspond to bandgaps in the dispersion relation of panel (c). The figure is adapted
from Ref. [56].

80
system allows us to write the Hamiltonian in terms of momentum-space operators,
ˆ = Í 𝑘 ( v̂ 𝑘 ) † h(𝑘) v̂ 𝑘 , where v̂ 𝑘 = ( 𝑎ˆ 𝑘 , 𝑏ˆ 𝑘 )𝑇 is a vector operator consisting of a
𝐻/ℏ
pair of A and B sublattice photonic mode operators, and the 𝑘-dependent kernel of
the Hamiltonian is given by,
𝜔0
𝑓 (𝑘)
h(𝑘) = ∗
(6.1)
𝑓 (𝑘) 𝜔0
Here, 𝑓 (𝑘) ≡ −𝐽 [(1 + 𝛿) + (1 − 𝛿)𝑒 −𝑖𝑘 𝑑 ] is the momentum-space coupling between
modes on different sublattice, which carries information about the topology of the
system. The eigenstates of this Hamiltonian form two symmetric bands centered
about the reference frequency 𝜔0 with dispersion relation
𝜔± (𝑘) = 𝜔0 ± 𝐽 2(1 + 𝛿2 ) + 2(1 − 𝛿2 ) cos (𝑘 𝑑),
where the + (−) branch corresponds to the upper (lower) frequency passband. While
the band structure is dependent only on the magnitude of 𝛿, and not on whether 𝛿 > 0
or 𝛿 < 0, deformation from one case to the other must be accompanied by the closing
of the middle bandgap (MBG), defining two topologically distinct phases. For a
finite system, it is well known that edge states localized on the boundary of the
waveguide at a 𝜔 = 𝜔0 only appear in the case of 𝛿 < 0, the so-called topological
phase [47, 246]. The case for which 𝛿 > 0 is the trivial phase with no edge states.
It should be noted that for an infinite system, the topological or trivial phase in the
SSH model depends on the choice of unit cell, resulting in an ambiguity in defining
the bulk properties. Despite this, considering the open boundary of a finite-sized
array or a particular section of the bulk, the topological character of the bands can
be uniquely defined and can give rise to observable effects.
We construct a circuit analog of this canonical model using an array of inductorcapacitor (LC) resonators with alternating coupling capacitance and mutual inductance as shown in Fig. 6.1a. The topological phase of the circuit model is determined
by the relative size of intra- and inter-cell coupling between neighboring resonators,
including both the capacitive and inductive contributions. Strictly speaking, this
circuit model breaks chiral symmetry of the original SSH Hamiltonian [47, 246],
which ensures the band spectrum to be symmetric with respect to 𝜔 = 𝜔0 . Nevertheless, the topological protection of the edge states under perturbation in the intra- and
inter-cell coupling strengths remains valid as long as the bare resonant frequencies

81
of resonators (diagonal elements of the Hamiltonian) are not perturbed, and the existence of edge states still persists due to the presence of inversion symmetry within
the unit cell of the circuit analog, leading to a quantized Zak phase [260]. For detailed analysis of the modeling, symmetry, and robustness of the circuit topological
waveguide see Apps. E.1 and E.2.
The circuit model is realized using fabrication techniques for superconducting metamaterials discussed in Refs. [109, 153], where the coupling between sites is controlled by the physical distance between neighboring resonators. Due to the nearfield nature, the coupling strength is larger (smaller) for smaller (larger) distance
between resonators on a device. An example unit cell of a fabricated device in
the topological phase is shown in Fig. 6.1b (the values of intra- and inter-cell distances are interchanged in the trivial phase). We find a good agreement between
the measured transmission spectrum and a theoretical curve calculated from a LC
lumped-element model of the test structures with 8 unit cells in both trivial and
topological configurations (Fig. 6.1c,d). For the topological configuration, the observed peak in the waveguide transmission spectrum at 6.636 GHz inside the MBG
signifies the associated edge state physics in our system.
6.3

Properties of quantum emitters coupled to the topological waveguide

The non-trivial properties of the topological waveguide can be accessed by coupling
quantum emitters to the engineered structure. To this end, we prepare Device I
consisting of a topological waveguide in the trivial phase with 9 unit cells, whose
boundary is tapered with specially designed resonators before connection to external
ports (see Fig. 6.2a). The tapering sections at both ends of the array are designed
to reduce the impedance mismatch to the external ports (𝑍0 = 50 Ω) at frequencies
in the upper passband (UPB). This is crucial for reducing ripples in the waveguide
transmission spectrum in the passbands [153]. Every site of the 7 unit cells in the
middle of the array is occupied by a single frequency-tunable transmon qubit [190]
(the device contains in total 14 qubits labeled Q𝑖𝛼 , where 𝑖 =1-7 and 𝛼=A,B are the
cell and sublattice indices, respectively). Properties of Device I and the tapering
section are discussed in further detail in Apps. E.3 and E.4, respectively.
Directional qubit-photon bound states
For qubits lying within the middle bandgap, the topology of the waveguide manifests
itself in the spatial profile of the resulting qubit-photon bound states. When the qubit
transition frequency is inside the bandgap, the emission of a propagating photon from

Q1B

Q7B

Q1A

6.65

6.7

|S22|
0.6
6.618 6.621 6.624

6.618

6.55

0.2

κe,2 /2π

0.1

6.7

QB4 Frequency (GHz)

30
20

0.6
0.4

AB AB AB AB AB ABAB

2 3 4

6.7

5 6 7

Qubit Index

j=4
j=6
j=7

6.65

κe,1 /2π

6.6

Frequency (GHz)

κe,1 /2π
κe,2 /2π

6.62

|S22|

0.2

αα

6.6

6.622

|gijAB| / 2π (MHz)

|S11|
0.9
0.8

Frequency (GHz)

6.6

6.624

QA5

6.6

Ext. Coupling Rate, κe,p /2π (MHz)

κe,2

QB4

QA4

6.55

6.5

QiA

6.55

0.3

6.65

0.4

QjB
… …

Q7A

QB4

6.7

κe,1

|gij | / 2π (MHz)

Bare QB4 Frequency (GHz)

−6 −5 −4 −3 −2 −1 0
AA, j = 7
BB, j = 1
BB, j = 2
BB, j = 4

1 2 3 4

|i − j|

i−j

1 2 3

Figure 6.2: Directionality of qubit-photon bound states. (a) Schematic of Device I,
consisting of 9 unit cells in the trivial phase with qubits (black lines terminated with
a square) coupled to every site on the 7 central unit cells. The ends of the array are
tapered with additional resonators (purple) with engineered couplings designed to minimize
impedance mismatch at upper passband frequencies. (b) Theoretical photonic envelope of
the directional qubit-photon bound states. At the reference frequency 𝜔0 , the qubit coupled
to site A (B) induces a photonic envelope to the right (left), colored in green (blue). The
bars along the envelope indicate photon occupation on the corresponding resonator sites.
(c) Measured coupling rate 𝜅 e, 𝑝 to external port numbers, 𝑝 = 1, 2, of qubit-photon bound
states. Left: external coupling rate of qubit QB4 to each port as a function of frequency inside
the MBG. Solid black curve is a model fit to the measured external coupling curves. The
frequency point of highest directionality is extracted from the fit curve, and is found to be
𝜔0 /2𝜋 = 6.621 GHz (vertical dashed orange line). Top (Bottom)-right: external coupling
rate of all qubits tuned to 𝜔 = 𝜔0 measured from port P1 (P2). The solid black curves in these
plots correspond to exponential fits to the measured external qubit coupling versus qubit
index. (d) Two-dimensional color intensity plot of the reflection spectrum under crossing
between a pair of qubits with frequency centered around 𝜔 = 𝜔0 . Left: reflection from P1
AB
(|𝑆11 |) while tuning QB4 across QA
4 (fixed). An avoided crossing of 2|𝑔44 |/2𝜋 = 65.7 MHz
is observed. Right: reflection from P2 (|𝑆22 |) while tuning Q4 across QA
5 (fixed), indicating
the absence of appreciable coupling. Inset to the right shows a zoomed-in region where a
AB |/2𝜋 = 967 kHz is measured. The bare qubit frequencies
small avoided crossing of 2|𝑔54
𝛼𝛽
from the fit are shown with dashed lines. (e) Coupling |𝑔𝑖 𝑗 | (𝛼, 𝛽 ∈ {A,B}) between various

qubit pairs (Q𝑖𝛼 ,Q 𝑗 ) at 𝜔 = 𝜔0 , extracted from the crossing experiments similar to panel (d).
Solid black curves are exponential fits to the measured qubit-qubit coupling rate versus qubit
index difference (spatial separation). Error bars in all figure panels indicate 95% confidence
interval, and are omitted on data points whose marker size is larger than the error itself. The
figure is adapted from Ref. [56].

83
the qubit is forbidden due to the absence of photonic modes at the qubit resonant
frequency. In this scenario, a stable bound state excitation forms, consisting of
a qubit in its excited state and a waveguide photon with exponentially localized
photonic envelope [126, 128]. Generally, bound states with a symmetric photonic
envelope emerge due to the inversion symmetry of the photonic bath with respect to
the qubit location [215]. In the case of the SSH photonic bath, however, a directional
envelope can be realized [55] for a qubit at the centre of the MBG (𝜔0 ), where the
presence of a qubit creates a domain wall in the SSH chain and the induced photonic
bound state is akin to an edge state (refer to App. E.5 for a detailed description).
For example, in the trivial phase, a qubit coupled to site A (B) acts as the last site
of a topological array extended to the right (left) while the subsystem consisting
of the remaining sites extended to the left (right) is interpreted as a trivial array.
Mimicking the topological edge state, the induced photonic envelope of the bound
state faces right (left) with photon occupation only on B (A) sites (Fig. 6.2b), while
across the trivial boundary on the left (right) there is no photon occupation. The
opposite directional character is expected in the case of the topological phase of the
waveguide. The directionality reduces away from the center of the MBG, and is
effectively absent inside the upper or lower bandgaps.
We experimentally probe the directionality of qubit-photon bound states by utilizing
the coupling of bound states to the external ports in the finite-length waveguide of
Device I (see Fig. 6.2c). The external coupling rate 𝜅e,𝑝 (𝑝 = 1, 2) is governed by the
overlap of modes in the external port 𝑝 with the tail of the exponentially attenuated
envelope of the bound state, and therefore serves as a useful measure to characterize
the localization [109, 215, 261]. To find the reference frequency 𝜔0 where the
bound state becomes most directional, we measure the reflection spectrum 𝑆11 (𝑆22 )
of the bound state seen from port 1 (2) as a function of qubit tuning. We extract the
external coupling rate 𝜅 𝑒,𝑝 by fitting the measured reflection spectrum with a Fano
lineshape [262]. For QB4 , which is located near the center of the array, we find 𝜅 e,1
to be much larger than 𝜅 e,2 at all frequencies inside MBG. At 𝜔0 /2𝜋 = 6.621 GHz,
𝜅e,2 completely vanishes, indicating a directionality of the QB4 bound state to the left.
Plotting the external coupling at this frequency to both ports against qubit index,
we observe a decaying envelope on every other site, signifying the directionality of
photonic bound states is correlated with the type of sublattice site a qubit is coupled
to. Similar measurements when qubits are tuned to other frequencies near the edge
of the MBG, or inside the upper bandgap (UBG), show the loss of directionality
away from 𝜔 = 𝜔0 (App. E.6).

84
A remarkable consequence of the distinctive shape of bound states is directiondependent photon-mediated interactions between qubits (Fig. 6.2d,e). Due to the
site-dependent shapes of qubit-photon bound states, the interaction between qubits
becomes substantial only when a qubit on sublattice A is on the left of the other
qubit on sublattice B, i.e., 𝑗 > 𝑖 for a qubit pair (Q𝑖A ,QB𝑗 ). From the avoided crossing
experiments centered at 𝜔 = 𝜔0 , we extract the qubit-qubit coupling as a function
of cell displacement 𝑖 − 𝑗. An exponential fit of the data gives the localization
length of 𝜉 = 1.7 (in units of lattice constant), close to the estimated value from
the circuit model of our system (see App. E.3). While theory predicts the coupling
between qubits in the remaining combinations to be zero, we report that coupling of
|𝑔𝑖AA,BB
|/2𝜋 . 0.66 MHz and |𝑔𝑖AB
𝑗 |/2𝜋 . 0.48 MHz (for 𝑖 > 𝑗) are observed, much
AB |/2𝜋 = 32.9 MHz. We
smaller than the bound-state-induced coupling, e.g., |𝑔45
attribute such spurious couplings to the unintended near-field interaction between
qubits. Note that we find consistent coupling strength of qubit pairs dependent only
on their relative displacement, not on the actual location in the array, suggesting that
physics inside MBG remains intact with the introduced waveguide boundaries. In
total, the avoided crossing and external linewidth experiments at 𝜔 = 𝜔0 provide
strong evidence of the shape of qubit-photon bound states, compatible with the
theoretical photon occupation illustrated in Fig. 6.2b.
Topology-dependent photon scattering
In the passband regime, i.e., when the qubit frequencies lie within the upper or
lower passbands, the topology of the waveguide is imprinted on cooperative interaction between qubits and the single-photon scattering response of the system.
The topology of the SSH model can be visualized by plotting the complex-valued
momentum-space coupling 𝑓 (𝑘) for 𝑘 values in the first Brillouin zone (Fig. 6.3a).
In the topological (trivial) phase, the contour of 𝑓 (𝑘) encloses (excludes) the origin
of the complex plane, resulting in the winding number of 1 (0) and the corresponding
Zak phase of 𝜋 (0) [260]. This is consistent with the earlier definition based on the
sign of 𝛿. It is known that for a regular waveguide with linear dispersion, the coherent exchange interaction 𝐽𝑖 𝑗 and correlated decay Γ𝑖 𝑗 between qubits at positions
𝑥𝑖 and 𝑥 𝑗 along the waveguide take the forms 𝐽𝑖 𝑗 ∝ sin 𝜑𝑖 𝑗 and Γ𝑖 𝑗 ∝ cos 𝜑𝑖 𝑗 [116,
120], where 𝜑𝑖 𝑗 = 𝑘 |𝑥𝑖 − 𝑥 𝑗 | is the phase length. In the case of our topological
waveguide, considering a pair of qubits coupled to A/B sublattice on 𝑖/ 𝑗-th unit
cell, this argument additionally collects the phase 𝜙(𝑘) ≡ arg 𝑓 (𝑘) [55]. This is an
important difference compared to the regular waveguide case, because the zeros of

ϕtr

Triv.

Topo.

Im[f(k)]

-J
-2 J

-J

ϕtp

Ji j / Γe

Re[f(k)]

-1

|i − j| = 2

Bare Qubit Frequency (GHz)

7.4
7.2
|S21|

7.4
7.2

ωmin

ωmax

6.8

7.2

Frequency (GHz)

7.4

6.8

7.2

Frequency (GHz)

7.4

Figure 6.3: Probing band topology with qubits. (a) 𝑓 (𝑘) in the complex plane for 𝑘 values
in the first Brillouin zone. 𝜙tr (𝜙tp ) is the phase angle of 𝑓 (𝑘) for a trivial (topological)
section of waveguide, which changes by 0 (𝜋) radians as 𝑘 𝑑 transitions from 0 to 𝜋 (arc in
upper plane following black arrowheads). (b) Coherent exchange interaction 𝐽𝑖 𝑗 between
a pair of coupled qubits as a function of frequency inside the passband, normalized to
individual qubit decay rate Γe (only 𝑘 𝑑 ∈ [0, 𝜋) branch is plotted). Here, one qubit is
coupled to the A sublattice of the 𝑖-th unit cell and the other qubit is coupled to the B
sublattice of the 𝑗-th unit cell, where |𝑖 − 𝑗 | = 2. Blue (green) curve corresponds to a
trivial (topological) intermediate section of waveguide between qubits. The intercepts at
𝐽𝑖 𝑗 = 0 (filled circles with arrows) correspond to points where perfect super-radiance occurs.
(c) Waveguide transmission spectrum |𝑆21 | as a qubit pair are resonantly tuned across the
B A
UPB of Device I [left: (QA
2 ,Q4 ), right: (Q2 ,Q5 )]. Top: schematic illustrating system
configuration during the experiment, with left (right) system corresponding to an interacting
qubit pair subtending a three-unit-cell section of waveguide in the trivial (topological)
phase. Middle and Bottom: two-dimensional color intensity plots of |𝑆21 | from theory and
experiment, respectively. Swirl patterns (highlighted by arrows) are observed in the vicinity
of perfectly super-radiant points, whose number of occurrences differ by one between trivial
and topological waveguide sections. The figure is adapted from Ref. [56].

86
equation
𝜑𝑖 𝑗 (𝑘) ≡ 𝑘 𝑑|𝑖 − 𝑗 | − 𝜙(𝑘) = 0

mod 𝜋

(6.2)

determine wavevectors (and corresponding frequencies) where perfect Dicke superradiance [42] occurs. Due to the properties of 𝑓 (𝑘) introduced above, for a fixed
cell-distance Δ𝑛 ≡ |𝑖 − 𝑗 | ≥ 1 between qubits there exists exactly Δ𝑛 − 1 (Δ𝑛)
frequency points inside the passband where perfect super-radiance occurs in the
trivial (topological) phase. An example for the Δ𝑛 = 2 case is shown in Fig. 6.3b.
Note that although Eq. (6.2) is satisfied at the band-edge frequencies 𝜔min and 𝜔max
(𝑘 𝑑 = {0, 𝜋}), they are excluded from the above counting due to breakdown of the
Born-Markov approximation (see App. E.7).
To experimentally probe signatures of perfect super-radiance, we tune the frequency
of a pair of qubits across the UPB of Device I while keeping the two qubits resonant
with each other. We measure the waveguide transmission spectrum 𝑆21 during this
tuning, keeping track of the lineshape of the two-qubit resonance as 𝐽𝑖 𝑗 and Γ𝑖 𝑗 varies
over the tuning. Drastic changes in the waveguide transmission spectrum occur
whenever the two-qubit resonance passes through the perfectly super-radiant points,
resulting in a swirl pattern in |𝑆21 |. Such patterns arise from the disappearance of
the peak in transmission associated with interference between photons scattered by
imperfect super- and sub-radiant states, resembling the electromagnetically-induced
transparency in a V-type atomic level structure [263]. As an example, we discuss
B A
the cases with qubit pairs (QA
2 ,Q4 ) and (Q2 ,Q5 ), which are shown in Fig. 6.3c.
Each qubit pair configuration encloses a three-unit-cell section of the waveguide;
however for the (QA
2 ,Q4 ) pair the waveguide section is in the trivial phase, whereas
for (QA
2 ,Q4 ) the waveguide section is in the topological phase. Both theory and
measurement indicate that the qubit pair (QA
2 ,Q4 ) has exactly one perfectly superradiant frequency point in the UPB. For the other qubit pair (QB2 ,QA
5 ), with waveguide
section in the topological phase, two such points occur (corresponding to Δ𝑛 = 2).
This observation highlights the fact that while the topological phase of the bulk in
the SSH model is ambiguous, a finite section of the array can still be interpreted
to have a definite topological phase. Apart from the unintended ripples near the
band-edges, the observed lineshapes are in good qualitative agreement with the
theoretical expectation in Ref. [55]. The frequency misalignment of swirl patterns
between the theory and the experiment is due to the slight discrepancy between the
realized circuit model and the ideal SSH model (see App. E.1 for details). Detailed
description of the swirl pattern and similar measurement results for other qubit

87
combinations with varying Δ𝑛 are reported in App. E.7.

EL
gL

Mod. Freq. (MHz)

c 1

270
245

100

200

Time (ns)

300

100

Population

e 1

0.5

220

d 1

Population

200

Time (ns)
ii.
iii.

300

0.5

100

200

Time (ns)

300

Time (ns)

Figure 6.4: Qubit interaction with topological edge modes. (a) Schematic of Device II,
consisting of 7 unit cells in the topological phase with qubits QL = Q𝑖𝛼 and QR = Q 𝑗 coupled
at sites (𝑖, 𝛼) = (2, A) and ( 𝑗, 𝛽) = (6, B), respectively. EL and ER are the left-localized and
right-localized edge modes which interact with each other at rate 𝐺 due to their overlap in
the center of the finite waveguide. (b) Chevron-shaped oscillation of QL population arising
from interaction with edge modes under variable frequency and duration of modulation
pulse. The oscillation is nearly symmetric with respect to optimal modulation frequency
242.5 MHz, apart from additional features at (219, 275) MHz due to spurious interaction of
unused sidebands with modes inside the passband. (c) Line-cut of panel b (indicated with
a dashed line) at the optimal modulation frequency. A population oscillation involving two
harmonics is observed due to coupling of EL to ER . (d) Vacuum Rabi oscillations between
QL and EL when QR is parked at the resonant frequency of edge modes to shift the frequency
of ER , during the process in panel (c) In panels (c) and (d) the filled orange circles (black
solid lines) are the data from experiment (theory). (e) Population transfer from QL to QR
composed of three consecutive swap transfers QL →EL →ER →QR . The population of QL
(QR ) during the process is colored dark red (dark blue), with filled circles and solid lines
showing the measured data and fit from theory, respectively. The light red (light blue) curve
indicates the expected population in EL (ER ) mode, calculated from theory. The figure is
adapted from Ref. [56].

88
6.4

Quantum state transfer via topological edge states

Finally, to explore the physics associated with topological edge modes, we fabricated
a second device, Device II, which realizes a closed quantum system with 7 unit cells
in the topological phase (Fig. 6.4a). We denote the photonic sites in the array by
(𝑖,𝛼), where 𝑖 =1-7 is the cell index and 𝛼 =A,B is the sublattice index. Due to
reflection at the boundary, the passbands on this device appear as sets of discrete
resonances. The system supports topological edge modes localized near the sites
(1,A) and (7,B) at the boundary, labeled EL and ER . The edge modes are spatially
distributed with exponentially attenuated tails directed toward the bulk. In a finite
system, the non-vanishing overlap between the envelopes of edge states generates
a coupling which depends on the localization length 𝜉 and the system size 𝐿 as
𝐺 ∼ 𝑒 −𝐿/𝜉 . In Device II, two qubits denoted QL and QR are coupled to the
topological waveguide at sites (2,A) and (6,B), respectively. Each qubit has a local
drive line and a flux-bias line, which are connected to room-temperature electronics
for control. The qubits are dispersively coupled to readout resonators, which are
coupled to a coplanar waveguide for time-domain measurement. The edge mode EL
(ER ) has photon occupation on sublattice A (B), inducing interaction 𝑔L (𝑔R ) with
QL (QR ). Due to the directional properties discussed earlier, bound states arising
from QL and QR have photonic envelopes facing away from each other inside the
MBG, and hence have no direct coupling to each other. For additional details on
Device II and qubit control, refer to App. E.8.
We probe the topological edge modes by utilizing the interaction with the qubits.
While parking QL at frequency 𝑓q = 6.835 GHz inside MBG, we initialize the
qubit into its excited state by applying a microwave 𝜋-pulse to the local drive line.
Then, the frequency of the qubit is parametrically modulated [77] such that the
first-order sideband of the qubit transition frequency is nearly resonant with EL .
After a variable duration of the frequency modulation pulse, the state of the qubit is
read out. From this measurement, we find a chevron-shaped oscillation of the qubit
population in time centered at modulation frequency 242.5 MHz (Fig. 6.4b). We
find the population oscillation at this modulation frequency to contain two harmonic
components as shown in Fig. 6.4c, a general feature of a system consisting of three
states with two exchange-type interactions 𝑔1 and 𝑔2 . In such cases, three singleexcitation eigenstates qexist at 0, ±𝑔 with respect to the bare resonant frequency
of the emitters (𝑔 ≡ 𝑔12 + 𝑔22 ), and since the only possible spacing between the
eigenstates in this case is 𝑔 and 2𝑔, the dynamics of the qubit population exhibits
two frequency components with a ratio of two. From fitting the QL population

89
oscillation data in Fig. 6.4c, the coupling between EL and ER is extracted to be
𝐺/2𝜋 = 5.05 MHz. Parking QR at the bare resonant frequency 𝜔E /2𝜋 = 6.601 GHz
of the edge modes, ER strongly hybridizes with QR and is spectrally distributed at
±𝑔R with respect to the original frequency (𝑔R /2𝜋 = 57.3 MHz). As this splitting
is much larger than the coupling of ER to EL , the interaction channel EL ↔ER is
effectively suppressed and the vacuum Rabi oscillation only involving QL and EL
is recovered (Fig. 6.4d) by applying the above-mentioned pulse sequence on QL .
The vacuum Rabi oscillation is a signature of strong coupling between the qubit and
the edge state, a bosonic mode, as described by cavity QED [29]. A similar result
was achieved by applying a simultaneous modulation pulse on QR to put its firstorder sideband near-resonance with the bare edge modes (instead of parking it near
resonance), which we call the double-modulation scheme. From the vacuum Rabi
oscillation QL ↔EL (QR ↔ER ) using the double-modulation scheme, we find the
effective qubit-edge mode coupling to be 𝑔˜ L /2𝜋 = 23.8 MHz (𝑔˜ R /2𝜋 = 22.5 MHz).
The half-period of vacuum Rabi oscillation corresponds to an iSWAP gate between
QL and EL (or QR and ER ), which enables control over the edge modes with singlephoton precision. As a demonstration of this tool, we perform remote population
transfer between QL and QR through the non-local coupling of topological edge
modes EL and ER . The qubit QL (QR ) is parked at frequency 6.829 GHz (6.835 GHz)
and prepared in its excited (ground) state. The transfer protocol, consisting of three
steps, is implemented as follows: i) an iSWAP gate between QL and EL is applied
by utilizing the vacuum Rabi oscillation during the double-modulation scheme
mentioned above, ii) the frequency modulation is turned off and population is
exchanged from EL to ER using the interaction 𝐺, iii) another iSWAP gate between
QR and ER is applied to map the population from ER to QR . The population of
both qubits at any time within the transfer process is measured using multiplexed
readout [264] (Fig. 6.4e). We find the final population in QR after the transfer
process to be 87 %. Numerical simulations suggest that (App. E.8) the infidelity in
preparing the initial excited state accounts for 1.6 % of the population decrease, the
leakage to the unintended edge mode due to ever-present interaction 𝐺 contributes
4.9 %, and the remaining 6.5 % is ascribed to the short coherence time of qubits
away from the flux-insensitive point [𝑇2∗ = 344 (539) ns for QL (QR ) at working
point].
We expect that a moderate improvement on the demonstrated population transfer
protocol could be achieved by careful enhancement of the excited state preparation

90
and the iSWAP gates, i.e. optimizing the shapes of the control pulses [265–268].
The coherence-limited infidelity can be mitigated by utilizing a less flux-sensitive
qubit design [269, 270] or by reducing the generic noise level of the experimental
setup [271]. Further, incorporating tunable couplers [214] into the existing metamaterial architecture to control the localization length of edge states in situ will fully
address the population leakage into unintended interaction channels, and more importantly, enable robust quantum state transfer over long distances [272]. Together
with many-body protection to enhance the robustness of topological states [255],
building blocks of quantum communication [26] under topological protection are
also conceivable.
6.5

Discussion and outlook

Looking forward, we envision several research directions to be explored beyond the
work presented here. First, the topology-dependent photon scattering in photonic
bands that is imprinted in the cooperative interaction of qubits can lead to new
ways of measuring topological invariants in photonic systems [273]. The directional and long-range photon-mediated interactions between qubits demonstrated in
our work also opens avenues to synthesize non-trivial quantum many-body states
of qubits, such as the double Néel state [55]. Even without technical advances
in fabrication [274–276], a natural scale-up of the current system will allow for
the construction of moderate to large-scale quantum many-body systems. Specifically, due to the on-chip wiring efficiency of a linear waveguide QED architecture,
with realistic refinements involving placement of local control lines on qubits and
compact readout resonators coupled to the tapered passband (intrinsically acting as
Purcell filters [184]), we expect that a fully controlled quantum many-body system
consisting of 100 qubits is realizable in the near future. In such systems, protocols
for preparing and stabilizing [101, 255, 277] quantum many-body states could be
utilized and tested. Additionally, the flexibility of superconducting metamaterial
architectures [109, 153] can be further exploited to realize other novel types of
topological photonic baths [46, 55, 259]. While the present work was limited to a
one-dimensional system, the state-of-the-art technologies in superconducting quantum circuits [107] utilizing flip-chip methods [275, 276] will enable integration
of qubits into two-dimensional metamaterial surfaces. It also remains to be explored whether topological models with broken time-reversal symmetry, an actively
pursued approach in systems consisting of arrays of three-dimensional microwave
cavities [253, 278], could be realized in compact chip-based architectures. Alto-

91
gether, our work sheds light on opportunities in superconducting circuits to explore
quantum many-body physics originating from novel types of photon-mediated interactions in topological waveguide QED, and paves the way for creating synthetic
quantum matter and performing quantum simulation [13, 82, 86, 93, 279].

92
Chapter 7

AN INTERMEDIATE-SCALE QUANTUM PROCESSOR BASED
ON DISPERSIVE WAVEGUIDE QED
The work discussed in the previous chapters laid the groundwork for realizing a
scalable waveguide QED-based architecture for quantum simulation and computation. In particular, we have for the first time demonstrated coherent multi-qubit
dynamics in waveguide QED in Chapter 4, developed and characterized a compact and extensible waveguide structure based on superconducting metamaterial in
Chapter 5, and interfaced a large number of qubits with a metamaterial waveguide
with limited control (only Z without XY and readout) and with spectroscopic characterization in Chapter 6. Integration of such building blocks of waveguide QED
with state-of-the-art control tools developed for quantum information processing is
predicted to enable new directions to perform quantum simulation of many-body
physics and quantum computation. In this chapter, we introduce the culmination
of our efforts, demonstrating our ongoing work on the construction of a scalable
quantum processor with tunable-range waveguide-mediated connectivity, which involves ten superconducting qubits with full individual addressing and high-fidelity
readout.
7.1

Introduction

Realizing a scalable architecture for quantum computation and simulation has been
a central goal in the field of quantum information science, with numerous physical
platforms, ranging from atom-based systems such as trapped ions and cold neutral
atoms to solid-state systems such as superconducting qubits, quantum dots, and
color centers in diamond, being actively investigated. While much of the attention
was devoted to systems with only nearest-neighbor coupling between quantum emitters, particularly for the realization of surface code [65, 66], achieving a quantum
processor with long-range connectivity remains an important subject. On the fundamental side, quantum many-body systems with high connectivity sheds new light
on important topics in contemporary physics such as quantum thermalization [84],
dynamics of quantum entanglement [86], and novel quantum phases of matter [15].
On the practical side, study of such long-range coupled systems can contribute to
the realization of quantum algorithms and quantum error correcting codes which

93
harness non-local quantum gate operations [57].
Increasing the size of a system often contradicts with the requirement for maintaining
a degree of connectivity that used to be present in a smaller-sized system. An
example is the cavity-mediated coupling between qubits which enables all-to-all
connectivity between qubits coupled to a common cavity [110–112]. There has been
recent demonstrations of intermediate-scale superconducting quantum processors
employing this principle, controlling about 20 qubits [105, 106], but it is evident that
such scheme is not scalable owing to the breakdown of the single-mode picture [280],
with the free-spectral range of cavity modes inversely proportional to the length scale
of the cavity. An alternative approach is to utilize an intrinsically one-dimensional
structure such as a waveguide [40], but operation in settings where emitters are nearly
resonant to a continuum of waveguide modes is subjected to substantial spontaneous
emission, requiring fine-tuned strategies to mitigate dissipation-induced loss of
quantum information [108, 120, 124, 281].
An intermediate regime is to utilize quantum emitters whose transition frequencies
are situated inside a bandgap of a photonic structure. As first predicted by Sajeev
John and Jian Wang in 1990 [126], the quantum emitters in such a scenario are
known to form emitter-photon bound states whose spontaneous emission to the
photonic structure is forbidden. The photonic tails of the bound states have been
shown to mediate tunable-range interactions between quantum emitters [127, 128],
recently revisited in the context of quantum simulation of many-body physics [125].
While there has been pioneering demonstrations of atoms interfaced with engineered nanophotonic waveguides [44] to achieve this feat, systematic control over
atom-photon bound states has been thwarted by technical difficulties associated
with trapping of atoms near nanophotonic structures [134, 143]. Parallel efforts
have been taken in superconducting circuits to implement this idea by coupling
superconducting qubits to a microwave analog of photonic crystal [215], created
with a impedance-modulated microwave transmission line, or a metamaterial composed of an array of deep sub-wavelength resonant elements, such as compact LC
resonators [56, 109] or high-impedance resonators employing Josephson superinductance [282]. However, investigations of such platforms were so far restricted
to cases involving only two qubits [199, 282] or a large number of qubits with
limited control [56, 283], with the long-standing goal of realizing a fully controllable waveguide-based quantum processor involving many qubits still waiting to be
explored.

C0 L0

2 µm

Frequency ω

···

UBG

ωc

200 µm

4J
LBG

-π/d

Wavevector k

π/d

Figure 7.1: Metamaterial waveguide. a, The metamaterial waveguide consists of a onedimensional array of LC resonators (parallel inductance 𝐿 0 and capacitance 𝐶0 to the ground)
capacitively coupled to nearest neighbors at capacitance 𝐶 𝑣 , whose electrical circuit model
is illustrated on the top. The bottom panel shows the realization of this model with compact
microwave resonators on chip, each consisting of a planar capacitor and a meander inductor
with a 2 𝜇m pitch. b, A simple representation of the metamaterial waveguide in terms of
an array of coupled cavities (resonant frequency 𝜔 𝑐 ) with nearest-neighbor tunneling rate
𝐽 (top) and the corresponding dispersion relation assuming a lattice constant of 𝑑 (bottom).
Two bandgaps—upper bandgap (UBG) and lower bandgap (LBG)—and a transmission band
at detunings [−2𝐽, 2𝐽] with respect to the bare cavity frequency 𝜔 𝑐 appears.

In this work, we demonstrate an intermediate-scale quantum processor based on an
extensible photonic waveguide design, realized with superconducting qubits coupled
to a compact, low-loss metamaterial waveguide on chip [56, 109, 153]. When the
transition frequencies of qubits are tuned inside the band gap, the metamaterial
waveguide acts as a bus for mediating tunable long-range coupling between distant
qubits. In addition, the dressing of qubits with waveguide photons is shown to modify
the spacing of energy levels, resulting in a tunable anharmonicity. The transmission
band of the metamaterial waveguide is simultaneously utilized as a channel for rapid
single-shot readout of qubits at a high fidelity without compromising the coherence
time of qubits [184]. Finally, we explore quantum many-body dynamics associated
with the long-range nature of coupling in this platform.
7.2

Metamaterial quantum processor

As a scalable photonic medium for inducing long-range interaction between qubits,
we consider a metamaterial waveguide consisting of an array of lumped-element
microwave resonators, illustrated in Fig. 7.1a. Each resonator forming the metamaterial can be described as a LC resonator of parallel inductance 𝐿 0 and capacitance

95
𝐶0 to the ground and is coupled to its nearest neighbors with capacitance 𝐶𝑣 . To
first order in the ratio of coupling capacitance to self capacitance, the metamaterial
waveguide can be simply described as a tight-binding lattice of microwave cavities
at a resonant frequency 𝜔𝑐 = 1/ 𝐿 0 (𝐶0 + 2𝐶𝑣 ) with a positive nearest-neighbor
tunnel coupling 𝐽 = 𝜔𝑐 𝐶𝑣 /[2(𝐶0 + 2𝐶𝑣 )] (Fig. 7.1b), whose Hamiltonian is written
as
Õh
𝐻ˆ wg /ℏ =
(7.1)
𝜔𝑐 𝑎ˆ †𝑛 𝑎ˆ 𝑛 + 𝐽 ( 𝑎ˆ †𝑛 𝑎ˆ 𝑛+1 + 𝑎ˆ †𝑛+1 𝑎ˆ 𝑛 ) .

Here, 𝑎ˆ 𝑛 (𝑎ˆ †𝑛 ) is the annihilation (creation) operator of the cavity at the 𝑛th site
of the metamaterial. In the momentum-space representation, this Hamiltonian is
transformed into one with a set of uncoupled modes 𝐻ˆ wg = ℏ 𝑘 𝜔 𝑘 𝑎ˆ †𝑘 𝑎ˆ 𝑘 with a
dispersion relation given by
𝜔 𝑘 = 𝜔𝑐 + 2𝐽 cos (𝑘 𝑑),

(7.2)

where 𝑘 is the wavevector and 𝑑 is the lattice constant of the metamaterial. Under
this description, the transmission band lies at detunings [−2𝐽, 2𝐽] with respect to
the resonant frequency 𝜔𝑐 of the cavity. The photonic bandgap lying at frequencies
lower (higher) than the transmission band is denoted as the lower (upper) bandgap.
In practice, the spatially extended character of charge operators of the metamaterial
canonically conjugate to the node fluxes [226] significantly alters the dispersion relation when the coupling capacitances are substantial, which is discussed in Refs. [56,
153].
Our quantum processor is realized by coupling ten transmon qubits [190] (labeled
as Q𝑖 , 𝑖 = 1–10) to a common metamaterial waveguide consisting of 42 resonator
sites with lattice constant 𝑑 = 292 𝜇m, as illustrated in Fig. 7.2. Each qubit, capacitively coupled to a resonator site of the metamaterial, is equipped with its own
charge drive line and flux bias line for individual XY and Z controls, respectively.
The metamaterial resonators are staggered to reduce stray near-field coupling between qubits coupled to the metamaterial waveguide. In addition, an additional
compact LC resonator capacitively couples to each qubit as well as the metamaterial
resonator at the corresponding site, allowing for dispersive readout of the qubit
utilizing the transmission band of the metamaterial waveguide. Such readout resonators are designed to have resonant frequencies inside the transmission band of
the metamaterial waveguide, whose sharp response in a finite-sized system owing
to impedance mismatch at the boundary is mitigated by tapering sections. The

···

1 mm
R2

R10

Q10

200 µm

Q1
R1

300 µm

Figure 7.2: Description of the metamaterial quantum processor. a, A cartoon representing an array of cavities with atoms, with tunnel coupling 𝐽 between photons of neighboring
cavities and atom-photon coupling 𝑔 inside each cavity. b, False-colored optical micrograph of the device, which realizes the scheme in panel a with superconducting circuits.
The device consists of a metamaterial waveguide formed by an array of compact microwave
resonators (light blue) to which superconducting transmon qubits (orange; labeled as Q𝑖 )
and readout resonators (green; labeled as R𝑖 ) are coupled (𝑖 = 1–10). The charge drive
lines and flux bias lines for single-qubit manipulation are colored in dark blue and pink,
respectively. The metamaterial waveguide transitions to the external input-output ports (red)
at the boundary via engineered tapering sections (violet), each consisting of four specifically
designed capacitively coupled compact microwave resonators. Two ancillary qubits, colored
yellow, are beyond the scope of this work.

Lifetime T1 (µs)

Anharmonicity α/2π (MHz)

Q10

7.4

7.5

7.6

101

10–1
–50

–100

–150

–200
4.5

4.6

4.7

4.8

4.9

7.2

7.3

Qubit Frequency ωge/2π (GHz)
Figure 7.3: Single-qubit characterization. a, The lifetime 𝑇1 of qubits plotted against the
|𝑔i–|𝑒i transition frequency 𝜔𝑔𝑒 inside the lower (left panel) and the upper (right panel)
bandgaps. b, The anharmonicity 𝛼 of qubits plotted against the |𝑔i–|𝑒i transition frequency
𝜔𝑔𝑒 inside the lower (left panel) and the upper (right panel) bandgaps. Throughout the
figure, the marker of each datapoint represents the qubit label, the meaning of which is
summarized on the top.

tapering section consists of a set of four specifically designed resonators at each
end of the metamaterial waveguide, prior to transition to coplanar waveguides with
𝑍0 = 50 Ω characteristic impedance connected to the input/output ports. Details on
the fabrication of the device and the cryogenic setup is outlined in Chapter 3.
7.3

Single-qubit characterization

We begin the characterization of the metamaterial quantum processor by investigating the properties of each qubit dependent on the frequency tuning. We measure
the lifetime 𝑇1 of each qubit across both the lower and the upper bandgaps while the

98
transition frequencies of the remaining qubits are far-detuned to be situated inside
the bandgap other than where the measured qubit lies in. The measurement result
illustrated in Fig. 7.3a shows that the qubits are sufficiently protected from radiative
decay deep inside the bandgap with a gradual reduction in lifetime as the transition
frequency approaches the edges of the transmission band. The additional decay
mechanism close to the band-edges is attributed to the overlap of long photonic tail
of qubit-photon bound states with the external ports close to the band edge, which
can be suppressed if a metamaterial long compared to the localization length of the
bound states is used.
We also perform measurement of the anharmonicity 𝛼 = 𝜔 𝑒 𝑓 − 𝜔𝑔𝑒 of qubit-photon
bound states in the bandgap regime by performing spectroscopy conditioned on
the qubit initialized in the first excited state |𝑒i (Fig. 7.3b). Deep inside the lower
bandgap near 𝜔𝑔𝑒 /2𝜋 ≈ 3.6 GHz, we find a nearly constant value of anharmonicity
𝛼/2𝜋 ≈ −220 MHz of bare transmons, the magnitude of which is reduced by almost
factor of two close to the lower band-edge (about 4.95 GHz) owing to the anomalous
Lamb shift [129] of the |𝑔i-|𝑒i transition frequency 𝜔𝑔𝑒 . Inside the upper bandgap,
even stronger hybridization of |𝑒i-| 𝑓 i transition with the photonic modes inside the
transmission band results in a change in anharmonicity sharper than inside the lower
bandgap. Such modification of anharmonicity, as noted in Ref. [199], can be utilized
as a knob for tuning the attractive interactions between photons on a transmon site, a
unique feature of qubit-photon bound states. The variation of anharmonicity values
between qubits is ascribed to the disorder in the fabrication.
7.4

Interaction between qubit-photon bound states

A notable property of the metamaterial waveguide is the interaction between qubitphoton bound states arising from the spatial overlap of photonic tails, which enables
qubit-qubit connectivity beyond nearest neighbors. In the simplest description of the
metamaterial waveguide in terms of coupled cavity array discussed in Eq. (7.1), it is
known that the coupling 𝐽𝑖 𝑗 between a pair of qubit-photon bound states associated
with qubits at sites (𝑖, 𝑗) of the metamaterial is given by [205]
𝑔𝑖 𝑔 𝑗
𝑠𝑖 𝑗 𝑒 −|𝑖− 𝑗 |/𝜉
(7.3)
𝐽𝑖 𝑗 = p
(𝜔𝑐 − 𝜔 𝑏 ) − 4𝐽
where 𝑔𝑖 is the coupling between the qubit and the cavity at the 𝑖th site, 𝜔 𝑏 is the
frequency of the qubit-photon bound state, and
−1
|𝜔𝑐 − 𝜔 𝑏 |
𝜉 = arccosh
(7.4)
2𝐽

···

i =1

i =2

i =3

i =4

i =5

i =6

i =7

i =8

i =9

|Ji,i+x |/2π (MHz)

x =1

101
x =1
100
x =9

x =9
10-1
4.3

4.5

4.7

Qubit Frequency ωge/2π (GHz)

4.9

7.2

7.4

Qubit Frequency ωge/2π (GHz)

7.6

Figure 7.4: Qubit-qubit interactions. a, Cartoons illustrating the characteristics of longrange coupling between qubits in the metamaterial quantum processor. Left: inside the
lower bandgap, the qubit-qubit coupling alternates between positive (red) and negative
(blue) signs with magnitude decaying exponentially with the distance. Right: inside the
upper bandgap, the sign of qubit-qubit coupling is positive (colored red) with magnitude
decaying exponentially with the distance. b, Chevron pattern obtained from interaction of
Q3 and Q4 , where Q3 is initialized in the excited state and both Q3 and Q4 are dynamically
tuned on resonance to 4.37 GHz. c, The magnitude of qubit-qubit interaction rate 𝐽𝑖 𝑗
measured between qubit pair (Q𝑖 , Q 𝑗 ) at frequencies in the lower bandgap (left panel) and
the upper bandgap (right panel). The colors of the datapoints indicate the distance 𝑥 = 𝑗 − 𝑖
between the pair of qubits (blue: 𝑥 = 1, orange: 𝑥 = 2, green: 𝑥 = 3, red: 𝑥 = 4, purple:
𝑥 = 5, brown: 𝑥 = 6, pink: 𝑥 = 7, gray: 𝑥 = 8, yellow: 𝑥 = 9) and the shapes of marker
represents the index of the first qubit Q𝑖 of the pair, whose correspondence is summarized
on the top of the panel.

100
is the localization length of the qubit-photon bound state. Here, the sign factor 𝑠𝑖 𝑗
takes the value 𝑠𝑖 𝑗 = −(−1) 𝑖− 𝑗 (𝑠𝑖 𝑗 = 1) inside the lower (upper) bandgap, reflecting
the wavevector 𝑘 = 𝜋/𝑑 (𝑘 = 0) at which the band-edge occurs (see Fig. 7.4a).
In practice, the modification of dispersion relation discussed earlier also affects
the exact form of the localization length 𝜉 of qubit-photon bound states and their
coupling 𝐽𝑖 𝑗 to be modified from Eqs. (7.3) and (7.4), which can be numerically
calculated.
We have characterized the strength of interaction between qubit-photon bound states
in the time domain by utilizing a pulse sequence described as follows: we first
prepare the qubit pair (𝑖, 𝑗) in state |𝑒i𝑖 |𝑔i 𝑗 by applying a microwave 𝜋-pulse on
Q𝑖 , followed by crosstalk- and distortion-corrected [284] fast flux-bias pulses with
a variable duration on both qubits to dynamically tune the qubits on resonance at
the desired frequency, which induces vacuum Rabi oscillation between the chosen
qubit pair. Here, the amplitude of the flux-bias pulse applied on one of the qubits is
also swept to find an exact resonance condition. After the end of the flux-bias pulse,
the qubits are returned to their original static frequencies and read out at a fixed
time relative to the beginning of the sequence. Then, the magnitude of coupling rate
|𝐽𝑖 𝑗 | between the qubit pair is determined by fitting the two-dimensional Chevron
pattern of qubit population plotted against the duration 𝜏 and the amplitude of the
flux-bias pulse, an example of which is illustrated in Fig. 7.4b. We have repeated
this experiment for all possible pairs of qubits in the system across a wide range of
dynamically tuned frequencies close to the band-edges, summarized in Fig. 7.4c.
7.5

High-fidelity single-shot readout

The metamaterial waveguide in our quantum processor, having qubits operating
inside the bandgaps and dispersive readout performed employing the transmission
band, naturally satisfies the requirements for a Purcell filter [184, 285]. The excess
spontaneous emission rate of a qubit caused by Purcell decay [114] through a readout
resonator is known to be proportional to the ratio 𝑟 𝑃 ≡ Re[𝑍ext (𝜔 𝑞 )]/Re[𝑍ext (𝜔𝑟 )]
associated with the external impedance 𝑍ext (𝜔) evaluated at the qubit frequency 𝜔 𝑞
and at the readout resonator frequency 𝜔𝑟 [286]. Utilizing a Purcell filter based
on periodic microwave structures, such as our metamaterial waveguide, will greatly
suppress this ratio 𝑟 𝑃 as the Bloch impedance of a periodic structure becomes purely
imaginary inside the bandgap while matching to the external 50 Ω network inside
the transmission band requires it to be purely real [180], resulting in a small number
𝑟 𝑃
1 arising only from parasitic intrinsic loss and finite-size effects. In such

101

Figure 7.5: Multi-qubit readout characterization. a, Assignment probability matrix
F ( 𝑗 |𝑖) for each prepared multi-qubit basis state index 𝑖 and the assigned multi-qubit basis state index 𝑗 (0 ≤ 𝑖, 𝑗 < 210 are decimal representations of 10-bitstrings representing
multi-qubit basis states), extracted from random preparation of multi-qubit basis states and
measurement repeated for 106 counts. The inset shows the diagonal elements F (𝑖|𝑖) of the
assignment probability matrix whose mean value 0.7566 (black dashed line) corresponds
to the multi-qubit readout fidelity. b, Average of off-diagonal elements of the assignment
probability matrix corresponding to each single-qubit and two-qubit error process. Top:
single-qubit bit-flip error rate (average of assignment probability matrix elements corresponding to preparation of state 𝑠𝑖 and assignment of state 𝑠¯𝑖 on qubit Q𝑖 ; the states of
remaining qubits are fixed) during the readout. It is observed that the single-qubit decay
|1i → |0i (few percent on average) is the dominant contributor to the infidelity of the
readout. Bottom: Two-qubit bit-flip error rate (average of assignment probability matrix
elements corresponding to preparation of state 𝑠𝑖 , 𝑠 𝑗 and assignment of state 𝑠¯𝑖 , 𝑠¯ 𝑗 on qubits
Q𝑖 , Q 𝑗 ; the states of remaining qubits are fixed) during the readout. The two-qubit error
rates are an order of magnitude smaller than the single-qubit error rates.

a regime, the coupling between readout resonator and qubit as well as the decay
rate 𝜅 of readout resonator can be made very large without compromising qubit
coherence, potentially enabling an ultrafast qubit state measurement. From the
design perspectives, this also allows us to perform resource-efficient without need
for placing additional large-footprint elements on a chip for a rapid, high fidelity
readout [185, 287].
From the experiment, we are able to find high single-qubit readout fidelity exceeding
98 % in most cases with only 100–200 ns long readout and integration with optimal
weights [186, 288], on par with the state-of-the-art in the field [185]. While the
naive approach of multiplexing the calibrated single-qubit readout pulses at different

102
frequencies was considered sufficient for long (about 1 𝜇s) multiplexed readout of
few qubits, it was observed that this method is no longer valid for short (about 400 ns
or shorter) multiplexed readout of all ten qubits in our system. This is ascribed to
high overall readout power sent into the system, resulting in unexpected non-linear
effects, the origin of which is yet to be confirmed. To overcome this, we have
successfully developed a routine to simultaneously calibrate the set of frequencies
and amplitudes of all ten readout signals based on linear discriminant analysis [186]
for determination of readout fidelity and outlier counting to prevent measurementinduced state transitions [289]. An example of the multiplexed readout benchmark
result after such calibration is illustrated in Fig. 7.5, where the fidelity of assignment
of 210 multi-qubit basis states of ten qubits is F = 75.66 %, corresponding to about
F1Q ≈ 97.25 % single-qubit readout fidelity. Our multiplexed readout is adversely
affected by correlated multi-qubit state preparation error as well as readout crosstalk
effects [287], which will be a subject of further investigations. Note that our method
for processing readout signals for state discrimination is fully implemented in real
time and therefore allows us to perform low-latency feedback operations based on
the outcome of ten-qubit measurement. An example use of this real-time feedback
is active reset where the state of all qubits is actively reset to the ground state based
on the measured bitstring, allowing us to perform multi-qubit experiments at an
ultrafast repetition rate exceeding 100 kHz.
7.6

Quantum many-body dynamics

Finally, as a demonstration of novel quantum many-body dynamics in our quantum
processor, we induce quantum walk of two photons along the qubit-photon bound
states inside the lower bandgap (Fig. 7.6), where the pair of qubits Q5 and Q6 located
at the center of the array are initialized in the excited state at their idle frequencies
after which a rapid simultaneous tuning of all qubits’ transition frequencies to the
desired interaction frequency takes place. After a variable interaction time, all
the qubits return to their original idle frequencies and are simultaneously read out.
The distribution of the bitstrings |𝑛1 𝑛2 · · · 𝑛10 i measured after such quenched time
evolution allows us to determine the population 𝑝𝑖 = h𝑛ˆ 𝑖 i at each site 𝑖 and the
correlation matrix 𝐶𝑖 𝑗 = h𝑛ˆ 𝑖 𝑛ˆ 𝑗 i for every pair (𝑖, 𝑗) of sites (𝑖, 𝑗 = 1–10). From
the time evolution of population 𝑝𝑖 shown in Fig. 7.6a, we find a gradual increase
in the speed of propagation as the qubit-photon bound states are tuned closer to
the band edge. Also, a more complicated interference pattern associated with
long-range coupling is observed near the band edge, suggesting a rapid buildup of

103

Time (ns)

800

0.8

600

0.6

400

0.4

200

0.2
•••

Site i

•••

Site i

10 1

•••

Site i

10 1

•••

Site i

0.0

Cij

0.8

•••

Site j

1.0

0.4

10 1

0.0

•••

0.1

10 1

Site j

0.2

0.0

•••

Site j

0.4
0.2

10 1

0.0

•••

0.1

Site j

0.2

0.0

•••

Site i

•••

Site i

10 1

•••

Site i

10 1

•••

Site i

Figure 7.6: Quantum walk of two photons along qubit-photon bound states. a, Qubit
population 𝑝 𝑖 plotted as a function of time during the quantum walk of two photons initialized
at Q5 and Q6 . The subpanels correspond to quantum walks at interaction frequencies
(4.50, 4.60, 4.71, 4.80)GHz from left to right. b, The particle-particle correlation matrix
𝐶𝑖 𝑗 is plotted at a few time steps during the quantum walk illustrated in panel a. The
subpanels on each column (corresponding to result from the interaction frequencies in
panel a) shows the correlation matrix at the instances soon after initialization 𝑡 = 12 ns,
approximately halfway to the first propagation to the boundary 𝑡 = 𝑡1 /2, approximately the
first major reflection at the boundary 𝑡 = 𝑡1 , and approximately the first major refocusing
𝑡 = 2𝑡1 , from top to bottom, which are indicated with black solid lines on the corresponding
subpanel in panel a. Here, 𝑡1 = (312, 240, 160, 120) ns from left to right columns.

104
quantum entanglement that would have occurred at a much slower rate in the case
of only nearest-neighbor coupling [100]. This is also observed in the pattern of the
correlation matrix shown in Fig. 7.6b where the initially localized particle-particle
correlation gets quickly delocalized with the presence of long-range coupling with
refocusing quantum walkers becoming quickly invisible. Such long-range character
of coupling gives rise to a non-integrable Hamiltonian in the two-excitation manifold,
resulting in a chaotic quantum many-body evolution.
7.7

Discussion and outlook

The quantum processor demonstrated in our work naturally opens up novel opportunities to explore quantum many-body physics. An extension of our work will enable
the study of emergent randomness from quantum many-body chaos [290, 291],
which sheds new light on the understanding of quantum thermalization. Employing
the capability to perform real-time measurement and feedforward operations at a
low latency demonstrated in this work, a novel hybrid quantum-classical protocol for
controlling quantum many-body states can be envisioned. While not scalable, qubit
readout utilizing the transmission band of the metamaterial waveguide discussed
in our work shows an interesting regime of Purcell filter, which can be harnessed
to perform ultrafast dispersive readout at a high fidelity. With the addition of an
individual filter between the metamaterial waveguide and each readout resonator,
vanishingly small readout crosstalk can be achieved [287], where experimental studies of measurement-induced phase transition [292–295] are expected to be within
reach [296]. On the practical hand, new ways of building up large-scale superconducting quantum processors can be conceived, which has been so far limited to
nearest-neighbor connectivity in most cases [76, 297–299]. In particular, the integration of tunable coupler [300, 301] between qubits and metamaterial waveguide
will enable high-fidelity metamaterial-mediated quantum gates, which will allow for
experimental demonstrations of quantum algorithms and quantum error correction
schemes requiring non-local quantum gate operations. Overall, our work opens up
new avenues for quantum information processing in the noisy intermediate-scale
quantum era [75].

105
Chapter 8

OUTLOOK AND FUTURE DIRECTIONS
Our work described in the thesis, the experimental studies of waveguide QED in
the platform of superconducting circuits, provides a unique ground for studying
quantum many-body physics and quantum computation with long-range photonmediated interactions between qubits. Below I will enumerate a few interesting
potential directions beyond the work described in the thesis.
8.1

Opportunities for studying many-body physics

Compared to traditional atom-based systems, superconducting qubits are readily equipped with full individual local qubit control (XY and Z), quantum nondemolition measurement, and real-time feedback operation. Without need for trapping atoms, experiments with superconducting qubits can be performed at ultra-high
repetition rates (up to ∼ 1 MHz with active qubit reset [302–305]), offering new
possibilities to study higher-order many-body effects that are often obscured by
statistical fluctuations. Utilizing the advanced control and calibration techniques
developed for building up practical quantum computers [76, 306], a novel set of
tools for controlling and measuring quantum many-body systems could be envisioned. In the following, I outline few research directions for exploring quantum
many-body physics with our quantum processor.
Extended Bose-Hubbard model
The Bose-Hubbard model [307, 308] describes the physics of interacting bosons
subjected to a shallow lattice potential and was at the heart of pioneering quantum simulation experiments with cold atoms in optical lattice [87–89, 309–311].
Superconducting transmon qubits [190] are also often described as sites of the
Bose-Hubbard model for photons due to their weak anharmonicity [101, 312, 313].
While most of the studies to date considered cases with inter-particle interactions
on the same site and the range of hopping limited to only nearest-neighbor sites,
investigating extended versions of the Bose-Hubbard model with longer-range interaction and hopping processes are expected to bring in new opportunities to study
strongly-correlated quantum phases of matter [314, 315]. For example, inclusion
of long-range interactions between particles is shown to enrich the phase diagram,

106
inducing Haldane insulator and density wave phases 1D [316] and supersolid phase
with both superfluid and crystalline order in 2D [317]. Also, extended range of hopping is known to amplify long-range and higher-order quantum correlations e.g.,
by creating correlated triplon-hole-hole pairs in Mott insulators [318] and greatly
influencing the phase boundary of Mott insulator-to-superfluid transition [319].
Our superconducting metamaterial quantum processor induces tunable long-range
coupling between transmon qubits, naturally realizing an extended Bose-Hubbard
model for photons. We envision experimental exploration of the canonical model
in a new regime with an unprecedented level of control and with new methods for
dissipative stabilizing quantum many-body states [101, 277].
Quantum information scrambling
Quantum information scrambling [320–323] refers to a general phenomenon where
initially localized quantum information is spread across many-body quantum degrees of freedom, resulting in a complex entanglement structure which prohibits
the recovery of originally encoded information with local measurements. This
topic has been a subject of extensive theoretical studies [324–327] in the last few
years. While there has been pioneering experimental efforts to measure out-oftime-ordered correlators [328, 329] and to probe quantum information scrambling
[330, 331] in state-of-the-art quantum platforms, scrambling effects arising from
a more generic scenario of thermalization in quantum many-body systems remain
yet to be observed. Also, an interesting question here is whether we can retrieve
the scrambled quantum information buried in multipartite quantum correlations by
utilizing high level of local control over a closed quantum system. We are currently
working on developing hybrid quantum-classical protocols to decode the scrambled
quantum information from chaotic quantum many-body evolution by measurement
and feed-forward. Our rapid high-fidelity multiplexed readout and the ability to
perform low-latency (< 1 𝜇s) feedback operations based on measurement outcomes
of a subsystem will play a crucial role in realizing this scheme.
Measurement-induced phase transition
There has been recent theoretical studies of competition between the growth of
entanglement in a quantum many-body system and local projective measurements
[292–295]. It was shown that if the rate of randomly interspersed local projective
measurement exceeds a certain threshold, the entanglement growth of the quantum
many-body system is restricted to follow the area-law. Such transition from the

107
entangling phase (volume law) to the disentangling phase (area law) is called the
measurement-induced phase transition. Experimentally probing the measurementinduced phase transition has been a formidable task as it requires ability to perform
random mid-circuit readout of arbitrary subsets of qubits without fully collapsing
the many-body wavefunction. A pioneering experiment conducted in a trapped ion
quantum simulator overcame this challenge by exploiting entangling gates between
system and ancilla qubits to defer the measurement until the end of the circuit [296].
However, this method limits the total number of measurements to the number of
ancillae and therefore is only applicable to short time evolution and small system size.
In superconducting circuits, it is known that the effect of measurement of one qubit
on other unmeasured qubits is mild, only adding parasitic dephasing rates on the
order of Γ̄/2𝜋 ∼ 102 kHz [287] during the on-time of readout pulses (𝜏𝑝 ∼ 102 ns),
and in principle can be avoided by careful design of feedlines and Purcell filters
[184, 285, 287, 332] for readout resonators [231]. Employing the superconducting
metamaterial waveguide as an efficient Purcell filter for readout, we look forward
to probing full regimes of the measurement-induced phase transition including its
criticality.
8.2

New directions for scaling up quantum processors

Scaling up quantum processors for conducting large-scale quantum computation
and simulation experiments entails significant technical challenges. On one hand,
realizing a large system often requires transition to a new scalable technology for
constructing hardware and extensive engineering of electronics. On the other hand,
achieving good systematic control over the large system involves serious efforts in
software development. The state-of-the-art large-scale quantum experiments are
being performed with ∼ 60 qubits in superconducting circuits [76, 297, 298, 333],
∼ 50 qubits in trapped ions [14, 69], and ∼ 200 qubits in neutral atom quantum
simulators [15, 16], all of which are expected to grow further in size over the next
few years.
In Chapter 7, we have successfully developed hardware and software to systematically control ten superconducting qubits with full individual addressing and longrange coupling. With established and new methods for cryogenics [159], microwave
packaging of device [152], and calibration [306], we expect to be able to run experiments on the scale of ∼ 100 qubits in the near future. Specifically, leveraging the
inherent 1D-scalability and long-range connectivity of the metamaterial quantum
processor will be a key goal in this direction.

108

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Appendix A

FABRICATION DETAILS
In this appendix, I provide details on the fabrication steps of the quantum devices
used in the thesis.
A.1

Cleaning the chip tray and glass beakers
• Cleaning the tray
Before placing cleaned chips inside the tray, rinse it with IPA and blow dry
with N2 . Note: Acetone attacks the tray (made of polyethylene) and should
not be used for cleaning it. It is advised to keep your tray away from fume
hood to avoid contamination of the tray. Moreover, you need to be cautious
about putting a wet chip on the tray since any remaining acetone on the chip
might dissolve the tray, and the chip may adhere to it.
• Cleaning glass beakers
Before using each beaker, rinse it with the solvent you would need to keep.
i.e., rinse TCE beaker with TCE, ACE beaker with ACE, and IPA beaker with
IPA.

A.2

Chip cleaning

The chip cleaning is the first thing you need to do after you take a chip from a diced
wafer, and this consists of three rinse steps (TCE-ACE-IPA) and blow dry.
1. Trichloroethlyne (TCE) rinse: > 5 min (usually 15 min) at 80◦ C.
Trichloroethlyne is an effective solvent for a variety of organic materials and
we use warm TCE at the very first step of chip cleaning. Warning: TCE is
toxic and you should try not to inhale the air above it by keeping yourself
outside the fume hood. Also, try to cover the beaker with Al foil when
carrying TCE in it, so that you can avoid smelling it.
2. Acetone (ACE) rinse: 5 min at room temperature.
Spray the chip with acetone while transferring the chip from TCE bath to
minimize solvent mixing. Sonicate for 5 minutes at half-maximum power and
maximum frequency setting of the sonicator.

141
3. Isopropyl alcohol (IPA) rinse: 3 min at room temperature.
Spray the chip with IPA while transferring the chip from acetone to minimize
solvent mixing. Sonicate for 3 minutes at half-maximum power and maximum
frequency setting of the sonicator.
4. Blow dry with N2 gun.
Hold the chip tightly on top of a sheet of Texwipe with a tweezer and use the
other hand to blow dry.
A.3

Pre-spin/bake or pre-evaporation cleaning

The following steps must be done immediately before spin/baking resist or evaporating Al, to clean the surface of the chip.
1. Oxygen plasma ashing:
5 minutes of conditioning run followed by the main run typically of 2 minutes.
The duration of main run can be varied depending of the level of ashing
you want. For the fabrication of Josephson junctions or airbridges, we do
20 seconds of light oxygen plasma since it eats away resist and can widen
dimensions significantly.
2. Hydrofluoric acid (HF) cleaning:
We perform two kinds of HF cleaning in our fabrication depending on the
processing step.
• Wet HF (BOE; Buffered Oxide Etch):
We perform wet HF cleaning prior to spinning and baking the resist
or before electron-beam evaporation as long as there is no aluminum
deposited on the chip. Extra precautions have to be taken in this step
as exposure to HF can be lethal. The process consists of the following
steps:
i. Swivel the chip inside buffered oxide etchant for 15 seconds.
ii. Rinse the chip for 10 seconds in the first DI water bath.
iii. Rinse the chip for 10 seconds in the second DI water bath.
iv. Careful blow-dry with a nitrogen gun.
• Vapor HF:
If wet HF cleaning is not possible (wet HF attacks aluminum on chip), we
use SPTS uEtch in the Painter lab cleanroom to perform selective etching

142
of oxides. A simple HF flash process exposes the chip to anhydrous HF
vapor to remove native oxides and create hydrogen-terminated silicon
surface.
A.4

Spin & bake

After chip cleaning, the chip is placed on the spinner and coated with a resist just
enough to cover the top surface of the chip (without overflowing), followed by
spinning at a calibrated rate. The chip is also baked on top of a hot plate either
before or after spinning the resist. We use the following recipes for spin and bake:
• ZEP520A resist for marker, ground plane, and bandage layers: spin at 3 krpm
with 1.5 krpm/s ramp rate, bake at 180 ◦ C
We first prebake the chip for 3 minutes. Then, we spin ZEP520A resist
followed by a 3-minute bake.
• Bilayer resist for Josephson junction layer: spin at 2.2 krpm with 1.5 krpm/s
ramp rate, bake at 170 ◦ C
We first prebake the chip for 3 minutes. Then, we spin copolymer MMA(8.5)MAA
EL11 resist followed by a 3-minute bake. The chip is then spun with 950
PMMA A4 resist and baked for another 3 minutes.
• Trilayer resist for airbridge layer: spin at 4 krpm with 1.5 krpm/s ramp rate,
bake at 170 ◦ C
We first prebake the chip for 3 minutes. Then, we spin 950 PMMA A9
resist (this resist is very viscous and it is recommended to use a positivedisplacement pipette for dropping this resist on the chip), followed by baking
for 3 minutes. After this, we spin copolymer MMA(8.5)MAA EL11 resist
and bake at the same condition as before. Finally, the chip is again spun with
950 PMMA A9 resist and baked with the identical settings as before.
A.5

Electron-beam lithography

After spinning and baking the resist, electron-beam lithography is performed with
Raith EBPG 5200 in Kavli Nanoscience Institute (KNI) to write patterns. We use
various set of beam current, aperture, and dose settings depending on the resist and
the pattern that is written, which are summarized below.
• Marker layer (ZEP520A resist):
Beam current: 10 nA, Aperture: 300 𝜇m, Dose: 210–245 𝜇C/cm2

143
• Ground plane layer (ZEP520A resist):
Beam current: 1 nA for planar coils and capacitors and 100 nA for ground
plane, Aperture: 300 𝜇m, Dose: 210–245 𝜇C/cm2
• Josephson junction layer (bilayer resist):
Beam current: 1 nA, Aperture: 300 𝜇m, Dose: 939 𝜇C/cm2
To reduce disorder, we have used smaller beam current and aperture settings
in later experiments to perform beamwriting with small spot size:
Beam current: 180 pA, Aperture: 200 𝜇m, Dose: 990 𝜇C/cm2
• Airbridge layer (trilayer resist):
Beam current: 4nA, Aperture: 300 𝜇m, Dose: 480 𝜇C/cm2
For details on the general workflow of electron-beam lithography, refer to the EBPG
manual.
A.6

Development

After writing patterns with electron-beam lithography, we perform development
depending on the type of the resist used in the spin and bake process introduced in
Sec. A.4, outlined as follows:
• ZEP520A resist for marker, ground plane, and bandage layers:
We develop the chip in ZED-N50 developer for 2 minutes and 30 seconds,
followed by rinsing in methyl isobutyl ketone (MIBK) for 30 seconds and a
gentle blow-dry with a nitrogen gun.
• Bilayer resist for Josephson junction layer:
We use a cold development procedure for developing patterns written on
the bilayer resists. This process is extremely temperature-sensitive (higher
temperature makes the development faster) and requires special attention to
1 Note that neither IPA nor water alone cannot dissolve electron-beam exposed resist, yet the

mixture of them acts as a cosolvent can develop the resist. The mechanism is uncertain, but it
has been suggested that the presence of highly polar water molecules modify the alcohol molecule,
improving the solvent action of IPA [151].
2 The temperature controller setup consists of a thermoelectric cooling plate that is thermally
anchored to a container with a bath of water, whose temperature is kept equilibrated by using an
electric stirrer operating at 500 rpm. The temperature of the water bath is monitored using a digital
thermometer. The beakers containing the developer and the rinsing solvent stay inside this water
bath throughout all the development stages.

144
the temperature of the developer and the duration of development. During
the preparation stage, a beaker containing a mixture of IPA and DI water in
a volume ratio of 3:1, used as a developer for this resist1, and another beaker
containing IPA are covered with aluminum foil and cooled down to 10.0 ◦ C
for at least an hour by using a custom-designed temperature controller setup2.
Once the developer and the rinsing solvent are cold, we develop the chip in
the developer for 90 seconds, quickly transfer the chip to the cold IPA bath
and rinse for 10 seconds, followed by an immediate blow dry with a nitrogen
gun.
• Trilayer resist for airbridge layer:
For the development of airbridge patterns on the trilayer resist written with
grayscale electron-beam lithography, we use the same procedure as the bilayer
resist but in a less stringent room-temperature setting. First, the chip is
developed in 3:1 IPA:water developer for about 2 minutes. Then the chip is
rinsed in IPA for 10 seconds, followed by blow-dry with nitrogen. We inspect
the chip with an optical microscope to check the progress. We repeat these
steps of development (with shorter duration)-rinsing-blow drying-inspection
until we find that the resist is fully developed. After this, we perform 2 hours
of resist reflow at 105 ◦ C.
A.7

Evaporation

We use the electron-beam evaporator Plassys MEB550S in the Painter lab cleanroom
for thin-film deposition of metals on chip. The evaporator consists of a loadlock
with a tiltable and rotatable stage on which samples are loaded and a chamber
where pre-loaded materials are melted with electron beam and evaporated. We use
titanium (Ti) as an absorbent to reduce oxygen and water vapor, conditioning the
loadlock before evaporating other metals. Aluminum (Al) is used for fabrication of
superconducting resonators and qubits and niobium (Nb) is used for fabrication of
alignment markers. The critical pressures for running a process is 10−6 mbar for
loadlock and 10−7 mbar for chamber. We use the following recipes for electron-beam
evaporation:
• Marker layer:
Pump out time is not important for this process. It is good to run the process
when the pressures are lower than the critical values.
i. Evaporation of Ti at a rate of 0.15 nm/s for 3 minutes.

145
ii. Evaporation of 150 nm of Nb at a rate of 0.4 nm/s.
• Ground plane layer:
It is good to pump out the loadlock for at least 4 hours after loading the
samples.
i. Evaporation of Ti at a rate of 0.2 nm/s rate for 3 minutes.
ii. Evaporation of 120 nm of Al at a rate of 1 nm/s.
iii. Exposure to static O2 at a pressure of 10 mbar for 2 minutes.
• Josephson junction layer:
It is necessary to pump out for at least 12 hours after loading the samples for
best reproducibility.
i. Evaporation of Ti at a rate of 0.2 nm/s rate for 3 minutes.
ii. Evaporation of 60 nm of Al at a rate of 1 nm/s with a tilt and rotation of
(60◦ , 90◦ ).
iii. Exposure to static O2 at a pressure of 5 mbar for 20 minutes.
iv. Evaporation of 120 nm of Al at 1 nm/s rate with a tilt and rotation of
(20◦ , 180◦ ).
v. Exposure to static O2 at a pressure of 10 mbar for 2 minutes.
• Bandage layer:
i. Ar ion milling at voltage and current (400 V, 21 mA) for 6 minutes.
ii. Evaporation of Ti at a rate of 0.2 nm/s for 3 minutes.
iii. Evaporation of 200 nm of Al at a rate of 1 nm/s.
iv. Exposure to static O2 at a pressure of 10 mbar for 2 minutes.
A.8

Liftoff

The liftoff after the electron-beam evaporation is performed by first placing the
chip inside a bath of N-methyl-2-pyrrolidone (NMP) heated to 150 ◦ C. If niobium
were evaporated on the chip, the niobium deposited on regions outside the desired
pattern immediately detaches from the chip upon touching the hot NMP bath and
we shoot bubbles of air to strip off the remaining pieces of metal by using a pipette.
If aluminum were evaporated on chip, we leave the chips in the hot NMP bath for at
least 2 hours, followed by sonication or pipetting to aid the liftoff process. Then, the

146
chip is moved to another clean heated bath of NMP and left for another hour or more
(depending on the quality of the first liftoff), followed by sonication or pipetting.
After this, we sonicate the chips in a bath of acetone for 5 minutes, followed by
3-minute sonication in IPA and blow drying with a nitrogen gun. When suspended
structures such as airbridges are fabricated on chip, one needs to be careful about
the choice of parameters for sonication.

147
Appendix B

DESIGNING JOSEPHSON JUNCTIONS FOR TRANSMON
QUBITS
In this note, I describe considerations for designing Josephson junctions for transmon
qubits and getting the frequencies of fabricated qubits (or “sweet spot” frequencies
for frequency-tunable qubits) close to the design values.
B.1

Basics

The frequency 𝜔q of transmon qubits are determined by two factors, the charging
energy 𝐸𝐶 = 𝑒 2 /(2𝐶Σ ) and the Josephson energy 𝐸 𝐽 , and is written as
ℏ𝜔q = 8𝐸 𝐽 𝐸𝐶 − 𝐸𝐶 .
Here, the effective capacitance 𝐶Σ is a sum of capacitance contributions that couples
to the qubit node, including the capacitance 𝐶𝐽 of the Josephson junction. Normally,
the contributions other than the junction capacitance 𝐶Σ − 𝐶𝐽 are thought to be
accurately estimated by using EM simulators such as Sonnet. It is important to note
that the anharmonicity of transmon qubit is given by 𝛼 = −𝐸𝐶 /ℏ, which can be
directly measured in the experiment by either a high-power spectroscopy that drives
two-photon excitation or a frequency sweep of XY drive conditioned on the first
excitation (by sending 𝜋-pulse at frequency resonant to first transition |𝑔i → |𝑒i).
Together with the sweet spot frequencies 𝜔q,max measured in the experiment, one
can fully determine the Josephson energies of fabricated junctions of qubits. This
allows us to test the accuracy of the design.
B.2

Design

In the design of qubits, both the Josephson energy 𝐸 𝐽 and junction capacitance
𝐶𝐽 depends on the area and oxide properties of Josephson junctions and hence
determining the properties of Josephson junctions are crucial to accurate design of
the experiments.
Josephson capacitance 𝐶𝐽
Assuming the Josephson junction is a parallel plate capacitor, with aluminum oxide
(typical relative permittivity 𝜖𝑟 = 10 for Al2 O3 ) as a dielectric and distance 𝑑 =
1.8 nm [334], the junction capacitance per area (𝐶𝐽 /𝐴𝐽 ) is calculated as

148
𝐶𝐽 𝜖 0 𝜖𝑟
≈ 0.049 F/m2 = 49 fF/𝜇m2
𝐴𝐽
This is very close to the value quoted in page 62 of John Teufel’s thesis [335] from
Prof. Robert Schoelkopf’s lab, although their junction fabrication process is not
necessarily identical to ours. Professor John Martinis lab’s typical value is reported
to be 𝐶𝐽 /𝐴𝐽 = 4 fF/(300 nm) 2 = 44.4 fF/𝜇m2 (see Supplementary Information of
Ref. [150]).
Assuming 𝐶𝐽 /𝐴𝐽 = 50 fF/𝜇m2 , for a typical junction area of 𝐴𝐽 = 0.232 𝜇m2 , the
capacitance arising from Josephson junctions is 𝐶𝐽 = 11.6 fF which can account
for > 10 % of effective qubit capacitance 𝐶Σ . After taking into account this factor,
the charging energy of transmon qubits we fabricated in the lab were consistently
measured to be close to the design values.
Josephson energy 𝐸 𝐽
For 𝐸 𝐽 /𝐴𝐽 , we find values fluctuating roughly by 10 %, somewhere in the 𝐸 𝐽 /𝐴𝐽 =
125–132 GHz/𝜇m2 range when the areas taken into account were determined by
high-resolution imaging with Nova 200/600 scanning electron microscope in KNI.
It is believed that this number is sensitive to conditions in the e-beam evaporation
chamber, changing after material refills.
B.3

Fabrication of test junctions

Accurate fabrication of superconducting qubits is a challenging task due to the fact
that a lot of factors can play roles in the fabrication of Josephson junctions, the size
of which is only few hundred nanometers. For example, about 30 nm error in each
dimension of Josephson junctions from design values can result in about 1 GHz
error in qubit frequency, which can make certain experiments infeasible. Prior to
fabrication of the device for experiments, we perform a few rounds of test fabrication
to accurately calibrate the area of Josephson junctions, which is crucial to accurate
prediction of qubit frequencies. We fabricate a test device consisting of an array
of Josephson junctions with a single-layer processing step for Josephson junctions
discussed in Appendix A. The junctions in the array have CAD dimensions swept in
both X and Y directions with a step of about 5–10 nm with enough repetitions and
have designs identical to the ones that will be used for qubits in the experiment. Once
fabricated, the Josephson junctions in the test devices are imaged with a scanning
electron microscope from which a set of CAD dimensions expected to realize the

149
desired fabricated dimensions can be determined. This process is repeated until we
get a fabricated junction dimensions close enough to the design values (within about
10 nm).
B.4

Imaging

Obtaining a scanning electron micrograph of good quality with calibrated dimensions is crucial to achieving accurate determination of junction area 𝐴𝐽 . Here, I
describe important tips for using Nova 200/600 SEM in KNI to image Josephson
junctions in a consistent fashion.
For imaging of Josephson junctions for dimension extraction, the scanning electron
micrographs are acquired with the following settings (developed with the help of
former KNI staff Dr. Matthew Hunt):
• High Voltage (HV): 10.00 kV
• Beam Current: 130 pA
• Working distance (WD): sample located at the eucentric height (reproducible
WD based only on chamber geometry, not on lens parameters)
• Horizontal Field Width (HFW): 1.00um
• Detector: through-the-lens detector (TLD) in the immersion mode, which
achieves ultra-high sub-10 nm resolution.
• Dwell time: 10 𝜇s
• If not in hurry, the sample is mounted to the stage with NIST-traceable standard
specimen1, which can be used to calibrate the dimensions measured in the
SEM more accurately (Nova is reported to be off by ∼ 1–2 %, which is within
the specification of the system).
• Save the images in .tif format, which saves all the configurations used for
imaging in the metadata.
An example of the image taken with these settings is illustrated in Fig. B.1a, in
which we can observe clear boundary of a Josephson junction accurate to within
about few nanometers.
1 Available from Ted Pella, Inc.

150

Figure B.1: Scanning electron micrograph of a Josephson junction. a, An image of a
Josephson junction taken with Nova 200 scanning electron microscope following the settings
in Sec. B.4. b, Determination of dimensions of the Josephson junction in panel a with image
processing.

Note that images taken with Hitachi S4300 SEM in Painter lab cleanroom didn’t
have enough resolution to accurately determine the critical dimensions of Josephson
junctions. Also, the dimension calibration in the Hitachi S4300 SEM seems to
be unreliable for accuracy within about 40 nm, which can often be larger than
10 % of dimensions of Josephson junctions. A careful set of calibrations must be
implemented to be able to use Hitachi S4300 SEM for Josephson junction calibration.
B.5

Determination of critical dimensions

The critical dimensions of Josephson junctions can be obtained by employing builtin softwares for scanning electron microscopes to draw lines on the image and
extract the corresponding dimensions. This manual process, however, can easily
add errors on the order of 10 nm which is larger than the resolution of the image.
For a fast and reliable determination of critical dimensions of Josephson junctions, I
have developed an imaging processing routine to detect the boundary of Josephson
junctions from an image and calculate the area. Thus routine involves the use of
Sobel filter for edge detection and fitting of points into lines. An example of the
result from this process is illustrated in Fig. B.1b.

151
Appendix C

SUPPLEMENTARY INFORMATION FOR CHAPTER 4
C.1

Spectroscopic measurement of individual qubits

The master equation of a qubit in a thermal bath at temperature 𝑇, driven by a
classical field is given by 𝜌¤̂ = −𝑖[ 𝐻/ℏ,
𝜌]
ˆ + L [ 𝜌],
ˆ where the Hamiltonian 𝐻ˆ and the
Liouvillian L is written as [336]
ˆ =−
𝐻/ℏ

𝜔p − 𝜔q
Ωp
ˆ𝑧 +
ˆ 𝑥,

(C.1)

L [ 𝜌]
ˆ = ( 𝑛¯ th + 1)Γ1 D [ 𝜎
ˆ − ] 𝜌ˆ + 𝑛¯ th Γ1 D [ 𝜎
ˆ + ] 𝜌ˆ +

Γ𝜑
D [𝜎
ˆ 𝑧 ] 𝜌.

(C.2)

Here, 𝜔p (𝜔q ) is the frequency of the drive (qubit), Ωp is the Rabi frequency of the
drive, 𝑛¯ th = 1/(𝑒 ℏ𝜔q /𝑘 B𝑇 − 1) is the thermal occupation of photons in the bath, Γ1
and Γ𝜑 are relaxation rate and pure dephasing rates of the qubit, respectively. The
superoperator
ˆ 𝜌ˆ = 𝐴ˆ 𝜌ˆ 𝐴ˆ † − 1 { 𝐴ˆ † 𝐴,
ˆ 𝜌}
D [ 𝐴]
(C.3)
denotes the Lindblad dissipator. The master equation can be rewritten in terms of
density matrix elements 𝜌𝑎,𝑏 ≡ h𝑎| 𝜌|𝑏i
as
𝑖Ωp
(𝜌e,g − 𝜌g,e ) − ( 𝑛¯ th + 1)Γ1 𝜌e,e + 𝑛¯ th Γ1 𝜌g,g
𝑖Ωp
(2𝑛¯ th + 1)Γ1 + 2Γ𝜑
𝜌¤ e,g = 𝑖(𝜔p − 𝜔q ) −
𝜌e,g +
(𝜌e,e − 𝜌g,g )
𝜌¤ e,e =

𝜌¤ g,e = 𝜌¤ e,g

𝜌¤ g,g = − 𝜌¤ e,e

(C.4)
(C.5)
(C.6)

With 𝜌e,e + 𝜌g,g = 1, the steady-state solution ( 𝜌¤̂ = 0) to the master equation can be
expressed as
ss
𝜌e,e

th
Ω2p /(Γth
1 + (𝛿𝜔/Γth
𝑛¯ th
1 Γ2 )
2)
th th
th th
2𝑛¯ th + 1 1 + (𝛿𝜔/Γth
2 1 + (𝛿𝜔/Γth
2 ) + Ωp /(Γ1 Γ2 )
2 ) + Ωp /(Γ1 Γ2 )

(C.7)
ss
𝜌e,g
= −𝑖

Ωp

1 + 𝑖 𝛿𝜔/Γth

th th
2Γth
¯ th + 1) 1 + (𝛿𝜔/Γth
2 (2𝑛
2 ) + Ωp /(Γ1 Γ2 )

(C.8)

where 𝛿𝜔 = 𝜔p − 𝜔q is the detuning of the drive from qubit frequency, Γth
1 =
th
th
(2𝑛¯ th + 1)Γ1 and Γ2 = Γ1 /2 + Γ𝜑 are the thermally enhanced decay rate and
dephasing rate of the qubit.

152
Now, let us consider the case where a qubit is coupled to the waveguide with decay
rate of Γ1D . If we send in a probe field 𝑎ˆ in from left to right along the waveguide,
the right-propagating output field 𝑎ˆ out after interaction with the qubit is written as
[116]
Γ1D
𝑎ˆ out = 𝑎ˆ in +
ˆ −.
The probe field creates a classical drive on the qubit with the rate of Ωp /2 =
−𝑖h𝑎ˆ in i Γ1D /2. With the steady-state solution of master equation (C.8) the transmission amplitude 𝑡 = h𝑎ˆ out i/h𝑎ˆ in i can be written as
1 + 𝑖 𝛿𝜔/Γth
Γ1D
𝑡 (𝛿𝜔) = 1 − th
th
th
2Γ2 (2𝑛¯ th + 1) 1 + (𝛿𝜔/Γ2 ) + Ωp /(Γth
1 Γ2 )

(C.9)

At zero temperature (𝑛¯ th = 0) Eq. (C.9) reduces to [146, 337]
𝑡 (𝛿𝜔) = 1 −

1 + 𝑖 𝛿𝜔/Γ2
Γ1D
2Γ2 1 + (𝛿𝜔/Γ2 ) 2 + Ω2p /(Γ1 Γ2 )

(C.10)

Here, Γ2 = Γ𝜑 + Γ1 /2 is the dephasing rate of the qubit in the absence of thermal
occupancy. In the following, we define the parasitic decoherence rate of the qubit
as Γ0 = 2Γ2 − Γ1D = Γloss + 2Γ𝜑 , where Γloss denotes the decay rate of qubit induced
by channels other than the waveguide. Examples of Γloss in superconducting qubits
include dielectric loss, decay into slotline mode, and loss from coupling to two-level
system (TLS) defects.
Effect of saturation
To discuss the effect of saturation on the extinction in transmission, we start with
the zero temperature case of Eq. (C.10). We introduce the saturation parameter
𝑠 ≡ Ω2p /Γ1 Γ2 to rewrite the on-resonance transmittivity as

Γ1D 1
Γ1D
Γ1D
Γ0
≈1−
(1 − 𝑠) = 1 + 𝑠 0
𝑡 (0) = 1 −
2Γ2 1 + 𝑠
2Γ2
Γ0 + Γ1D

(C.11)

where the low-power assumption 𝑠
1 has been made in the last step. For
the extinction to get negligible effect from saturation, the power-dependent part
in Eq. (C.11) should be small compared to the power-independent part. This is
equivalent to 𝑠 < Γ0/Γ1D . Using the relation
2Γ1D 𝑃p
Ωp =
ℏ𝜔q

153
between the driven Rabi frequency and the power 𝑃p of the probe and assuming
Γ0
Γ1D , this reduces to
ℏ𝜔q Γ0
𝑃p .
(C.12)
In the experiment, the probe power used to resolve the extinction was -150 dBm
(10−18 W), which gives a limit to the observable Γ0 due to our coherent drive of
Γ0/2𝜋 ≈ 150 kHz.
Effect of thermal occupation
To take into account the effect of thermal occupancy, we take the limit where the
saturation is very small (Ωp ≈ 0). On resonance, the transmission amplitude is
expressed as
𝑡 (0) = 1 −

Γ1D (Γ1 + Γ𝜑 )Γ1D
Γ1D
≈1−
𝑛¯ th , (C.13)
[(2𝑛¯ th + 1)Γ1 + 2Γ𝜑 ] (2𝑛¯ th + 1)
2Γ2
Γ22

where we have assumed the thermal occupation is very small, 𝑛¯ th
1. In the limit
where Γ1D is dominating spurious loss and pure dephasing rates (Γ2 ≈ Γ1D /2), this
reduces to
𝑡 (0) ≈ 𝑡 (0)|𝑇=0 + 4𝑛¯ th
(C.14)
and hence the thermal contribution dominates the transmission amplitude unless
𝑛¯ th < Γ0/4Γ1D .
Using this relation, we can estimate the upper bound on the temperature of the
environment based on our measurement of extinction. We have measured the

Transmittance, |t|2

10-1
10-2
10-3
10-4
10-5
-10

Before
After

-5

Detuning (MHz)

Figure C.1: Effect of thermal occupancy on extinction. The transmittance of Q1 is
measured at the flux-insensitive point before and after installation of customized microwave
attenuator. We observe an order-of-magnitude enhancement in extinction after the installation, indicating a better thermalization of input signals to the chip.

|e〉p|E〉

Γ1D,p

|e〉p|D〉

ΩXY

Interaction
Decay
Drive

|g〉p|E〉
Γ1D,p XY 2Γ1D WG

154
|e〉a|2〉 |g〉a|3〉
3g
3κ
|e〉a|1〉 |g〉a|2〉
2g
2κ

|e〉p|G〉 J |g〉p|D〉 |g〉p|B〉
Ω 2Γ1D
Γ1D,p XY
ΩWG

|e〉a|0〉 g |g〉a|1〉

|g〉p|G〉

|g〉a|0〉

Figure C.2: Level structure of the atomic cavity and linear cavity. a, Level structure of
the three-qubit system of probe qubit and atomic cavity. Γ1D,p and 2Γ1D denotes the decay
rates into the waveguide channel, ΩXY is the local drive on the probe qubit, and ΩWG is the
drive from the waveguide. The coupling strength 𝐽 is the same for the first excitation and
second excitation levels, b, Level structure of an atom coupled to a linear cavity. |𝑒ia (|𝑔ia )
denotes the excited state (ground state) of the atom, while |𝑛i is the 𝑛-photon Fock state of
the cavity field. 𝑔 is the coupling, 𝛾 is the decay rate of the atom, and 𝜅 is the photon loss
rate of the cavity.

transmittance of Q1 at its maximum frequency (Figure C.1) before and after installing
a thin-film microwave attenuator, which is customized for proper thermalization of
the input signals sent into the waveguide with the mixing chamber plate of the
dilution refrigerator [157]. The minimum transmittance was measured to be |𝑡| 2 ≈
1.7 × 10−4 (2.1 × 10−5 ) before (after) installation of the attenuator, corresponding to
the upper bound on thermal photon number of 𝑛¯ th . 3.3 × 10−3 (1.1 × 10−3 ). With
the attenuator, this corresponds to temperature of 43 mK, close to the temperature
values reported in Ref. [157].
C.2

Detailed modeling of the atomic cavity

In this section, we analyze the atomic cavity discussed in the Chapter 4 in more
detail, taking into account its higher excitation levels. The atomic cavity is formed
by two identical mirror qubits [frequency 𝜔q , decay rate Γ1D (Γ0) to waveguide
(spurious loss) channel placed at 𝜆/2 distance along the waveguide (Figure 1a).
From the 𝜆/2 spacing, the correlated decay of the two qubits is maximized to −Γ1D ,
while the exchange interaction is zero. This results in formation of dark state |𝐷i
and bright state |𝐵i
|𝐷i =

|𝑒𝑔i + |𝑔𝑒i

|𝐵i =

|𝑒𝑔i − |𝑔𝑒i

(C.15)

155
which are single-excitation states of two qubits with suppressed and enhanced waveguide decay rates Γ1D,D = 0, Γ1D,B = 2Γ1D to the waveguide. Here, g (e) denotes
the ground (excited) state of each qubit. Other than the ground state |𝐺i ≡ |𝑔𝑔i,
there also exists a second excited state |𝐸i ≡ |𝑒𝑒i of two qubits, completing 22 = 4
eigenstates in the Hilbert space of two qubits. We can alternatively define |𝐷i and
|𝐵i in terms of collective annihilation operators
1 (1)
1 (1)
(2)
(2)
𝑆D = √ 𝜎
ˆ− + 𝜎
ˆ − , 𝑆B = √ 𝜎
ˆ− − 𝜎
ˆ−
(C.16)
as |𝐷i = 𝑆ˆD
|𝐺i and |𝐵i = 𝑆ˆB† |𝐺i. Here, 𝜎
ˆ −(𝑖) de-excites the state of 𝑖-th mirror qubit.
Note that the doubly-excited state |𝐸i can be obtained by successive application of
either 𝑆ˆD
or 𝑆ˆB† twice on the ground state |𝐺i.

The interaction of qubits with the field in the waveguide is written in the form of
𝐻ˆ WG ∝ ( 𝑆ˆB + 𝑆ˆB† ), and hence the state transfer via classical drive on the waveguide
can be achieved only between states of non-vanishing transition dipole h 𝑓 | 𝑆ˆB |𝑖i. In
the present case, only |𝐺i ↔ |𝐵i and |𝐵i ↔ |𝐸i transitions are available via the
waveguide with the same transition dipole. This implies that the waveguide decay
rate of |𝐸i is equal to that of |𝐵i, Γ1D,E = 2Γ1D .
To investigate the level structure of the dark state, which is not accessible via the
waveguide channel, we introduce an ancilla probe qubit [frequency 𝜔q , decay rate
Γ1D,p (Γ0p ) to waveguide (loss) channel] at the center of mirror qubits. The probe
qubit is separated by 𝜆/4 from mirror qubits, maximizing the exchange interaction
to Γ1D,p Γ1D /2 with zero correlated decay. This creates an interaction of excited
state of probe qubit to the dark state of mirror qubits |𝑒ip |𝐺i ↔ |𝑔ip |𝐷i, while the
bright state remains decoupled from this dynamics.
The master equation of the three-qubit system reads 𝜌¤̂ = −𝑖[ 𝐻/ℏ,
𝜌]
ˆ + L [ 𝜌],
ˆ where
the Hamiltonian 𝐻ˆ and the Liouvillian L are given by
(p) ˆ
(p) ˆ†
𝐻 = ℏ𝐽 𝜎
ˆ − 𝑆D + 𝜎
ˆ + 𝑆D
(C.17)
L [ 𝜌]
ˆ = (Γ1D,p + Γ0p ) D 𝜎
ˆ −(p) 𝜌ˆ + (2Γ1D + Γ0) D 𝑆ˆB 𝜌ˆ + Γ0 D 𝑆ˆD 𝜌ˆ (C.18)
(p)
Here, 𝜎
ˆ ± are the Pauli operators for the probe qubit, 2𝐽 = 2Γ1D,p Γ1D is the
interaction between probe qubit and dark state, and D [·] is the Lindblad dissipator
defined in Eq. (C.3). The full level structure of the 23 = 8 states of three qubits
and the rates in the system are summarized in Fig. C.2a. Note that the effective
(non-Hermitian) Hamiltonian 𝐻ˆ eff in Eq. (4.2) can be obtained from absorbing part
of the Liouvillian in Eq. (C.18) excluding terms associated with quantum jumps.

156
To reach the dark state of the atomic cavity, we first apply a local gate |𝑔ip |𝐺i →
|𝑒ip |𝐺i on the probe qubit (ΩXY in Fig. C.2a) to prepare the state in the firstexcitation manifold. Then, the Rabi oscillation |𝑒ip |𝐺i ↔ |𝑔ip |𝐷i takes place with
the rate of 𝐽. We can identify 𝑔 = 𝐽, 𝛾 = Γ1D,p + Γ0p , 𝜅 = Γ0 in analogy to cavity
QED (Fig. 4.1a and Fig. C.2b) and calculate cooperativity as
C=

2Γ1D,p Γ1D
(2𝐽) 2
2Γ1D
Γ1,p Γ1,D (Γ1D,p + Γp )Γ
Γ0

when the spurious loss rate Γ0 is small. A high cooperativity can be achieved in
this case due to collective suppression of radiation in atomic cavity and cooperative
enhancement in the interaction, scaling linearly with the Purcell factor 𝑃1D = Γ1D /Γ0.
Thus, we can successfully map the population from the excited state of probe qubit
to dark state of mirror qubits with the interaction time of (2𝐽/𝜋) −1 .
† 2
Going further, we attempt to reach the second-excited state |𝐸i = ( 𝑆ˆD
) |Gi of atomic
cavity. After the state preparation of |𝑔ip |𝐷i mentioned above, we apply another
local gate |𝑔ip |𝐷i → |𝑒ip |𝐷i on the probe qubit and prepare the state in the secondexcitation manifold. In this case, the second excited states |𝑒ip |𝐷i ↔ |𝑔ip |𝐸i have
interaction strength 𝐽, same as the first excitation, while the |𝐸i state becomes highly
radiative to waveguide channel. The cooperativity C is calculated as

2Γ1D,p Γ1D
(2𝐽) 2
< 1,
Γ1,p Γ1,E (Γ1D,p + Γ0p )(2Γ1D + Γ0)

which is always smaller than unity. Therefore, the state |𝑔ip |𝐸i is only virtually
populated and the interaction maps the population in |𝑒ip |𝐷i to |𝑔ip |𝐵i with the
rate of (2𝐽) 2 /(2Γ1D ) = Γ1D,p . This process competes with radiative decay (at a rate
of Γ1D,p ) of probe qubit |𝑒ip |𝐷i → |𝑔ip |𝐷i followed by the Rabi oscillation in the
first-excitation manifold, giving rise to damped Rabi oscillation in Fig. 4.5e.
Effect of phase length mismatch
Deviation of phase length between mirror qubits from 𝜆/2 along the waveguide can
act as a non-ideal contribution in the dynamics of atomic cavity. The waveguide
decay rate of dark state can be written as Γ1D,D = Γ1D (1 − | cos 𝜙|), where 𝜙 = 𝑘 1D 𝑑
is the phase separation between mirror qubits [116]. Here, 𝑘 1D is the wavenumber
and 𝑑 is the distance between mirror qubits.
We consider the case where the phase mismatch Δ𝜙 = 𝜙 − 𝜋 of mirror qubits is
small. The decay rate of the dark state scales as Γ1D,D ≈ Γ1D (Δ𝜙) 2 /2 only adding

157
a small contribution to the decay rate of dark state. Based on the decay rate of dark
states from time-domain measurement in Table C.2, we estimate the upper bound
on the phase mismatch Δ𝜙/𝜋 to be 5% for type I and 3.5% for type II.
Effect of asymmetry in Γ1D
So far we have assumed that the waveguide decay rate Γ1D of mirror qubits are
identical and neglected the asymmetry. If the waveguide decay rates of mirror
qubits are given by Γ1D,1 ≠ Γ1D,2 , the dark state and bright state are redefined as
Γ1D,2 |𝑒𝑔i + Γ1D,1 |𝑔𝑒i
Γ1D,1 |𝑒𝑔i − Γ1D,2 |𝑔𝑒i
|Di =
, |Bi =
(C.19)
Γ1D,1 + Γ1D,2
Γ1D,1 + Γ1D,2
with collectively suppressed and enhanced waveguide decay rates of Γ1D,D = 0,
Γ1D,B = Γ1D,1 + Γ1D,2 , remaining fully dark and fully bright even in the presence of
asymmetry. We also generalize Eq. (C.16) as
Γ1D,2 𝜎
ˆ −(1) + Γ1D,1 𝜎
ˆ −(2)
Γ1D,1 𝜎
ˆ −(1) − Γ1D,2 𝜎
ˆ −(2)
𝑆ˆD =
, 𝑆ˆB =
(C.20)
Γ1D,1 + Γ1D,2
Γ1D,1 + Γ1D,2
With this basis, the Hamiltonian can be written as
(p) ˆ
(p) ˆ
(p) ˆ†
(p) ˆ†
ˆ − 𝑆B + 𝜎
ˆ + 𝑆B ,
ˆ − 𝑆D + 𝜎
ˆ + 𝑆D + ℏ𝐽B 𝜎
𝐻 = ℏ𝐽D 𝜎
where

Γ1D,p Γ1D,1 Γ1D,2
𝐽D = p
Γ1D,1 + Γ1D,2

𝐽B =

(C.21)

Γ1D,p (Γ1D,1 − Γ1D,2 )
2 Γ1D,1 + Γ1D,2

Thus, the probe qubit interacts with both the dark state and bright state with the ratio
of 𝐽D : 𝐽B = 2 Γ1D,1 Γ1D,2 : (Γ1D,1 − Γ1D,2 ), and thus for a small asymmetry in the
waveguide decay rate, the coupling to the dark state dominates the dynamics. In
addition, we note that the bright state superradiantly decays to the waveguide, and it
follows that coupling of probe qubit to the bright state manifest only as contribution
of
Γ1D,1 − Γ1D,2 2
(2𝐽B ) 2
= Γ1D,p
Γ1D,1 + Γ1D,2
Γ1D,1 + Γ1D,2
to the probe qubit decay rate into spurious loss channel. In our experiment, the
|Γ
−Γ1D,2 |
maximum asymmetry 𝑑 = Γ1D,1
in waveguide decay rate between qubits is
1D,1 +Γ1D,2
0.14 (0.03) for type I (type II) from Table C.1, and this affects the decay rate of
probe qubit by at most ∼ 2%.

158
Fitting of Rabi oscillation curves
The Rabi oscillation curves in Fig. 4.4 and Fig. 4.6d are modeled using a numerical
master equation solver [338, 339]. The qubit parameters used for fitting the Rabi
oscillation curves are summarized in Table C.1. For all the qubits, Γ1D was found
from spectroscopy. In addition, we have done a time-domain population decay
measurement on the probe qubit to find the total decay rate of Γ1 /2𝜋 = 1.1946 MHz
(95% confidence interval [1.1644, 1.2263] MHz, measured at 6.55 GHz). Using the
value of Γ1D /2𝜋 = 1.1881 MHz (95% confidence interval [1.1550, 1.2211] MHz,
measured at 6.6 GHz) from spectroscopy, we find the spurious population decay
rate Γloss /2𝜋 = Γ1 /2𝜋 − Γ1D /2𝜋 = 6.5 kHz (with uncertainty of 45.3 kHz) for the
probe qubit. The value of spurious population decay rate is assumed to be identical
for all the qubits in the experiment. Note that the decaying rate of the envelope in
the Rabi oscillation curve is primarily set by the waveguide decay rate of the probe
qubit Γ1D,p , and the large relative uncertainty in Γloss does not substantially affect
the fit curve.
The dephasing rate of the probe qubit is derived from time-domain population decay
and Ramsey sequence measurements Γ𝜑 = Γ2 − Γ1 /2. In the case of the mirror
qubits, the table shows effective single qubit parameters inferred from measurements
of the dark state lifetime. We calculate single mirror qubit dephasing rates that
theoretically yield the corresponding measured collective value. Assuming an
uncorrelated Markovian dephasing for the mirror qubits forming the cavity we find
Γ𝜑,m = Γ𝜑,D (See Sec. C.3). Similarly, the waveguide decay rate of the mirror qubits
is found from the spectroscopy of the bright collective state as Γ1D,m = Γ1D,B /2.
The detuning between probe qubit and the atomic cavity (Δ) is treated as the only
free parameter in our model. The value of Δ sets the visibility and frequency of the
Rabi oscillation, and is found from the the fitting algorithm.
C.3

Lifetime (𝑇1 ) and coherence time (𝑇2∗ ) of dark state

The dark state of mirror qubits belongs to the decoherence-free subspace in the
system due to its collectively suppressed radiation to the waveguide channel. However, there exists non-ideal channels that each qubit is coupled to, and such channels
contribute to the finite lifetime (𝑇1 ) and coherence time (𝑇2∗ ) of the dark state (See
Table C.2). In the experiment, we have measured the decoherence rate Γ2,D of the
dark state to be always larger than the decay rate Γ1,D , which cannot be explained by
simple Markovian model of two qubits subject to their own independent noise. We
discuss possible scenarios that can give rise to this situation of Γ2,D > Γ1,D , with

159
Type
II
Dark
compound
Bright
compound

Γ1D,p /2𝜋 Γ1D,m /2𝜋 Γ𝜑,p /2𝜋
(MHz)
(MHz)
(kHz)
1.19
13.4
191
0.87
96.7
332

Qubits
involved
Q2 , Q6
Q 1 , Q7

Γ𝜑,m /2𝜋
(kHz)
210
581

Δ/2𝜋
(MHz)
1.0
5.9

Q2 Q3 , Q5 Q6

1.19

4.3

191

146

0.9

Q2 Q3 , Q5 Q6

1.19

20.2

191

253

1.4

Table C.1: Parameters used for fitting Rabi oscillation curves. The first and second row
are the data for 2-qubit dark states, the third and fourth row are the data for 4-qubit dark
states made of compound mirrors. Here, Γ1D,p (Γ1D,m ) is the waveguide decay rate and Γ 𝜑,p
(Γ 𝜑,m ) is the pure dephasing rate of probe (mirror) qubit, Δ is the detuning between probe
qubit and mirror qubits used for fitting the data.

distinction of the Markovian and non-Markovian noise contributions.
There are two major channels that can affect the coherence of the dark state. First,
coupling of a qubit to dissipative channels other than the waveguide can give rise to
additional decay rate Γloss = Γ1 − Γ1D (so-called non-radiative decay rate). This type
of decoherence is uncorrelated between qubits and is well understood in terms of the
Lindblad form of master equation, whose contribution to lifetime and coherence time
of dark state is similar as in individual qubit case. Another type of contribution that
severely affects the dark state coherence arises from fluctuations in qubit frequency,
which manifest as pure dephasing rate Γ𝜑 in the individual qubit case. This can
affect the decoherence of the dark state in two ways: (i) By accumulating a relative
phase between different qubit states, this act as a channel to map the dark state into
the bright state with short lifetime, and hence contributes to loss of population in
Type
II
Dark
compound
Bright
compound

Qubits
involved
Q2 , Q6
Q1 , Q7

Γ1,D /2𝜋
(kHz)
210
581

Γ2,D /2𝜋
(kHz)
366
838

Q2 Q3 , Q5 Q6

146

215

Q2 Q3 , Q5 Q6

253

376

Table C.2: Decay rate and decoherence rate of dark states. The first and second row are
the data for 2-qubit dark states, the third and fourth row are the data for 4-qubit dark states
made of compound mirrors. Here, Γ1,D (Γ2,D ) is the decay (decoherence) rate of the dark
state.

160
the dark state; (ii) fluctuations in qubit frequency also induces the frequency jitter
of the dark state and therefore contributes to the dephasing of dark state.
In the following, we model the aforementioned contributions to the decoherence of
dark state. Let us consider two qubits separated by 𝜆/2 along the waveguide on
resonance, in the presence of fluctuations Δ̃ 𝑗 (𝑡) in the qubit frequency. The master
equation can be written as 𝜌¤̂ = −𝑖[ 𝐻/ℏ,
𝜌]
ˆ + L [ 𝜌],
ˆ where the Hamiltonian 𝐻ˆ and
the Liouvillian L are given by
( 𝑗) ( 𝑗)
𝐻ˆ (𝑡) = ℏ
Δ̃ 𝑗 (𝑡) 𝜎
ˆ+ 𝜎
ˆ− ,
(C.22)
𝑗=1,2

L [𝜌] =

Õ
𝑗,𝑘=1,2

(−1)

𝑗−𝑘

Γ1D + 𝛿 𝑗 𝑘 Γloss

1 (𝑘) ( 𝑗)
ˆ+ 𝜎
ˆ − , 𝜌}
ˆ −( 𝑗) 𝜌ˆ 𝜎
ˆ +(𝑘) − { 𝜎

(C.23)

Here, Γ1D (Γloss ) is the decay rate of qubits into waveguide (spurious loss) channel.
Note that we have assumed the magnitude of fluctuation Δ̃ 𝑗 (𝑡) in qubit frequency
is small and neglected its effect on exchange interaction and correlated decay. We
investigate two scenarios in the following subsections depending on the correlation
of noise that gives rise to qubit frequency fluctuations.
Markovian noise
If the frequency fluctuations of the individual qubits satisfy the conditions for Born
and Markov approximations, i.e. the noise is weakly coupled to the qubit and has
short correlation time, the frequency jitter can be described in terms of the standard
Lindblad form of dephasing [336].
More generally, we also consider the correlation between frequency jitter of different
qubits. Such contribution can arise when different qubits are coupled to a single
fluctuating noise source. For instance, if two qubits in a system couple to a magnetic
˜ that is global to the chip with 𝐷 𝑗 ≡ 𝜕 Δ̃ 𝑗 /𝜕 𝐵,
˜ the correlation between
field 𝐵0 + 𝐵(𝑡)
detuning of different qubits follows correlation of the fluctuations in magnetic field,
˜ 𝐵(𝑡
˜ + 𝜏)i ≠ 0. The Liouvillian associated
giving hΔ̃1 (𝑡) Δ̃2 (𝑡 + 𝜏)i = 𝐷 1 𝐷 2 h 𝐵(𝑡)
with dephasing can be written as [340]
Γ𝜑, 𝑗 𝑘
1 n (𝑘) ( 𝑗) o
( 𝑗)
(𝑘)
L 𝜑, 𝑗 𝑘 [ 𝜌]
ˆ =
ˆ 𝑧 𝜌ˆ 𝜎
ˆ𝑧 −
ˆ 𝜎
ˆ , 𝜌ˆ ,
(C.24)
2 𝑧 𝑧
where the dephasing rate Γ𝜑, 𝑗 𝑘 between qubit 𝑗 and qubit 𝑘 ( 𝑗 = 𝑘 denotes individual
qubit dephasing and 𝑗 ≠ 𝑘 is the correlated dephasing) is given by
1 +∞
Γ𝜑, 𝑗 𝑘 ≡
d𝜏 hΔ̃ 𝑗 (0) Δ̃ 𝑘 (𝜏)i.
(C.25)
2 −∞

161
Here, the average h·i is taken over an ensemble of fluctuators in the environment.
Note that the correlated dephasing rate Γ𝜑, 𝑗 𝑘 can be either positive or negative
depending on the sign of noise correlation, while the individual pure dephasing rate
Γ𝜑, 𝑗 𝑗 is always positive.
After we incorporate the frequency jitter as the dephasing contributions to the
Liouvillian, the master equation takes the form
( 𝑗) (𝑘) 1 (𝑘) ( 𝑗)
𝑗−𝑘
𝜌¤̂ =
ˆ − 𝜌ˆ 𝜎
ˆ + − {𝜎
(−1) Γ1D + 𝛿 𝑗 𝑘 Γloss 𝜎
ˆ 𝜎
ˆ , 𝜌}
2 + −
𝑗,𝑘=1,2
o
Γ𝜑, 𝑗 𝑘
( 𝑗)
( 𝑗)
ˆ 𝑧 𝜌ˆ 𝜎
ˆ 𝑧(𝑘) −
ˆ (𝑘) 𝜎
ˆ , 𝜌ˆ
(C.26)
2 𝑧 𝑧
We diagonalize the correlated decay part of the Liouvillian describe the two-qubit
system in terms of bright and dark states defined in Eq. (C.15). From now on,
we assume the pure dephasing rate and the correlated dephasing rate are identical
for the two qubits, and write Γ𝜑 ≡ Γ𝜑,11 = Γ𝜑,22 , Γ𝜑,c ≡ Γ𝜑,12 = Γ𝜑,21 . For
qubits with a large Purcell factor (Γ1D
Γ𝜑 , |Γ𝜑,c |, Γloss ), we can assume that the
superradiant states |𝐵i and |𝐸i are only virtually populated [123] and neglect the
density matrix elements associated with |𝐵i and |𝐸i. Rewriting Eq. (C.26) in the
basis of {|𝐺i, |𝐵i, |𝐷i, |𝐸i}, the dynamics related to dark state can be expressed as
𝜌¤ D,D ≈ −Γ1,D 𝜌D,D and 𝜌¤ D,G ≈ −Γ2,D 𝜌D,G , where
Γloss
+ Γ𝜑 .
(C.27)
Note that if the correlated dephasing rate Γ𝜑,c is zero, Γ1,D is always larger than Γ2,D ,
which is in contradiction to our measurement result.
Γ1,D = Γloss + Γ𝜑 − Γ𝜑,c ,

Γ2,D =

We estimate the decay rate into non-ideal channels to be Γloss /2𝜋 = 6.5 kHz from
the difference in Γ1 and Γ1D of the probe qubit, and assume Γloss to be similar for
all the qubits. Applying Eq. (C.27) to measured values of Γ2,D listed in Table C.2,
we expect that the pure dephasing of the individual qubit is the dominant decay and
decoherence source for the dark state. In addition, we compare the decay rate Γ1,D
and decoherence rate Γ2,D of dark states in the Markovian noise model and infer that
the correlated dephasing rate Γ𝜑,c is positive and is around a third of the individual
dephasing rate Γ𝜑 for all types of mirror qubits.
Non-Markovian noise
In a realistic experimental setup, there also exists non-Markovian noise sources
contributing to the dephasing of the qubits, e.g. 1/ 𝑓 -noise or quasi-static noise

162
[190, 341, 342]. In such cases, the frequency jitter cannot be simply put into the
Lindblad form as described above. In this subsection, we consider how the individual
qubit dephasing induced by non-Markovian noise influences the decoherence of dark
state. As shown below, we find that a non-Markovian noise source can lead to a
shorter coherence time to lifetime ratio for the dark states, in a similar fashion to
correlated dephasing. However, we find that the functional form of the visibility of
Ramsey fringes is not necessarily an exponential for a non-Markovian noise source.
We start from the master equation introduced in Eqs. (C.22)-(C.23) can be written
in terms of the basis of {|𝐺i, |𝐵i, |𝐷i, |𝐸i},
𝑖 ˆ
𝜌¤̂ = − [ 𝐻,
𝜌]
ˆ + (2Γ1D + Γloss ) D [ 𝑆ˆB ] 𝜌ˆ + Γloss D [ 𝑆ˆD ] 𝜌,

(C.28)

where the Hamiltonian is written using the common frequency jitter Δ̃𝑐 (𝑡) ≡ [Δ̃1 (𝑡)+
Δ̃2 (𝑡)]/2 and differential frequency jitter Δ̃𝑑 (𝑡) ≡ [Δ̃1 (𝑡) − Δ̃2 (𝑡)]/2
𝐻ˆ (𝑡)/ℏ = Δ̃𝑐 (𝑡) (2|𝐸ih𝐸 | + |𝐷ih𝐷 | + |𝐵ih𝐵|) + Δ̃𝑑 (𝑡) (|𝐵ih𝐷| + |𝐷ih𝐵|) . (C.29)
Here, 𝑆ˆB and 𝑆ˆD are defined in Eq. (C.16). From the Hamiltonian in Eq. (C.29),
we see that the common part of frequency fluctuation Δ̃𝑐 (𝑡) results in the frequency
jitter of the dark state while the differential part of frequency fluctuation Δ̃𝑑 (𝑡) drives
the transition between |𝐷i and |𝐵i, which acts as a decay channel for the dark state.
For uncorrelated low-frequency noise on the two qubits, the decoherence rate is
approximately the standard deviation of the common frequency jitter hΔ̃𝑐 (𝑡) 2 i.
The decay rate in this model can be found by modeling the bright state as a cavity in
the Purcell regime, and calculate the damping rate of the dark state using the Purcell
factor as h4Δ̃𝑑 (𝑡) 2 /ΓB i. As evident, in this model the dark state’s population decay
rate is strongly suppressed by the large damping rate of bright state ΓB , while the
dark state’s coherence time can be sharply reduced due to dephasing.
C.4

Shelving

We consider the case of two identical mirror qubits of frequency 𝜔q , separated by
distance 𝜆/2 along the waveguide. In addition to free evolution of qubits, we include
a coherent probe signal from the waveguide in the analysis. In the absence of pure
dephasing (Γ𝜑 = 0) and thermal occupancy (𝑛¯ th = 0), the master equation in the
rotating frame of the probe signal takes the same form as Eq. (C.28), where the
Hamiltonian containing the drive from the probe signal is written as

Õ
Ω𝜇
† ˆ
𝐻/ℏ =
−𝛿𝜔 𝑆 𝜇 𝑆 𝜇 +
𝑆𝜇 + 𝑆𝜇 ,
(C.30)
𝜇=B,D

163
where 𝑆ˆB and 𝑆ˆD are defined in Eq. (C.16), 𝛿𝜔 = 𝜔p − 𝜔q is the detuning of the
probe signal from the mirror qubit frequency, Ω 𝜇 is the corresponding driven Rabi
frequency. Note that due to the symmetry of the excitations with respect to the
waveguide, we see that ΩD = 0 and ΩB = 2Ω1 , where Ω1 is the Rabi frequency of
one of the mirror qubits from the probe signal along the waveguide.
Let us consider the limit where the Purcell factor 𝑃1D = Γ1D /Γ0 of qubits is much
larger than unity (equivalent to ΓD = Γ0
ΓB = 2Γ1D + Γ0) and the drive applied to
the qubits is weak ΩB
ΓB . Then, we can effectively remove some of the density
matrix elements1,
𝜌E,E , 𝜌B,E , 𝜌E,B , 𝜌G,E , 𝜌E,G ≈ 0
and restrict the analysis to ones involved with three levels {|𝐺i, |𝐷i, |𝐵i}. In
addition, the dark state |𝐷i is effectively decoupled from |𝐺i and |𝐵i, acting as
a metastable state. Therefore, we only consider the following set of the master
equation:
𝑖ΩB
(𝜌B,G − 𝜌G,B )
𝜌¤ B,B ≈ −ΓB 𝜌B,B +
2
ΓB
𝑖ΩB
𝜌¤ B,G ≈ 𝑖𝛿𝜔 −
(𝜌B,B − 𝜌G,G )
𝜌B,G +
𝜌¤ G,G ≈ − 𝜌¤ B,B ;

𝜌¤ G,B = 𝜌¤ B,G

(C.31)
(C.32)
(C.33)

1 From the master equation, the time-evolution of part of the density matrix elements are approx-

imately written as
𝑖ΩB
𝜌¤ E,E = −(ΓB + ΓD ) 𝜌E,E +
(𝜌B,E − 𝜌E,B ),
2
ΓD
𝑖ΩB
𝜌¤ E,B = 𝑖𝛿𝜔 − ΓB +
𝜌E,B +
(𝜌B,B − 𝜌E,E + 𝜌E,G ),
ΓB + ΓD
𝑖ΩB
𝜌¤ E,G = 2𝑖𝛿𝜔 −
(𝜌B,G + 𝜌E,B ),
𝜌E,G +
𝜌¤ E,B = 𝜌¤ B,E

𝜌¤ E,G = 𝜌¤ G,E

In the steady state, it can be shown that
𝜌E,E ∼ O (𝑥 2 ) 𝜌B,B + O (𝑥 3 ) (𝜌B,G − 𝜌G,B )
𝜌B,E ∼ O (𝑥) 𝜌B,B + O (𝑥 2 ) 𝜌G,B
𝜌G,E ∼ O (𝑥 2 ) 𝜌B,B + O (𝑥) 𝜌G,B

to leading order in 𝑥 ≡ ΩB /ΓB < 1, and hence we can neglect the contributions from 𝜌E,E , 𝜌B,E ,
𝜌E,B , 𝜌G,E , 𝜌E,G from the analysis in the weak driving limit. The probe power we have used in the
experiment corresponds to 𝑥 ∼ 0.15, which makes this approximation valid.

164
Using the normalization of total population 𝜌G,G + 𝜌D,D + 𝜌B,B ≈ 1 with Eqs. (C.31)(C.33), we obtain the approximate steady-state solution
h𝑆ˆB i ≈ 𝜌B,G ≈ −

𝑖ΩB (1 − 𝜌D,D )
ΓB − 2𝑖𝛿𝜔

The input-output relation [116] is given as
Γ1D (1)
Γ1D (2)
𝑎ˆ out = 𝑎ˆ in +
ˆ− −
ˆ − = 𝑎ˆ in + Γ1D 𝑆ˆB ,

(C.34)

(C.35)

where 𝑎ˆ in is the input field operator and 𝑎ˆ out is the operator for output field propagating in the same direction as the input field (here, the input field is assumed to be
incident from only one direction). The transmission amplitude is calculated as
(1 − 𝜌D,D )Γ1D
h𝑎ˆ out i
=1−
h𝑎ˆ in i
−𝑖𝛿𝜔 + ΓB /2
where the relation Ω1 /2 = −𝑖h𝑎ˆ in i Γ1D /2 has been used.
𝑡=

(C.36)

In the measurement, we use the state transfer protocol to transfer part of the ground
state population into the dark state. Following this, we drive the |𝐺i ↔ |𝐵i transition
by sending a weak coherent pulse with a duration 260 ns into the waveguide,
and recording the transmission spectrum. As a comparison, we also measure
the transmission spectrum when the mirror qubits are in the ground state, which
corresponds to having 𝜌D,D = 0. The transmittance in the two cases (Figure 3d)
are fitted with identical parameters for Γ1D and ΓB . The dark state population 𝜌D,D
following the iSWAP gate is extracted from the data as 0.58, which is lower than the
value (0.68) found from the Rabi oscillation peaks (Fig. 4.4). The lower value of the
dark state population can be understood considering the finite lifetime of dark state
(757ns), which leads to a partial population decay during the measurement time (the
single-shot measurement time is set by the duration of the input pulse). It should be
noted that the input pulse has a transform-limited bandwidth of ∼ 3.8 MHz, which
results in frequency averaging of the spectral response over this bandwidth. For this
reason, the on-resonance transmission extinction measured in the pulsed scheme is
lower than the value found from continuous wave (CW) measurement (Fig. 4.2).

165
Appendix D

SUPPLEMENTARY INFORMATION FOR CHAPTER 5
D.1

Band structure analysis

Quantization of a periodic resonator-loaded waveguide
We consider the case of a waveguide that is periodically loaded with microwave
resonators. Figure D.1 depicts a unit cell for this configuration. The Lagrangian for
this system can be written as [226]
2
Õ 1
(Φa𝑛 − Φa𝑛−1 ) 2 1
(Φb𝑛 )
a 2
b 2
b 2
𝐶0 ( Φ𝑛 ) −
𝐿=
+ 𝐶r ( Φ𝑛 ) + 𝐶g ( Φ𝑛 − Φ𝑛 ) −
2𝐿 0
2𝐿 r
(D.1)
In order to find solutions in form of traveling waves, it is easier to work with the
Fourier transform of node fluxes. We use the following convention for defining the
(discrete) Fourier transformation

1 Õ −𝑖2𝜋(𝜅/𝑀)𝑛 a,b
Φa,b
Φ𝑛 ,
𝑀 𝑛=−𝑁
Φan−1
···

(D.2)

Φan+1

Φan
L0

Φbn
Cr

C0
Lr

Φbn+1
Cr

C0
Lr

···

L0
C0

···

L0
Lg

···

Figure D.1: Circuit diagram of metamaterial waveguide. The waveguide can be made
from periodic arrays of transmission line sections loaded with capacitively coupled resonators (top), or inductively loaded resonators (bottom).

166
where 𝑀 = 2𝑁 + 1 is the total number of periods in the waveguide. Using the
Fourier relation we find the Lagrangian in 𝑘-space as
a 2
Õ 1
¤ a𝜅 | 2 − 1 − 𝑒 −𝑖2𝜋(𝜅/𝑀) |Φ𝜅 |
𝐿=
(𝐶0 + 𝐶g )| Φ
2𝐿 0
b |2
¤ b𝜅 Φ
¤ a−𝜅 + Φ
¤ b−𝜅 Φ
¤ a𝜅
|Φ
¤ 𝜅| −
+ (𝐶g + 𝐶r )| Φ
− 𝐶g
2𝐿 r
To proceed further, we need to find the canonical node charges which are defined
𝜕𝐿
as 𝑄 a,b
, and subsequently derive the Hamiltonian of the system by using a
𝜅 = 𝜕Φ
¤ a,b
Legendre transformation. Doing so we find
Õ 𝑄a 𝑄a
2 a a
𝑄 b𝜅 𝑄 b−𝜅 Φb𝜅 Φb−𝜅 𝑄 a𝜅 𝑄 b−𝜅 + 𝑄 a−𝜅 𝑄 b𝜅
𝜅 −𝜅
−𝑖2𝜋(𝜅/𝑀) Φ𝜅 Φ−𝜅
𝐻=
0 + 1−𝑒
0 + 2𝐿
2𝐶
2𝐿
2𝐶
2𝐶
Here, we have defined the following quantities
𝐶00 =

𝐶g𝐶r + 𝐶g𝐶0 + 𝐶0𝐶r 0 𝐶g𝐶r + 𝐶g𝐶0 + 𝐶0𝐶r 0 𝐶g𝐶r + 𝐶g𝐶0 + 𝐶0𝐶r
, 𝐶r =
, 𝐶g =
𝐶g + 𝐶r
𝐶g + 𝐶0
𝐶g

The canonical commutation relation [Φ𝑖𝜅 , 𝑄 −𝜅 0 ] = 𝑖ℏ𝛿𝑖, 𝑗 𝛿 𝜅,𝜅 0 allows us to define the
following annihilation operators as a function of charge and flux operators
0𝜔
𝐶00 Ω 𝑘
𝑎ˆ 𝜅 =
Φ𝜅 + 0 𝑄 𝜅 , 𝑏ˆ 𝜅 =
Φ𝜅 + 0 𝑄 𝜅 .
(D.3)
2ℏ
𝐶0 Ω 𝑘
2ℏ
𝐶r 𝜔0
Here, we have defined the resonance frequency for each mode as
4sin2 (𝑘 𝑑/2)
, 𝜔0 = p
Ω𝑘 =
𝐿 0𝐶0
𝐿 r𝐶r0

(D.4)

where 𝑘 = 2𝜋𝜅/(𝑀 𝑑) is the wavenumber. It is evident that Ω 𝑘 has the expected
dispersion relation of a discrete periodic transmission line and 𝜔0 is the resonance
frequency of the loaded microwave resonators. Using the above definitions for 𝑎ˆ 𝜅 , 𝑏ˆ 𝜅
Õ
𝐻ˆ =
Ω 𝑘 𝑎ˆ †𝑘 𝑎ˆ 𝑘 + 𝑎ˆ −𝑘 𝑎ˆ †−𝑘 + 𝜔0 𝑏ˆ †𝑘 𝑏ˆ 𝑘 + 𝑏ˆ −𝑘 𝑏ˆ †−𝑘
2 𝑘

− 𝑔 𝑘 𝑏ˆ −𝑘 − 𝑏ˆ 𝑘 𝑎ˆ 𝑘 − 𝑎ˆ −𝑘 − 𝑔 𝑘 𝑎ˆ 𝑘 − 𝑎ˆ −𝑘 𝑏ˆ −𝑘 − 𝑏ˆ 𝑘 ,
(D.5)
along with the coupling coefficient
p 0 0
𝐶0𝐶r p
𝐶g 𝜔0 Ω 𝑘
𝜔0 Ω 𝑘 = p
𝑔𝑘 =
2𝐶g0
2 (𝐶0 + 𝐶g )(𝐶r + 𝐶g )

(D.6)

167
An alternative structure for coupling microwave resonators is depicted in the bottom
panel of Fig. D.1. In this geometry, the coupling is controlled by the inductive
element 𝐿 g . Repeating the analysis above for this case, we find
p 0 0
𝐿0 𝐿r p
4sin2 (𝑘 𝑑/2)
Ω𝑘 =
𝜔0 Ω 𝑘 .
(D.7)
𝐶0 𝐿 00
2𝐿 0g
𝐶r 𝐿 0r
We have defined the modified inductance values as
𝐿g 𝐿r + 𝐿g 𝐿0 + 𝐿0 𝐿r 0 𝐿g 𝐿r + 𝐿g 𝐿0 + 𝐿0 𝐿r 0 𝐿g 𝐿r + 𝐿g 𝐿0 + 𝐿0 𝐿r
𝐿 00 =
, 𝐿r =
, 𝐿g =
𝐿g + 𝐿r
𝐿g + 𝐿0
𝐿g
Band structure calculation with RWA
Using the rotating wave approximation, the Hamiltonian in Eq. (D.5) can be simplified to

Õ
†ˆ
†ˆ
(D.8)
Ω 𝑘 𝑎ˆ 𝑘 𝑎ˆ 𝑘 + 𝜔0 𝑏 𝑘 𝑏 𝑘 + 𝑔 𝑘 𝑏 𝑘 𝑎ˆ 𝑘 + 𝑎ˆ 𝑘 𝑏 𝑘 .
𝐻=ℏ

Note that this approximation is applicable only when the coupling is sufficiently
weak, 𝑔 𝑘
min(𝜔0 , Ω 𝑘 ), and the detuning is sufficiently small |𝜔0 − Ω 𝑘 |

(𝜔0 + Ω 𝑘 ). Assuming Ω 𝑘 and 𝜔0 are of the same order, this condition is satisfied
when 𝐶g
2 (𝐶0𝐶r ).
The simplified Hamiltonian can be written in the compact form
𝐻=ℏ
x†𝑘 H 𝑘 x 𝑘 ,

(D.9)

where

Ω𝑘 𝑔 𝑘
H𝑘 =
𝑔 𝑘 𝜔0

x𝑘 =

𝑎ˆ 𝑘
𝑏ˆ 𝑘

(D.10)

We desire to transform the Hamiltonian to a diagonalized form
𝜔+,𝑘
H̃ 𝑘 =
0 𝜔−,𝑘

(D.11)

It is straightforward to use the eigenvalue decomposition to find 𝜔±,𝑘 as
(Ω 𝑘 + 𝜔0 ) ± (Ω 𝑘 − 𝜔0 ) + 4𝑔 𝑘 ,
𝜔±,𝑘 =

(D.12)

along with the corresponding eigenstates |±, 𝑘i = 𝛼ˆ ±,𝑘 |0i, where
𝛼ˆ ±,𝑘 = q

(𝜔±,𝑘 − 𝜔0 )
(𝜔±,𝑘 − 𝜔0 )

+ 𝑔 2𝑘

𝑎ˆ 𝑘 + q

𝑔𝑘
(𝜔±,𝑘 − 𝜔0 )

𝑏ˆ 𝑘 .

+ 𝑔 2𝑘

(D.13)

168
Band structure calculation beyond RWA
The exact Hamiltonian in Eq. (D.5) can be written in the compact form
𝐻ˆ =

x†𝑘 H 𝑘 x 𝑘 ,

(D.14)

where
Ω
𝑔 𝑘 −𝑔 𝑘 
 𝑘
 0
Ω 𝑘 −𝑔 𝑘 𝑔 𝑘 
H𝑘 = 
0 
 𝑔 𝑘 −𝑔 𝑘 𝜔0
−𝑔 𝑘 𝑔 𝑘
𝜔0 

 𝑎ˆ 
𝑘 
 † 
 𝑎ˆ 
x 𝑘 =  −𝑘  .
 𝑏ˆ 𝑘 
 † 
 𝑏ˆ 
 −𝑘 

(D.15)

To find the eigenstates of the system, we can use a linear transform to map the
state vector x̃ 𝑘 = S 𝑘 x 𝑘 such that x†𝑘 H 𝑘 x 𝑘 = x̃†𝑘 H̃ 𝑘 x̃ 𝑘 with the transformed diagonal
Hamiltonian matrix
𝜔
0 
 +,𝑘
 0 𝜔+,𝑘
H̃ 𝑘 = 
−,𝑘
 0
−,𝑘

(D.16)

In order to preserve the canonical commutation relations, the matrix S 𝑘 has to
be symplectic, i.e. J = S 𝑘 JS†𝑘 , with the matrix J = diag(1, −1, 1, −1). A linear
transformation (such as S 𝑘 ) that diagonalizes a set of quadratically coupled boson
fields while preserving their canonical commutation relations is often referred to
as a Bogoliubov-Valatin transformation. While it is generally difficult to find the
transform matrix S 𝑘 , it is easy to find the eigenvalues of the diagonalized Hamiltonian
by exploiting some of the properties of S 𝑘 . Note that since J = S 𝑘 JS†𝑘 , the matrices
JH̃ 𝑘 and JH 𝑘 share the same set of eigenvalues. The eigenvalues of JH̃ 𝑘 are the two
frequencies 𝜔±,𝑘 , and thus we have
2
Ω2𝑘 + 𝜔20 ±
Ω2𝑘 − 𝜔20 + 16𝜔0 Ω 𝑘 𝑔 2𝑘 .
(D.17)
𝜔2±,𝑘 =
Circuit theory derivation of the band structure
Consider the pair of equations that describe the propagation of a monochromatic
electromagnetic wave of the form 𝑣(𝑥, 𝑡) = 𝑉 (𝑥)𝑒 −𝑖𝑘𝑥 𝑒𝑖𝜔𝑡 (along with the corresponding current relation) inside a transmission line
𝑉 (𝑥) = −𝑍 (𝜔)𝐼 (𝑥),
d𝑥

𝐼 (𝑥) = −𝑌 (𝜔)𝑉 (𝑥).
d𝑥

(D.18)

169
Here, 𝑍 (𝜔) and 𝑌 (𝜔) are frequency dependent impedance and admittance functions
that model the linear response of the series and parallel portions of a transmission
line with length 𝑑. It is straightforward to check that the solutions to these equation
satisfy 𝑘 (𝜔) = 𝑛𝜔/𝑐 = −𝑍 (𝜔)𝑌 (𝜔)/𝑑. For a loss-less waveguide and in the
absence of dispersion we have 𝑍 (𝜔) = 𝑖𝜔𝐿 0 and 𝑌 (𝜔) = 𝑖𝜔𝐶0 , and thus we find the
familiar dispersion relation 𝑘 (𝜔) = 𝜔 𝐿 0𝐶0 /𝑑. Nevertheless, the pair of equations
above remain valid for arbitrary impedance and admittance functions 𝑍 (𝜔) and
𝑌 (𝜔), provided that the dimension of the model circuit remains much smaller than
the wavelength under consideration. In this model, a real and negative quantity
for the product 𝑍𝑌 results in an imaginary wavenumber and subsequently creates a
stop band in the dispersion relation. This situation can be achieved by periodically
loading a transmission line with an array of resonators [343, 344]. Assuming a unit
length of 𝑑 we find
𝜔 2
2𝑐𝛾
𝑛2 1 +
(D.19)
𝑘2 =
𝑛𝑑 𝜔20 − 𝜔2
Here, 𝜔0 is the resonance frequency, and 𝛾e is the external coupling decay rate of an
individual resonator in the array. For moderate values of gap-midgap ratio (Δ/𝜔𝑚 ),
the frequency gap can be found as

𝑐 𝛾e
Δ=
(D.20)
𝑛𝑑 𝜔0
and 𝜔𝑚 = 𝜔0 + Δ/2. We have defined the gap as the range of frequencies where the
wavenumber is imaginary.
Although a microwave resonator can be realized by using a two-element LC-circuit,
the three-element circuits in Fig. D.1 provide an additional degree of freedom which
enables setting the coupling 𝛾e independent of the resonance frequency 𝜔0 . Using
circuit theory, it is straightforward to show
2
𝐶g
𝑍0
𝜔0 = p
, 𝛾e =
(D.21)
2𝐿 r 𝐶r + 𝐶g
𝐿 r (𝐶r + 𝐶g )
Here, 𝑍0 is the characteristic impedance of the unloaded waveguide. It is easy
to check that for small values of 𝐶g /𝐶r , the resonance frequency is only a weak
function of 𝐶g . As a result, it is possible to adjust the coupling rate 𝛾e by setting the
capacitor 𝐶g while keeping the resonance frequency almost constant. Fig. D.1 also
depicts an alternative strategy for coupling microwave resonators to the waveguide.
In this design, the inductive element 𝐿 g is used to set the coupling in a “current

170
divider" geometry. We provide experimental results for implementation of bandgap
waveguide based on both designs in the next section.
While the “continuum" model described above provides a heuristic explanation for
formation of bandgap in a waveguide loaded with resonators, its results remains valid
as far as 𝑘
2𝜋/𝑑. To avoid this approximation, we can use the transfer matrix
method to find the exact dispersion relation for a system with discrete periodic
symmetry [180]. In this case, Equation (D.19) is modified to
𝜔 2 𝑛2 𝑑 2 𝑛𝑑𝛾
𝜔2
cos (𝑘 𝑑) = 1 −
(D.22)
𝑐 𝜔20 − 𝜔2
Note that this relation still requires 𝑑 to be much smaller than the wavelength of the
unloaded waveguide 𝜆 = 2𝜋𝑐/(𝑛𝜔).
Dispersion and group index near the band-edges
Equation (D.17) can be reversed to find the wavenumber 𝑘 as a function of frequency.
Assuming, a linear dispersion relation of the form 𝑘 = 𝑛Ω 𝑘 /𝑐 for the bare waveguide
we find
𝑛𝜔 𝜔2 − 𝜔2𝑐+
(D.23)
𝑘=
𝜔2 − 𝜔2𝑐−
Here, 𝜔𝑐+ = 𝜔0 and 𝜔𝑐− = 𝜔0 1 − 4𝑔 2𝑘 /(Ω 𝑘 𝜔0 ) are the upper and lower cutoff frequencies, respectively. The quantity 𝑔 2𝑘 /(Ω 𝑘 𝜔0 ) is a unit-less parameter
quantifying the size of the bandgap and is independent of the wavenumber 𝑘.
The dispersion relation can be written in simpler forms by expanding the wavenumber in the vicinity of the two band-edges
𝑛𝜔 𝑐−
 𝑐
for 𝜔 ≈ 𝜔𝑐− ,
q −𝛿−
𝑘=
(D.24)
 𝑛𝜔𝑐𝑐+ 𝛿Δ+
for 𝜔 ≈ 𝜔𝑐+ .
Here, Δ = 𝜔𝑐+ − 𝜔𝑐− is the frequency span of the bandgap and 𝛿± = 𝜔 − 𝜔𝑐± are
the detunings from the band-edges.
The form of the dispersion relation Eq. (D.17) suggests that the maxima of the group
index happens near the band-edges. Having the wavenumber, we can evaluate the
group velocity 𝑣 𝑔 = 𝜕𝜔/𝜕𝑘 and find the group index 𝑛𝑔 = 𝑐/𝑣 𝑔 as
𝑛𝑔 =

𝑛𝜔 𝑐− Δ
 −4(𝛿 −𝑖𝛾 ) 3

for 𝜔 ≈ 𝜔𝑐− ,
(D.25)

𝑛𝜔 𝑐+
 √4Δ(𝛿 −𝑖𝛾 )

for 𝜔 ≈ 𝜔𝑐+ .

171
Note that we have replaced 𝛿± with 𝛿± − 𝑖𝛾𝑖 to account for finite internal quality
factor of the resonators in the structure.
Coupling a Josephson junction qubit to a metamaterial waveguide
We consider the coupling of a Josephson junction qubit to the metamaterial waveguide. Assuming rotating wave approximation (valid for weak coupling 𝑓 𝑘

𝜔 𝑘 , 𝜔q ), the Hamiltonian of this system can be written as

Õ
𝜔q
ˆ + 𝑎ˆ 𝑘 𝜎
ˆ .
(D.26)
𝐻ˆ = ℏ
𝜔 𝑘 𝑎ˆ 𝑘 𝑎ˆ 𝑘 +
ˆ 𝑧 + 𝑓 𝑘 𝑎ˆ 𝑘 𝜎
Here 𝑓 𝑘 is the coupling factor of the qubit to the waveguide photons, and 𝜔 𝑘 = 𝜔±,𝑘 ,
where the plus or minus sign is chosen such that the qubit frequency 𝜔q lies within
the band. Without loss of generality, we assume 𝑓 𝑘 to be a real number. The
Heisenberg equations of motions for the qubit and the photon operators can be
written as
𝑎ˆ 𝑘 = −𝑖𝜔 𝑘 𝑎ˆ 𝑘 − 𝑖 𝑓 𝑘 𝜎
ˆ−
𝜕𝑡
𝜕 −
ˆ = −𝑖𝜔q 𝜎
ˆ− −𝑖
𝑓 𝑘 𝑎ˆ 𝑘
𝜕𝑡

(D.27)
(D.28)

The equation for 𝑎ˆ 𝑘 can be formally integrated and substituted in the equation for
ˆ − to find
𝜕 −
ˆ = −𝑖𝜔q 𝜎
ˆ− −𝑖
𝑓 𝑘 𝑒 −𝑖𝜔 𝑘 (𝑡−𝑡0 ) 𝑎ˆ 𝑘 (𝑡 0 )
𝜕𝑡
Õ ∫ 𝑡
𝑓 𝑘2
𝑒 −𝑖(𝜔 𝑘 ) (𝑡−𝜏) 𝜎
ˆ − (𝜏)d𝜏.
(D.29)
𝑡0

We now use the Markov approximation to write 𝜎
ˆ − (𝜏) ≈ 𝜎
ˆ − (𝑡)𝑒 −𝑖(𝜔q ) (𝜏−𝑡) , and thus
𝜕 −
𝑓 𝑘 𝑒 −𝑖𝜔 𝑘 (𝑡−𝑡0 ) 𝑎ˆ 𝑘 (𝑡 0 )
ˆ = −𝑖𝜔q 𝜎
ˆ− −𝑖
𝜕𝑡
Õ ∫ 𝑡
−𝑖(𝜔 𝑘 −𝜔q ) (𝑡−𝜏)
d𝜏 𝜎
𝑓𝑘
ˆ − (𝑡).

(D.30)

𝑡0

Considering the generic equation of motion for a linearly decaying qubit, (𝜕/𝜕𝑡) 𝜎
ˆ−=
−𝑖𝜔q 𝜎
ˆ − − (𝛾/2) 𝜎
ˆ − , we can identify real part of the last term in the equation above
as the decay rate due to radiation of the qubit into the waveguide. We can extend the
integral’s bound to approximately evaluate this term as 𝛾 ≈ 2𝜋 𝑘 𝑓 𝑘2 𝛿(𝜔 𝑘 − 𝜔q ).

172
Band
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Upper
Upper

Frequency (GHz)
4.2131
4.6012
4.7395
4.8044
4.8373
4.856
4.8654
6.6768
7.309

𝑔/2𝜋 (MHz)
15.3
19.74
18.14
16.53
14.03
9.73
4.48
39.44
58.06

𝑄 e (×103 )
49.47
35.09
43.58
94.17
152.06
455.8
2100
15.74
12.02

𝑄 i (×103 )
74.99
76.25
75
75.59
73.77
76.47
72
68.06
70.44

Table D.1: Measured resonance parameters for metamaterial waveguide. The values
are measured for the waveguide of Figs. 5.2-5.4. The resonances are measured in reflection
from the input 50-Ω CPW port. The qubit-resonance coupling, 𝑔, is inferred from the
anti-crossing observed as the qubit is tuned through each waveguide resonance.

Assuming the coupling rate 𝑓 𝑘 is a smooth function of the 𝑘-vector, we can evaluate
this some in the continuum limit as
𝛾 = 2𝜋
𝑓 𝑘 𝛿(𝜔 𝑘 − 𝜔q ) ≈ 𝑀 𝑑
d𝑘 𝑓 𝑘2 𝛿(𝜔 𝑘 − 𝜔q )

=𝐿

𝜕𝑘
d𝜔
𝑓 𝑘2 𝛿(𝜔 𝑘 − 𝜔q ) = 𝑓 (𝜔q ) 2 𝑛g (𝜔q ).
𝜕𝜔

It is evident that reducing the group velocity increases the radiation decay rate of
the qubit. A similar analysis can be applied to find the decay rate of a linear cavity
with resonance frequency of 𝜔0 (i.e. a harmonic oscillator) that has been coupled to
the waveguide with coupling constant 𝑔(𝜔). In this case we find
𝛾=

D.2

𝑔(𝜔0 ) 2 𝑛g (𝜔0 ),

𝑄 e = 𝜔0 /𝛾 =

𝜔0 𝑐
𝐿 𝑔(𝜔0 ) 𝑛g (𝜔0 )

Characterization and modeling of the metamaterial waveguide

Several competing effects in the design of the metamaterial waveguide influence
its utility within a waveguide QED setting. We desire a compact waveguide unit
cell to reduce the required real estate in a chip-scale platform. This should be
combined with a large bandgap to provide more spectral bandwidth and tighter
localization of photon bound states. We also require efficient qubit-waveguide
coupling. These attributes allow for denser integration of qubits both in space and
frequency, enabling larger-scale and more complex quantum circuits. They can be
obtained in a single metamaterial design provided both the resonator elements and
the waveguide sections are of high impedance, and that the waveguide section has

173
large inductance. The logic for this is as follows. The bandgap of the metamaterial
waveguide scales roughly with the product of the coupling capacitance, the zeropoint voltage of the resonator, and the zero-point voltage of the waveguide section,
CPW𝑉 res . The zero-point voltage fluctuations scale with the impedance.
Δ ∝ 𝐶 𝑘 𝑉zpf
zpf
Additionally, a large bandgap requires that the inductance of the waveguide section
be large so that the resonant frequency of the bare waveguide section at the 𝑋-point
(𝑘 𝑑 = 𝜋/𝑎) is not too far detuned from that of the bare resonators, Ω 𝑘=𝜋/𝑑𝑎 =
(𝐿 0𝐶00 ) −1/2 ∼ 𝜔r .
We obtain a large resonator impedance by using spiral inductors made from narrow
cross-section wires of long coil length, with the impedance of the resonator roughly
scaling as the inverse of the square root of the coil width (𝑤), 𝑍res ∼ 1/ 𝑤.
The impedance of the CPW line can be set by adjusting the ratio of the center
conductor width to the physical gap between center conductor and the ground plane
(smaller ratio yields higher impedance). In order to also realize a large inductance
at the same time, without dramatically increasing the length of the waveguide
section, we meander the center conductor of the waveguide section to give it more
effective path length and larger inductance. In the devices presented in this work the
period of the metamaterial waveguide is 350 𝜇m, with the length of the waveguide
section corresponding to 210 𝜇m of this length. The wire width of the CPW center
conductor (resonator coil) was chosen conservatively to be 5 𝜇m (1 𝜇m) to limit
the potential disorder arising from fluctuations in the kinetic inductance due to wire
width inhomogeneity.
Estimates of the resulting lumped element parameters obtained from fits to the
measured transmission data of the fabricated metamaterial waveguide in Fig. 5.1
are: 𝐶𝑟 = 345 fF, 𝐿 𝑟 = 1.43 nH, 𝐶g = 389 fF, 𝐶0 = 50.5 fF, 𝐿 0 = 0.7885 nH.
These values are very close to the design values. From the measured widths of
the waveguide resonances in the transmission bands, we find a good to fit to the
loss in the waveguide by assuming a resistance 𝑅𝑟 = 8 × 10−4 Ω in series with
𝐿 𝑟 and a resistance 𝑅0 = 4 × 10−4 Ω in series with 𝐿 0 . For the metamaterial
waveguide coupled to the qubit of Figs. 5.2-5.4 we used a slightly different design,
with estimated lumped-element parameters equal to: 𝐶𝑟 = 240 fF, 𝐿 𝑟 = 2.10 nH,
𝑅𝑟 = 1.1 × 10−3 Ω, 𝐶g = 252 fF, 𝐶0 = 52.0 fF, 𝐿 0 = 1.19 nH, 𝑅0 = 6 × 10−4 Ω.
Here, the fit parameters were inferred from the frequencies and linewidths of the
lower and upper band resonances (within the 4-8 GHz circulator bandwidth of our
set-up), measured in reflection from the input 50-Ω CPW port. The read-out of the

174

b 0
-10

arg(S21)

S21 (dB)

0.5
-20

-30

-40

-0.5

Q e =1098

-1

Q i = 256332

6.098

6.1

6.102

6.104

6.106

6.098

Frequency (GHz)

0.005

-20

-0.01
5.2

-30

-1

Q e =325

5.6

5.8

6.2

Frequency (G Hz)

6.4

6.6

6.8

-0.5

-50

5.4

6.106

-40

-0.005

6.104

0.5

arg(S21)

-10

6.102

1.5

0.01

S21 (dB)

6.1

Frequency (GHz)

5.88

Q i = 485562

5.89

5.9

5.91

5.92

-1.5
5.88

Frequency (GHz)

5.89

5.9

5.91

5.92

Frequency (GHz)

Figure D.2: Characterization of lumped element resonators a, Optical and scanning
electron micrographs of microwave resonator array chip. Middle: optical image of the
chip with two arrays of coupled resonators on a 1 × 1 cm silicon chip. Left and Right:
SEM image (false-color) of the fabricated inductively (left) and capacitively (right) coupled
microwave resonator pairs. The resonator region is colored red and the waveguide central
conductor is colored blue. b-c, Amplitude and phase response of two capacitively-coupled
microwave resonator pairs measured at the fridge temperature 𝑇 𝑓 ≈ 7 mK. The legends
show the intrinsic (𝑄 i = 𝜔0 /𝛾i ) and extrinsic (𝑄 e = 𝜔0 /𝛾e ) quality factors extracted from
a Fano line shape fit. d, Difference between the measured and the expected design value
of the resonance frequencies for 9 resonators with similar geometries and wire widths of
500 nm. The dashed lines mark the standard deviation of the frequency difference, which is
equivalent to a normalized value of 𝜎 = 0.3%.

qubit state has been performed using one of the two upper band modes (f = 6.67
GHz, and f = 7.3 GHz), depending on the frequency of the qubit at each flux bias
point. The measured resonance parameters, along with their coupling (𝑔) to the
qubit, are tabulated in Table D.1. The qubit-to-waveguide coupling was designed
and simulated to be given by a coupling capacitance of 𝐶𝑔 = 4.8 fF. In all our circuit
model fits this coupling capacitance was fixed at the design value, and not needed
as a fitting parameter.
D.3

Characterization of lumped-element microwave resonators

We have achieved a characteristic size of 𝜆0 /150 (130 𝜇m × 76 𝜇m for 𝜔0 /2𝜋 ≈
6 GHz) and 𝜆 0 /76 (155 𝜇m × 92 𝜇m for 𝜔0 /2𝜋 ≈ 10 GHz), using a wire width of
500 nm and 1 𝜇m, respectively.
Figure D.2 shows the typical amplitude and phase of measured for a waveguide
coupled to a pair of identical resonators. Microwave spectroscopy of the fabricated
resonators is performed in a dilution refrigerator cooled-down to a temperature of

175
𝑇 𝑓 ≈ 7 mK. The input microwave is launched onto the chip via a 50-Ω CPW. The
output microwave signal is subsequently amplified and analyzed using a network
analyzer (for more details regarding the measurement setup, refer to Ref. [345]).
We have extracted the intrinsic and extrinsic decay rates of the cavity by fitting the
transmission data to a Fano line shape of the form
𝑆21 (𝜔) = 1 −

𝛾e 𝑒𝑖𝜙0
𝛾i + 𝛾e + 2𝑖(𝜔 − 𝜔0 )

(D.31)

Here 𝛾e and 𝛾i are the extrinsic and intrinsic decay rates of the resonator, respectively. The phase 𝜙0 is a parameter that sets the asymmetry of the Fano line shape
[346]. The data demonstrates that it is possible to adjust the external coupling to the
resonator in a wide range without much degradation in the internal quality factor (it
is straightforward to convert the extrinsic quality factor 𝑄 e to the coupling constants
𝑔 𝑘 used in our theoretical analysis above). We have compared the measured resonance frequency with the resonance frequency found from numerical simulations
in Fig. D.2d. We find that the measured resonance frequencies are in agreement
with the simulated values, with a multiplicative scaling factor of 0.85. Using this
scale factor, we have measured a random variation 0.3 % in the resonance frequency.
It has been previously suggested that the shift in the resonance frequency and its
statistical variation can be attributed to the kinetic inductance of the free charge
carriers in the superconductor, and the variations can be mitigated by increasing the
wire width [228].
D.4

Disorder and Anderson localization

Propagation of electron waves in a one dimensional quasi-periodic potential is
described by
𝜕2 Õ
(𝑈 + 𝑈𝑛 )𝛿(𝑥 − 𝑎𝑛) 𝜓 𝑞 (𝑥) = 𝑞 2 𝜓 𝑞 (𝑥).
(D.32)
− 2+
𝜕𝑥
Here, 𝑞 is the quasi momentum and 𝑈𝑛 is the random variable that models compositional disorder at position 𝑥 = 𝑛𝑎. Disorder leads to localization of waves with a
characteristic length defined as
* 𝑁−1
Õ 𝜓𝑛+1
ℓ −1 = lim
ln
(D.33)
𝑁→∞ 𝑁
𝑛=0
Here, the brackets represent averaging over different realization of the disorder,
whereas the summation accounts for spatial/temporal averaging for traveling waves.

176
For this model, previous authors have found the localization length to be [225, 347,
348]
ℓ 2Γ(1/6) −2/3
= 1/3 √ 𝜎
≈ 3.45𝜎 −2/3 .

(D.34)

In this model 𝜎 2 = h𝑈𝑛2 i sin2 (𝑞 0 𝑎)/𝑞 20 is a parameter that quantifies the strength of
disorder, and 𝑞 0 is the value of quasi-momentum at the band-edge.
Now, we consider the propagation of current waves in a one dimensional waveguide
that has been periodically loaded with resonators (a similar analysis can be applied
to the voltage waves for the case of inductively coupled resonators). Starting from
Eq. (D.19), it is straightforward to find
𝑑Δ𝛿(𝑥 − 𝑎𝑛)
𝜕 𝐼 (𝑥)
𝑛2 1 +
= 0.
(D.35)
+ 𝐼 (𝑥)
𝜔0,𝑛 − 𝜔 + 𝑖𝛾i
𝜕𝑥
By comparing this equation with the Schrodinger equation for the Kronig-Penny
model Eq. (D.32) we find
𝜔 2
𝜔 2
𝑑Δ
𝑞 →
𝑛 , 𝑈 + 𝑈𝑛 → −
(D.36)
𝜔0,𝑛 − 𝜔 + 𝑖𝛾i
For small variation in resonance frequencies, 𝛿𝜔0 , we can expand the resonance
potential term to find
𝜔 2
𝑑Δ
2 𝜕
𝑈𝑛 = −
𝛿𝜔0
(D.37)
𝜕𝜔0,𝑛 𝜔0,𝑛 − 𝜔 + 𝑖𝛾i
By evaluating the expression for 𝑈𝑛 and substituting it in the relation above for 𝜎 2 ,
we find
2
4
𝛾 4 𝛿𝜔 2
𝛿𝜔0
𝛾e
, 𝜎high =
(D.38)
𝜎low =
𝛾i
The analysis above gives us ℓdis . In addition to disorder, absorption loss in the metamaterial waveguide components (specifically the resonators) leads to an exponential
extinction of the wave’s amplitude. An effective localization length incorporating
absorption loss, ℓloss , can be found by solving for the complex band structure and
setting ℓloss = 1/Im(𝑘). For propagation of a classical wave through the waveguide
both loss and disorder contribute to exponential extinction of the wave with a total
localization length of
ℓtotal

ℓdis ℓloss

(D.39)

177
metamaterial waveguide
Vq
Cg
EJ

Zline

Figure D.3: Circuit diagram for a transmon qubit coupled to a metamaterial waveguide. The resistive termination is used to model radiation into the 50Ω coplanar waveguide.

Two important points should be made here. First, ℓloss as defined is purely an
absorption loss effect only outside any photonic bandgap region. Inside a photonic
bandgap the periodic loading of the waveguide gives rise to an imaginary 𝑘-vector
as well. As such, ℓloss inside the gap will contain both periodic loading effects
and absorption loss effects. Second, the exponential localization of the photonic
wavefunction caused by the periodic loading of the waveguide and the localization
caused by structural disorder are coherent (unitary) effects. On the contrary, the
exponential attenuation of a traveling wavepacket due to the loss in the resonators
is a dissipative effect. When considering photon-mediated interactions between
qubits, these two effects for the most part need to be addressed separately. In this
context, the value of ℓtotal as a single parameter is limited to primarily estimating the
spatial extent over which strong coherent interactions can be obtained.
D.5

Qubit frequency shift and lifetime

Circuit theory modeling
The qubit frequency shift can be derived from circuit theory by modeling the qubit
as a linear resonator. Consider the circuit diagram in Fig. D.3. The load impedance
seen from the qubit port can be written as
𝑍L (𝜔) =

+ 𝑍line (𝜔),
𝑖𝜔𝐶g

(D.40)

𝑌L (𝜔) =

𝑖𝜔𝐶g
1 + 𝑍line (𝜔)𝑖𝜔𝐶g

(D.41)

and

178
a 10 -1

× 10

-5

-2

-3

-4

-5

0. 8

T1 lifetime (s)

0. 9

0. 7
0. 6
0. 5
0. 4
0. 3
0. 2
0. 1

10 -6

Bare qubit frequency (GHz)

6. 4

6.6

6.8

Bare qubit frequency (GHz)

Figure D.4: Qubit lifetime as a function of resonance frequency. a, Simulated qubit
lifetime set by radiation into the output CPW port (blue), and structural loss in the waveguide
(red). b, Comparison of the experimental results (open circles) with the simulated qubit
lifetime (solid and dashed lines) near the first resonance dip in the upper transmission band.
The lifetime set by radiation into the output port and structural loss in the waveguide are
shown as blue and red solid lines, respectively. Both of these contributions have been
adjusted to include a frequency independent intrinsic qubit life time of 10.86 𝜇s. The black
dashed line shows the theoretical qubit excited state lifetime including all contributions.

For weak coupling, the decay rate can be found using the real part of the load
impedance as
𝜅 ≃ 𝜔2q 𝐿 J Re 𝑌L (𝜔q ) .
(D.42)
Here, 𝜔q is the resonance frequency of the qubit. Similarly, the shift in qubit
frequency is found as
Δ𝜔q ≃ −

𝜔2q 𝐿 J

Im 𝑌L (𝜔q ) .

(D.43)

For a transmon qubit, we have the following relation that approximate its behavior
in the linear regime
2
𝐿J =

Φ0
2𝜋

𝐸J

𝜔q = p

(D.44)

𝐿 J𝐶q

We first use the simplified continuum model to find the input impedance 𝑍line
𝑍line (𝜔) = 𝑍B (𝜔)

𝑅L + 𝑍B (𝜔) tanh {Im[𝑘 (𝜔)]𝑥}
𝑍B (𝜔) + 𝑅L tanh {Im[𝑘 (𝜔)]𝑥}

(D.45)

179
Here, Im[𝑘 (𝜔)] is the attenuation constant (we are assuming Re[𝑘 (𝜔)] = 0, i.e.
valid when the value of 𝜔 is within the bandgap), 𝑍B (𝜔) is the Bloch impedance of
the periodic structure, and 𝑥 is the length of the waveguide. Assuming Im[𝑘 (𝜔)]𝑥

1, this expression can be simplified as
𝑍line (𝜔) ≈𝑍B (𝜔) +

4𝑅L |𝑍B (𝜔)| 2

𝑅L + |𝑍B (𝜔)|

𝑒 −2Im[𝑘 (𝜔)]𝑥

≈𝑍B (𝜔) + 4𝑅L 𝑒 −2Im[𝑘 (𝜔)]𝑥 .

(D.46)

Note that we have assumed 𝑅L
|𝑍B (𝜔)| to make the last approximation. For
weak coupling, the qubit coupling capacitance, 𝐶g , should be chosen such that the
magnitude of impedance 𝑍g = 1/(𝑖𝜔𝐶g ) is much larger than |𝑍line (𝜔)|. In this
situation, we use Eq. (D.43) and Eq. (D.46) to find
Δ𝜔q
= − (𝐿 J 𝜔q )(𝐶g 𝜔q ) − (𝐿 J 𝜔q )(𝐶g 𝜔q ) 2 Im[𝑍B (𝜔q )]
𝜔q
𝐶g
𝐶g
Im[𝑍B (𝜔q )]𝐶g 𝜔q .
=−
2𝐶q 2𝐶q

(D.47)

Note that the first term in the frequency shift is merely caused by addition of the
coupling capacitor to the overall qubit capacitance.
We find the qubit’s decay rate caused by radiation into the output port by substituting
Eq. (D.46) in Eq. (D.42)
𝜅rad =

4𝜔2q𝐶g 2
𝐶q

𝑅L 𝑒 −2Im[𝑘 (𝜔)]𝑥 .

(D.48)

Subsequently, the lifetime of the qubit can be written as
𝑇1,𝑡𝑒𝑥𝑡𝑟𝑎𝑑 =

𝐶q
4𝜔2q𝐶g 2 𝑅L

𝑒 2𝑥/ℓ(𝜔q ) ,

(D.49)

where ℓ(𝜔) = 1/Im[𝑘 (𝜔)] is the localization length in the bandgap. We note that
the analysis from circuit theory is only valid for weak qubit-waveguide coupling
rates, where the Markov approximation can be applied. In the strong coupling
regime, the qubit frequency and lifetime can be found by numerically finding the
zeros of the circuit’s admittance function 𝑌 (𝜔) = 𝑌L (𝜔) + 𝑌q (𝜔), where 𝑌q (𝜔) =
𝑖𝜔q𝐶q + 1/(𝑖𝜔q 𝐿 J ).
Effect of structural loss in the waveguide on the qubit lifetime
Equation (D.48) gives the decay rate of qubit’s excited state caused by radiation
into the output CPW port. In addition to this radiative component, the loss in the

180
waveguide also contributes to the decay rate of the qubit excited state. The effect
of loss in the waveguide can be modeled as the (incoherent) sum of contributions
to the decay rate from the individual resonances in the transmission bands of the
waveguide:
𝜅loss =

2𝜅
𝑔𝑚
𝑖,𝑚

𝑚 2𝑔𝑚 + |𝜅𝑖,𝑚 − 𝑖(𝜔q − 𝜔0,𝑚 )|

(D.50)

Here, 𝑚 denotes the index of each waveguide resonance, 𝑔𝑚 is the coupling rate
of the qubit to the waveguide resonance, and 𝜅𝑖,𝑚 is the intrinsic decay rate for
each waveguide resonance. The parameters 𝜔q and 𝜔0,𝑚 denote the fundamental
transition frequency of the qubit and the resonance frequency of the waveguide
mode, respectively. For a finite waveguide made from 9 unit cells we expect a total
of 18 resonances, with half of them distributed in the lower frequency transmission
band and half of them distributed in the upper transmission band. Table D.1 presents
the measured 𝜔0,𝑚 , 𝑄 𝑖,𝑚 = 𝜔0,𝑚 /𝜅𝑖,𝑚 , and 𝑔𝑚 parameters for the 9 resonances closest
to the waveguide bandgap that are observable in the frequency band of the circulators
used in our experiment. The total lifetime of the qubit excited state can be determined
from 𝑇1 = 1/(𝜅rad + 𝜅loss + 𝜅i ), where 𝜅𝑖 represents a third decay channel for the qubit
corresponding to coupling to all other degrees of freedom (two-level systems, etc.).
From a fit to the measured qubit 𝑇1 data deep in the bandgap we find an intrinsic
lifetime of 𝜅i−1 = 10.86 𝜇s.
To identify regions of frequency space where the qubit lifetime is limited by output port radiation, and therefore may tell us about the finite extent of the photon
wavepacket coupled to the qubit, we plot in Fig. D.4a the estimated loss and output
radiation contributions to the qubit lifetime as a function of frequency in and around
the waveguide bandgap for our experiment. It is evident the internal waveguide
loss contribution is the dominant factor deep in the band gap and near the lower
transmission band-edge, where the localization length of the photon wavepacket
coupled to the qubit is much smaller than the finite length of the waveguide. On
the contrary, the radiation into the output CPW port is the dominant factor near the
upper band-edge frequency in the band gap and inside the upper transmission band,
where the localization length becomes comparable and larger than the finite length
of the waveguide. A zoomed-in plot around the upper band-edge frequency of the
measured qubit excited state lifetime along with the different estimated components
of the qubit decay are shown in Fig. D.4b. The asymmetric profile of the measured
lifetime near the first resonance in the upper transmission band is a clear sign of

181
the radiation into the output port, where the shorter lifetime for frequencies above
the resonance frequency can be attributed to coherent (constructive) interference
from multiple waveguide resonances. These subtle features help differentiate single
mode and incoherent multi-mode cavity-QED effects, from true waveguide-QED
effects in which multi-mode interference leads to radiative dynamics governed by a
localized photon wavepacket.
D.6

Group delay and the qubit lifetime profile

Equation (D.49) demonstrates the relation between the qubit lifetime and the localization length. Moving the qubit frequency beyond the gap, results in a drastic
increase in the localization length and subsequently reduces the qubit lifetime. The
normalized slope of the lifetime profile in the vicinity of the band-edge can be
written as
𝜕Im(𝑘)
1 𝜕𝑇1,rad
= 𝑥
= 𝑥Im(𝑛g )/𝑐 .
𝑇1,rad 𝜕𝜔
𝜕𝜔

(D.51)

We now evaluate Eq. (D.25) to find the group index at the upper and lower bandedges 𝛿± = 0

|Re(𝑛g )| = |Im(𝑛g )| =

 𝑛𝜔c− 8𝛾i 3
 𝑛𝜔c+ √8Δ𝛾i

for 𝜔 = 𝜔c− ,
(D.52)
for 𝜔 = 𝜔c+ .

Consequently, we can write the normalized slope of the lifetime profile at the bandedge as
1 𝜕𝑇1,rad
= 𝑥Im 𝑛g (𝜔𝑐± ) /𝑐 = 𝑥Re 𝑛g (𝜔𝑐± ) /𝑐 = 𝜏delay . (D.53)
𝑇1,rad 𝜕𝜔
𝜔=𝜔 𝑐±
This result has a simple description: the normalized slope of the lifetime profile at
the band-edge is equal to the (maximum) group delay.
D.7

Scaling the waveguide length

Scaling the length of the waveguide to the extreme limits requires dealing with a
number of technical challenges. Below, we outline a number of these challenges
and possible strategies for addressing them. A systematic study of these challenges
and efficient strategies for overcoming them will be the subject of a future study.
Resonator size: The size of lumped-element resonators is ultimately limited by
the fabrication considerations for thin-film aluminum nano-wires. A pitch size of

182
60 nm can be achieved in these structures by using electron beam lithography for
patterning the wires [349]. Assuming a quarter-wave resonator geometry, and a
wire-to-airgap ratio of unity, we find the characteristic size for the resonator as
𝑑 = (60 nm × 𝜆/4). Using 𝜆 = 2 cm for a 6 GHz resonator (on Si substrate), we
have 𝑑 = 𝜆/1100. An alternative strategy for miniaturizing the resonators is to use
kinetic inductance of disordered superconductors [350].
We emphasize that using smaller resonators requires a careful study of disorder in
the resonance frequency, and possible strategies for reducing it.
Bandgap size: The lumped-element nature of the components in our device allows
for achieving a larger bandgap-to-midgap ratio by simply increasing the coupling
capacitor, 𝐶g , and reducing the internal capacitance, 𝐶r . Ultimately, however, the
gap size will be limited by the minimum value of 𝐶r which is itself set by the parasitic
capacitance of the inductor in the resonator (𝐿 r ). Considering the numerical values
of these quantities listed in the Sec. D.2, we anticipate that a two-fold increase in
the bandgap-to-midgap is feasible.
Parasitic modes: The placement of the qubit with respect to the metamaterial
waveguide in our device relies on the symmetry of the waveguide modes to eliminate
coupling to the parasitic modes of the structure. Alternatively, Aluminum air bridges
can be implemented in the coplanar waveguide (CPW) sections of our device in
order to suppress the slot-line modes of the waveguide. Suppressing the slot-line
modes allows for realizing a more flexible geometry where multiple qubits can be
capacitively coupled to resonators along the waveguide.
Making turns with the metamaterial waveguide: Effective use of chip’s area
requires the ability to make 90◦ /180◦ turns along the path for long waveguides. In
our device, turns can be implemented by modifying the CPW sections between the
cavities. To this end, the meandered coplanar waveguide between the cavities can
be unwrapped to realize turns with a radius of curvature 50–200 𝜇m. The resulting
asymmetry and local frequency shift caused by the bend can be compensated by numerical modeling of these effects and making proper adjustments to the neighboring
resonators.

183
Appendix E

SUPPLEMENTARY INFORMATION FOR CHAPTER 6
E.1

Modeling of the topological waveguide

In this section we provide a theoretical description of the topological waveguide
discussed in Chapter 6, an analog to the Su-Schrieffer-Heeger model [245]. An
approximate form of the physically realized waveguide is given by an array of
coupled LC resonators, a unit cell of which is illustrated in Fig. E.1. Each unit
cell of the topological waveguide has two sites A and B whose intra- and inter-cell
coupling capacitance (mutual inductance) are given by 𝐶𝑣 (𝑀𝑣 ) and 𝐶𝑤 (𝑀𝑤 ). We
∫𝑡
denote the flux variable of each node as Φ𝑛𝛼 (𝑡) ≡ −∞ d𝑡 0 𝑉𝑛𝛼 (𝑡 0) and the current
going through each inductor as 𝑖 𝑛𝛼 (𝛼 = {A, B}). The Lagrangian in position space
reads
2 𝐶
2 𝐶 2 2
Õ 𝐶𝑣
𝑤 ¤A
¤A
¤ B𝑛
Φ𝑛 − Φ𝑛 +
Φ𝑛+1 − Φ𝑛 +
+ Φ
L=
𝐿0 A 2 B 2
− 𝑀𝑣 𝑖 𝑛A𝑖 𝑛B − 𝑀𝑤 𝑖 𝑛B𝑖 𝑛+1
(E.1)
𝑖𝑛 + 𝑖𝑛
The node flux variables are written in terms of current through the inductors as
ΦA
𝑛 = 𝐿 0𝑖 𝑛 + 𝑀𝑣 𝑖 𝑛 + 𝑀𝑤 𝑖 𝑛−1 ,

VnA InA

ΦB𝑛 = 𝐿 0𝑖 𝑛B + 𝑀𝑣 𝑖 𝑛A + 𝑀𝑤 𝑖 𝑛+1

VnB InB

C0
iA

Vn+1

Mv
L0

(E.2)

C0
iB

Figure E.1: Modeling of the topological waveguide. LC resonators of inductance 𝐿 0
and capacitance 𝐶0 are coupled with alternating coupling capacitance 𝐶 𝑣 , 𝐶 𝑤 and mutual
inductance 𝑀 𝑣 , 𝑀 𝑤 . The voltage and current at each resonator node A (B) are denoted as
𝑉𝑛A , 𝐼𝑛A (𝑉𝑛B , 𝐼𝑛B ).

184
Considering the discrete translational symmetry in our system, we can rewrite the
variables in terms of Fourier components as
1 Õ 𝑖𝑛𝑘 𝑑 𝛼
Φ𝑛𝛼 = √
Φ𝑘 ,
𝑁 𝑘

1 Õ 𝑖𝑛𝑘 𝑑 𝛼
𝑖 𝑛𝛼 = √
𝑒 𝑖𝑘 ,
𝑁 𝑘

(E.3)

where 𝛼 = A, B, 𝑁 is the number of unit cells, and 𝑘 = 2𝜋𝑚
𝑁 𝑑 (𝑚 = −𝑁/2, · · · , 𝑁/2−1)
are points in the first Brillouin zone. Equation (E.2) is written as
𝑖𝑛𝑘 0 𝑑 A
𝑖𝑛𝑘 0 𝑑
−𝑖𝑘 0 𝑑
Φ𝑘 0 =
𝐿 0 𝑖 𝑘 0 + 𝑀𝑣 𝑖 𝑘 0 + 𝑒
𝑀𝑤 𝑖 𝑘 0
𝑘0

𝑘0

under this transform. Multiplying the above equation with 𝑒 −𝑖𝑛𝑘 𝑑 and summing over
all 𝑛, we get a linear relation between Φ𝛼𝑘 and 𝑖 𝛼𝑘 :
ΦA𝑘
𝐿0
𝑀𝑣 + 𝑀𝑤 𝑒 −𝑖𝑘 𝑑 𝑖 A
𝑘 .
𝑖𝑘
Φ𝑘
𝑀𝑣 + 𝑀 𝑤 𝑒
𝐿0
𝑖𝑘
By calculating the inverse of this relation, the Lagrangian of the system (E.1) can
be rewritten in 𝑘-space as
Õ 𝐶0 + 𝐶𝑣 + 𝐶𝑤
¤A Φ
¤A+Φ
¤B Φ
¤ B − 𝐶𝑔 (𝑘) Φ
¤A Φ
¤B
L=
−𝑘 𝑘
−𝑘 𝑘
−𝑘 𝑘
𝐿0
A ΦA + ΦB ΦB − 𝑀 (𝑘)ΦA ΦB
−𝑘 𝑘
−𝑘 𝑘
−𝑘 𝑘
(E.4)
− 2
𝐿 0 − 𝑀𝑔 (−𝑘) 𝑀𝑔 (𝑘)
where 𝐶𝑔 (𝑘) ≡ 𝐶𝑣 + 𝐶𝑤 𝑒 −𝑖𝑘 𝑑 and 𝑀𝑔 (𝑘) ≡ 𝑀𝑣 + 𝑀𝑤 𝑒 −𝑖𝑘 𝑑 . The node charge
¤ 𝛼 canonically conjugate to node flux Φ𝛼 are
variables 𝑄 𝛼𝑘 ≡ 𝜕L/𝜕 Φ
¤A
𝑄 A𝑘
𝐶0 + 𝐶 𝑣 + 𝐶 𝑤
−𝐶𝑔 (−𝑘)
−𝑘 .
−𝐶𝑔 (𝑘)
𝐶0 + 𝐶𝑣 + 𝐶𝑤 ΦB−𝑘
𝑄𝑘
Note that due to the Fourier transform implemented on flux variables, the canonical
charge in momentum space is related to that in real space by
𝑄 𝑛𝛼 =

¤𝛼
Õ 𝜕L 𝜕 Φ
1 Õ −𝑖𝑛𝑘 𝑑 𝛼
𝜕L
𝑄𝑘 ,
¤ 𝑛𝛼
¤ 𝛼 𝜕Φ
¤ 𝑛𝛼
𝜕Φ
𝜕Φ
𝑁 𝑘

which is in the opposite sense of regular Fourier transform in Eq. (E.3). Also,
due to the Fourier-transform properties, the constraint that Φ𝑛𝛼 and 𝑄 𝑛𝛼 are real
𝛼 and (𝑄 𝛼 ) ∗ = 𝑄 𝛼 . Applying the Legendre transformation
reduces to (Φ𝛼𝑘 ) ∗ = Φ−𝑘
−𝑘

185
𝐻=

𝛼¤𝛼
𝑘,𝛼 𝑄 𝑘 Φ 𝑘 − L, the Hamiltonian takes the form

𝐻=

Õ 𝐶Σ (𝑄 A 𝑄 A + 𝑄 B 𝑄 B ) + 𝐶𝑔 (−𝑘)𝑄 A 𝑄 B + 𝐶𝑔 (𝑘)𝑄 B 𝑄 A
−𝑘

−𝑘 𝑘
−𝑘 𝑘
2𝐶𝑑 (𝑘)
A
𝐿 0 (ΦA
−𝑘 Φ 𝑘 + Φ−𝑘 Φ 𝑘 ) − 𝑀𝑔 (𝑘)Φ−𝑘 Φ 𝑘 − 𝑀𝑔 (−𝑘)Φ−𝑘 Φ 𝑘
2𝐿 2𝑑 (𝑘)

−𝑘

where
𝐶Σ ≡ 𝐶0 + 𝐶𝑣 + 𝐶𝑤 ,

𝐶𝑑2 (𝑘) ≡ 𝐶Σ2 − 𝐶𝑔 (−𝑘)𝐶𝑔 (𝑘),

𝐿 2𝑑 (𝑘) ≡ 𝐿 02 − 𝑀𝑔 (−𝑘)𝑀𝑔 (𝑘).

Note that 𝐶𝑑2 (𝑘) and 𝐿 2𝑑 (𝑘) are real and even function in 𝑘. We impose the
canonical commutation relation between real-space conjugate variables [Φ̂𝑛𝛼 , 𝑄ˆ 𝑛 0 ] =
𝑖ℏ𝛿𝛼,𝛽 𝛿𝑛,𝑛 0 to promote the flux and charge variables to quantum operators. This re𝛽
duces to [ Φ̂𝛼𝑘 , 𝑄ˆ 𝑘 0 ] = 𝑖ℏ𝛿𝛼,𝛽 𝛿 𝑘,𝑘 0 in the momentum space [Note that due to the
𝛼 and ( 𝑄
ˆ 𝛼 ) † = 𝑄ˆ 𝛼 , meaning flux and charge opFourier transform, ( Φ̂𝛼𝑘 ) † = Φ̂−𝑘
−𝑘
erators in momentum space are non-Hermitian since the Hermitian conjugate flips
ˆ where the
the sign of 𝑘]. The Hamiltonian can be written as a sum 𝐻ˆ = 𝐻ˆ 0 + 𝑉,
“uncoupled” part 𝐻ˆ 0 and coupling terms 𝑉ˆ are written as
𝛼 Φ̂𝛼
A Φ̂B
Õ 𝑄ˆ A 𝑄ˆ B
Õ 𝑄ˆ 𝛼 𝑄ˆ 𝛼
Φ̂
Φ̂
−𝑘 𝑘
−𝑘 𝑘
+ −𝑘
, 𝑉ˆ =
+ −𝑘
+ H.c. , (E.5)
𝐻ˆ 0 =
eff
eff
eff
eff
2𝐶
(𝑘)
2𝐿
(𝑘)
2𝐶
(𝑘)
2𝐿
(𝑘)
𝑘,𝛼
with the effective self-capacitance 𝐶0eff (𝑘), self-inductance 𝐿 0eff (𝑘), coupling capacitance 𝐶𝑔eff (𝑘), and coupling inductance 𝐿 eff
𝑔 (𝑘) given by
𝐶0eff (𝑘) =

𝐶𝑑2 (𝑘)
𝐶Σ

𝐿 0eff (𝑘) =

𝐿 2𝑑 (𝑘)
𝐿0

𝐶𝑔eff (𝑘) =

𝐶𝑑2 (𝑘)
𝐶𝑔 (−𝑘)

𝐿 eff
𝑔 (𝑘) = −

𝐿 2𝑑 (𝑘)
𝑀𝑔 (𝑘)
(E.6)

The diagonal part 𝐻ˆ 0 of the Hamiltonian can be written in a second-quantized form
by introducing annihilation operators 𝑎ˆ 𝑘 and 𝑏ˆ 𝑘 , which are operators of the Bloch
waves on A and B sublattice, respectively:
 Φ̂B
1  Φ̂ 𝑘
eff (𝑘) 𝑄
ˆ 𝑘 ≡ √  q 𝑘
ˆ B  .
𝑎ˆ 𝑘 ≡ √  q
+ 𝑖 𝑍0eff (𝑘) 𝑄ˆ A
−𝑘 
−𝑘 
2ℏ  𝑍 eff (𝑘)
2ℏ  𝑍 eff (𝑘)
Here, 𝑍0eff (𝑘) ≡ 𝐿 0eff (𝑘)/𝐶0eff (𝑘) is the effective impedance of the oscillator at
wavevector 𝑘. Unlike the Fourier transform notation, for bosonic modes 𝑎ˆ 𝑘 and
𝑏ˆ 𝑘 , we use the notation ( 𝑎ˆ 𝑘 ) † ≡ 𝑎ˆ †𝑘 and ( 𝑏ˆ 𝑘 ) † ≡ 𝑏ˆ †𝑘 . Under this definition, the

186
commutation relation is rewritten as [ 𝑎ˆ 𝑘 , 𝑎ˆ †𝑘 0 ] = [ 𝑏ˆ 𝑘 , 𝑏ˆ †𝑘 0 ] = 𝛿 𝑘,𝑘 0 . Note that the flux
and charge operators are written in terms of mode operators as
eff (𝑘)
ℏ𝑍
Φ̂A𝑘 =
𝑎ˆ 𝑘 + 𝑎ˆ †−𝑘 , 𝑄ˆ A𝑘 =
−𝑘
𝑘 ,
𝑖 2𝑍0eff (𝑘)
ℏ𝑍0eff (𝑘)
Φ̂ 𝑘 =
𝑏 𝑘 + 𝑏 −𝑘 , 𝑄 𝑘 =
𝑏 −𝑘 − 𝑏 𝑘 .
𝑖 2𝑍0eff (𝑘)
The uncoupled Hamiltonian is written as
𝐻ˆ 0 =

Õ ℏ𝜔0 (𝑘)

𝑎ˆ †𝑘 𝑎ˆ 𝑘 + 𝑎ˆ −𝑘 𝑎ˆ †−𝑘 + 𝑏ˆ †𝑘 𝑏ˆ 𝑘 + 𝑏ˆ −𝑘 𝑏ˆ †−𝑘 ,

(E.7)

where the “uncoupled” oscillator frequency is given by 𝜔0 (𝑘) ≡ [𝐿 0eff (𝑘)𝐶0eff (𝑘)] −1/2 ,
which ranges between values
𝐿 0𝐶Σ
𝜔0 (𝑘 = 0) =
[𝐿 0 − (𝑀𝑣 + 𝑀𝑤 ) 2 ] [𝐶Σ2 − (𝐶𝑣 + 𝐶𝑤 ) 2 ]
𝐿 0𝐶Σ
𝜋
𝜔0 𝑘 =
(𝐿 02 − |𝑀𝑣 − 𝑀𝑤 | 2 )(𝐶Σ2 − |𝐶𝑣 − 𝐶𝑤 | 2 )
The coupling Hamiltonian 𝑉ˆ is rewritten as
Õ ℏ𝑔𝐶 (𝑘)
𝑉ˆ = −
𝑎ˆ −𝑘 𝑏ˆ 𝑘 − 𝑎ˆ −𝑘 𝑏ˆ †−𝑘 − 𝑎ˆ †𝑘 𝑏ˆ 𝑘 + 𝑎ˆ †𝑘 𝑏ˆ †−𝑘
ℏ𝑔 𝐿 (𝑘)
𝑎ˆ −𝑘 𝑏ˆ 𝑘 + 𝑎ˆ −𝑘 𝑏ˆ †−𝑘 + 𝑎ˆ †𝑘 𝑏ˆ 𝑘 + 𝑎ˆ †𝑘 𝑏ˆ †−𝑘 + H.c. ,

(E.8)

where the capacitive coupling 𝑔𝐶 (𝑘) and inductive coupling 𝑔 𝐿 (𝑘) are simply written
as
𝜔0 (𝑘)𝐶𝑔 (𝑘)
𝜔0 (𝑘)𝑀𝑔 (𝑘)
𝑔𝐶 (𝑘) =
, 𝑔 𝐿 (𝑘) =
(E.9)
2𝐶Σ
2𝐿 0
respectively. Note that 𝑔𝐶∗ (𝑘) = 𝑔𝐶 (−𝑘) and 𝑔 ∗𝐿 (𝑘) = 𝑔 𝐿 (−𝑘). In the following,
we discuss the diagonalization of this Hamiltonian to explain the dispersion relation
and band topology.
Band structure within the rotating-wave approximation
We first consider the band structure of the system within the rotating-wave approximation (RWA), where we discard the counter-rotating terms 𝑎ˆ 𝑏ˆ and 𝑎ˆ † 𝑏ˆ † in the

187
Hamiltonian. This assumption is known to be valid when the strength of the couplings |𝑔 𝐿 (𝑘)|, |𝑔𝐶 (𝑘)| are small compared to the uncoupled oscillator frequency
𝜔0 (𝑘). Under this approximation, the Hamiltonian in Eqs. (E.7)-(E.8) reduces
to a simple form 𝐻ˆ = ℏ 𝑘 ( v̂ 𝑘 ) † h(𝑘) v̂ 𝑘 , where the single-particle kernel of the
Hamiltonian is,
𝜔0 (𝑘) 𝑓 (𝑘)
h(𝑘) = ∗
(E.10)
𝑓 (𝑘) 𝜔0 (𝑘)
Here, v̂ 𝑘 = ( 𝑎ˆ 𝑘 , 𝑏ˆ 𝑘 )𝑇 is the vector of annihilation operators at wavevector 𝑘 and
𝑓 (𝑘) ≡ 𝑔𝐶 (𝑘) − 𝑔 𝐿 (𝑘). In this case, the Hamiltonian is diagonalized to the form
Õh
𝐻=ℏ
𝜔+ (𝑘) 𝑎ˆ +,𝑘 𝑎ˆ +,𝑘 + 𝜔− (𝑘) 𝑎ˆ −,𝑘 𝑎ˆ −,𝑘 ,
(E.11)

where two bands 𝜔± (𝑘) = 𝜔0 (𝑘) ± | 𝑓 (𝑘)| symmetric with respect to 𝜔0 (𝑘) at each
wavevector 𝑘 appear [here, note that 𝑎ˆ †±,𝑘 ≡ ( 𝑎ˆ ±,𝑘 ) † ]. The supermodes 𝑎ˆ ±,𝑘 are
written as 𝑎ˆ ±,𝑘 = [±𝑒 −𝑖𝜙(𝑘) 𝑎ˆ 𝑘 + 𝑏ˆ 𝑘 ]/ 2, where 𝜙(𝑘) ≡ arg 𝑓 (𝑘) is the phase of
coupling term. The Bloch states in the single-excitation bands are written as
1
|𝜓 𝑘,± i = 𝑎ˆ †±,𝑘 |0i = √ ±𝑒𝑖𝜙(𝑘) |1 𝑘 , 0 𝑘 i + |0 𝑘 , 1 𝑘 i ,
where |𝑛 𝑘 , 𝑛0𝑘 i denotes a state with 𝑛 (𝑛0) photons in mode 𝑎ˆ 𝑘 (𝑏ˆ 𝑘 ).
As discussed below in App. E.2, the kernel of the Hamiltonian in Eq. (E.10) has
an inversion symmetry in the sublattice unit cell which is known to result in bands
with quantized Zak phase [260]. In our system the Zak phase of the two bands are
evaluated as
!#
1 ±𝑒𝑖𝜙(𝑘)
𝜕𝜙(𝑘)
1 −𝑖𝜙(𝑘) 𝜕
=−
d𝑘
Z=𝑖
d𝑘 √ ±𝑒
𝜕𝑘
2 B.Z.
𝜕𝑘
B.Z.
The Zak phase of photonic bands is determined by the behavior of 𝑓 (𝑘) in the
complex plane. If the contour of 𝑓 (𝑘) for 𝑘 values in the first Brillouin zone excludes
(encloses) the origin, the Zak phase is given by Z = 0 (Z = 𝜋) corresponding to
the trivial (topological) phase.
Band structure beyond the rotating-wave approximation
Considering all the terms in the Hamiltonian in Eqs. (E.7)-(E.8), the Hamiltonian
can be written in a compact form 𝐻ˆ = 2ℏ 𝑘 ( v̂ 𝑘 ) † h (𝑘) v̂ 𝑘 with a vector composed

188
of mode operators v̂ 𝑘 = 𝑎ˆ 𝑘 , 𝑏ˆ 𝑘 , 𝑎ˆ †−𝑘 , 𝑏ˆ †−𝑘

𝑇

and

𝑔(𝑘) ª
© 𝑐∗ −𝑙 ∗
©𝜔0 (𝑘) 𝑓 (𝑘)
𝑓 ∗ (𝑘) 𝜔 (𝑘) 𝑔 ∗ (𝑘)
𝑘 𝑘
0 ®
h (𝑘) =
® = 𝜔0 (𝑘) 2
0
𝑔(𝑘) 𝜔0 (𝑘) 𝑓 (𝑘) ®
∗ ∗
0
−𝑐 𝑘 −𝑙 𝑘
∗ (𝑘)
∗ (𝑘) 𝜔 (𝑘)
« 2

𝑐 𝑘 −𝑙 𝑘

−𝑐 𝑘 −𝑙 𝑘

−𝑐∗𝑘 −𝑙 ∗𝑘

𝑐∗𝑘 −𝑙 ∗𝑘

−𝑐 𝑘 −𝑙 𝑘
2 ª

0 ®®
𝑐 𝑘 −𝑙 𝑘 ®
2 ®
1 ¬
(E.12)

where 𝑓 (𝑘) ≡ 𝑔𝐶 (𝑘) − 𝑔 𝐿 (𝑘) as before and 𝑔(𝑘) ≡ −𝑔𝐶 (𝑘) − 𝑔 𝐿 (𝑘). Here, 𝑙 𝑘 ≡
𝑀𝑔 (𝑘)/𝐿 0 and 𝑐 𝑘 ≡ 𝐶𝑔 (𝑘)/𝐶Σ are inductive and capacitive coupling normalized to
frequency. The dispersion relation can be found by diagonalizing the kernel of the
Hamiltonian in Eq. (E.12) with the Bogoliubov transformation
U 𝑘 V∗−𝑘
ŵ𝑘 = S𝑘 v̂𝑘 ,
S𝑘 =
(E.13)
V 𝑘 U∗−𝑘
where ŵ 𝑘 ≡ ( 𝑎ˆ +,𝑘 , 𝑎ˆ −,𝑘 , 𝑎ˆ †+,−𝑘 , 𝑎ˆ †−,−𝑘 )𝑇 is the vector composed of supermode operators and U 𝑘 , V 𝑘 are 2 × 2 matrices forming blocks in the transformation S 𝑘 . We
want to find S 𝑘 such that ( v̂ 𝑘 ) † h (𝑘) v̂ 𝑘 = ( ŵ 𝑘 ) † h̃ (𝑘) ŵ 𝑘 , where h̃ (𝑘) is diagonal.
To preserve the commutation relations, the matrix S 𝑘 has to be symplectic, satisfying J = S 𝑘 J(S 𝑘 ) † , with J = diag(1, 1, −1, −1). Due to this symplecticity, it can
be shown that the matrices Jh (𝑘) and Jh̃ (𝑘) are similar under transformation S 𝑘 .
Thus, finding the eigenvalues and eigenvectors of the coefficient matrix
© 𝑐∗ −𝑙 ∗
Jh (𝑘) 𝑘 2 𝑘
=
m (𝑘) ≡
𝜔0 (𝑘) 0

𝑐 𝑘 −𝑙 𝑘

−𝑐∗𝑘 −𝑙 ∗𝑘

−𝑐 𝑘 −𝑙 𝑘
2 ª

0 ®®
(E.14)
𝑐 𝑘 +𝑙 𝑘
−𝑐 𝑘 +𝑙 𝑘 ®
−1
2 ®
−𝑐∗𝑘 +𝑙 ∗𝑘
𝑐∗𝑘 +𝑙 ∗𝑘
−1
« 2
is sufficient to obtain the dispersion relation and supermodes of the system. The
eigenvalues of matrix m (𝑘) are evaluated as
𝑙 𝑘 𝑐∗𝑘 + 𝑙 𝑘∗ 𝑐 𝑘
𝑙 𝑘 𝑐∗𝑘 + 𝑙 𝑘∗ 𝑐 𝑘 2
± 1−
1−
− (1 − |𝑙 𝑘 | 2 )(1 − |𝑐 𝑘 | 2 )

and hence the dispersion relation of the system taking into account all terms in
Hamiltonian (E.12) is
2

𝐿 0 − 𝑀𝑔 (−𝑘)𝑀𝑔 (𝑘) 𝐶Σ2 − 𝐶𝑔 (−𝑘)𝐶𝑔 (𝑘)
𝜔˜ ± (𝑘) = 𝜔˜ 0 (𝑘) 1 ± 1 −
2
𝐿 0𝐶Σ − 12 𝑀𝑔 (−𝑘)𝐶𝑔 (𝑘) + 𝐶𝑔 (−𝑘) 𝑀𝑔 (𝑘)
(E.15)

189
where
𝜔˜ 0 (𝑘) ≡ 𝜔0 (𝑘) 1 −

𝑀𝑔 (𝑘)𝐶𝑔 (−𝑘) + 𝑀𝑔 (−𝑘)𝐶𝑔 (𝑘)
2𝐿 0𝐶Σ

max
min
max
The two passbands range over frequencies [𝜔min
+ , 𝜔+ ] and [𝜔− , 𝜔− ], where
the band-edge frequencies are written as

𝜔min
+ = p
𝜔max
=p
𝜔min
− = p
𝜔max
=p

(E.16a)

(E.16b)

(E.16c)

(E.16d)

[𝐿 0 + 𝑝 2 (𝑀 𝑣 − 𝑀 𝑤 )] [𝐶Σ − 𝑝 2 (𝐶 𝑣 − 𝐶 𝑤 )]
[𝐿 0 + 𝑝 1 (𝑀 𝑣 + 𝑀 𝑤 )] [𝐶Σ − 𝑝 1 (𝐶 𝑣 + 𝐶 𝑤 )]
[𝐿 0 − 𝑝 1 (𝑀 𝑣 + 𝑀 𝑤 )] [𝐶Σ + 𝑝 1 (𝐶 𝑣 + 𝐶 𝑤 )]
[𝐿 0 − 𝑝 2 (𝑀 𝑣 − 𝑀 𝑤 )] [𝐶Σ + 𝑝 2 (𝐶 𝑣 − 𝐶 𝑤 )]

Here, 𝑝 1 ≡ sgn[𝐿 0 (𝐶𝑣 +𝐶𝑤 ) −𝐶Σ (𝑀𝑣 + 𝑀𝑤 )] and 𝑝 2 ≡ sgn[𝐿 0 (𝐶𝑣 −𝐶𝑤 ) −𝐶Σ (𝑀𝑣 −
𝑀𝑤 )] are sign factors. In principle, the eigenvectors of the matrix m (𝑘) in Eq. (E.14)
can be analytically calculated to find the transformation S 𝑘 of the original modes to
supermodes 𝑎ˆ ±,𝑘 . For the sake of brevity, we perform the calculation in the limit of
vanishing mutual inductance (𝑀𝑣 = 𝑀𝑤 = 0), where the matrix m (𝑘) reduces to
𝑐 𝑘 /2
−𝑐 𝑘 /2ª
© 1
𝑐∗ /2
−𝑐∗𝑘 /2
0 ®®
m𝐶 (𝑘) ≡ 𝑘
®.
/2
−1
−𝑐
/2
∗ /2
∗ /2
−𝑐
−1
« 𝑘

(E.17)

In this case, the block matrices U 𝑘 , V 𝑘 in the transformation in Eq. (E.13) are written
as
−𝑖𝜙(𝑘) 𝑥
−𝑖𝜙(𝑘) 𝑦
+,𝑘
+,𝑘
+,𝑘
+,𝑘
, V𝑘 = √
U𝑘 = √
−𝑖𝜙(𝑘)
−𝑖𝜙(𝑘)
𝑥 −,𝑘 𝑥 −,𝑘
𝑦 −,𝑘 𝑦 −,𝑘
2 2 −𝑒
2 2 −𝑒
p4
1 ± |𝑐 𝑘 | − √
where 𝑥±,𝑘 = 4 1 ± |𝑐 𝑘 | + √
, and 𝜙(𝑘) = arg 𝑐 𝑘 .
±,𝑘
1±|𝑐 𝑘 |

1±|𝑐 𝑘 |
Note that the constants are normalized by relation 𝑥 ±,𝑘 − 𝑦 2±,𝑘 = 4.

The knowledge of the transformation S 𝑘 allows us to evaluate the Zak phase of
photonic bands. In the Bogoliubov transformation, the Zak phase can be evaluated

190
as [351]
±𝑒𝑖𝜙(𝑘) 𝑥 ±,𝑘 ª
®
 1 𝑥±,𝑘
1 −𝑖𝜙(𝑘)
®
 √
Z=𝑖
d𝑘 √ ±𝑒
®
𝑥 ±,𝑘 𝑥 ±,𝑘 ±𝑒 −𝑖𝜙(𝑘) 𝑦 ±,𝑘 𝑦 ±,𝑘 · J ·
𝑖𝜙(𝑘)
®
𝜕𝑘  2 2 ±𝑒
B.Z.
2 2
±,𝑘
®
±,𝑘
¬
∮ 
1 𝜕𝜙(𝑘) 2
𝜕 2
𝜕𝜙(𝑘)
=𝑖
d𝑘
(𝑥 ±,𝑘 − 𝑦 2±,𝑘 ) +
(𝑥 ±,𝑘 − 𝑦 2±,𝑘 ) = −
d𝑘
𝜕𝑘
𝜕𝑘
2 B.Z.
𝜕𝑘
B.Z.
identical to the expression within the RWA. Again, the Zak phase of photonic bands
is determined by the winding of 𝑓 (𝑘) around the origin in complex plane, leading
to Z = 0 in the trivial phase and Z = 𝜋 in the topological phase.
Extraction of circuit parameters and the breakdown of the circuit model
As discussed in Fig. 6.1d, the parameters in the circuit model of the topological
waveguide is found by fitting the waveguide transmission spectrum of the test
structures. We find that two lowest-frequency modes inside the lower passband fail
to be captured according to our model with capacitively and inductively coupled
LC resonators. We believe that this is due to the broad range of frequencies (about
1.5 GHz) covered in the spectrum compared to the bare resonator frequency ∼
6.6 GHz and the distributed nature of the coupling, which can cause our simple
model based on frequency-independent lumped elements (inductor, capacitor, and
mutual inductance) to break down. Such deviation is also observed in the fitting of
waveguide transmission data of Device I (Fig. E.7).
E.2

Mapping of the system to the SSH model and discussion on robustness of
edge modes

Mapping of the topological waveguide to the SSH model
We discuss how the physical model of topological waveguide in App. E.1 could
be mapped to the photonic SSH model, whose Hamiltonian is given as Eq. (6.1).
Throughout this section, we consider the realistic circuit parameters extracted from
fitting of test structures given in Fig. 6.1: resonator inductance and resonator capacitance, 𝐿 0 = 1.9 nH and 𝐶0 = 253 fF, and coupling capacitance and parasitic
mutual inductance, (𝐶𝑣 , 𝐶𝑤 ) = (33, 17) fF and (𝑀𝑣 , 𝑀𝑤 ) = (−38, −32) pH in the
trivial phase (the values are interchanged in the topological phase).
To most directly and simply link the Hamiltonian described in Eqs. (E.7)-(E.8) to the
SSH model, here we impose a few approximations. First, the counter-rotating terms
in the Hamiltonian are discarded such that only photon-number-conserving terms

191

Frequency (GHz)

7.5
Full Model

Within RWA
Final Mapping
to SSH Model

6.5

−π

Figure E.2: Band structure of the realized topological waveguide under various assumptions discussed in App. E.2. The solid lines show the dispersion relation in the upper
(lower) passband, 𝜔± (𝑘): full model without any assumptions (red), model within RWA
(blue), and the final mapping to SSH model (black) in the weak coupling limit. The dashed
lines indicate the uncoupled resonator frequency 𝜔0 (𝑘) under corresponding assumptions.

are left. To achieve this, the RWA is applied to reduce the kernel of the Hamiltonian
into one involving a 2 × 2 matrix as in Eq. (E.10). Such an assumption is known
to be valid when the coupling terms in the Hamiltonian are much smaller than the
frequency scale of the uncoupled Hamiltonian 𝐻ˆ 0 [352]. According to the coupling
terms derived in Eq. (E.9), this is a valid approximation given that
𝑔𝐶 (𝑘)
|𝐶𝑣 + 𝐶𝑤 |
≈ 0.083,
𝜔0 (𝑘)
2𝐶Σ

𝑔 𝐿 (𝑘)
|𝑀𝑣 | + |𝑀𝑤 |
≈ 0.018.
𝜔0 (𝑘)
2𝐿 0

and the RWA affects the dispersion relation by less than 0.3 % in frequency.
Also different than in the original SSH Hamiltonian, are the 𝑘-dependent diagonal elements 𝜔0 (𝑘) of the single-particle kernel of the Hamiltonian for the circuit
model. This 𝑘-dependence can be understood as arising from the coupling between resonators beyond nearest-neighbor pairs, which is inherent in the canonical
quantization of capacitively coupled LC resonator array (due to circuit topology)
as discussed in Ref. [153]. The variation in 𝜔0 (𝑘) can be effectively suppressed
in the limit of 𝐶𝑣 , 𝐶𝑤
𝐶Σ and 𝑀𝑣 , 𝑀𝑤
𝐿 0 as derived in Eq. (E.6). We note
that while our coupling capacitances are small compared to 𝐶Σ (𝐶𝑣 /𝐶Σ ≈ 0.109,
𝐶𝑤 /𝐶Σ ≈ 0.056 in the trivial phase), we find that they are sufficient to cause the
𝜔0 (𝑘) to vary by ∼1.2 % in the first Brillouin zone. Considering this limit of small
coupling capacitance and mutual inductance, the effective capacitance and inductance of (E.6) become quantities independent of 𝑘, 𝐶0eff (𝑘) ≈ 𝐶Σ , 𝐿 0eff (𝑘) ≈ 𝐿 0 , and

7.5

∆fedge (MHz)

Frequency (GHz)

192

6.5

Mode Index

100
10-10
10-20
10 20 30 40 50

# of unit cells, N

Figure E.3: Eigenspectrum of the finite-sized topological circuit. a, Resonant frequencies of a finite system with 𝑁 = 40 unit cells, calculated from eigenmodes of Eq. (E.19). The
bandgap regions calculated from dispersion relation are shaded in gray (green) for upper
and lower bandgaps (middle bandgap). The two data points inside the middle bandgap
(mode indices 40 and 41) correspond to edge modes. b, Frequency splitting Δ 𝑓edge of edge
modes with no disorder in the system are plotted against the of number of unit cells 𝑁. The
black solid curve indicates exponential fit to the edge mode splitting, with decay constant of
𝜉 = 1.76.

the kernel of the Hamiltonian under RWA reduces to
𝜔0
𝑓 (𝑘)
h(𝑘) = ∗
𝑓 (𝑘) 𝜔0
Here,
𝜔0 = √
𝐿 0𝐶Σ

𝜔0
𝑓 (𝑘) =

𝐶 𝑣 𝑀𝑣
𝐶𝑤 𝑀𝑤 −𝑖𝑘 𝑑
𝐶Σ 𝐿 0
𝐶Σ
𝐿0

This is equivalent to the photonic SSH Hamiltonian in Eq. (6.1) under redefinition of
gauge which transforms operators as ( 𝑎ˆ 𝑘 , 𝑏ˆ 𝑘 ) → ( 𝑎ˆ 𝑘 , −𝑏ˆ 𝑘 ). Here, we can identify
the parameters 𝐽 and 𝛿 as
𝜔 0 𝐶 𝑣 + 𝐶 𝑤 𝑀𝑣 + 𝑀 𝑤
𝐿 0 (𝐶𝑣 − 𝐶𝑤 ) − 𝐶Σ (𝑀𝑣 − 𝑀𝑤 )
𝐽=
, 𝛿=
, (E.18)
𝐶Σ
𝐿0
𝐿 0 (𝐶𝑣 + 𝐶𝑤 ) − 𝐶Σ (𝑀𝑣 + 𝑀𝑤 )
where 𝐽 (1 ± 𝛿) is defined as intra-cell and inter-cell coupling, respectively. The
dispersion relations under different stages of approximations mentioned above are
plotted in Fig. E.2, where we find a clear deviation of our system from the original
SSH model due to the 𝑘-dependent reference frequency.
Robustness of edge modes under perturbation in circuit parameters
While we have linked our system to the SSH Hamiltonian in Eq. (6.1), we find that
our system fails to strictly satisfy chiral symmetry Ch(𝑘)C −1 = −h(𝑘) (C = 𝜎
ˆ 𝑧 is
the chiral symmetry operator in the sublattice space). This is due to the 𝑘-dependent

193
diagonal 𝜔0 (𝑘) terms in h(𝑘), resulting from the non-local nature of the quantized
charge and nodal flux in the circuit model which results in next-nearest-neighbor
coupling terms between sublattices of the same type. Despite this, an inversion
symmetry, Ih(𝑘)I −1 = h(−𝑘) (I = 𝜎
ˆ 𝑥 in the sublattice space), still holds for the
circuit model. This ensures the quantization of the Zak phase (Z) and the existence
of an invariant band winding number (𝜈 = Z/𝜋) for perturbations that maintain
the inversion symmetry. However, as shown in Refs. [353, 354], the inversion
symmetry does not protect the edge states for highly delocalized coupling along the
dimer resonator chain, and the correspondence between winding number and the
number of localized edge states at the boundary of a finite section of waveguide is
not guaranteed.
For weak breaking of the chiral symmetry (i.e., beyond nearest-neighbor coupling
much smaller than nearest neighbor coupling) the correspondence between winding
number and the number of pairs of gapped edge states is preserved, with winding
number 𝜈 = 0 in the trivial phase (𝛿 > 0) and 𝜈 = 1 in the topological (𝛿 < 0) phase.
Beyond just the existence of the edge states and their locality at the boundaries, chiral
symmetry is special in that it pins the edge mode frequencies at the center of the
middle bandgap (𝜔0 ). Chiral symmetry is maintained in the presence of disorder
in the coupling between the different sublattice types along the chain, providing
stability to the frequency of the edge modes. In order to study the robustness of the
edge mode frequencies in our circuit model, we perform a simulation over different
types of disorder realizations in the circuit illustrated in Fig. E.1. As the original
SSH Hamiltonian with chiral symmetry gives rise to topological edge states which
are robust against the disorder in coupling, not in on-site energies [246], it is natural
to consider disorder in circuit elements that induce coupling between resonators:
𝐶 𝑣 , 𝐶 𝑤 , 𝑀𝑣 , 𝑀 𝑤 .
The classical equations of motion of a circuit consisting of 𝑁 unit cells is written as
d𝑖 𝑛A
(𝑛) d𝑖 𝑛
(𝑛) d𝑖 𝑛−1
+ 𝑀𝑣
+ 𝑀𝑤
𝑉𝑛 = 𝐿 0

d𝑡
d𝑡
d𝑡
d𝑖
d𝑖
(𝑛)
(𝑛) d𝑖 𝑛
𝑛+1
𝑉𝑛 = 𝐿 0
+ 𝑀𝑤 ,
+ 𝑀𝑣
d𝑡
d𝑡
d𝑡

(𝑛) d𝑉𝑛
(𝑛) d𝑉𝑛
(𝑛−1) d𝑉𝑛−1
𝑖 𝑛 = −𝐶Σ,A
+ 𝐶𝑣
+ 𝐶𝑤

d𝑡
d𝑡
d𝑡
d𝑉
d𝑉
d𝑉
(𝑛)
𝑖 𝑛B = −𝐶Σ,B
+ 𝐶𝑣(𝑛) 𝑛 + 𝐶𝑤(𝑛) 𝑛+1 ,
d𝑡
d𝑡
d𝑡

where the superscripts indicate index of cell of each circuit element and
(𝑛)
𝐶Σ,A
= 𝐶0 + 𝐶𝑣(𝑛) + 𝐶𝑤(𝑛−1) ,

(𝑛)
𝐶Σ,B
= 𝐶0 + 𝐶𝑣(𝑛) + 𝐶𝑤(𝑛) .

194
The 4𝑁 coupled differential equations are rewritten in a compact form as
© 1ª
© 1ª
u ®
d u2 ®
−1 2 ®
® = C . ®,
.. ®
d𝑡 ... ®
®
®
«u 𝑁 ¬
«u𝑁 ¬

(E.19)

where the coefficient matrix C is given by

𝑀 (1)

© (1)
(1)
−𝐶
Σ,A
𝑀 𝑣(1)
𝐿0
𝑀 𝑤(1)
(1)
(1)
(1)
..
𝐶𝑣
−𝐶
Σ,B
..
..
(1)
𝑀𝑤
C≡
®.
..
..
..
(1)
𝐶𝑤
..
..
..
𝑀 𝑣( 𝑁 ) ®
..
..
(𝑁)
..
(𝑁)
0 ®
(𝑁)
𝐶 𝑣( 𝑁 )
−𝐶Σ,B
0 ¬

Here, the matrix elements not specified are all zero. The resonant frequencies of the
system can be determined by finding the positive eigenvalues of 𝑖C−1 . Considering
the model without any disorder, we find the eigenfrequencies of the finite system to
be distributed according to the passband and bandgap frequencies from dispersion
relation in Eq. (E.15), as illustrated in Fig. E.3. Also, we observe the presence of a
pair of coupled edge mode resonances inside the middle bandgap in the topological
phase, whose splitting due to finite system size scales as Δ 𝑓edge ∼ 𝑒 −𝑁/𝜉 with
𝜉 = 1.76.
To discuss the topological protection of the edge modes, we keep track of the set
of eigenfrequencies for different disorder realizations of the coupling capacitance
and mutual inductance for a system with 𝑁 = 50 unit cells. First, we consider the
case when the mutual inductance 𝑀𝑣 and 𝑀𝑤 between resonators are subject to
disorder. The values of 𝑀𝑣(𝑛) , 𝑀𝑤(𝑛) are assumed to be sampled uniformly on an
interval covering a fraction ±𝑟 of the original values, i.e.,
(𝑛)
(𝑛)
(𝑛)
(𝑛)
𝑀𝑣 = 𝑀𝑣 1 + 𝑟 𝛿 𝑀 𝑣 , 𝑀 𝑤 = 𝑀 𝑤 1 + 𝑟 𝛿 𝑀 𝑤 ,
(𝑛) ˜ (𝑛)
where 𝛿˜𝑀
, 𝛿 𝑀𝑤 are independent random numbers uniformly sampled from an inter𝑣
val [−1, 1]. Figure E.4a illustrates an example with a strong disorder with 𝑟 = 0.5

195
7.4

7.4

7.2

6.8
6.6
6.4

Frequency (GHz)

c 7.6

Frequency (GHz)

b 7.6

Frequency (GHz)

a 7.6

6.8
6.6
6.4

6.2

80 100

Disorder Realization

80 100

Disorder Realization

80 100

Disorder Realization

Figure E.4: Eigenfrequencies of the system under 100 disorder realizations in coupling
elements. Each disorder realization is achieved by uniformly sampling the parameters within
fraction ±𝑟 of the original value. a, Disorder in mutual inductance 𝑀 𝑣 and 𝑀 𝑤 between
neighboring resonators with the strength 𝑟 = 0.5. b, Disorder in coupling capacitance 𝐶 𝑣
and 𝐶 𝑤 between neighboring resonators with the strength 𝑟 = 0.1. c, The same disorder
as panel b with 𝑟 = 0.5, while keeping the bare self-capacitance 𝐶Σ of each resonator fixed
(correlated disorder between coupling capacitances and resonator 𝐶0 ).

under 100 independent realizations, where we find the frequencies of the edge
modes to be stable, while frequencies of modes in the passbands fluctuate to a much
larger extent. This suggests that the frequencies of edge modes have some sort of
added robustness against disorder in the mutual inductance between neighboring
resonators despite the fact that our circuit model does not satisfy chiral symmetry.
The reduction in sensitivity results from the fact that the effective self-inductance
𝐿 0eff (𝑘) of the resonators, which influences the on-site resonator frequency, depends
on the mutual inductances only to second-order in small parameter (𝑀𝑣,𝑤 /𝐿 0 ). It
is this second-order fluctuation in the resonator frequencies, causing shifts in the
diagonal elements of the Hamiltonian, which results in fluctuations in the edge mode
frequencies. The direct fluctuation in the mutual inductance couplings themselves,
corresponding to off-diagonal Hamiltonian elements, do not cause the edge modes
to fluctuate due to chiral symmetry protection (the off-diagonal part of the kernel of
the Hamiltonian is chiral symmetric).
Disorder in coupling capacitance 𝐶𝑣 and 𝐶𝑤 are also investigated using a similar
model, where the values of 𝐶𝑣(𝑛) , 𝐶𝑤(𝑛) are allowed to vary by a fraction ±𝑟 of the
original values (uniformly sampled), while the remaining circuit parameters are
kept constant. From Fig. E.4b we observe severe fluctuations in the frequencies
of the edge modes even under a mild disorder level of 𝑟 = 0.1. This is due to
the fact that the coupling capacitance 𝐶𝑣 and 𝐶𝑤 contribute to the effective self-

196
Γ0/2𝜋
(kHz)

QB1

QB2

QB3

QB4

QB5

QB6

QB7

326

150

247

105a 268

183

221

224

193

263

206

333

347

a Measured in a separate cooldown

Table E.1: Qubit coherence in the middle bandgap. The parasitic decoherence rate Γ0 of
qubits on Device I at 6.621 GHz inside the MBG. The data for QB2 was taken in a separate
cooldown due to coupling to a two-level system defect.

capacitance of each resonator 𝐶0eff (𝑘) to first-order in small parameter (𝐶𝑣,𝑤 /𝐶0 ),
thus directly breaking chiral symmetry and causing the edge modes to fluctuate.
An interesting observation in Fig. E.4b is the stability of frequencies of modes in
the upper passband with respect to disorder in 𝐶𝑣 and 𝐶𝑤 . This can be explained
by noting the expressions for band-edge frequencies in Eqs. (E.16a)-(E.16d), where
the dependence on coupling capacitance gets weaker close to the upper band-edge
frequency 𝜔max
= 1/ (𝐿 0 + 𝑀𝑣 + 𝑀𝑤 )𝐶0 of the upper passband.
Finally, we consider a special type of disorder where we keep the bare selfcapacitance 𝐶Σ of each resonator fixed. Although unrealistic, we allow 𝐶𝑣 and
𝐶𝑤 to fluctuate and compensate for the disorder in 𝐶Σ by subtracting the deviation
in 𝐶𝑣 and 𝐶𝑤 from 𝐶0 . This suppresses the lowest-order resonator frequency fluctuations, and hence helps stabilize the edge mode frequencies even under strong
disorder 𝑟 = 0.5, as illustrated in Fig. E.4c. While being an unrealistic model for
disorder in our physical system, this observation sheds light on the fact that the
circuit must be carefully designed to take advantage of the topological protection.
It should also be noted that in all of the above examples, the standard deviation in
the edge mode frequencies scale linearly to lowest order with the standard deviation of the disorder in the inter- and intra-cell coupling circuit elements (only the
pre-coefficient changes). Exponential suppression of edge mode fluctuations due to
disorder in the coupling elements as afforded by the SSH model with chiral symmetry would require a redesign of the circuit to eliminate the next-nearest-neighbor
coupling present in the current circuit layout.
E.3

Device I characterization and Experimental setup

In this section, we provide a detailed description of elements on Device I, where
the directional qubit-photon bound state and passband topology experiments are
performed. The optical micrograph of Device I is shown in Fig. E.5.

197
QB1

QB2

QB3

QB4

QB5

QB6

QB7
P2

P1
QA1

QA2

QA3

QA4

QA5

QA6

QA7
500 μm

Figure E.5: Optical micrograph of Device I (false-colored). The device consists of a
topological waveguide with 9 unit cells (resonators corresponding to A/B sublattice colored
red/blue) in the trivial phase, where the intra-cell coupling is larger than the inter-cell
coupling. Qubits (cyan, labeled Q 𝛼𝑗 where 𝑖=1-7 and 𝛼=A,B) are coupled to every site of the
seven inner unit cells of the topological waveguide, each connected to on-chip flux-bias lines
(orange) for individual frequency control. At the boundary of the topological waveguide are
tapering sections (purple), which provide impedance matching to the external waveguides
(green) at upper bandgap frequencies. P1 (P2) denotes port 1 (port 2) of the device.

Qubits
All 14 qubits on Device I are designed to be nominally identical with asymmetric
Josephson junctions (JJs) on superconducting quantum interference device (SQUID)
loop to reduce the sensitivity to flux noise away from maximum and minimum
frequencies, referred to as “sweet spots”. The sweet spots of all qubits lie deep
inside the upper and lower bandgaps, where the coupling of qubits to external
ports are small due to strong localization. This makes it challenging to access
the qubits with direct spectroscopic methods near the sweet spots. Alternatively,
a strong drive tone near resonance with a given qubit frequency was sent into the
waveguide to excite the qubit, and a passband mode dispersively coupled to the
qubit is simultaneously monitored with a second probe tone. With this method,
the lower (upper) sweet spot of QA
1 is found to be at 5.22 GHz (8.38 GHz), and
the anharmonicity near the upper sweet spot is measured to be 297 MHz (effective
qubit capacitance of 𝐶q = 65 fF). The Josephson energies of two JJs of QA
1 are
extracted to be (𝐸 𝐽1 , 𝐸 𝐽2 )/ℎ = (21.85, 9.26) GHz giving the junction asymmetry of
𝐽2
𝑑 = 𝐸𝐸 𝐽𝐽 11−𝐸
+𝐸 𝐽 2 = 0.405.
The coherence of qubits is characterized using spectroscopy inside the middle
bandgap (MBG). Here, the parasitic decoherence rate is defined as Γ0 ≡ 2Γ2 − 𝜅 𝑒,1 −
𝜅 𝑒,2 , where 2Γ2 is the total linewidth of qubit, and 𝜅 𝑒,1 (𝜅 𝑒,2 ) is the external coupling
rate to port 1 (2) (see Supplementary Note 1 of Ref. [108] for a detailed discussion).
Here, Γ0 contains contributions from both qubit decay to spurious channels other
than the desired external waveguide as well as pure dephasing. Table E.1 shows the

198
parasitic decoherence rate of all 14 qubits at 6.621 GHz extracted from spectroscopic
measurement at a power at least 5 dB below the single-photon level (defined as
ℏ𝜔𝜅 𝑒,𝑝 with 𝑝 = 1, 2) from both ports.
Utilizing the dispersive coupling between the qubit and a resonator mode in the
passband, we have also performed time-domain characterization of qubits. The
measurement on QB4 at 6.605 GHz in the MBG gives 𝑇1 = 1.23 𝜇s and 𝑇2∗ = 783 ns
corresponding to Γ0/2𝜋 = 281.3 kHz, consistent with the result from spectroscopy
in Table E.1. At the upper sweet spot, QB4 was hard to access due to the small
coupling to external ports arising from short localization length and a large physical
distance from the external ports. Instead, QB1 is characterized to be 𝑇1 = 9.197 𝜇s
and 𝑇2∗ = 11.57 𝜇s at its upper sweet spot (8.569 GHz).
Waveguide
Input 1

Z control

64
kHz

20 dB

Waveguide
Input 2

HEMT 1

20 dB

40 dB

Waveguide Waveguide
Output 1 Output 2

HEMT 2

300 K
50 K plate
4 K plate
Mixing
chamber
plate

40 dB

3.5
GHz

DC block
Attenuator
Thin-film attenuator
Eccosorb filter
Low-pass filter
High-pass filter

1.9
MHz

3.5
GHz

20 dB

Band-pass filter
2×2 switch

30 dB

Circulator
TWPA
4-8
GHz

HEMT amplifier

4-8
GHz

Figure E.6: Schematic of the measurement setup inside the dilution refrigerator for
Device I. The meaning of each symbol in the schematic on the left is enumerated on the right.
The level of attenuation of each attenuator is indicated with number next to the symbol. The
cutoff frequencies of each filter is specified with numbers inside the symbol. Small squares
attached to circulator symbols indicate port termination with 𝑍0 = 50 Ω, allowing us to use
the 3-port circulator as a 2-port isolator. The input pump line for TWPA is not shown in the
diagram for simplicity.

Metamaterial waveguide and coupling to qubits
As shown in Fig. E.5, the metamaterial waveguide consists of a SSH array in the
trivial configuration and tapering sections at the boundary (the design of tapering
sections is discussed in App. E.4). The array contains 18 identical LC resonators,
whose design is slightly different from the one in test structures shown in Fig. 6.1b.
Namely, the “claw” used to couple qubits to resonators on each site is extended to

199
generate a larger coupling capacitance of 𝐶𝑔 = 5.6 fF and the resonator capacitance
to ground was reduced accordingly to maintain the designed reference frequency.
On resonator sites where no qubit is present, an island with shape identical to
that of a qubit was patterned and shorted to ground plane in order to mimic the
self-capacitance contribution from a qubit to the resonator. The fitting of the whole
structure to the waveguide transmission spectrum results in a set of circuit parameters
similar yet slightly different from ones of the test structures quoted in Fig. 6.1:
(𝐶𝑣 , 𝐶𝑤 ) = (35, 19.2) fF, (𝑀𝑣 , 𝑀𝑤 ) = (−38, −32) pH, 𝐶0 = 250 fF, 𝐿 0 = 1.9 nH.
Here, the definition of 𝐶0 includes contributions from coupling capacitance between
qubit and resonator, but excludes the contribution to the resonator self-capacitance
from the coupling capacitances 𝐶𝑣 , 𝐶𝑤 between resonators in the array. With these
parameters we calculate the corresponding parameters in the SSH model to be
𝐽/2𝜋 = 356 MHz and 𝛿 = 0.256 following Eq. (E.18), resulting in the localization
1+𝛿 −1
)] = 1.91 at the reference frequency. From the measured
length 𝜉 = [ln( 1−𝛿
AB
avoided crossing 𝑔45 /2𝜋 = 32.9 MHz between qubit-photon bound states facing
toward each other on nearest-neighboring sites
q together with 𝐽 and 𝛿, we infer the
AB 𝐽 (1 + 𝛿) = 2𝜋×121.3MHz [55],
qubit coupling to each resonator site to be 𝑔 = 𝑔45
close to the value
𝐶𝑔
𝜔0 = 2𝜋 × 132 MHz
2 𝐶q𝐶Σ
expected from designed coupling capacitance [355]. Note that we find an inconsistent set of values 𝐽/2𝜋 = 368 MHz and 𝛿 = 0.282 (with 𝜉 = 1.73 and
𝑔/2𝜋 = 124.6 MHz accordingly) from calculation based on the difference in observed band-edge frequencies, where the frequency difference between the highest
frequency in the UPB and the lowest frequency in the LPB equals 4𝐽 and the size
of the MBG equals 4𝐽 |𝛿|. The inconsistency indicates the deviation of our system
from the proposed circuit model (see App. E.1 for discussion), which accounts for
the difference between theoretical curves and the experimental data in Fig. 6.1d and
left sub-panel of Fig. 6.2c. The values of 𝐽, 𝛿 and 𝑔 from the band-edge frequencies
are used to generate the theoretical curves in Fig. 6.3 as well as in Fig. E.11. The
intrinsic quality factor of one of the normal modes (resonant frequency 6.158 GHz)
of the metamaterial waveguide was measured to be 𝑄 𝑖 = 9.8 × 104 at power below
the single-photon level, similar to typical values reported in Refs. [109, 153].
Experimental setup
The measurement setup inside the dilution refrigerator is illustrated in Fig. E.6. All
the 14 qubits on Device I are DC-biased with individual flux-bias (Z control) lines,

200

C1g

C2g

C0
Mw

100 μm

|S21| (dB)

Tapering section

...

-20
-40
-60

6.5

Frequency (GHz)

7.5

Figure E.7: Tapering section of Device I. a, The circuit diagram of the tapering section
connecting a coplanar waveguide to the topological waveguide. The coplanar waveguide,
first tapering resonator, and second tapering resonator are shaded in purple, yellow, and
green, respectively. b, Optical micrograph (false colored) of the tapering section on Device
I. The tapering section is colored in the same manner as the corresponding components
in panel a. c, Red: measured waveguide transmission spectrum |𝑆21 | for Device I. Black:
fit to the data with parameters (𝐶 𝑣 , 𝐶 𝑤 ) = (35, 19.2) fF, (𝑀 𝑣 , 𝑀 𝑤 ) = (−38, −32) pH,
(𝐶1𝑔 , 𝐶2𝑔 ) = (141, 35) fF, (𝐶1 , 𝐶2 ) = (128.2, 230) fF, 𝐶0 = 250 fF, 𝐿 0 = 1.9 nH.

filtered by a 64 kHz low-pass filter at the 4K plate and a 1.9 MHz low-pass filter
at the mixing chamber plate. The Waveguide Input 1 (2) passes through a series
of attenuators and filters including a 20 dB (30 dB) thin-film attenuator developed
in B. Palmer’s group [157]. It connects via a circulator to port 1 (2) of Device I,
which is enclosed in two layers of magnetic shielding. The output signals from
Device I are routed by the same circulator to the output lines containing a series of
circulators and filters. The pair of 2×2 switches in the amplification chain allows
us to choose the branch to be further amplified in the first stage by a traveling-wave
parametric amplifier (TWPA) from MIT Lincoln Laboratories. Both of the output
lines are amplified by an individual high electron mobility transistor (HEMT) at the
4K plate, followed by room-temperature amplifiers at 300 K. All four S-parameters
𝑆𝑖 𝑗 (𝑖, 𝑗 ∈ {1, 2}) involving port 1 and 2 on Device I can be measured with this
setup by choosing one of the waveguide input ports and one of the waveguide output
ports, e.g. 𝑆11 can be measured by sending the input signal into Waveguide Input 1
and collecting the output signal from Waveguide Output 2 with both 2×2 switches
in the cross (×) configuration.

201
E.4

Tapering sections on Device I

The finite system size of metamaterial waveguide gives rise to sharp resonances
inside the passband associated with reflection at the boundary (Fig. 6.1d). Also, the
decay rate of qubits to external ports inside the middle bandgap (MBG) is small,
making the spectroscopic measurement of qubits inside the MBG hard to achieve.
In order to reduce ripples in transmission spectrum inside the upper passband and
increase the decay rates of qubits to external ports comparable to their intrinsic
contributions inside the middle bandgap, we added two resonators at each end of
the metamaterial waveguide in Device I as tapering section.
Similar to the procedure described in Appendix C of Ref. [153], the idea is to increase the coupling capacitance gradually across the two resonators while keeping
the resonator frequency the same as other resonators by changing the self capacitance as well. However, unlike the simple case of an array of LC resonators with
uniform coupling capacitance, the SSH waveguide consists of alternating coupling
capacitance between neighboring resonators and two separate passbands form as
a result. In this particular work, the passband experiments are designed to take
place at the upper passband frequencies and hence we have slightly modified the
resonant frequencies of tapering resonators to perform impedance-matching inside
the upper passband. The circuit diagram shown in Fig. E.7a was used to model the
tapering section in our system. While designing of tapering sections involves empirical trials, microwave filter design software, e.g. iFilter module in AWR Microwave
Office [356], can be used to aid the choice of circuit parameters and optimization
method.
Figure E.7b shows the optical micrograph of a tapering section on Device I. The
circuit parameters are extracted by fitting the normalized waveguide transmission
spectrum (𝑆21 ) data from measurement with theoretical circuit models. We find a
good agreement in the frequency of normal modes and the level of ripples between
the theoretical model and the experiment as illustrated in Fig. E.7c. The level of
ripples in the transmission spectrum of the entire upper passband is about 8 dB
and decreases to below 2 dB near the center of the band, allowing us to probe the
cooperative interaction between qubits at these frequencies.
E.5

Directional shape of qubit-photon bound state

In this section, we provide detailed explanations on the directional shape of qubitphoton bound states discussed in Chapter 6. As an example, we consider a system

202

δ>0

nth unit cell

A B

Jw = J(1−δ)
A B

ω0
S1

Jv = J(1+δ)

S´1

S´2

ω0

S´1

S´2

Figure E.8: Understanding the directionality of qubit-photon bound states. a,
Schematic of the full system consisting of an infinite SSH waveguide with a qubit coupled to the A sublattice of the 𝑛-th unit cell and tuned to frequency 𝜔0 in the center of the
MBG. Here we make the unit cell choice in which the waveguide is in the trivial phase
(𝛿 > 0). b, Division of system in panel a into two subsystems S1 and S2 in Description I.
c Division of system in panel a into three subsystems [qubit (Q), S10 , S20 ] in Description II.
For panels b and c, the left side shows the schematic of the division into subsystems and the
right side illustrates the mode spectrum of the subsystems and the coupling between them.

consisting of a topological waveguide in the trivial phase and a qubit coupled to
the A sublattice of the 𝑛-th unit cell (Fig. E.8a). Our descriptions are based on
partitioning the system into subsystems under two alternative pictures (Fig. E.8b,c),
where the array is divided on the left (Description I) or the right (Description II) of
the site (𝑛, A) where the qubit is coupled to.
Description I
We divide the array into two parts by breaking the inter-cell coupling 𝐽𝑤 = 𝐽 (1 − 𝛿)
that exists on the left of the site (𝑛, A) where the qubit is coupled to, i.e., between
sites (𝑛 − 1, B) and (𝑛, A). The system is described in terms of two subsystems
S1 and S2 as shown in Fig. E.8b. The subsystem S1 is a semi-infinite array in the
trivial phase extended from the (𝑛 − 1)-th unit cell to the left and the subsystem S2
comprising a qubit and a semi-infinite array in the trivial phase extended from the
𝑛-th unit cell to the right. The coupling between the two subsystems is interpreted to
take place at a boundary site with coupling strength 𝐽𝑤 . When the qubit frequency
is resonant to the reference frequency 𝜔0 , the subsystem S2 can be viewed as a semiinfinite array in the topological phase, where the qubit effectively acts as an edge
site. Here, the resulting topological edge mode of subsystem S2 is the qubit-photon
bound state, with photon occupation mostly on the qubit itself and on every B site
with a decaying envelope. Coupling of subsystem S2 to S1 only has a minor effect

203
Ext. Coupling Rate κe,p /2π (MHz)

κe,1 /2π

κe,2 /2π

A B A B A B A B A B A B A B

Qubit Index

A B A B A B A B A B A B A B

Qubit Index

Figure E.9: External coupling rate of qubit-photon bound states away from the reference frequency. a, Upper (Lower) plots: external coupling rate of the qubit-photon bound
states to port 1 (2) at 6.72 GHz in the middle bandgap. Exponential fit (black curve) on the
data gives the localization length of 𝜉 = 2. b, Upper (Lower) plots: external coupling rate of
the qubit-photon bound states to port 1 (2) at 7.485 GHz in the upper bandgap. Exponential
fit (black curve) on the data gives the localization length of 𝜉 = 1.8. The localization
lengths are represented in units of lattice constant. For all panels, the error bars show 95%
confidence interval and are removed on data points whose error is smaller than the marker
size.

on the edge mode of S2 as the modes in subsystem S1 are concentrated at passband
frequencies, far-detuned from 𝜔 = 𝜔0 . Also, the presence of an edge state of S2
at 𝜔 = 𝜔0 cannot induce an additional occupation on S1 by this coupling in a way
that resembles an edge state since the edge mode of S2 does not occupy sites on
the A sublattice. The passband modes S1 and S2 near-resonantly couple to each
other, whose net effect is redistribution of modes within the passband frequencies.
Therefore, the qubit-photon bound state can be viewed as a topological edge mode for
subsystem S2 which is unperturbed by coupling to subsystem S1 . The directionality
and photon occupation distribution along the resonator chain of the qubit-photon
bound state can be naturally explained according to this picture.
Description II
In this alternate description, we divide the array into two parts by breaking the
intra-cell coupling 𝐽𝑣 = 𝐽 (1 + 𝛿) that exists on the right of the site (𝑛, A) where
the qubit is coupled to, i.e., between sites (𝑛, A) and (𝑛, B). We consider the
division of the system into three parts: the qubit, subsystem S01 , and subsystem S02
as illustrated in Fig. E.8c. Here, the subsystem S01 (S02 ) is a semi-infinite array in
the topological phase extended to the left (right), where the last site hosting the
topological edge mode E01 (E02 ) at 𝜔 = 𝜔0 is the A (B) sublattice of the 𝑛-th unit cell.
The subsystem S01 is coupled to both the qubit and the subsystem S02 with coupling

204
strength 𝑔 and 𝐽𝑣 = 𝐽 (1 + 𝛿), respectively. Similar to Description I, the result of
coupling between subsystem modes inside the passband is the reorganization of
modes without significant change in the spectrum inside the middle bandgap. On
the other hand, modes of the subsystems at 𝜔 = 𝜔0 (qubit, E01 , and E02 ) can be
viewed as emitters coupled in a linear chain configuration, whose eigenfrequencies
and corresponding eigenstates in the single-excitation manifold are given by
˜𝑣
𝜔˜ ± = 𝜔0 ± 𝑔˜ 2 + 𝐽˜𝑣2 , |𝜓± i = √ p
|100i ± |010i + p
|001i ,
𝑔˜ 2 + 𝐽˜𝑣2
𝑔˜ 2 + 𝐽˜𝑣2
and
𝜔˜ 0 = 𝜔0 ,

|𝜓0 i = p

𝑔˜ 2 + 𝐽˜𝑣2

𝐽˜𝑣 |100i − 𝑔|001i

where |𝑛1 𝑛2 𝑛3 i denotes a state with (𝑛1 , 𝑛2 , 𝑛3 ) photons in the (qubit, E01 , E02 ),
respectively. Here, 𝑔˜ (𝐽˜𝑣 ) is the coupling between edge mode E01 and the qubit (edge
mode E02 ), diluted from 𝑔 (𝐽𝑣 ) due to the admixture of photonic occupation on sites
other than the boundary in the edge modes. Note that in the limit of short localization
length, we recover 𝑔˜ ≈ 𝑔 and 𝐽˜𝑣 ≈ 𝐽𝑣 . Among the three single-excitation eigenstates,
the states |𝜓± i lie at frequencies of approximately 𝜔0 ± 𝐽, and are absorbed into the
passbands. The only remaining state inside the middle bandgap is the state |𝜓0 i,
existing exactly at 𝜔 = 𝜔0 , which is an anti-symmetric superposition of qubit excited
state and the single-photon state of E02 , whose photonic envelope is directed to the
right with occupation on every B site. This accounts for the directional qubit-photon
bound state emerging in this scenario.
E.6

Coupling of qubit-photon bound states to external ports at different frequencies

As noted in Chapter 6 (Fig. 6.2), the perfect directionality of the qubit-photon bound
states is achieved only at the reference frequency 𝜔0 inside the middle bandgap. In
this section, we discuss the breakdown of the observed perfect directionality when
qubits are tuned to different frequencies inside the middle bandgap by showing the
behavior of the external coupling 𝜅 e,𝑝 (𝑝 = 1, 2) to the ports.
Inside the middle bandgap, detuned from the reference frequency
Figure E.9a shows the external coupling rate of qubits to the ports at 6.72 GHz, a
frequency in the middle bandgap close to band-edge. The alternating behavior of
external coupling rate is still observed, but with a smaller contrast than in Fig. 6.2
of Chapter 6. The dependence of external linewidth on qubit index still exhibits the

7.1

|S21|

|D’〉

7 iii.

ii.
i.

6.9

|S21|

Qubit Frequency (GHz)

205

|gg〉

|D〉

|B’〉

ii.

|B〉
|gg〉

|B’’〉

|D’’〉

iii.

|gg〉

0.5

6.9

Frequency (GHz)

7.1

6.9

6.95

Frequency (GHz)

6.9

6.95

Frequency (GHz)

6.9

6.95

Frequency (GHz)

Figure E.10: Understanding the swirl pattern. a, Zoomed-in view of the swirl feature
near 6.95 GHz of the experimental data illustrated in Fig. 6.3c. b, Transmission spectrum
across two-qubit resonance for three different frequency tunings, corresponding to line cuts
marked with green dashed lines on panel a. The insets to panel b show the corresponding
level diagram with |ggi denoting both qubits in ground states and |Bi (|Di) representing
the perfect bright (dark) state. The state notation with prime (double prime) in sub-panel
i. (iii.) denotes the imperfect super-radiant bright state and sub-radiant dark state, with the
width of orange arrows specifying the strength of the coupling of states to the waveguide
channel. The sub-panel ii. occurs at the center of the swirl, where perfect super-radiance
and sub-radiance takes place (i.e., bright state waveguide coupling is maximum and dark
state waveguide coupling is zero). The black and red curves correspond to experimental
data and theoretical fit, respectively.

remaining directionality with qubits on A (B) sublattice maintaining large coupling
to port 2 (1), while showing small non-zero coupling to the opposite port.
Inside the upper bandgap
Inside the upper bandgap (7.485 GHz), the coupling of qubit-photon bound states to
external ports decreases monotonically with the distance of the qubit site to the port,
regardless of which sublattice the qubit is coupled to (Fig. E.9b). This behavior
is similar to that of qubit-photon bound states formed in a structure with uniform
coupling, where bound states exhibit a symmetric photonic envelope surrounding
the qubit. Note that we find the external coupling to port 2 (𝜅e,2 ) to be generally
smaller than that to port 1 (𝜅e,1 ), which may arise from a slight impedance mismatch
on the connection of the device to the external wiring.
E.7

Probing band topology with qubits

Signature of perfect super-radiance
Here we take a closer look at the swirl pattern in the waveguide transmission
spectrum—a signature of perfect super-radiance—which is discussed in Fig. 6.3c of
Chapter 6. In Fig. E.10 we zoom in to the observed swirl pattern near 6.95 GHz, and
three horizontal line cuts. At the center of this pattern (sub-panel ii. of Fig. E.10b),

206
the two qubits form perfect super-/sub-radiant states with maximized correlated
decay and zero coherent exchange interaction [116, 147]. At this point, the transmission spectrum shows a single Lorentzian lineshape (perfect super-radiant state
and bright state) with linewidth equal to the sum of individual linewidths of the
coupled qubits. The perfect sub-radiant state (dark state), which has no external
coupling, cannot be accessed from the waveguide channel here and is absent in
the spectrum. Slightly away from this frequency, the coherent exchange interaction
starts to show up, making hybridized states |B0i, |D0i formed by the interaction of
the two qubits. In this case, both of the hybridized states have non-zero decay rate
to the waveguide, forming a V-type level structure [55]. The interference between
photons scattering off the two hybridized states gives rise to the peak in the middle
of sub-panels (i.) and (iii.) in Fig. E.10b.
The fitting of lineshapes starts with the subtraction of transmission spectrum of
the background, which are taken in the same frequency window but with qubits
detuned away. Note that the background subtraction in this case cannot be perfect
due to the frequency shift of the upper passband modes under the presence of
qubits. Such imperfection accounts for most of the discrepancy between the fit
and the experimental data. The fit employs the transfer matrix method discussed
in Refs. [118, 119, 147]. Here, the transfer matrix of the two qubits takes into
account the pure dephasing, which causes the sharp peaks in sub-panels (i.) and
(iii.) of Fig. E.10b to stay below perfect transmission level (unity) as opposed to the
prediction from the ideal case of electromagnetically induced transparency [263].
Topology-dependent photon scattering on various qubit pairs
As mentioned in Chapter 6, when two qubits are separated by Δ𝑛 (Δ𝑛 > 0) unit cells,
the emergence of perfect super-radiance (vanishing of coherent exchange interaction)
is governed by Eq. (6.2). Although Eq. (6.2) is satisfied at the band-edges it does
not lead to additional point of super-radiance because the non-Markovianity at these
points do not lead to effective correlated decay [205]. Therefore, the perfect superradiance takes place exactly Δ𝑛 − 1 times in the trivial phase and Δ𝑛 times in the
topological phase across the entire passband. Chapter 6 shows the case of Δ𝑛 = 2.
Here we report similar measurements on other qubit pairs with different cell distance
Δ𝑛 between the qubits. Figure E.11 shows good qualitative agreement between the
experiment and theoretical result in Ref. [55]. The small avoided-crossing-like
features in the experimental data are due to coupling of one of the qubits with a
local two-level system defect. An example of this is seen near 6.85 GHz of Δ𝑛 = 3

207

c 7.4

Δn

Δn = 0

|S21|

7.2
|S21|

7.4 Δn = 1
7.2

Qubit Frequency (GHz)

|S21|

7.4 Δn = 3
7.2

|S21|

7.4 Δn = 4
7.2

|S21|

7.4 Δn = 5
7.2

7.2

Frequency (GHz)

7.4

6.8

7.2

Frequency (GHz)

7.4

7.2

Frequency (GHz)

7.4

6.8

7.2

Frequency (GHz)

7.4

Figure E.11: Topology-dependent photon scattering on various qubit pairs. a,
Schematic showing two qubits separated by Δ𝑛 unit cells in the trivial configuration. b,
Corresponding schematic for topological phase configuration. c, Waveguide transmission spectrum |𝑆21 | when frequencies of two qubits are resonantly tuned across the upper
passband in the trivial configuration. d, Waveguide transmission spectrum |𝑆21 | for the
topological configuration. For both trivial and topological spectra, the left spectrum illustrates theoretical expectations based on Ref. [55] whereas the right shows the experimental
data.

in the topological configuration. For Δ𝑛 = 0, there is no perfect super-radiant point
throughout the passband for both trivial and topological configurations. For all the
other combinations in Fig. E.11, the number of swirl patterns indicating perfect
super-radiance agrees with the theoretical model.

208

QL
300 μm

Figure E.12: Optical micrograph of Device II (false-colored). The device consists of a
topological waveguide with 7 unit cells (resonators corresponding to A/B sublattice colored
red/blue) in the topological phase, where the inter-cell coupling is larger than the intra-cell
coupling. Two qubits QL (dark red) and QR (dark blue) are coupled to A sublattice of the
second unit cell and B sublattice of sixth unit cell, respectively. Each qubit is coupled to
a 𝜆/4 coplanar waveguide resonator (purple) for dispersive readout, flux-bias line (orange)
for frequency control, and charge line (yellow) for local excitation control.

Qubit
QL
QR

𝑓max
(GHz)
8.23
7.99

𝐸𝐶 /ℎ
(MHz)
294
296

𝐸 𝐽Σ /ℎ
(GHz)
30.89
28.98

𝑔 𝐸 /2𝜋
(MHz)
58.1
57.3

𝑓RO
(GHz)
5.30
5.39

𝑔RO /2𝜋 𝑇1
(MHz) (𝜇s)
43.5
4.73
43.4
13.9

𝑇2∗
(𝜇s)
4.04
8.3

Table E.2: Qubit parameters on Device II. 𝑓max is the maximum frequency (sweet spot)
and 𝐸𝐶 (𝐸 𝐽 Σ ) is the charging (Josephson) energy of the qubit. 𝑔 𝐸 is the coupling of qubit
to the corresponding edge state. The readout resonator at frequency 𝑓RO is coupled to the
qubit with coupling strength 𝑔RO . 𝑇1 (𝑇2∗ ) is the lifetime (Ramsey coherence time) of a qubit
measured at the sweet spot.

E.8

Device II characterization and experimental setup

In this section we provide a detailed description of the elements making up Device
II, in which the edge mode experiments are performed. The optical micrograph of
Device II is illustrated in Fig. E.12.
Qubits
The parameters of qubits on Device II are summarized in Table E.2. The two qubits
are designed to have identical SQUID loops with symmetric JJs. The lifetime and
Ramsey coherence times in the table are measured when qubits are tuned to their
sweet spot. Qubit coherence at the working frequency in the middle bandgap is
also characterized, with the lifetime and Ramsey coherence times of QL (QR ) at
6.829 (6.835) GHz measured to be 𝑇1 = 6.435 (5.803) 𝜇s and 𝑇2∗ = 344 (539) ns,

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RO
Waveguide
Input

20 dB
40 dB
20 dB

Z control
DC

64
kHz

Z control
RF

XY drive

20 dB

40 dB

500
MHz

RO
Waveguide
Output
HEMT

DC block
Attenuator
300 K
50 K plate
4 K plate

Thin-film attenuator

Mixing
chamber
plate

Band-pass filter

4-8
GHz

Low-pass filter

Bias tee
2×2 switch
Circulator
TWPA
HEMT amplifier
50 Ω termination

Figure E.13: Schematic of the measurement setup inside the dilution refrigerator for
Device II. The meaning of each symbol in the schematic on the left is enumerated on the
right. The level of attenuation of each attenuator is indicated with number next to the
symbol. The cutoff frequencies of each filter is specified with numbers inside the symbol.
Small squares attached to circulator symbols indicate port termination with 𝑍0 = 50 Ω,
allowing us to use the 3-port circulator as a 2-port isolator. The pump line for the TWPA is
not shown in the diagram for simplicity.

respectively.
Metamaterial waveguide and coupling to qubits
The resonators in the metamaterial waveguide and their coupling to qubits are
designed to be nominally identical to those in Device I. The last resonators of the
array are terminated with a wing-shape patterned ground plane region in order to
maintain the bare self-capacitance identical to other resonators.
Edge modes
The coherence of the edge modes is characterized by using qubits to control and
measure the excitation with single-photon precision. Taking EL as an example, we
define the iSWAP gate as a half-cycle of the vacuum Rabi oscillation in Fig. 6.4d.
For measurement of the lifetime of the edge state EL , the qubit QL is initially
prepared in its excited state with a microwave 𝜋-pulse, and an iSWAP gate is applied
to transfer the population from QL to EL . After waiting for a variable delay, we
perform the second iSWAP to retrieve the population from EL back to QL , followed
by the readout of QL . In order to measure the Ramsey coherence time, the qubit
QL is instead prepared in an equal superposition of ground and excited states with
a microwave 𝜋/2-pulse, followed by an iSWAP gate. After a variable delay, we
perform the second iSWAP and another 𝜋/2-pulse on QL , followed by the readout

210
of QL . An equivalent pulse sequence for QR is used to characterize the coherence
of ER . The lifetime and Ramsey coherence time of EL (ER ) are extracted to be
𝑇1 = 3.68 (2.96) 𝜇s and 𝑇2∗ = 4.08 (2.91) 𝜇s, respectively, when QL (QR ) is parked
at 6.829 (6.835) GHz. Due to the considerable amount of coupling 𝑔 𝐸 between
the qubit and the edge mode compared to the detuning at park frequency, the edge
modes are hybridized with the qubits during the delay time in the above-mentioned
pulse sequences. As a result, the measured coherence time of the edge modes is
likely limited here by the dephasing of the qubits.
Experimental setup
The measurement setup inside the dilution refrigerator is illustrated in Fig. E.13.
The excitation of the two qubits is controlled by capacitively-coupled individual XY
microwave drive lines. The frequency of qubits are controlled by individual DC
bias (Z control DC) and RF signals (Z control RF), which are combined using a bias
tee at the mixing chamber plate. The readout signals are sent into RO Waveguide
Input, passing through a series of attenuators including a 20 dB thin-film attenuator
developed in B. Palmer’s group [157]. The output signals go through an optional
TWPA, a series of circulators and a band-pass filter, which are then amplified by a
HEMT amplifier (RO Waveguide Output).
Details on the population transfer process
In step i) of the double-modulation scheme described in Chapter 6, the frequency
modulation pulse on QR (control modulation) is set to be 2 ns longer than that on QL
(transfer modulation). The interaction strength induced by the control modulation is
21.1 MHz, smaller than that induced by the transfer modulation in order to decrease
the population leakage between the two edge states. For step iii), the interaction
strength induced by the control modulation on QL is 22.4 MHz, much closer to
interaction strength for the transfer than expected (this was due to a poor calibration
of the modulation efficiency of qubit sideband). The interaction strengths being too
close between QL ↔ EL and QR ↔ ER gives rise to unwanted leakage and decreases
the required interaction time in step ii). We expect that a careful optimization on
the frequency modulation pulses would have better addressed this leakage problem
and increase the transfer fidelity (see below).
The fit to the curves in Fig. 6.4e of Chapter 6 are based on numerical simulation
with QuTiP [338, 339], assuming the values of lifetime (𝑇1 ) and coherence time
(𝑇2∗ ) from the characterization measurements. The free parameters in the simulation

211
are the coupling strengths 𝑔˜ L , 𝑔˜ R between qubits and edge states, whose values are
extracted from the best fit of the experimental data.
The detailed contributions to the infidelity of the as-implemented population transfer
protocol are also analyzed by utilizing QuTiP. The initial left-side qubit population
probability is measured to be only 98.4 %, corresponding to an infidelity of 1.6 %
in the 𝜋-pulse qubit excitation in this transfer experiment (compared to a previously
calibrated ‘optimized’ pulse). In the following steps, we remove the leakage between edge modes and the decoherence process sequentially to see their individual
contributions to infidelity. First, we set the coupling strength between the two edge
modes to zero during the two iSWAP gates while keeping the above-mentioned initial population probability, coupling strengths, lifetimes, and coherence times. The
elimination of unintended leakage during the left and right side iSWAP steps between the edge modes gives the final transferred population probability of 91.9 %,
suggesting 91.9 % − 87 % = 4.9 % of the infidelity comes from the unintended
leakage between edge modes. Also, as expected, setting the population decay and
decoherence of the qubits and the edge modes to zero, the final population is found
to be identical to the initial value, indicating that 98.4 % − 91.9 % = 6.5 % of loss
arises from the decoherence processes.