ABSTRACT

Title of Dissertation: IMPROVEMENTS AND STUDIES
OF PLANAR TRANSMON QUBITS

Yizhou Huang
Doctor of Philosophy, 2023

Dissertation Directed by: Professor Christopher J. Lobb
Department of Physics

Dr. Benjamin S. Palmer
Laboratory for Physical Sciences

This dissertation describes three main projects focused on characterizing and im-

proving superconducting transmon qubits operating nominally at temperature of 20

mK. The first topic I discuss is characterization of ground state fidelity of a passively

cooled 3D transmon qubit using two techniques. The first technique was counting

the number of false counts when performing single-shot read-out measurements of the

weak resonator signal using a nearly quantum limited traveling wave parametric am-

plifier. Over about a million shots, only 772 counts were found with the system in the

excited state, corresponding to a residual excited state population of Pe = 0.083%.

The second technique used was performing Rabi oscillations between the first and

second excited state levels of the transmon qubit. By fitting the data, the residual

excited state population was shown to be Pe = 0.088%±0.018%. These state of the art

low values for the infidelity of the ground state suggest that the effective temperature

of the transmon qubit with fundamental transition frequency of 3.6 GHz was T < 25

mK.

The second topic I discuss is improvements in the coherence of our planar trans-



mon qubits such that I was able to measure energy relaxation times and coherence

times up to tens of microseconds. There were two main improvements that I achieved

during this research. First, I identified a significant loss mechanism associated with

the way that the planar transmon qubits were packaged. I did this by measuring

the internal quality factors (Qi) for a series of thin-film Al quarter-wave resonators

with fundamental resonant frequencies varying between 4.9 and 5.8 GHz. By utilizing

resonators with different widths and gaps, I sampled different electromagnetic energy

volumes that affected Qi. When the backside of the sapphire substrate of the res-

onator device was adhered to a Cu package with a conducting silver glue, a monotonic

decrease in the maximum achievable Qi was found as the electromagnetic sampling

volume was increased. Simulations and modeling showed that the observed dissipa-

tion was a result of induced currents in large surface resistance regions underneath

the substrate. By placing a hole underneath the substrate and using superconducting

material for the package, I was able to decrease the Ohmic losses and increase the

maximum Qi by an order of magnitude for the larger size resonators. The second

improvement I made to achieve improvements to our planar transmon qubits was de-

veloping a new fabrication process to improve the quality of the interface between the

substrate and the superconducting shunting capacitor of the transmon. For this new

process, the large features (> 1 µm) of the thin superconducting film are subtractively

defined by etching the film. The small Al/AlOx/Al junction is added in a second step

by defining the junction in electron-beam lithography, an ion mill step, and standard

double-angle evaporation.

Finally, I discuss my study of the coherence recovery time after injecting quasi-

particles in several transmon qubits. Quasiparticles in the transmon junction were

created by applying a large-amplitude microwave pulse resonant with the readout res-

onator. Immediately after generating the quasiparticles, a significant decrease of the



energy relaxation time of the transmon qubit was observed. By performing relaxation

time T1 and coherence time T2 measurements over the course of several milliseconds,

I tracked the recovery T1 and T2, which I then used as metrics of the quasiparticle

density at the junction. I fitted the recovery data with a numerical model involving

differential equations and extracted the quasiparticle trapping rates around 1ms−1

and recombination rates around 1/25ns−1 at the sites of a few transmon qubits. I

measured transmons that were either galvanically connected to ground or isolated,

fabricated with aluminum (with superconducting gap ∆Al ≃ 200 µV) or tantalum

(with ∆Ta ≃ 600 µV) for the shunting capacitor. Finally, with a larger quasiparti-

cle injection power and by measuring two transmons on the same chip, I observed a

phenomenon that was consistent with phonon-assisted quasiparticle poisoning. I dis-

cussed my quantitative modelling of such data, and how this effect presents challenges

to further improving the coherence times of transmon qubits.



IMPROVEMENTS AND STUDIES OF PLANAR TRANSMON QUBITS

by

Yizhou Huang

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2023

Advisory Committee:
Professor Christopher J. Lobb, Chair/Co-Advisor
Dr. Benjamin S. Palmer, Primary-Advisor
Professor Frederick C. Wellstood
Professor Steven Anlage
Professor Thomas Murphy, Dean’s Representative



ACKNOWLEDGMENTS

I wish to express my sincere appreciation to the individuals and institutions who

have been instrumental in guiding me through the journey to such degree.

Foremost, my heartfelt thanks go to my advisor, Dr. Benjamin S. Palmer, for his

unwavering mentorship and dedication to my academic development. His expertise,

thoughtful feedback, and patient guidance have significantly influenced the course of

my research.

I extend my gratitude to the members of my dissertation committee, Profes-

sor Frederick C. Wellstood, Professor Christopher J. Lobb, Professor Steven Anlage,

and Professor Thomas Murphy, for their valuable insights and constructive critiques,

which have enriched the quality of this work.

I would also like to acknowledge the camaraderie and support of my colleagues

and fellow researchers at LPS and UMD - Jen-Hao Yeh, Haozhi Wang, Shavindra Pre-

maratne, Rui Zhang, Tamin Tai, Zachary Steffen, Yi-Hsiang Huang, Jonathan Cripe,

Sudeep Dutta, Kungang Li, Neda Forouzani, Chih-Chiao Hung, Martin Ritter. The

collaborative environment we’ve shared, the stimulating discussions, and the shared

enthusiasm for discovery have been fundamental in fostering creativity showcased in

this dissertation.

I want to thank the LPS cleanroom, machine shop, research, and administrative

staff members, that provided crucial support to ensure the smooth operation of our

research facilities. Your tireless efforts often go unnoticed, but they are deeply valued

and have significantly contributed to the success of my research.

I wish this dissertation shall serve as a testament to the collective effort of those

who have been part of this journey with me. Thank you.

ii



TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Increasing Qubit Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 THEORY OF TRANSMON AND READOUT RESONATOR . . . . . . . . . . . 7

2.1 Coplanar Waveguide Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Theory of the Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Transmon Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Transmon Coupled to Readout Resonator . . . . . . . . . . . . . . . . . . 15

2.2.4 Measuring the State of the Transmon. . . . . . . . . . . . . . . . . . . . . . . 17

2.2.5 Numerical Simulation of Jaynes Cummings Readout . . . . . . . . 19

2.2.6 Estimating the Voltage across the Junction . . . . . . . . . . . . . . . . . 26

2.2.7 Driving Transmon Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.8 Bloch Sphere Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.9 Density Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.10 Lindblad-Kossakowski Master Equation and Decoherence . . . 35

2.2.11 Determining Optimal Readout Length . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Loss Mechanisms of Transmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 Quasiparticle Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.2 Two-Level System Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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CHAPTER Page

2.3.3 Purcell Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.4 Package Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.5 Vortex Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 DEVICE DESIGN AND FABRICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Finite Element Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 Black Box Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Photolithography and Wet Etching of Large Superconduct-

ing Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.2 Photolithography of Alignment Marks . . . . . . . . . . . . . . . . . . . . . . 59

3.2.3 Electron-Beam Lithography of the Josephson junction . . . . . . 61

3.3 Tuning Oxidation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 SYSTEM CHARACTERIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1 Dilution Refrigerator Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Vector Network Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Pulsed Heterodyne Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.3 Generating Microwave Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Overview of Measured Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Resonator Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5 Transmon Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6 Transmon Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.6.1 Transmon Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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CHAPTER Page

4.6.2 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.6.3 Energy Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6.4 Ramsey Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6.5 Spin Echo Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6.6 Photon Number Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6.7 Residual Excited State Population. . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 REDUCING LOSS FROM THE PACKAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.1 Overview of Package Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Measuring Resonator Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3 Modelling Resonator Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4 Finite Element Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.4.1 Surface Simulation with COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.4.2 Dielectric Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.5 Extension to Transmon Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.5.1 Improvements in Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.5.2 Backside Metalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6 MEASURING QUASIPARTICLE LIFETIMES . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.1 Generating Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2 Probing Quasiparticle Dynamics with Qubit Coherence . . . . . . . . . . . . 157

6.3 Extracting Quasiparticle Dynamics from Experiment Data . . . . . . . . . 160

6.3.1 Simulation Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.5 Quasiparticle-Phonon Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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CHAPTER Page

7 CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.1 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.1.1 Residual Excited State Population. . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.1.2 Improvements in Microwave Packaging . . . . . . . . . . . . . . . . . . . . . 178

7.1.3 Quasiparticle Lifetime Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

A APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

A.1 Code for Numerical Simulation of Jaynes Cummings Readout . . . . . . 181

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

vi



LIST OF TABLES

Table Page

4.1 Parameters of measured qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Parameters of measured resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Comparison of observed P∣e⟩ and Teff to values reported in the literature 127

5.1 Resonator Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.2 Measured resistivity of normal metal present in the packaging . . . . . . . . . 138

5.3 HFSS simulated magnetic field geometric factors γ . . . . . . . . . . . . . . . . . . . . 141

5.4 Package limited T1 of various qubit designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.5 Contribution from various PCB surfaces to qubit loss . . . . . . . . . . . . . . . . . 147

5.6 Qubit T1 limit due to a grid backside metalization . . . . . . . . . . . . . . . . . . . . . 152

6.1 QP injection and fitting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

vii



LIST OF FIGURES

Figure Page

1.1 Different types of superconducting qubits distinguished by energy scales 3

1.2 Superconducting qubit coherence time versus year . . . . . . . . . . . . . . . . . . . . . 4

2.1 Cross-sectional schematics of a coplanar waveguide and the associated

electric and magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 3D rendering and circuit symbol of a Josephson junction . . . . . . . . . . . . . . 10

2.3 Circuit diagram of transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Transmon qubit energy levels for four different Ej/Ec ratios . . . . . . . . . . . 13

2.5 Schematics of angular resonant frequency for a notch resonator coupled

to a transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Relative frequency of resonator vs stored photon number n for qubit

in ∣g⟩ and ∣e⟩ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Stored resonator photon number vs input power for qubit in ∣g⟩ and

∣e⟩ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Relationship between transmitted voltage and input voltage . . . . . . . . . . . 23

2.9 Simulated ∣S21∣ vs input power for qubit in ∣g⟩ and ∣e⟩ states . . . . . . . . . . . 25

2.10 Simulated transmission ∣S21∣ of the readout resonator vs frequency and

power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Lumped-element circuit model to estimate the voltage across the Joseph-

son junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.12 False color plot of numerically simulated voltage across a Josephson

junction during high power Jaynes Cummings readout. . . . . . . . . . . . . . . . . 30

2.13 Bloch sphere representation of qubit state and rotation . . . . . . . . . . . . . . . . 33

2.14 Bloch sphere representation of qubit decoherence . . . . . . . . . . . . . . . . . . . . . . 38

2.15 Figurative representation of SNR with respect to readout length . . . . . . . 40

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CHAPTER Page

2.16 Double-well potential model of TLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 2D design layout of multiplexed resonators and transmon qubits with

annotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 2D design layout of single transmon qubit with annotations . . . . . . . . . . . 49

3.3 2D design layout of structures for e-beam lithography and Josephson

junction fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Setup of HFSS simulation to help develop device design . . . . . . . . . . . . . . . 52

3.5 Example of simulation mesh used in HFSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Simulated transmission S21 versus frequency as simulated by HFSS . . . . 54

3.7 Results from black-box quantization showing the imaginary part of

admittance Y versus frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8 Photolithography and wet etching process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.9 Mask design for photolithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 Photolithography for alignment marks process. . . . . . . . . . . . . . . . . . . . . . . . . 60

3.11 Micrograph of alignment marks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.12 E-beam lithography for Josephson junction fabrication . . . . . . . . . . . . . . . . 63

3.13 Picture of alignment process on JEOL 6500F SEM screen . . . . . . . . . . . . . 65

3.14 Confocal laser micrograph during different stages of junction processing 66

3.15 SEM image of cross-section of ZEP-MMA bilayer e-beam resist . . . . . . . . 67

3.16 Chip loaded into an e-beam evaporator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.17 SEM image of Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.18 2D design layout of structures for testing the resistance of Josephson

junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.19 Measured Josephson junction resistance versus oxidation pressure. . . . . . 71

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CHAPTER Page

4.1 Photograph of the dilution refrigerators used in this dissertation . . . . . . . 74

4.2 Dilution refrigerator setup diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 VNA measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Setup for heterodyne measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5 Pulse sequences of digital delay generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Micrographs of packaged qubit devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Sample of VNA measured resonator spectroscopy and fit . . . . . . . . . . . . . . 87

4.8 S21 versus power for device G3-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9 Pulse sequence for S21 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.10 Pulsed heterodyne measurement of ∣S21∣2 vs power and frequency for

different qubit states for qubit C-i in high power Jaynes Cummings

regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.11 Pulsed heterodyne measurement of ∣S21∣2 vs power and frequency for

different qubit states for qubit C-i in low power dispersive regime . . . . . . 91

4.12 Pulse sequence for qubit spectroscopy measurements . . . . . . . . . . . . . . . . . . 92

4.13 Transmon spectroscopy for qubit C-i with Lorentzian fits . . . . . . . . . . . . . . 94

4.14 Repeated spectroscopy measurement for qubit C-i . . . . . . . . . . . . . . . . . . . . . 95

4.15 Repeated spectroscopy measurement for qubit A1 . . . . . . . . . . . . . . . . . . . . . 95

4.16 Pulse sequence for Rabi oscillation measurements . . . . . . . . . . . . . . . . . . . . . 96

4.17 Sample Rabi oscillation data for qubit C-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.18 Measured Rabi oscillation and fit for qubit C-i . . . . . . . . . . . . . . . . . . . . . . . . 99

4.19 Pulse sequence for T1 measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.20 T1 measurement and fit for qubit C-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.21 Pulse sequence for Ramsey oscillation measurements . . . . . . . . . . . . . . . . . . 102

x



CHAPTER Page

4.22 Bloch sphere representation of Ramsey oscillation measurements . . . . . . . 102

4.23 Ramsey oscillation measurement and fit for qubit D . . . . . . . . . . . . . . . . . . . 103

4.24 Frequency jumps for qubit H-i as observed from Ramsey measurements 105

4.25 Pulse sequence for spin echo measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.26 Bloch sphere representation of spin echo measurements . . . . . . . . . . . . . . . . 107

4.27 Spin echo measurement and fit for qubit J-i . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.28 Spin echo measurement and fit for qubit C-i . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.29 Pulse sequence for photon splitting measurement . . . . . . . . . . . . . . . . . . . . . . 111

4.30 Data and fit for photon number splitting measurements of qubit C-iii . . 112

4.31 Pulse sequence for qubit ∣g⟩/∣e⟩/∣f⟩ state single shot measurement . . . . . 115

4.32 False color map of ∣g⟩, ∣e⟩, and ∣f⟩ states from single shot measurement

for qubit J-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.33 Fit of false color map of ∣g⟩, ∣e⟩, and ∣f⟩ states from single shot mea-

surement for qubit J-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.34 Pulse sequence for Rabi ∣e⟩↔ ∣f⟩ oscillation measurement . . . . . . . . . . . . . 120

4.35 Illustration of Rabi ∣e⟩↔ ∣f⟩ oscillations in IQ plane. . . . . . . . . . . . . . . . . . . 121

4.36 Data and fit for Rabi ∣e⟩ ↔ ∣f⟩ oscillations at base temperature for

qubit J-iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.37 Data and fit for Rabi ∣e⟩↔ ∣f⟩ oscillations at elevated temperature for

qubit J-iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.38 Effective qubit temperature Teff vs mixing chamber temperature Tmix

for qubit J-iii’s ∣e⟩↔ ∣f⟩ Rabi oscillation measurements . . . . . . . . . . . . . . . . 125

xi



CHAPTER Page

4.39 Data and fit for Rabi ∣e⟩ ↔ ∣f⟩ oscillations at base temperature for

qubit E-ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.40 Data and fit for Rabi ∣e⟩ ↔ ∣f⟩ oscillations at base temperature for

qubit H-ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.1 Stitched optical micrograph of the test resonator chip . . . . . . . . . . . . . . . . . 130

5.2 Photos of two device packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3 Log-log plot of the fitted resonator Qi versus stored photon number for

all four packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 Measured maximum Qi,m on a log scale for each resonator and package 134

5.5 Photographs of 3D cavity made of OFHC copper . . . . . . . . . . . . . . . . . . . . . . 136

5.6 Plot of quality factor for 3D cavity and extracted resistivity versus

temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.7 Picture of half-wave PCB resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.8 Confocal microscope image of silver impregnated glue used for DC

resistivity measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.9 Simulation setup of resonator chip and surrounding PCB . . . . . . . . . . . . . . 140

5.10 Comparison of simulated surface current density between packages. . . . . 142

5.11 Comparison of estimated and measured resonator loss 1/Qi,m . . . . . . . . . . 143

5.12 2D COMSOL simulation of the ∣H ∣ field of a TEM wave in a CPW . . . . 144

5.13 Common planar qubit designs studied for package loss . . . . . . . . . . . . . . . . . 146

5.14 Cross section view of PCB and substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.15 Photographs of device A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.16 Setup of HFSS simulation to study qubit T1 limit due to backside

metalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xii



CHAPTER Page

5.17 Magnetic field distribution on back side of substrate . . . . . . . . . . . . . . . . . . . 152

5.18 Sample meshing for HFSS simulation of backside metalization . . . . . . . . . 153

6.1 Semiconductor model energy diagram of a S-I-S Josephson junction at

zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2 Pulse sequence for measuring qubit’s energy relaxation time T1 after

quasiparticle injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3 Sample T1 measurement data with and without quasiparticle injection . 158

6.4 Additional qubit decay rate δΓ plotted against separation time t after

quasiparticle injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.5 Partition of transmon geometry for numerical simulation . . . . . . . . . . . . . . 162

6.6 Measurement of T1 recovery after quasiparticle injection for qubit A1-

xmon and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.7 Measurement of T1 recovery after quasiparticle injection for qubit A2-

xmon and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.8 Measurement of T1 recovery after quasiparticle injection for qubit A3-

float and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.9 Measurement of T1 recovery after quasiparticle injection for qubit A4-

float and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.10 Measurement of T1 recovery after quasiparticle injection for qubit B-

Ta-float and simulations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.11 Simulated T1 recovery after quasiparticle injection for different geomet-

rical variations of qubit A1-xmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.12 Additional quasiparticle density as calculated from measured fractional

frequency shift and change in the decay rate for qubit B-Ta-float . . . . . . 170

xiii



CHAPTER Page

6.13 Measured coherence recovery after quasiparticle injection for qubit B-

Ta-float . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.14 Measured coherence recovery after quasiparticle injection for qubit C-

xmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.15 Measurement of T1 recovery after quasiparticle injection for qubit H-

xmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.16 Measurements and Simulations of T1 recovery of qubit A3-float after

quasiparticle injection at qubit A2-xmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.17 Measurements and Simulations of T1 recovery of qubit A1-xmon after

quasiparticle injection at qubit A2-xmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A.1 Simulated transmission ∣S21∣ of the readout resonator vs frequency and

power, with qubit frequency above resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 187

xiv



Chapter 1

INTRODUCTION

1.1 Quantum Computation

The exponential growth of information has fueled an unceasing pursuit for more

potent computing systems capable of addressing the escalating demands of modern

society. Computation, as we have come to understand it, has predominantly relied

on classical computers that manipulate discrete bits represented by the binary digits

of 0 and 1. However, parallel to these advances in classical computation, our deepest

understanding of the physical world has been shaped by the principles of quantum

mechanics, a field of study that unveils the astonishing behavior of matter and energy

at the quantum level.

Quantum mechanics, formulated in the early 20th century, elucidates the funda-

mental laws that govern the behavior of particles at microscopic scales. It reveals

a reality that is probabilistic, often non-intuitive, and characterized by phenomena

such as superposition and entanglement. Quantum mechanics lay the foundation for

the concept of a “qubit”, the fundamental unit of quantum information.

In contrast to a classical bit, which has only two discrete states (0 or 1), a qubit

can be in a simultaneous superposition of both 0 and 1 states. If ∣ψ⟩ represents a

qubit state, it can be expressed as

∣ψ⟩ = α∣0⟩ + β∣1⟩, (1.1)

with complex numbers α and β satisfying ∣α∣2 + ∣β∣2 = 1.

1



Moreover, two or more qubits can also be entangled, a phenomenon where the

quantum states of multiple qubits become inherently interconnected, regardless of

their physical separation. As a result, in general it requires ∼ 2n complex numbers

on a classical computer to completely store the information of the Hilbert space

spanned by n qubits. Exploiting entanglement allows for the execution of parallel

computations across multiple qubits, providing an exponential speedup in certain

problem-solving scenarios compared to classical computers.

Several algorithms for quantum computation have been proposed that would yield

a speedup against best-known classical algorithms, including Shor’s algorithm for

discrete factoring [125], Grover’s algorithm for unobstructed search [49], and the HHL

algorithm for solving linear systems [54]. Experimentally, there has been reports

[3, 153, 148] over the past few years of quantum computers achieving “quantum

supremacy”, solving problems faster than all existing classical computers, although

the problems solved in these demonstrations were of little predicted use.

There are many physical implementations for a qubit, including neutral atoms

[121], trapped ions [91, 105], nuclear spins [62, 64], quantum dots [76] and various

superconducting circuits [93]. In particular, qubits based on superconducting circuits

can be made from capacitors, inductors and Josephson junctions. The Josephson

junction provides energy levels with nonlinearity, which is critical for constructing a

qubit (see Sec. 2.2.1). Depending on the choice of other reactive circuit components,

superconducting qubits can be roughly categorized into charge qubit [141], phase

qubit [83, 130], flux qubit [38, 138], transmon [68], fluxonium [78, 106], unimon [57],

and many more. Fig. 1.1 show different types of superconducting qubits, as distin-

guished by the relations between the Josephson energy EJ , the inductive energy EL

and the charging energy Ec. Among these, this dissertation in particular is focused

on the study of transmon qubits.

2



Figure 1.1: Different types of superconducting qubits distinguished by the ratio
of the Josephson energy EJ and inductive energy EL to charging energy Ec. See
Sec. 2.2.2 for details of some of the energy definitions. Figure from Ref. [57] under
Creative Commons CC by license.

1.2 Increasing Qubit Lifetime

Unlike a classical bit, the continuous nature of quantum states means a qubit is

inherently vulnerable to the problem of decoherence. This means a qubit may only

preserve quantum information for a limited lifetime before such information is lost

due to its interaction with the environment. Two common metrics characterizing the

coherent behavior of a qubit are its energy relaxation time T1 and coherence time T2

(see Sec. 2.2.10 and 4.6.3-4.6.5 for details).

As shown in Fig. 1.2, over the past two decades,T1 and T2 for superconducting

qubits has increased significantly. Despite that, lifetime in these systems cannot

support arbitrarily long quantum computations, contrary to classical computation

with very low bit error rates. To address this issue, quantum error correction, such

3



Figure 1.2: Superconducting qubit coherence time versus year, figure from Ref. [127]
under license number 5574400977925.

as Shor’s code [126] and the surface code [37], has been proposed. Due to the no-

cloning theorem [100], an arbitrary quantum state cannot be “copied” in a classical

sense. Instead, quantum error correction works by spreading the information of one

(logical) qubit onto a highly entangled state of several (physical) qubits, and multi-

qubit measurements are intermittently performed, which do not disturb the quantum

information in the encoded state, but retrieve information about potential errors on a

subset of the physical qubits (usually one), and then correct for those errors. Recently,

a quantum error correction scheme extending the lifetime of a logical qubit beyond

the longest lifetime of its constituent physical qubits has been demonstrated [97].

For quantum error correction schemes to work, the error rate per quantum gate op-

eration has to be below a certain threshold (typically below 1%) [113, 37]. Lower error

4



rates require a smaller number of auxiliary qubits. Currently, the fidelities for some

qubit gates for superconducting qubits are limited by decoherence times [132, 35].

In this case, improving the lifetime of qubits could lead to meaningful expansion of

the capabilities of near-term quantum computers. In Chapter 5, I discuss improving

the coherence of my superconducting planar resonators and transmon qubits by im-

proving the package environment. In most error correction schemes, the error rate is

assumed to be Markovian and uncorrelated between different qubits. In contrast, it

has been shown that bursts of cosmic-ray muons and γ-rays could generate a large

amount of quasiparticles, resulting in correlated errors of multiple qubits on the same

chip [145]. In Chapter 6, I present my studies on the quasiparticle dynamics that

affects superconducting qubits.

1.3 Overview of the Dissertation

In Chapter 2 of this dissertation, I discuss the theory of transmon qubits and

the Hamiltonian when it’s coupled to a readout resonator. I lay out the numerical

simulations of the energy levels that qualitatively captures the characteristics of the

high-power Jaynes Cummings readout [116]. These simulations were needed for the

studies of quasiparticle dynamics in Chapter 6, where quasiparticles were inadver-

tently generated during readout.

In Chapter 3, I discuss how I designed my transmon qubits and coplanar waveguide

resonators, and the steps I took to fabricate them.

In Chapter 4, I discuss the experimental setup that I used to measure and char-

acterize the devices, determining parameters such as resonant transition frequencies,

coupling strengths, as well as T1 and T2 of the transmon qubits and Qi of the res-

onators. In particular, in Sec. 4.6.7 I discuss how I measured the residual excited

5



state population of some of my qubits, which are among the lowest observed in su-

perconducting qubits.

In Chapter 5, I discuss my measurement of the internal quality factors Qi of

five resonators on a single chip and how such values were related to the packaging

environment. Using finite element simulations, I identified a conducting adhesive to

be the dominant source of loss for some of the resonators. I extended these simulations

to show how the T1 of a transmon qubit could be limited by the packaging environment

in the absence of such conducting adhesive. These studies led to the design of a new

package that greatly improved the maximum observed T1 for planar transmons in our

group.

In Chapter 6, I discuss my investigations of quasiparticle diffusion, trapping and

recombination rates. Quasiparticles were injected into the qubit junction using a

high power microwave pulse at the frequency of the readout resonator. By fitting a

numerical model to my result, I found the extracted quasiparticle dynamics are similar

to what other groups have reported. In particular, I observed an effect consistent

with phonon-assisted quasiparticle poisoning across a long distance on the chip, with

a somewhat longer recovery time constant than reported in the literature.

Finally, I conclude in Chapter 7 with a summary of my main results together with

possible directions for future studies.

6



Chapter 2

THEORY OF TRANSMON AND READOUT

RESONATOR

In this chapter, I discuss the theory of transmon qubits, their physical realization

with Josephson junctions, their Hamiltonian when coupled to a readout resonator for

state measurement, the microwave manipulation of transmon states, and predominant

channels of loss. By numerically solving the eigenenergies of a transmon coupled

to a readout resonator, I obtained qualitative insight into the high-power Jaynes

Cummings Readout in Sec. 2.2.5. For some of my qubits, I found the quasiparticles

were generated during the high-power Jaynes Cummings, which inspired my studies

into their dynamics, as discussed in Chapter 6.

2.1 Coplanar Waveguide Resonator

A coplanar waveguide (CPW) is a planar transmission line fabricated on a sub-

strate, consisting of a center conductor stripe and a pair of ground planes on either

side [143]. This geometry supports a quasi-TEM (transverse electromagnetic) mode

of transmission. Fig. 2.1 shows a cross-sectional view of the components of a CPW,

and the distribution of electric and magnetic fields of quasi-TEM propagation.

A CPW can be modelled with a distributed capacitance Cgeo and distributed

inductance Lgeo. Let the width of the center trace be w and the gap between the center

trace and ground plane be g. I assume w and g are much larger than the thickness t

of the metal layers, but much smaller than the thickness h of substrate underneath

them, with substrate relative dielectric constant ϵr. These are valid assumptions for

7



air

electric field
magnetic field

metal

dielectric

w g

h

t

Figure 2.1: Cross-sectional schematics of a coplanar waveguide and the associated
electric and magnetic fields. Figure adopted from Spinningspark at Wikipedia under
CC BY-SA 3.0 license.

the devices studied in this dissertation. Then [128]

Cgeo =2ϵ0(ϵr + 1)
K(k0)
K(k′0)

(2.1)

Lgeo =
µ0

4

K(k′0)
K(k0)

(2.2)

Z0 =
√

Lgeo

Cgeo

=
√

µ0

8ϵ0(ϵr + 1)
K(k′0)
K(k0)

, (2.3)

where K(k) is a complete elliptic integral with modulus k, k0 = w/(w + 2g) and

k′0 =
√
1 − k20. For impedance matching, the values of w and g were designed to give

Z0 = 50 Ω.

To create a resonant structure out of a CPW, typically two discontinuities along

the waveguide are introduced, associated with impedances different from Z0. Typical

physical realizations of discontinuities are a gap in the center trace or an inductive

short to the ground. The discontinuities prevent travelling wave structures and thus

create local storage of energy in the form of oscillating electromagnetic fields. Su-

perconducting CPW resonators have been used as optical detectors for astronomical

applications [29] and for readout in quantum information systems [45].

All of the CPW resonators that I studied in this dissertation were quarter-wave

resonators, created by shorting the center trace to the surrounding ground plane on

one end (voltage node, current anti-node), and making the other end open (current

8



node, voltage anti-node), with the distance between the two ends l related to the

wavelength of the fundamental resonant mode λ by l = λ/4, hence the name “quarter”

wave.

2.2 Theory of the Transmon

2.2.1 Josephson Junction

A Josephson junction consists of two superconducting electrodes with a barrier or

restriction between them (see Fig. 2.2). For superconductor-insulator-superconductor

(S-I-S) Josephson junction, which is the focus of this dissertation, the barrier is a thin

layer of insulator in between the two superconductors. In contrast, a thin layer of

normal metal can serve as a weak barrier to create a superconductor-normal metal-

superconductor (S-N-S) Josephson junction. It was predicted by Brian Josephson

in 1962 that a current I can flow through the junction with zero DC voltage drop

[63]. This is a result of the macroscopic wavefunction of the superconductor, which

extends through the barrier and allows Cooper pairs to tunnel as a coherent quantum

mechanical object. In particular, the current-voltage characteristics for an ideal bare

Josephson junction can be obtained from the following two relations:

I =Ic sin (δϕ) (2.4)

d(δϕ)
dt
=2eV
h̵
, (2.5)

where V is the voltage across the junction, I is the current that flows through the

junction, δϕ is the phase difference of the two superconducting wave functions across

the junction, and Ic is the junction’s critical current.

From Eq. 2.5, V = 0 as long as δϕ is stationary. According to Eq. 2.4, a non-

9



Figure 2.2: 3D rendering (left) and circuit symbol (right, a cross) of a Josephson
junction.

zero δϕ results in a constant supercurrent Ic sin (δϕ) flowing across the junction, this

is called the DC Josephson Effect. In addition, if V ≠ 0, then δϕ as well as the

current through the junction I oscillates at a frequency of 2eV /h, this is called the

AC Josephson Effect.

Transmon qubits typically operate with small I and V biases. Performing a first

order Taylor series expansion (sinx = x +O (x3)) on Eq. 2.4 gives

I

Ic
≃ δϕ. (2.6)

Taking the time derivative of Eq. 2.6 and plugging into Eq. 2.5 leads to

V = h̵

2eIc

dI

dt
= LdI

dt
, (2.7)

where L = h̵
2eIc

is identified as the Josephson inductance. From Eq. 2.7 one can see

that to the lowest order, a Josephson junction behaves like an inductor, where the

inductance is determined by the tunneling of Cooper pairs and not a magnetic flux.

The higher-order terms in the Taylor series expansion give the a non-linear Josephson

inductance. As I will discuss later, this nonlinearity is essential for constructing a

qubit.

10



2.2.2 Transmon Qubit

A transmon qubit, short for transmission line shunted plasma oscillation qubit,

consists of a Josephson junction shunted by a capacitor [68]. Fig. 2.3 shows a circuit

diagram of a transmon that’s voltage biased via coupling to gate capacitance Cg. The

Hamiltonian for this circuit is given by

H = 4Ec (n̂ − ng)2 −Ej cos ϕ̂, (2.8)

where n̂ is the number operator for excess Cooper pairs that tunnel from one side of the

capacitor to the other side, ng = CgVg/e is the reduced gate charge, ϕ̂ is the operator

of the difference of superconducting phase across the junction, Ec = e2/2(Cj +Cg) is

the electrostatic charging energy, Ej = IcΦ0/2π is the Josephson energy and Φ0 = h/2e

is the flux quantum. For transmon qubits, typically Cj ≫ Cg, where Cj contains

both the shunting capacitance and the self-capacitance of the junction. In the low

temperature limit, the Josephson energy Ej is related to the normal state resistance

Rj of the barrier by [136]

Ej =
RQ

Rj

∆

8
, (2.9)

where RQ = h/e2 is the resistance quantum and ∆ is the superconducting gap for the

two superconducting leads, which were aluminum in this dissertation.

While Eq. 2.8 can be rigorously solved, I provide some insight into the structure

of the energy levels for the transmon. Using the conjugate relationship

[ϕ̂, n̂] = i (2.10)

and assuming ng = 0, the Hamiltonian given by Eq. 2.8 can be rewritten as

H = 4Ec (
1

i

∂

∂ϕ̂
)
2

−Ej cos ϕ̂. (2.11)

11



Figure 2.3: Circuit diagram of gate-biased transmon qubit.

Taking a first order expansion of the cosine function, one finds

H ≈ 4Ec (
1

i

∂

∂ϕ̂
)
2

+ 1

2
Ejϕ̂

2 −Ej. (2.12)

This resembles the Hamiltonian for a simple harmonic oscillator

H = p̂2

2m
+ 1

2
kx̂2 = 1

2m
h̵2 (1

i

∂

∂x̂
)
2

+ 1

2
kx̂2 (2.13)

with energy eigenvalues given by

En = (n +
1

2
) h̵
√

k

m
. (2.14)

Using this as an analogy, to the lowest order of the cosine function, the energy

eigenvalues of Eq. 2.12 are

En ≈ (n +
1

2
)
√
8EjEc −Ej. (2.15)

For the harmonic oscillator, in Eq. 2.13, the classical amplitude related to energy E

is xmax =
√

2E
k . Similarly for Eq. 2.12, one can think of an equivalent “amplitude” of

ϕ as ϕmax =
√

2E
Ej
. The approximation in Eq. 2.12 only includes terms up to O (ϕ̂2),

so for the same energy E, the larger the Ej, the smaller the “amplitude” ϕmax, the

12



-2 -1 0 1 2

0

5

10

15

ng

E
n
(n
g
)/
E
0
1
(0
.2
5
)

E j/Ec=0.1

-2 -1 0 1 2

0

5

10

15

ng

E j/Ec=1

-2 -1 0 1 2

0

1

2

3

4

5

ng

E j/Ec=10

-2 -1 0 1 2

0

1

2

3

4

ng

E j/Ec=100

Figure 2.4: Transmon qubit energy levels for four different Ej/Ec ratios from 0.1 to
100. The six lowest energy levels normalized by E01(0.25) = E1(0.25) −E0(0.25) are
plotted.

smaller the approximation error with Eq. 2.12. Thus, the transmon energy levels

resemble a linear harmonic oscillator, but only to lowest order in n. To conclude, in a

qualitative manner, the anharmonicity of the transmon is negatively correlated with

the ratio Ej/Ec.

The exact solution for the eigenvalues of Eq. 2.8 is given by the Mathieu’s char-

acteristic value functions[26]

En(ng) = Eca2[ng+k(n,ng)] (−
Ej

2Ec

) , (2.16)

where av(q) is the v-th Mathieu’s function and k(n,ng) is a sorting function given by

k(n,ng) = ∑
l=±1

[int(2ng +
l

2
) mod 2] (int(ng) + l(−1)n [(n + 1) div 2]) . (2.17)

where int(x) is a function rounding x to the nearest integer, and (x div y) gives the

integer quotient of x and y.

The exact transmon’s energies (Eq. 2.16) are plotted in Fig. 2.4 for a variety of

Ej/Ec values. As the ratio Ej/Ec increases, the charge dispersion (i.e. the dependence

13



of En on reduced gate charge ng) decreases. Also note that for a fixed Ej/Ec, the

charge dispersion of a level increases as the energy level increases.

For sufficiently large Ej/Ec and small enough level number n where ng can be

neglected, the Hamiltonian (Eq. 2.11) can be Taylor expanded to the ϕ̂4 order in the

cosine term. The energy levels are then

En ≃ (n +
1

2
)
√
8EjEc −

Ec

4
(2n2 + 2n + 1) −Ej. (2.18)

The equation above has an additional term of −Ec (2n2 + 2n + 1) /4 when compared

to Eq. 2.15. Eq. 2.18 gives

E01 =
√
8EjEc −Ec (2.19)

for the transition energy between the ground state (∣g⟩) and the first excited state

(∣e⟩) of a transmon. From Eq. 2.18, the transition energy between the first (∣e⟩) and

second (∣f⟩) excited state of a transmon is given by

E12 =
√
8EjEc − 2Ec. (2.20)

As I discuss in Sec. 4.6.7, I used the E12 transition to measure the residual excited

state population for some of my qubits. The anharmonicity of a transmon qubit is

given by

α = E12 −E01 ≃ −Ec. (2.21)

From Fig. 2.4 and Eq. 2.19-2.21, an offset-charge insensitive transmon is a par-

ticular kind of qubit with sufficiently large Ej/Ec ratio such that its E01 transition

does not have a noticeable dependence on ng. I designed my devices to be in this

regime because the local voltage and charge environment drifts and fluctuates, result-

ing in changes in ng, instabilities of E01, and therefore dephasing of the transmon.

[68] At the same time, a transmon has its Ej/Ec ratio not too large such that the

relative anharmonicity, α/E01 ≈
√
Ec/8Ej is not too small. This ensures that it can

14



be treated as an effective two-level system with sufficient energy spacing to reduce

spurious transitions being driven to the second excited state.

2.2.3 Transmon Coupled to Readout Resonator

A common way to measure the state of a transmon qubit is to couple it to a

bosonic field with frequency ωr/2π. For the case of a 3D transmon qubit, the bosonic

field is formed from a physical 3D cavity. [90] While for a 2D or planar transmon

qubit, the field is typically formed from a CPW resonator. [12] In general, when a

two level system (ideal qubit) is coupled to a single-mode bosonic field, the undriven

system can be described by the Jaynes-Cummings Hamiltonian [60]

HJC = h̵ωr (â†â + 1

2
) + 1

2
h̵ωqσ̂z + h̵g (âσ̂+ + â†σ̂−) , (2.22)

where σ̂± = (σ̂x ± iσ̂y)/2, σ̂x, σ̂y, σ̂z are Pauli operators on the two-level system, â†

and â are creation and annilation operators on the bosonic field, ωq is the transition

frequency of the two-level system, and g is the coupling strength between the two-

level system and the bosonic field. Since most of my transmon qubits studied in this

dissertation (Table 4.1) were 2D transmon qubits, I use the language of a transmon

qubit coupled to a resonator for the remainder of this chapter.

The Jaynes-Cummings Hamiltonian for a multi-level transmon with nonuniform

energy spacing can be generalized to [11]

HJC,multi-level = h̵ωr (â†â + 1

2
) +∑

j=0

h̵ωj ∣j⟩⟨j∣ +∑
j=0

h̵gj,j+1 (â∣j + 1⟩⟨j∣ + â†∣j⟩⟨j + 1∣) ,

(2.23)

where the level coupling strengths are gj,j+1 ≈
√
j + 1g0,1. The detuning ∆j,j+1 between

the resonator and the j ↔ j + 1 transition of the multi-level system is defined as

∆j,j+1 = (ωj+1 − ωj) − ωr. (2.24)

15



I typically operated the system in the dispersive limit where ∆j,j+1/gj,j+1 ≫ 1. In

this limit, the following unitary transformation

T ≈ exp(∑
j=0

gj,j+1
∆j,j+1

(â†∣j⟩⟨j + 1∣ − â∣j + 1⟩⟨j∣)) (2.25)

can be used to diagonalize the Hamiltonian. If I only keep the two lowest levels of

the transmon, the unitary transformation of the Hamiltonian in Eq. 2.22 is

THJCT
† ≈Hdisp

JC = h̵ωr (â†â + 1

2
) + 1

2
h̵ωqσ̂z + h̵χâ†âσ̂z. (2.26)

Here, χ = g2/∆ is the dispersive shift between the two-level system (transmon) and

the readout resonator.

The dispersive shift for the j ↔ j + 1 transition of the multi-level system is

χj,j+1 ≃
g2j,j+1
∆j,j+1

. (2.27)

To distinguish from number states in the resonator, in this dissertation, I follow

the tradition of the transmon community by using the notation ∣g⟩, ∣e⟩, and ∣f⟩ to

denote the lowest three coupled transmon states. Here “g” stands for ground state,

“e” stands for (first) excited state, and “f” is for second excited state. Considering

multiple transmon levels, applying the unitary transformation on Eq. 2.23 results in

THJC,multi-levelT
† ≈Hdisp

JC = h̵ω̃r (â†â + 1

2
) + 1

2
h̵ω̃qσ̂z + h̵χâ†âσ̂z, (2.28)

where χ = χge−χef/2, ω̃q = ωge+χge, and ω̃r = ωr−χef/2. Here χge and χef are defined

by taking j = 0 and j = 1 in Eq. 2.27. Among these parameters, ωr/2π corresponds

to the “bare” frequency of the resonator, which is the frequency of the resonator if it

weren’t coupled to the qubit. The two frequencies (ω̃r ± χ)/2π are called “dressed”

frequencies of the resonator, since they are the frequencies of the resonator coupled

to the qubit. Fig. 2.5 shows an illustration of the relationship between these various

frequencies, assuming ωq < ωr.

16



ωr ω̃r

−χge

−1
2χef −χ−χ

Angular Frequency

|S
2
1
|

Figure 2.5: Schematics of angular resonant frequency for a notch resonator coupled
to a transmon qubit. The solid black curve is the bare resonance at ωr. The two
dressed resonances correspond to the qubit in the ∣g⟩ state (blue curve) and the qubit
in the ∣e⟩ state (red curve) at ω̃r ± χ.

2.2.4 Measuring the State of the Transmon

Eq. 2.28 shows that when the qubit is in an eigenstate of the σ̂z operator (with

eigenvalues ±1), the third term causes the frequency of the resonator to differ by

±χ/2π. Thus by probing the resonator, the state of the qubit can be inferred. For

example, Fig. 2.5 shows the transmission across a notch resonator that’s used for qubit

state readout, which is the case for most of the qubits studied in this dissertation.

If a microwave pulse at (ω̃r + ∣χ∣)/2π frequency is applied and low transmission is

measured, that means the pulse is on resonance with the readout resonator and thus

the qubit is in the ∣g⟩ state. If instead high transmission is measured, then the readout

resonator’s frequency is not at (ω̃r + ∣χ∣)/2π and thus the qubit is not in the ∣g⟩ state.

This method is called “low power dispersive readout” [13, 141]. In addition, for each

additional photon in the resonator, the frequency of the qubit is shifted additionally

by 2χ/2π. Thus by performing qubit spectroscopy, the number of photon stored in

17



the resonator can be accessed (see Sec. 4.6.6 for details).

This “Low power dispersive readout” has the advantage of being quantum nonde-

molition (QND) [12]. That is, the qubit eigenstates (∣g⟩, ∣e⟩) are preserved after the

measurement.

Another way to measure the state of the qubit is the “high power Jaynes Cum-

mings readout”[15, 116]. To understand it, note the transformation from Eq. 2.23

into Eq. 2.28 is only valid if the number of photons stored in the resonator n satisfies

n≪ ncrit =
∆2

ge

4g2ge
, (2.29)

where ncrit is the “critical photon number”. Otherwise for n ≳ ncrit and Eq. 2.28

breaks down, the frequency of the resonator eventually goes to ωr/2π, which differs

from the low-photon case of (ω̃r ± χ)/2π (see Fig. 2.5).

In the next subsection, I provide a detailed numerical analysis of the “high power

Jaynes Cummings readout”. To understand it in simple terms, according to Fig. 2.5,

if a microwave pulse of large enough power is applied at the bare resonator frequency

ωr/2π, the dressed resonator frequency for qubit ∣e⟩ state at (ω̃r − ∣χ∣)/2π is closer to

ωr/2π than the dressed frequency (ω̃r + ∣χ∣)/2π if the qubit is in ∣g⟩ state. This means

during photon buildup in the resonator, the pulse detuning is smaller if the qubit is

in ∣e⟩ state and thus it’s easier for photon number to reach ncrit and for the dispersive

condition to break down. As a consequence, there is a particular range in power

(usually 2-4 dB for my measurements) that a microwave pulse at ωr/2π frequency

would create enough photons so that the resonator is at the bare frequency of ωr/2π,

if the qubit is in ∣e⟩, in which case there would be low transmission (following the

notch resonator example earlier); but the resonator would not be at ωr/2π frequency

if the qubit is in ∣g⟩ state, in which case there would be high transmission. The

difference in the transmission can be used to distinguish the state of the qubit.

18



The behavior is somewhat different if qubit frequency is higher than the resonator

frequency, i.e. ωq > ωr. In this case, the dressed resonator frequency is closer to

the bare resonator frequency if the qubit is in the ∣g⟩ state at low photon numbers.

Nonetheless, as photon number builds up in the resonator, there would still be a range

of photon numbers where this relationship flips, and qubit ∣e⟩ state corresponds to

smaller detuning. See the program in the appendix of this dissertation for details.

Since a high photon number is used with the “high power Jaynes Cummings

readout”, it has the benefit of a relatively large signal-to-noise ratio. This is helpful

if the first stage amplifier has a small quantum efficiency. Furthermore, it’s only

sensitive to qubit decay during the photon buildup phase, which is helpful if T1 of the

qubit is small. The most significant drawback is that it’s not a QND measurement

as the qubit state is destroyed by hybridization with a large number of photons. In

addition, I found for some of my transmons, this “high power Jaynes Cummings

readout” generates quasiparticles in the transmon (see Sec. 2.2.6) and decreases its

T1 (see Chapter 6).

2.2.5 Numerical Simulation of Jaynes Cummings

Readout

If the cosine term in Eq. 2.11 is expanded to order ϕ̂6 and constants are ignored,

Eq. 2.23 can be rewritten as

H = h̵ωra
†a +
√
8EjEcb

†b − Ec

12
(6 (b†b)2 + 6 (b†b)) + h̵g (ab† + a†b)

+ E2
c

45
√
8EjEc

(20 (b†b)3 + 30 (b†b)2 + 40 (b†b)) , (2.30)

where a†(a) and b†(b) are the raising (and lowering) operators for the resonator and

transmon respectively. The last term on the second line is from the expansion of order

19



ϕ̂6. I will use ∣α,β⟩ to represent a state with α excitations in the resonator and β

excitations in the transmon. The n-excitation manifold for the lowest 5 energy levels

of the transmon is spanned by the set of basis states

(∣n − 4,4⟩, ∣n − 3,3⟩, ∣n − 2,2⟩, ∣n − 1,1⟩, ∣n − 0,0⟩)T .

The Hamiltonian in Eq. 2.30 is then expressed in matrix form as

⎛
⎜⎜⎜⎜⎜⎜⎜⎜
⎝

(n−4)ωr+4ωj−10Ec+
128
3

E2
c

ωj
g
√

n−3
√

4

g
√

n−3
√

4 (n−3)ωr+3ωj−6Ec+
62
3

E2
c

ωj
g
√

n−2
√

3

g
√

n−2
√

3 (n−2)ωr+2ωj−3Ec+8
E2
c

ωj
g
√

n−1
√

2

g
√

n−1
√

2 (n−1)ωr+1ωj−1Ec+2
E2
c

ωj
g
√
n

g
√
n nωr

⎞
⎟⎟⎟⎟⎟⎟⎟⎟
⎠

(2.31)

whose eigenvalues can be solved numerically. Here I have defined ωj =
√
8EjEc and

assumed h̵ = 1 for simplicity. In this way, the qubit frequency is approximated as

ωq = (ωj −Ec)/2π. Considering the common case where the qubit frequency is smaller

than the resonator frequency {ωj, ωq} < ωr, let vn,i be the i-th largest eigenvalue of

Eq. 2.31 of the n-excitation manifold, then vn,1 would correspond to the energy levels

of the transmon-resonator system when the transmon is in the ∣g⟩ state, and vn,2

for ∣e⟩ state. The transition frequency of the resonator between the n-photon and

(n + 1)-photon states is

f̃r(n) = (vn+i,i − vn+i−1,i) /2π. (2.32)

Figure 2.6 shows the difference between the transition frequency of the resonator

f̃r and its bare frequency ωr/2π versus photon number n, for the transmon in ∣g⟩

and ∣e⟩ states. The transition frequency f̃r was obtained by numerically solving

Eq. 2.32, setting i = 1 for transmon ∣g⟩ state and i = 2 for transmon ∣e⟩ state. The

nominal parameters were: bare resonator frequency ωr/2π = 8 GHz, qubit frequency

ωq/2π = 5.8 GHz, charging energy Ec/h = 200 MHz, and resonator-qubit coupling

g/2π = 60 MHz. At low photon numbers with the qubit in the ∣g⟩ state, the dressed

20



100 101 102 103 104 105 106 107 108
0

0.5

1

1.5

−χge

−2χ

ncrit

n

f̃ r
−

(ω
r
/2
π
)
(M

H
z)

Qubit in |g⟩
Qubit in |e⟩

Figure 2.6: Relative frequency of resonator vs stored photon number n for qubit in
∣g⟩ (blue) and ∣e⟩ (red) states. The dashed line represents critical photon numbers
ncrit ≈ 336. The arrows and texts correspond to some of shifts in frequencies defined
in Sec. 2.2.3 and shown in Fig. 2.5.

resonator frequency is shifted from the bare resonator frequency by an amount ∣χge∣ ≃

g2/∣ωr − ωq ∣ ≈ 1.64 MHz(×2π). The difference in the dressed resonator frequency

between the qubit in the ∣g⟩ state and ∣e⟩ state is given by ∣2χ∣, with ∣χ∣ = ∣χge−χef/2∣ ≈

g2 ×Ec/(ωr − ωq)/(ωr − ωq −Ec) ≈ 0.14 MHz(×2π). The quantities of χge and 2χ are

represented by arrows in Fig. 2.6. As the photon number n in the resonator increases,

the dispersive approximation breaks down, and the resonator frequency f̃r gradually

shifts back to the bare frequency ωr/2π.

For the high power Jaynes Cummings readout, a microwave pulse of power Pin is

applied at the bare resonator frequency ωr/2π. Depending on the qubit state, such a

pulse would result in a different average photon number in the resonator and different

transmitted power Ptrans. Neglecting the transmon, the relationship between Pin and

the average photon number n in a resonator is

Pin =
2hQe

π
× (κ

2

4
+ 4π2∆f 2)n, (2.33)

21



where κ is the photon relaxation rate of the resonator, given by κ = 2πf/QL, Q−1L =

Q−1e + Q−1i where QL is the loaded quality factor, Qe is the external quality factor

and Qi is the internal quality factor, and ∆f is frequency the detuning between the

applied microwave pulse and the resonant frequency. Most of the readout resonators

that I studied in this dissertation are over-coupled, i.e. the internal quality factor Qi

was much larger than the external quality factor Qe, and thus QL ≃ Qe (see Table

4.1). For the following numerical simulations, I used Qe = 3 × 104,Qi = 3 × 105 and

assumed these values did not depend on the number of photons n.

As Eq. 2.33 shows, the average photon number n depends on ∆f and Pin. Fur-

thermore, ∆f depends on qubit state and photon number n (see Fig. 2.6). Combining

these, Fig. 2.7 shows a plot of photon number n versus microwave input power Pin

applied at the bare resonator frequency, for the qubit in the ∣g⟩ and ∣e⟩ states. In

Fig. 2.7, at the lowest and highest input power / photon number, there’s an approxi-

mately linear relationship between these two parameters, which doesn’t have obvious

dependence on the qubit state. However, at intermediate powers, a small increase in

input power leads to a huge increase in the average photon number. The location in

power where this occurs depends on qubit state.

The ratio of the magnitude of the transmitted signal to the magnitude of the input

signal is

∣S21∣ = ∣
Vt
Vin
∣ =
√

Pt

Pin

, (2.34)

where V is the complex voltage, P stands for power. Fig. 2.8 illustrates the relation-

ship between Vin and Vt for a notch resonator. The ratio of the minimum transmitted

voltage Vt,min (which is achieved when ∆f = 0) to the input voltage Vin is

∣Vt,min/Vin∣ = QL/Qi. (2.35)

As Fig. 2.8 shows, the actual transmitted voltage Vt lies on a circle whose diameter

22



10−15 10−14 10−13 10−12 10−11 10−10 10−9

102

104

106

108

Pin (Watt)

n

Qubit in |g⟩
Qubit in |e⟩

Figure 2.7: Stored resonator photon number n vs input power Pin for qubit in
∣g⟩ (blue) and ∣e⟩ (red) states, when applying a microwave pulse at bare resonator
frequency ωr/2π. The same parameters as Fig. 2.6 were used for this simulation.

VinVt,min

Vt

θ

in-phase

quadrature

Figure 2.8: Relationship between the complex transmitted voltage Vt and the com-
plex input voltage Vin, see main text for description.

23



connects Vt,min and Vin. In the limit ∆f/f̃r ≪ 1, the angle θ in Fig. 2.8 is given by

tan θ =
4QL

∆f

f̃r

1 − 4 (QL
∆f

f̃r
)
2 , (2.36)

where −π < θ < π, and that θ and ∆f have the same sign. From Fig. 2.8, it can be

shown that

∣S21∣ =

¿
ÁÁÀ( QL

2Qc

sin θ)
2

+ (1 − QL

2Qc

(cos θ + 1))
2

. (2.37)

θ is given by Eq. 2.36 and it depends on ∆f and f̃r, both of which can be obtained

from Fig. 2.6 and 2.7. As a result, the ratio ∣S21∣ can be calculated as a function of

input power.

Figure 2.9 shows ∣S21∣ versus input power. At low input powers, the applied

microwave signal at ωr is not on resonance with the dressed resonator, so a high

transmission is observed. As the input power increases, the frequency of the readout

resonator shifts to ωr (Fig. 2.6), reducing ∣S21∣ as the detuning between the applied

microwave signal and resonator decreases. This transition happens at a lower input

power when the qubit is in the ∣e⟩ state (see red curve in Fig. 2.7 and 2.9). The

dashed line in Fig. 2.9 represents input power that gives maximum difference in S21

between the two qubit states. At this power, when the qubit is in the ∣e⟩ state, the

microwave pulse is in resonance with the (notch) resonator, resulting in a small ∣S21∣,

but when the qubit is in ∣g⟩, it is not in resonance, resulting in a larger ∣S21∣. It is this

difference between the transmitted power that I used to distinguish the qubit states

as high-power Jaynes Cummings readout.

It’s also worth noting that the two curves diverge at roughly the same input power

in Fig. 2.7 and Fig. 2.9. In Fig. 2.7, the range of photon numbers where the two curves

diverge is roughly the same range of photon numbers in Fig.2.6 where the resonator

transitions from dressed frequency to bare frequency. In addition, the lower end of

24



10−14 10−13 10−12 10−11 10−10 10−9
0

0.2

0.4

0.6

0.8

1

Pin (Watt)

|S
21
|

Qubit in |g⟩
Qubit in |e⟩

Figure 2.9: Simulated ∣S21∣ vs input power Pin for qubit in ∣g⟩ (blue) and ∣e⟩ (red)
states, when applying a microwave pulse at bare resonator frequency ωr/2π. The
same parameters as Fig. 2.6 and Fig. 2.7 were used for this simulation. The dashed
line at 2.6 pW represents the power of maximum difference in S21 between the two
qubit states, which is used for high-power Jaynes Cummings readout.

this photon number range is close to the critical photon number ncrit = ∆2/4g2 ≈ 336

for the parameters I used in these simulations.

While Fig. 2.9 shows ∣S21∣ versus input power for a microwave signal applied at

the bare resonator frequency ωr/2π, the calculation can be extended to microwave

signals applied at any frequency and power. For example, in Fig. 2.10 I show false

color plots of ∣S21∣ vs frequency and input power, for the qubit in the ∣g⟩ and ∣e⟩ states.

It’s worth noting this simple numerical model does not consider coupling to higher

modes of the readout resonator, the transient behavior during photon buildup, the

Poisson distribution of photon numbers, higher energy levels of the transmon which

may have significant charge dispersion, and higher order expansions of the cosine term

in Eq. 2.11. For these reasons, these simulation results appear somewhat differently

compared to the experimentally measured spectra presented in later chapters (see

25



|g

7.999 8    8.001 8.002 8.003

Frequency (GHz)

10
-11

10
-12

10
-13

10
-14

10
-15

In
p
u
t 
P

o
w

e
r 

(W
)

|S
21

|

|e

7.999 8    8.001 8.002 8.003

Frequency (GHz)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.10: Simulated transmission ∣S21∣ of the readout resonator vs frequency and
power, with the qubit in the ∣g⟩ (left) and the ∣e⟩ (right) state. The same parameters
as Fig. 2.6 were used for simulation. Areas in white are not covered by simulation
due to small photon numbers. At approximately 2 × 10−12 Watt input power and
8.00016GHz frequency, ∣S21∣ ∼ 1 for the ∣g⟩ state and ∣S21∣≪ 1 for the ∣e⟩ state, which
could be used for high power Jaynes Cummings readout. In the transition region
(around 10−13 and 10−12), the simulated S21 is very sensitive to input power and
frequencies, due to the simplifications of the model discussed in the main text.

Fig. 4.10 and 4.11). This is particularly evident in the transition between the low

power dispersive region and the high power bare resonator region (around 10−13 and

10−12 W input power in Fig. 2.10).

2.2.6 Estimating the Voltage across the Junction

While performing characterization on some of the transmons (e.g. qubit H-i, see

Sec. 4.3 for naming nomenclature), I observed a reduced T1 (see Sec. 4.6.3 for mea-

surement setup) when measuring the state using the high-power Jaynes Cummings

readout, as compared to measuring the state with the low-power dispersive readout.

I suspect that an AC voltage V was being generated across the Josephson junction

during the readout process, such that V exceeds 2∆/e where ∆ is the the super-

26



Figure 2.11: Lumped-element circuit model to estimate the voltage across the
Josephson junction.

conducting gap of aluminum. Such a large voltage can generate quasiparticles (see

Fig. 6.1) in the superconducting leads and thus reduces the qubit T1. Note that if

“multiple Andreev reflection” is present, a voltage of 2∆/ne with integer n may also

generate quasiparticles. [5]

To estimate the AC voltage across the Josephson junction when a microwave signal

was applied at the input port of the package, I used a linear lumped-element circuit

model (see Fig. 2.11).

In this model, Z0 = 50 Ω is the characteristic impedance of the transmission

line and the distributed impedance of the CPW resonator. For this calculation, the

Josephson junction is approximated as a linear inductor with inductance Lq, and the

transmon qubit includes a shunting capacitance Cq = e2/2Ec. The (angular) resonant

frequency of the qubit is ωq = 1/
√
Lq ×Cq. The LRC resonator is coupled to the

transmon with a coupling capacitance Cg, with a value obtained either from finite

27



element simulations or the qubit-resonator coupling g by [133]

g ≈
√

ωq

CqZr

Cg

Cr

= Cg

√
ωqZr

Cq

ωr (2.38)

after some approximations. Here ωr = 1/
√
Lr ×Cr is the angular frequency of the read-

out resonator, and Zr ≈
√

Lr

Cr
is the characteristic impedance of the lumped element

resonator. For a quarter-wave resonator, which is what I used in my experiments, Zr

is related to the distributed impedance by [41]

Zr =
4

π
Z0. (2.39)

The resistance Rr in the parallel RLC circuit of the readout resonator is related to

the internal quality factor of the resonator by

Rr = QiZr. (2.40)

In addition, Cs is the coupling capacitance between the resonator and the transmission

line. It is related to the external quality factor Qe of the resonator by

Qe =
2

Z0 ×Zr × ω2
r ×C2

s

. (2.41)

Figure 2.9 shows the (simulated) resonator transmission ∣S21∣ when applying mi-

crowave signal at bare resonator frequency ωr/2π. From this plot, I can determine

the input power resulting in the maximum difference of S21 between qubit states ∣g⟩

and ∣e⟩ (dashed line in Fig. 2.9); this is the power one would use for readout. At

this input power and assuming the qubit is in the ∣e⟩ state, the average number of

photons stored in the readout resonator would be n = 1.7 × 105 for the parameters

used in Fig. 2.7 and 2.9. In the lumped circuit model of Fig. 2.11, the relationship

between the voltage across the readout resonator VRLC and photon number n is given

by

VRLC =
√
n × h̵ × ω2

r ×Zr. (2.42)

28



This voltage is split between the coupling capacitor with impedance 1/iωrCg and the

transmon with impedance (iωrCq +1/iωrLq)−1, so the voltage across the junction can

be expressed as

Vj = VRLC ×
(ωrCq − 1/ωrLq)−1

(ωrCq − 1/ωrLq)−1 + 1/ωrCg

. (2.43)

For the parameters used in Fig. 2.9 in Sec. 2.2.5, this corresponds to 55 µV -RMS

across the junction for the high power Jaynes Cummings readout.

Next, I fixed the parameters of the readout resonator from Sec. 2.2.5 (ωr/2π = 8

GHz, Qe = 3 × 104,Qi = 3 × 105), and Ec/h = 200 MHz of the qubit. I then varied the

frequency of the qubit ωq/2π and the coupling g in the simulation. Fig. 2.12 shows

how the estimated voltage across the junction during high power Jaynes Cummings

readout depends on frequency detuning and coupling strength. As expected, a smaller

detuning and larger g both yield a larger voltage across the junction. An important

observation from Fig. 2.12 is that, if the frequency detuning between the resonator

and the qubit exceeds some threshold (about 2.5 GHz), the voltage across the junction

during the readout process only has a small dependence on detuning.

To avoid quasiparticle generation when performing the high power Jaynes Cum-

mings readout, the peak voltage (a factor of
√
2 of the RMS voltage plotted in

Fig. 2.12) should be smaller than two times the superconducting gap. (This ig-

nores the possibility of Andreev processes.) Given aluminum’s superconducting gap

∆Al ∼ 200 µeV , this puts an upper bound of g ∼ 140 MHz of coupling strength between

the qubit and the readout resonator.

I note that this discussion only involved the resonator and transmon lumped circuit

model in Fig. 2.11, and not the input/output transmission line. In Sec. 6.1 of this

dissertation, I discuss a similar way of estimating Vj by solving the entire circuit in

Fig. 2.11 for a ratio of Vj/Vin. In addition to the limitations of the numerical model

(outlined at the end of Sec. 2.2.5), quasiparticle generation at a Josephson junction

29



Estimated Voltage across junction ( V-RMS)

50 100 150 200 250

Coupling g/2  (MHz)

1

2

3

4

5

D
e
tu

n
in

g
 (

r -
 

q
)/

2
 (

G
H

z
)

200

400

600

800

1000

1200

1400

1600

Figure 2.12: False color plot of numerically simulated (RMS) voltage across a
Josephson junction during high power Jaynes Cummings readout, versus detun-
ing between readout resonator and qubit (y-axis) and coupling g/2π between read-
out resonator and qubit (x-axis), assuming qubit is in the ∣e⟩ state. The solid
black line indicates the contour where the estimated voltage across the junction is
2∆Al/

√
2 ∼ 283 µV . Beyond a certain detuning (∼ 2.5GHz), the voltage across the

junction predominantly depends on coupling strength g.

is a highly nonlinear process. Thus the model in this section is highly simplified.

Nonetheless it sheds some light on the measurement of quasiparticle dynamics, to be

discussed further in Chapter 6 of this dissertation.

2.2.7 Driving Transmon Transitions

Eq. 2.28 describes the Hamiltonian of the dressed transmon-resonator system in

the absence of an applied time-dependent microwave pulse. In practice, applied mi-

crowaves resonant with the transmon are used to stimulate transitions and manipulate

30



its state. Suppose the qubit has an electric dipole moment operator given by [48]

ˆ⃗
d = ⟨g∣d⃗∣e⟩ (∣g⟩⟨e∣ + ∣e⟩⟨g∣) = ⟨g∣d⃗∣e⟩ (σ− + σ+) , (2.44)

and the electric field of the applied microwave pulse is given by

E⃗(t) = ϵ⃗E0 cosωdt =
1

2
ϵ⃗E0 (eiωdt + e−iωdt) . (2.45)

Here ωd/2π is the frequency of the microwave pulse, and E0 is the amplitude of the

electric field (which could have a time dependent envelope). The interaction between

the qubit and the oscillating electric field of microwave pulse can be expressed as

Hint,q = − ˆ⃗d ⋅ E⃗(t) = −
E0

2
⟨g∣ϵ⃗ ⋅ d⃗∣e⟩ (σ− + σ+) (eiωdt + e−iωdt) (2.46)

≃ − E0

2
⟨g∣ϵ⃗ ⋅ d⃗∣e⟩ (σ−eiωdt + σ+e−iωdt) , (2.47)

where the rotating wave approximation (RWA) [59] was used to derive Eq. 2.47.

Defining

Ωq = −
E0

h̵
⟨g∣ϵ⃗ ⋅ d⃗∣e⟩ (2.48)

as the Rabi frequency (a metric of the strength of the drive), the interaction Hamil-

tonian can be written as

Hint,q =
h̵Ωq

2
(σ−eiωdt + σ+e−iωdt) . (2.49)

Following a unitary transformation U = exp (iωdσz

2 t), one can remove the explicit time

dependence in Eq. 2.49 to arrive at

Hint,q =
h̵Ωq

2
(σ− + σ+) . (2.50)

In a similar manner, if a microwave pulse with frequency ωp/2π is applied and

interacts with the resonator, one can change the Pauli operators to creation/annihi-

lation operators of the resonator and transform Eq. 2.50 into

Hint,r =
h̵Ωr

2
(a + a†) . (2.51)

31



The combined Hamiltonian is thus

H = h̵∆r (â†â + 1

2
) + 1

2
h̵∆qσz + h̵χâ†âσz +

h̵Ωr

2
(a + a†) +

h̵Ωq

2
σx, (2.52)

where ∆r = ω̃r−ωp is the detuning between the resonator and its drive, and ∆q = ω̃q−ωd

is the detuning between the qubit and its drive.

2.2.8 Bloch Sphere Representation

The Hamiltonian in Eq. 2.52 includes Pauli operators that act on the qubit. A

pure state of an isolated qubit can be expressed in the form of

∣ψ⟩ =
⎛
⎜⎜
⎝

β

α

⎞
⎟⎟
⎠
, ∣α∣2 + ∣β∣2 = 1, (2.53)

or in Dirac notation as

∣ψ⟩ = cos θ
2
∣g⟩ + eiϕ sin θ

2
∣e⟩, (2.54)

with these two expressions related by

α = cos θ
2
, β = eiϕ sin θ

2
. (2.55)

A Bloch sphere [14] is a powerful way to graphically represent the state of a qubit.

The two angles (ϕ and θ in Eq. 2.54) that define the qubit state also define the location

of the state on a Bloch sphere surface(see Fig. 2.13a).

Assuming σj is a generalized Pauli matrix that operates on a two-level system with

Rabi frequency Ω (Hamiltonian is thus H = h̵Ωσj/2), solving Schrödinger equation

yields the evolution of the state

∣ψ(t)⟩ = exp (−iσj(Ωt)/2)∣ψ(0)⟩ = (I cos(Ωt/2) − iσj sin(Ωt/2)) ∣ψ(0)⟩. (2.56)

The state evolution in Eq. 2.56 can be visually interpreted as rotation of state ∣ψ⟩ on

the Bloch sphere, about an axis defined by the components of σj, and with rotational

angular velocity Ω (see Fig. 2.13b and the following discussions).

32



Φ

θ

(a) (b)

Figure 2.13: Bloch sphere representation of qubit state and rotation. (a) Bloch
sphere with states labelled. I label the south pole of the sphere as ∣g⟩ state, the north
pole as ∣e⟩ state, and the states on the equator are equal superposition of them with

different phases. The green arrow is a sample state ( ∣g⟩√
2
+ (1+i)∣e⟩2 ) uniquely identified

by two angles θ = π/2 and ϕ = π/4. (b) Evolution of qubit state starting from ∣g⟩
under Pauli operator σx (green) and operator (σx + 0.5σz) /

√
1.25 (red). The x, y and

z axes are labelled. The sphere is rotated 90 degrees along z axis from (a) to better
present the rotation of qubit states. The qubit states go through different rotations
(see text) with the arrows indicating the axes of rotation.

To illustrate the state rotation of a qubit on the Bloch sphere, assume the system

described by Eq. 2.52 starts in the ground state, i.e. there are no photons in the

resonator (⟨â†â⟩ = 0) and the qubit is in the ∣g⟩ state, and there is no resonator

drive (Ωr = 0), but there’s a microwave drive on resonance with the qubit frequency

(∆q = 0). The remaining (qubit) Hamiltonian in the rotating wave approximation is

simply

H = h̵Ωqσx/2. (2.57)

The evolution of the qubit state is a rotation about the x-axis on the Bloch sphere

(i.e. around the (∣g⟩ + ∣e⟩)/
√
2 state), indicated by the green points in Fig. 2.13b.

33



Starting from ∣g⟩ state, after time π/Ωq the qubit is “flipped” and reaches the ∣e⟩

state. This is called a “π-pulse” as it rotates the state of the qubit on a Bloch Sphere

by an angle of π about the x-axis.

If the qubit drive is not on resonance with the qubit (∆q ≠ 0), the (qubit) Hamil-

tonian would be H = h̵Ωqσx/2+ h̵∆qσz/2. The red points in Fig. 2.13b show the qubit

state evolution starting from ∣g⟩ for the case ∆q/Ωq = 0.5. Here, the qubit state ro-

tates about an axis (red arrow) that’s an angle of (arctan (∆q/Ωq))) from the x-axis.

As a result of this detuning, the qubit never reaches the ∣e⟩ state.

As a result, when driving qubit transitions I typically sent microwave pulses on

resonance with the qubit frequency. In rare occasions when a small detuning was

needed (such as Ramsey oscillation measurements), I made sure ∆q ≪ Ωq so that

the transitions had good fidelity. On the other hand, due to the limited anharmonic-

ity of the transmon, the microwave pulse could also couple to unwanted higher-level

transitions of the transmon (e.g. ∣e⟩ ↔ ∣f⟩) which drive the transmon out of the

computational basis. The ∣e⟩↔ ∣f⟩ transition differs from the desired ∣g⟩↔ ∣e⟩ tran-

sition by the qubit anharmonicity ∣α∣. To decrease the spurious ∣e⟩↔ ∣f⟩ transition,

I typically required Ωq ≪ ∣α∣. For example, for typical ∣α∣ ∼ 200 MHz, the maximum

drive strength Ωq I used in my experiments was on the order of a few MHz.

2.2.9 Density Matrix Representation

In addition to pure quantum states, which can be represented by Eq. 2.53, a more

general quantum state can be represented by a density matrix operator ρ̂ as [94]

ρ̂ =∑
i

pi∣ψi⟩⟨ψi∣, (2.58)

where ∣ψi⟩ are pure quantum states, and pi indicates the probability that the system

is in state ∣ψi⟩. For a two level system such as a qubit, ρ̂ in general can be expressed

34



by a 2 × 2 matrix

ρ̂ =
⎛
⎜⎜
⎝

∣β∣2 η

η∗ ∣α∣2

⎞
⎟⎟
⎠
, (2.59)

where both ∣α∣2 and ∣β∣2 are non-negative real numbers satisfying ∣α∣2 + ∣β∣2 = 1. In

addition,

tr (ρ̂ ⋅ ρ̂) ≤ 1, (2.60)

where “tr” stands for trace as the summation of all diagonal terms of a matrix.

Eq. 2.60 takes ‘=’ sign if ρ̂ is a pure state that can be expressed without the summation

in Eq. 2.58. Eq. 2.60 takes ‘<’ sign if ρ̂ is a mixed state. [94]

If Ô is an observable of the system, then its expected value can be expressed as

⟨Ô⟩ = tr (Ôρ̂) . (2.61)

In particular, all Pauli matrices are physical observables, and thus can have expec-

tation values ⟨σi⟩, i ∈ {x, y, z}. Since the Bloch sphere in Fig. 2.13 has unit radius, a

general qubit state ρ̂ can be represented by coordinates (⟨σx⟩, ⟨σy⟩, ⟨σz⟩). This point

lies on the surface of the Bloch sphere if it corresponds to a pure state, and inside

the Bloch sphere if it’s a mixed state.

For an isolated system described by Hamiltonian H, the density matrix evolves in

time according to

ih̵
∂ρ̂

∂t
= −[ρ̂,H] =Hρ̂ − ρ̂H. (2.62)

2.2.10 Lindblad-Kossakowski Master Equation and

Decoherence

There are many ways for a quantum system to be in a mixed state, such as being

the result of a partial trace from a larger Hilbert Space. For my qubits, a mixed

35



quantum state often stems from decoherence.

While Eq. 2.62 describes the density matrix evolution of an isolated quantum

system, a typical quantum system would interact with the surrounding environment.

For example, a qubit may be coupled to and exchange excitation with a heat bath.

In general, it’s not possible to keep track of coherent evolution of the environment.

Instead, the effects of the intersection are treated as decoherence of the qubit system.

Assuming the environment can be treated as Markovian [79], that is, it retains no

memory of its interaction with the smaller system, then the evolution of the small

system can be described by the Lindblad-Kossakowski master equation [69, 73]

ih̵
∂ρ̂

∂t
= −[ρ̂,H] +L[ρ̂]. (2.63)

Compared to Eq. 2.62, there is an additional term L[ρ̂], which is called a Liouvillian

and is given by

L[ρ̂]/h̵ =∑
i

ΓiD[Âi]ρ̂ =∑
i

Γi (Âiρ̂†
i −

1

2
†

i Âiρ̂ −
1

2
ρ̂†

i Âi) . (2.64)

In Eq. 2.64, the summation is over the decoherence channels, each with decoherence

rate Γi, and Âi is the “Lindblad operator” of each decoherence channel [89].

One decoherence channel for a qubit is energy relaxation, with a Lindblad operator

given by

σ− =
⎛
⎜⎜
⎝

0 0

1 0

⎞
⎟⎟
⎠
. (2.65)

Using the matrix notation of Eq. 2.59, one can see

D[σ−]ρ̂ =
⎛
⎜⎜
⎝

−∣β∣2 −0.5η

−0.5η∗ ∣β∣2

⎞
⎟⎟
⎠
, (2.66)

which can be interpreted as transferring population from the upper-left cell of ρ̂ (i.e.

∣e⟩ state) to the lower-right cell (i.e. ∣g⟩ state), while proportionally reducing the

diagonal terms. This is an incoherent process.

36



Figure 2.14a illustrates qubit energy relaxation in a graphical way on a Bloch

sphere, where the qubit decays from ∣e⟩ to ∣g⟩. The blue points represent sampling

of density matrix ρ̂ with time interval 0.1/Γ−, where Γ− is the decay rate for the

Liouvillian. It can be shown that the qubit’s population in ∣e⟩ (P∣e⟩) decays in an

exponential fashion P∣e⟩ ∼ exp (−Γ−t). In addition, a qubit may be excited by its

environment, with Lindblad operator σ+ and rate Γ+, although usually Γ+ ≪ Γ−.

Adding these two rates gives the energy relaxation rate, the inverse of which is a

qubit’s energy relaxation time constant T1,

T1 =
1

Γ+ + Γ−
≈ 1

Γ−
. (2.67)

Another typical decoherence channel for a qubit is pure dephasing, with a Lindblad

operator given by

σz =
⎛
⎜⎜
⎝

1 0

0 −1

⎞
⎟⎟
⎠

(2.68)

such that

D[σz]ρ̂ =
⎛
⎜⎜
⎝

0 −2η

−2η∗ 0

⎞
⎟⎟
⎠
. (2.69)

This can be interpreted as removing phase information, which is encoded in the off-

diagonal terms of ρ̂, while keeping the “population” of ∣g⟩ and ∣e⟩ constant.

Figure 2.14b illustrates qubit dephasing in a graphical way on the Bloch sphere,

where qubit state evolves from (∣g⟩ + ∣e⟩) /
√
2 to an evenly mixed state of ∣g⟩ and ∣e⟩.

The blue points represent sampling of density matrix ρ̂ with time interval 0.1/Γz,

where Γz/2 is the dephasing rate for the Liouvillian (due to the extra factor of 2 in

Eq. 2.69). It can be shown that the qubit’s population evolves in an exponential

faction towards the central z-axis in the Bloch Sphere, whose time constant is defined

as the pure dephasing time

Tϕ = Γ−1z . (2.70)

37



(a) (b)

Figure 2.14: Bloch sphere representation of qubit decoherence, (a) qubit energy

relaxation from ∣e⟩ to ∣g⟩ and (b) qubit pure phase relaxation from (∣g⟩ + ∣e⟩) /
√
2 to

an evenly mixed state of ∣g⟩ and ∣e⟩.

2.2.11 Determining Optimal Readout Length

As discussed in Sec. 2.2.4, the “low power dispersive readout” is a common way to

measure the state of the qubit. This technuqie uses the dispersive frequency shift of

the readout resonator due to changes in the qubit state. Due to vacuum fluctuations

and noise in the amplifier chain, the readout signal from the resonator inherently

contains some noise. If the qubit has infinite T1, then a longer readout pulse would

give better signal-to-noise ratio: for a readout pulse of length t ≪ T1, the ‘signal’ is

proportional to t while the (averaged) ‘noise’ is proportional to
√
t. On the other

hand, a qubit only has a finite lifetime T1, as a result, at time τ < t during the

readout pulse, the qubit only has probability exp (−τ/T1) of being in the original

state, scaling the ‘signal’ by a similar factor. Most of the measurements discussed in

this dissertation are averaged over many repetitions. As a result, I typically aimed for

a readout length t that was a balance between these two aspects, giving me the best

38



signal-to-noise ratio of the averaged data (instead of single-shot readout fidelity).

To simplify the readout process, I ignore resonator ringup and ringdown for the

moment. In addition, for my measurements, microwave signals throughout the read-

out length t were integrated using equal weights. To quantitatively study the optimal

readout length t, suppose ν is the ‘signal’ per unit time, characterized by the (voltage)

difference in the transmitted signal between the qubit ∣g⟩ and ∣e⟩ state (See Fig. 4.32

for a sample of single-shot readout data in the IQ plane). Considering time τ < t

where t is the readout pulse length, during the small time interval from τ to τ + dτ ,

the qubit only has probability exp (−τ/T1) of being in the original state, thus the

contribution to the overall ‘signal’ from this small time interval is exp (−τ/T1)dτ × ν.

The overall readout ‘signal’ is an integration of such term

signal = ∫
t

0
exp (−τ/T1)dτ × ν = T1ν (1 − exp (−t/T1)) . (2.71)

On the other hand, suppose µ is the ‘noise’ per square root of unit time, then the

readout ‘noise’ is thus µ
√
t. This corresponds to the spread of single-shot readout

clusters on the IQ plane in Fig. 4.32. In this way, signal-to-noise ratio (SNR) of the

readout can be expressed as

SNR = ∣T1ν (1 − exp (−t/T1))
µ
√
t

∣
2

(2.72)

The SNR in Eq. 2.72 is maximized respect to readout length t by setting d(SNR)/dt =

0, which is satisfied when

exp (−t/T1) − (1 − exp (−t/T1)) ×
T1
2t
= 0. (2.73)

One finds at t ≈ 1.26T1. It’s worth noting this optimal readout length does not depend

on either µ or ν, but only on the qubit’s T1.

It’s worth noting that discussions in this section correspond to my measurements

where microwave signals throughout readout length t were integrated using equal

39



readout length t

Signal

Non-Signal

0

1

P
e

Figure 2.15: Figurative representation of SNR with respect to readout length t.
The readout ‘signal’ is proportional to the area in red, which, due to qubit energy
relaxation has an exponential shape with time constant T1. The readout ‘noise’ is
proportional to

√
t.

weights. It’s possible to assign different weights to different portions of the signal,

e.g. assign more weights to the initial part of the signal where the qubit is more likely

to be in its original state, and less weight to the later part where the qubit likely

has experienced decay. In other words, the the ideal length of the “box car” filter

function was discussed in this section where the filter function could be of any shape,

which could result in higher SNR than discussed in this section. [40]

2.3 Loss Mechanisms of Transmon

2.3.1 Quasiparticle Loss

In a superconductor, electrons with opposite spins and momenta form Cooper

pairs [22]. A Bogoliubov quasiparticle, or quasiparticle for short, is an elementary

excitation above the ground state of a superconductor, which can be formed as a

40



result of broken Cooper pairs. Assuming a symmetric (equal) superconducting gap

∆ on two sides of an ideal Josephson junction, it can be shown that due to single

quasiparticle tunneling, the transition rate from qubit state i to (a different) qubit

state f can be expressed as [43, 20]

Γif =
16Ej

h̵π∆ ∫
∞

∆
dϵf(ϵ) (1 − f(ϵ + h̵ωif))

ϵ(ϵ + h̵ωif) +∆2

√
ϵ2 −∆2

√
(ϵ + h̵ωif)2 −∆2

∣⟨f ∣ sin ϕ̂
2
∣i⟩∣

2

(2.74)

where ϕ̂ is the operator for the difference of superconducting phase across the junction,

f is the quasiparticle distribution function, h̵ωif = Ei − Ef is the energy difference

between the initial and final states. This equation assumes the low-energy limit

∣h̵ωif ∣≪∆ and ignores Andreev processes [5].

For fixed-frequency transmons with no external flux bias (ϕ0 = 0), assuming i = ∣e⟩

and f = ∣g⟩, it can be shown that

∣⟨f ∣ sin ϕ̂
2
∣i⟩∣

2

= Ec

h̵ωeg

1 + cosϕ0

2
= Ec

h̵ωeg

. (2.75)

Assuming the quasiparticles have a distribution function f that follows Fermi-Dirac

distribution at an effective temperature T that’s much smaller than the supercon-

ducting gap (T ≪∆/kB), it can be shown that

Γeg ≃
√

2ωeg∆

π2h̵
xqp, (2.76)

where xqp is the normalized quasiparticle density, defined by xqp = nqp/ncp, the ratio

between absolute quasiparticle volume density nqp and Cooper pair density, which

is ncp = 4 × 106µm−3 for aluminum (when T ≪ ∆/kB). At low temperatures and in

thermal equilibrium, xqp is related to T by [19]

xqp ≃
√
2πkBT /∆exp (−∆/kBT ) . (2.77)

Assuming T ≃ 20 mK for the temperature of the dilution fridge, ∆ = 200 µeV as

the superconducting gap of aluminum, one would expect xqp ∼ 10−52. However, ex-

41



perimentally measured quasiparticle densities are in the range of xqp = 10−8 to 10−4

[4, 124, 81, 140, 142, 31]. Possible sources of such excess quasiparticle densities are

the absorption of stray infrared radiation [9] and ionizing radiation such as muons

and gamma rays [139, 85, 17].

2.3.2 Two-Level System Loss

Dielectric two-level system (TLS) defects are a another source of loss for super-

conducting qubits and resonators. A simple model for a TLS defect is a charged ion

that can tunnel between two asymmetrical potential wells (see Fig. 2.16) [103, 32]. I

assume the minima of energy of the two potential wells differ by ϵ, and the tunneling

between the two wells has a strength ∆, which is related to the distance d between

the wells, the tunneling barrier height V and the effective mass of the TLS m, such

that ∆∝ exp (−d
√
2mV /h̵2). The energy difference between the ground and excited

state of the TLS is then E =
√
ϵ2 +∆2.

If the detuning between a TLS and a qubit is small and the coupling between them

is significant, excitations can be exchanged between the TLS and the qubit, leading to

the qubit’s energy relaxation and dephasing. Although the exact microscopic identity

of TLS causing decoherence in superconducting qubits is not completely understood

[92], possible candidates are atomic defects such as O-H bonds [82, 123], chemical

residuals [110], oxygen defects [1, 72] among others[92].

Given that TLS loss often occurs in dielectric materials and interfaces, on a macro-

scopic level, this yields a loss tangent tan δi. A common characteristic of TLS loss in

superconducting resonators is that the loss decreases with increasing applied electric

field. On a microscopic level, once a TLS is driven into a mixed state with equal

probability, it first has to dissipate its energy before it can absorb more energy from

42



Figure 2.16: Double-well potential model of TLS, reproduced from Ref. [127] under
license number 5574400977925.

the external field. In this way, the TLS limiting Qi of a resonator can be expressed

as [104]

1

Qi,TLS

=∑
i

pi tan δi
tanh ( h̵ωr

2kBT
)

√
1 + ( n

nc
)α

(2.78)

where i indicates different loss regions, pi are their participation ratios (ratio of electric

field energy in that region compared to total electric field energy), tan δi are loss

tangents characterizing TLS loss of each region, ωr is the resonant frequency of the

resonator, nc is a critical photon number characterizing the saturation of TLS loss,

and n is the photon number stored in the resonator. For a single TLS under uniform

field, α = 1 [104]. From Eq. 2.78, one can see that as n increases, the loss from TLS

decreases. In Fig. 5.3 of this dissertation, the measured Qi versus photon number

is plotted for a series of coplaner waveguide resonators, most of which displayed an

increase of Qi with increasing photon number n, a trend consistent with TLS loss.

While the loss for superconducting resonators can be reduced by saturating the

TLS, transmon qubits typically operate in the single-excitation condition (n ≤ 1 in

Eq. 2.78), thus TLS loss can be a significant loss mechanism for transmon qubits.

43



2.3.3 Purcell Loss

Discovered by Edward M. Purcell, the Purcell effect [109] refers to enhancement

(or suppression) of the spontaneous emission rate of an excited atom or quantum

emitter that is coupled to a resonant optical cavity or nanostructure. For the case

of a transmon coupled to a readout resonator with resonator decay rate κ, the qubit

can decay due to such coupling, with a decay rate given by [114]

ΓPurcell ≃ (g/∆)2κ, (2.79)

where g is the coupling strength between the resonator and the qubit, and ∆ is the

detuning between the two. Eq. 2.79 is only valid in the dispersive limit g ≪∆.

Loss from the Purcell effect can be reduced by decreasing κ (i.e. increasing QL)

or by reducing the effective resonator-qubit coupling (g/∆). Reducing κ increases

the ring-up time of the resonator, preventing a fast measurement of the qubit states

while reducing (g/∆) reduces the signal-to-noise ratio of qubit state measurement.

In practice, the qubit may also couple to higher modes of a resonator structure,

which introduces additional Purcell loss. For transmons, additional modes are most

obvious when the qubit frequency is above the fundamental frequency of the readout

resonator.

For the qubits that I studied (see Table 4.1), the qubit frequencies were sometimes

significantly different than the design value, due to variations in the device fabrication.

However, the qubit-resonator detuning ∆ was typically sufficiently greater than the

coupling g so that most of the qubits’ coherence was not limited by the Purcell effect.

Using qubit H-i as an example, κ ∼ 1.3 × 106 s−1, g/2π = 157 MHz, ∆ = 2.05 GHz, the

Purcell limited qubit T1 is 130 µs, much larger than the observed value of 18 µs.

One way to mitigate Purcell loss without affecting measurement speed is the use

of a Purcell filter [122], which is an additional low quality-factor resonator between

44



the transmission line and the readout resonator. However, fast and high-fidelity

measurements of the qubit state were not a focus of this dissertation. As a result, I

did not incorporate Purcell filters into the designs of my qubits.

2.3.4 Package Loss

My 2D superconducting devices were mounted in a package that could introduce

additional loss. For example, the qubit could couple to spurious low-Q resonance

modes [119, 56] of the package. This can be understood as another case of Purcell

loss discussed in the previous subsection. A common way to mitigate such loss is to

increase the minimum frequencies of spurious modes of the package, typically above 10

GHz for conventional superconducting transmon qubits. Reducing the dipole moment

of transmon qubits and thus reducing coupling to spurious modes is another way to

mitigate this loss. In addition, if normal metal is used in the package, it can cause

loss due to eddy currents. This loss mechanism is discussed in Chapter 5.

2.3.5 Vortex Loss

When a magnetic field is applied to a superconductor, screening currents flow on

the surface of the superconductor so as to cancel the magnetic field inside the super-

conductor. If a superconducting sample is cooled through its critical temperature Tc,

any applied magnetic field will be expelled. This is called the Meissner effect [88]. For

type-I superconductors (which include aluminum), such expulsion of magnetic fields

happens up to a critical field strength Hc. If H > Hc, the material becomes normal.

For type-II superconductors, such expulsion is only up to a lower critical