THE UNIVERSITY OF CHICAGO

MILLIMETER-WAVE SUPERCONDUCTING QUANTUM DEVICES

A DISSERTATION SUBMITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES

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IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
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DEPARTMENT OF PHYSICS
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BY
ALEXANDER VLADIMIR ANFEROV

CHICAGO, ILLINOIS
MARCH 2024
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Copyright © 2024 by Alexander Vladimir Anferov

All Rights Reserved
 To Sophie and Miso

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"If at first you don’t succeed Mr. Wint...?
Try, try again Mr. Kid."
- Diamonds are Forever 1971

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 TABLE OF CONTENTS

LIST OF PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 SUPERCONDUCTORS AND THEIR PROPERTIES . . . . . . . . . . . . . . . . 7

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2.1 Magnetic Field Explusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Drude Conductivity and London Equations . . . . . . . . . . . . . . . 9
2.1.2 Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Kinetic Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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2.2.1 Currents in a superconducting wire . . . . . . . . . . . . . . . . . . . 12
2.2.2 Kinetic energy of a supercurrent . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Total Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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2.2.4 Nonlinearity in kinetic inductance . . . . . . . . . . . . . . . . . . . . 16
2.3 Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Wavefunction and Velocity . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Vortices in the superconductor . . . . . . . . . . . . . . . . . . . . . . 19
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2.4 Coherence Length and Proximity Effect . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Superconductor Boundaries . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Aluminum Proximitized by Niobium . . . . . . . . . . . . . . . . . . 24
2.4.3 Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 BCS theory and Superconducting Energy Gap . . . . . . . . . . . . . . . . . 26
2.5.1 Quasiparticle Excitations . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Mattis-Bardeen Theory of Complex conductivity . . . . . . . . . . . . . . . . 31
2.6.1 Conductivity Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.2 Resistance and Reactance . . . . . . . . . . . . . . . . . . . . . . . . 32

3 HIGH-FREQUENCY CIRCUIT COMPONENTS . . . . . . . . . . . . . . . . . . 34

3.1 Circuit Components at High Frequencies . . . . . . . . . . . . . . . . . . . . 35
3.2 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Hollow Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Coaxial Lines at High Frequency . . . . . . . . . . . . . . . . . . . . . . . . 45
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 3.4 Planar Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.1 Rectangular Cavity (Box) Resonator . . . . . . . . . . . . . . . . . . 51
3.5.2 Transmission Line Resonator . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.3 Simple LCR Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.4 Real Resonator Measured In Reflection . . . . . . . . . . . . . . . . . 57
3.5.5 Side-coupled Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.6 Asymmetry in Side-coupled Resonators . . . . . . . . . . . . . . . . . 62
3.6 Sources of Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.1 Multiple Loss Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.2 Disentangling Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 NONLINEAR QUANTUM SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Resolving Single Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Kerr Nonlinear Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Side-coupled Nonlinear Resonator . . . . . . . . . . . . . . . . . . . . 75

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4.3 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Critical Current from Superconducting Properties . . . . . . . . . . . 79
4.3.2 Junction Capacitance . . . . . . . . . . . . . . . .
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4.4 Josephson Nonlinear Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.1 Cooper Pair Box in the Transmon Limit . . . . . . . . . . . . . . . . 86
4.4.2 Energy Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Experiments with a Two Level System . . . . . . . . . . . . . . . . . . . . . 89
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4.5.1 Qubit Resonator Interactions . . . . . . . . . . . . . . . . . . . . . . 90
4.5.2 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6 Dissipation for a Two-Level System . . . . . . . . . . . . . . . . . . . . . . . 95

5 MILLIMETER-WAVE FOUR-WAVE MIXING . . . . . . . . . . . . . . . . . . . 97

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5.1 High-KI Niobium Nitride Films . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Quantum Measurements at Millimeter-wave Frequencies . . . . . . . . . . . 101
5.3 Exploring loss with Millimeter-wave Loss with Resonators . . . . . . . . . . 103
5.3.1 Thermal Losses from Complex Conductivity . . . . . . . . . . . . . . 107
5.4 Single-Tone Kerr Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 Controlling nonlinearity in the presence of additional losses . . . . . . 111
5.5 Degenerate Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.6.1 Nonlinearity Relative to Linewidth . . . . . . . . . . . . . . . . . . . 116

6 IMPEDANCE-ENHANCED KERR NONLINEARITY . . . . . . . . . . . . . . . 118

6.1 Increasing Nonlinearity from Kinetic Inductance . . . . . . . . . . . . . . . . 119
6.2 Parasitic Inductance in a Finger Capacitor . . . . . . . . . . . . . . . . . . . 121
6.3 High Capacitance Density with Fractals . . . . . . . . . . . . . . . . . . . . . 127
6.4 Microwave Titanium Nitride Fractal Resonators . . . . . . . . . . . . . . . . 132
6.5 Nonlinearity Relative to Linewidth . . . . . . . . . . . . . . . . . . . . . . . 134
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 6.6 Nonlinear Decay of a Coherent State . . . . . . . . . . . . . . . . . . . . . . 136
6.6.1 Heterodyne Measurements . . . . . . . . . . . . . . . . . . . . . . . . 138
6.7 Impedance Enhanced Kerr at 100 GHz . . . . . . . . . . . . . . . . . . . . . 140

7 NIOBIUM TRILAYER JUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.1 Trilayer Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.1.1 Wafer Preparation and Trilayer deposition . . . . . . . . . . . . . . . 145
7.1.2 Trilayer Etch 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.1.3 Spacer Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.1.4 Spacer Etch 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.1.5 Wiring Layer Deposition . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.1.6 Wiring Etch 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1.7 Spacer removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Junction DC Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Junction Superconductor Properties . . . . . . . . . . . . . . . . . . . . . . . 152
7.4 Lossy Plasma Etch Residues and Treatment . . . . . . . . . . . . . . . . . . 156

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7.4.1 Optimizing Etch Conditions to Reduce Residue Formation . . . . . . 158
7.4.2 Residue Defluorination with Sodium and Potassium . . . . . . . . . . 159
7.4.3 Dissolving Fluorinated Organometallic Compounds . . . .
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7.5 Junction Area Dependence, Variation and Stability . . . . . . . . . . . . . . 163
7.6 Junction Annealing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.6.1 High Jc with Low Temperature PECVD . . . . . . . . . . . . . . . . 167
7.7 Qubit Geometry and Experimental Setup . . . . . . . . . . . . . . . . . . . . 168
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7.8 Microwave Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.8.1 Measuring Losses with Qubit Coherence . . . . . . . . . . . . . . . . 172
7.8.2 Material Loss Probed by Resonator Quality Factor . . . . . . . . . . 174
7.8.3 Detailed Model of Junction Losses . . . . . . . . . . . . . . . . . . . . 176
7.8.4 Temperature dependence of decoherence . . . . . . . . . . . . . . . . 180
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7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8 K BAND QUBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.1 K Band Qubit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.1.1 High Frequency Trilayer Junction . . . . . . . . . . . . . . . . . . . . 183
8.1.2 Post Fabrication Residue Removal . . . . . . . . . . . . . . . . . . . . 185
8.1.3 Qubit Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2 K Band Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.3 Qubit Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.3.1 Number Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.4 Qubit Coherence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.5 Thermal Dependence of Decoherence . . . . . . . . . . . . . . . . . . . . . . 193
8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

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9 IMPROVED MILLIMETER-WAVE MEASUREMENTS . . . . . . . . . . . . . . 196
9.1 Tapered Waveguide Transition Design . . . . . . . . . . . . . . . . . . . . . . 198
9.2 Cryogenic Measurements and Calibration . . . . . . . . . . . . . . . . . . . . 202
9.2.1 Experimental Measurement Setup . . . . . . . . . . . . . . . . . . . . 202
9.2.2 Cryogenic Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.3 Waveguide Transition Characterization . . . . . . . . . . . . . . . . . . . . . 206
9.3.1 Leakage Bypassing the Transition . . . . . . . . . . . . . . . . . . . . 208
9.4 Ground-Shielded Resonator Design . . . . . . . . . . . . . . . . . . . . . . . 209
9.5 Millimeter-wave Resonator Measurements . . . . . . . . . . . . . . . . . . . 211
9.5.1 Reducing Losses With Surface Oxide Etch . . . . . . . . . . . . . . . 214
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10 MILLIMETER-WAVE QUBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.1 The Millimeter Wave Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.1.1 Helium-4 Experiment Refrigeration . . . . . . . . . . . . . . . . . . . 220
10.2 Continuous Wave Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 221

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10.2.1 Identifying Qubits with Two Tone Spectroscopy . . . . . . . . . . . . 223
10.2.2 Resolved Qubit Transitions . . . . . . . . . . . . . . . . . . . . . . . 226
10.2.3 Drive Power Calibration via AC Stark Shift . . . . . .
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10.2.4 Qubit Properties From Transition Linewidth . . . . . . . . . . . . . . 229
10.3 Pulsed Qubit Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.3.1 Sub Nanosecond Pulse Measurements . . . . . . . . . . . . . . . . . . 231
10.3.2 Pulse Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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10.3.3 Flat Top Pulse Envelope . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.3.4 Frequency Dependence of Rabi Oscillations . . . . . . . . . . . . . . . 237
10.3.5 Optimized Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . 240
10.3.6 Millimeter-wave Qubit Coherence in the Time Domain . . . . . . . . 242
10.4 Qubit Performance Properties in Context . . . . . . . . . . . . . . . . . . . . 244
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11 CONCLUSIONS AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . 246

11.1 Coherence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
11.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
11.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

A DERIVATION OF RESONATOR RESPONSES . . . . . . . . . . . . . . . . . . . 277

A.1 1-Port Shunt Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
A.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
A.1.2 2-Port Shunt Resonator . . . . . . . . . . . . . . . . . . . . . . . . . 280
A.2 Side Coupled LRC Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A.2.1 2-Port Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

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B FABRICATION RECIPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
B.1 Substrate Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
B.2 Atomic Layer Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
B.2.1 Niobium Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
B.2.2 Titanium Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
B.3 Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
B.4 Reactive Ion Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
B.5 Resist Removal and Dicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

C DISENTANGLING HETERODYNE QUBIT IMAGES . . . . . . . . . . . . . . . 290

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 LIST OF PUBLICATIONS
This thesis is based in part on the following works:

[1] Alexander Anferov, Aziza Suleymanzade, Andrew Oriani, Jonathan Simon, and
David I. Schuster. Millimeter-Wave Four-Wave Mixing via Kinetic Inductance
for Quantum Devices. Physical Review Applied, 13(2):024056, February 2020.
doi:10.1103/PhysRevApplied.13.024056.

[2] Alexander Anferov, Shannon P. Harvey, Fanghui Wan, Kan-Heng Lee, Jonathan Simon,
and David I. Schuster. Low-loss Millimeter-wave Resonators with an Improved Coupling
Structure. arXiv, November 2023. doi:10.48550/arXiv.2311.01670.

[3] Alexander Anferov, Kan-Heng Lee, Fang Zhao, Jonathan Simon, and David I. Schuster.
Improved Coherence in Optically-Defined Niobium Trilayer Junction Qubits. arXiv, June
2023. doi:10.48550/arXiv.2306.05883.

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[4] Alexander Anferov, Shannon P. Harvey, Fanghui Wan, Jonathan Simon, and David I.
Schuster. Superconducting Qubits Above 20 GHz Operating Over 200 mk. In prepara-
tion. IE
[5] Alexander Anferov, Shannon P. Harvey, Fanghui Wan, Jonathan Simon, and David I.
Schuster. A Millimeter-wave Artificial Atom Cooled with Helium-4. In preparation.

[6] Aziza Suleymanzade, Alexander Anferov, Mark Stone, Ravi K. Naik, Andrew Oriani,
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Jonathan Simon, and David Schuster. A tunable high-Q millimeter wave cavity for
hybrid circuit and cavity QED experiments. Appl. Phys. Lett., 116(10), March 2020.
ISSN 0003-6951. doi:10.1063/1.5137900. (Cited on page 68).
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 LIST OF FIGURES
1.1 Processor clock speed in MHz over time. After an initial exponential increase,
the clock speeds plateau after 2005 reaching a limit of 4-5 GHz. (manufacturers
not labelled for brevity). Data from [51, 52] . . . . . . . . . . . . . . . . . . . . 1
1.2 Electromagnetic spectrum in terms of frequency, along with atmospheric atten-
uation which highlighting regions with low attenuation where communication
technology helped develop instrumentation and components. . . . . . . . . . . . 3
1.3 Average number of photons in a harmonic oscillator as a function of temperature,
plotted for varying resonant frequencies. . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Periodic table highlighting elements with superconducting transitions (labeled in

Kelvin) and colored by transition temperature TC . Data taken from [68]. . . . . 7
2.2 Illustration of the Meissner Effect. An external magnetic field outside of a su-
perconductor results in a magnetic field that decays exponentially on the length
scale λL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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2.3 The Meissner Effect in action: an external applied magnetic field B, which would
penetrate a normal conductor is expelled from a superconductor, inside which
B = 0 (other than a small distance λL from the surface). . . . . . . . . . . . . . 11
2.4 A cylindrical wire carrying a current generates a magnetic field inside the wire.
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For a normal conductor this current depends linearly on radius, however for a
superconductor the current is localized on the surface in order to satisfy the
London equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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2.5 Cross sections of a rectangular wire in the thick and thin-film limits (with respect
with the London length). In the thin-film limit, the entire area of the wire carries
the current, while in the thick-film limit only the surface does. . . . . . . . . . . 14
2.6 Kinetic and magnetic (or geometric) inductance, plotted as a function of film
thickness relative to the London length. While for thick films the two are com-
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parable, for thin films we find that kinetic inductance becomes significantly higher. 16
2.7 Superconducting vortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Illustration of the proximity effect at the boundary between a superconductor
and normal metal, showing that superconductivity extends some distance into
the normal metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9 Illustration of the Josephson effect: two superconductors separated by a very thin
barrier d ≪ ξ have two separate phases on either side . . . . . . . . . . . . . . . 25
2.10 Left: Energy spectrum of unpaired electrons for a normal conductor (dashed)
with hole and electron solutions for energy. In a superconductor (solid lines), the
excitations are increased by a gap energy ∆. Right: Normalized density of states
for quasiparticles and quasiholes, which is split around the Fermi energy by ∆,
so forming either kind of excitation requires an energy of at least ∆. . . . . . . . 27
2.11 Exact solution for the temperature dependence of the superconducting energy
gap plotted in reduced units with respect to T /Tc . . . . . . . . . . . . . . . . . 29

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2.12 Numerically solved complex conductivities, solved for various frequencies (for
niobium this range roughly corresponds to 10-100 GHz). Real conductivity σ1
from quasiparticles is shown on the left, and complex conductivity σ2 from Cooper
pairs is shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Illustration of a realistic non-ideal capacitor (left) and inductor (right). Para-
sitic reactances (inductive and capacitive respectively) are shown with dashed
lines. Approximate circuits modelling the parasitics are shown below. Since the
parasitics should be small we should be able to take some shortcuts. . . . . . . . 36
3.2 Reactance plotted as a function of frequency for a realistic capacitor (left) and
inductor (right). Even up to microwave frequencies (10 GHz), the parasitics have
little effect and can safely be ignored, but by 100 GHz their effects become much
more significant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Rectangular waveguide (left) showing the electric fields of its lowest frequency
T E10 mode, and cylindrical waveguide (right) showing its lowest frequency T E11
mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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3.4 Dispersion relation of a WR10 rectangular waveguide (left) showing the cutoff
frequency, and the single-moded frequency range (59–117 GHz). The dispersion
relation for a cylindrical wavguide is shown on the right. The lowest frequency
mode behaves fairly similar to the rectangular waveguide, however since the struc-
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ture is symmetric the next highest mode cutoff is much closer. . . . . . . . . . . 43
3.5 Transmission through a rectangular waveguide (left) and circular waveguide (right)
below their cutoff frequencies, plotted for different lengths of waveguide section,
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showing the drastic attenuation resulting from the evanescent propagation. . . . 44
3.6 Single-moded operation range of a coaxial cable with a 50 Ω impedance, plotted
as a function of outer diameter D for different dielectric constants (For example
Teflon has ϵr = 2.02). Standard coax connector and cable dimensions are marked
in gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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3.7 Photograph of a partially assembled 1 mm coaxial connector. The outer diameter

of the dielectric (white ring) is less than 1 mm, so these parts are very fragile! . 47
3.8 Illustration of common types of planar transmission lines, with electric fields
labelled with dashed lines. Other than the stripline, all the surface transmission
lines shown here can be operated with a backside or external ground. . . . . . . 48
3.9 Illustration of a transmission line showing the electric and magnetic fields of its
supported TEM mode (left) along with charges and currents, and the lossless
model (right) of the transmission line made up of ideal components. . . . . . . . 49
3.10 Diagram of a rectangular resonance (or box mode). Did I mention it looks like a
box? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 Mode frequencies of a 100 µm-thick box made from vacuum, quartz and sapphire
respectively, assuming the other two dimensions are equal. . . . . . . . . . . . . 53
3.12 Mode frequencies of a 1 mm-thick box with varying square dimensions containing
a sapphire chip of varying thickness. . . . . . . . . . . . . . . . . . . . . . . . . 54
3.13 Circuit diagram of the simplest LCR resonator . . . . . . . . . . . . . . . . . . . 55

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3.14 Complex reflection response of a resonator, plotted for various strengths of inter-
nal losses Qi . The ambiguity of the magnitude response between under-coupled
and over-coupled cases necessitates a full measurement of both quadratures. . . 56
3.15 LCR resonator with multiple loss channels Ri . . . . . . . . . . . . . . . . . . . . 57
3.16 Complex asymmetric reflection response of a resonator, plotted for various strengths
of asymmetry ϕ and internal losses Qi . . . . . . . . . . . . . . . . . . . . . . . . 59
3.17 Circuit diagram for a side-coupled resonator. . . . . . . . . . . . . . . . . . . . . 59
3.18 Side-coupled resonator model (a) 3 port network of an H-plane splitter coupled
to a black box resonator, showing corresponding transmission coefficients with
input and output fields labelled by port. The inset shows the equivalent circuit
network. (b) Analogous configuration for an optical cavity, adjusted for boundary
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.19 Complex transmission response of a side-coupled resonator, plotted for various
strengths of internal losses Qi relative to the coupling Qe . . . . . . . . . . . . . 61
3.20 Complex transmission response of an asymmetric side-coupled resonator, plotted
for various values of asymmetry ϕ (for negative values the effect is simply flipped).

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Here we have fixed Qi = 10Qe : notice however that the depth of the transmission
dip increases as asymmetry grows making it easy to overestimate Qi without
taking into account the asymmetric response. . . . . . . . . . . . . . . . . . . .
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3.21 LCR resonator with multiple loss channels Ri . . . . . . . . . . . . . . . . . . . . 65

4.1 Energy levels of a harmonic oscillator (left) which are all evenly spaced, and an
anharmonic oscillator (right) whose potential isolates the energy levels making
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them individually addressable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Nonlinearity strength relative to linewidth (or loss). For large linewidths or weak
nonlinearity, individual photon levels are indistinguishable from one another and
the system is governed by classical behavior, and many photons are required to
observe the nonlinear effects. Only when the anharmonicity exceeds the linewidth
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can we address energy levels individually. . . . . . . . . . . . . . . . . . . . . . . 71

4.3 a) Circuit model of a nonlinear resonator described by coupling κ, loss γ, fre-
quency ω0 and nonlinearity K, with input and output fields. b) Optics model
of the same system where the resonator is a Kerr-nonlinear optical cavity. c)
Normalized intracavity photon number as a function √ of reduced pump frequency
δ for reduced drive strengths ξ, where ξcrit = −1/ 27. . . . . . . . . . . . . . . 72
4.4 Nonlinear reflection response of a Kerr nonlinear resonator in the complex plane
(left) and in terms of magnitude (right). We chose κ = 2γ to make the magnitude
response apparent. Notably the circle profile in the complex plane is unaffected
by the nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Nonlinear transmission response of a side-coupled Kerr nonlinear resonator in the
complex plane (left) and in terms of magnitude (right). Here κ is chosen to be 8γ. 76
4.6 Nonlinear transmission response of an asymmetric side-coupled Kerr nonlinear
resonator shown for significant asymmetries ϕ = −0.5 (left) and ϕ = 0.5. Here κ
is also chosen to be 8γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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4.7 Josephson junction critical current and Josephson inductance as a function of
room temperature resistance for a junction made with niobium and aluminum.
Note the nearly order of magnitude difference between the two materials. . . . . 80
4.8 Josephson junction critical current and Josephson inductance as a function of
junction area for varying junction critical current density Jc . Since the super-
conducting properties of the junction are encoded in Jc , these relationships are
agnostic to junction materials. For the junctions used in microwave qubits, typi-
cally LJ ∼ 1–10nH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.9 Illustration of a generic Josephson junction, highlighting the parallel plate ca-
pacitance formed on the barrier. For this reason, junctions are modelled with a
capacitance CJ , sometimes abbreviated with a single circuit element. . . . . . . 82
4.10 Josephson junction plasma frequency, plotted as a function of critical current
density Jc , highlighting the need for high-Jc junctions for millimeter-wave ap-
plications. Here we assumed that a 1 nm and 2 nm barrier result in 3 kA and
30 A/cm2 respectively: neglecting the effects of barrier thickness on capacitance
(grey dashed lines corresponding to fixed 1 and 2 nm) still gives a pretty good

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estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.11 Circuit diagram for a simple anharmonic (nonlinear) circuit consisting of a Joseph-
son junction shunted by a capacitance. A microscope image of one of such circuits
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is shown on the right, where the shunting capacitor has a large cross shape. This
particular circuit has a frequency of 19 GHz, and a nonlinearity around 200 MHz. 84
4.12 Circuit diagram for a cooper pair box, where the capacitively shunted Josephson
junction is capacitively coupled to a charging voltage. . . . . . . . . . . . . . . . 86
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4.13 Energy levels in the transmon limit. For sufficiently large EJ ≫ Ec the energy
levels have nearly no dependence on the gate charge ng making the qubit. . . . 87
4.14 Left: sorting function k(m, ng ) used to organize the energy solutions of the trans-
mon. Right: variation of each energy level ϵm with respect to the charging energy. 88
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4.15 Illustration of the interacting cavity (or resonator) and two level system (atom
or qubit) characterized by the Jaynes Cummings hamiltonian . . . . . . . . . . 90
4.16 Excited state population after being driven by a signal with amplitude Ω detuned
from the qubit by ∆ for some time t. a) Amplitude-freqency Rabi oscillations
increase in frequency off-resonance while also changing in contrast as a function
of amplitude, resulting in a WiFi pattern. b) The most popular chevron pattern
appears from time-frequency Rabi oscillations, for which contrast is a fixed func-
tion of detuning. c) At ∆ = 0, the oscillation rate can be expressed as a function
of pulse area Ωt, so we observe constant population fringes for curves of constant
Ωt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.17 Rabi oscillations, shown by plotting the excited state population as a function of
drive amplitude Ω (in MHz), detuning from the qubit ∆ (in MHz), and time t
(in ns). Section cuts from respective axes yield the plots in Figure 4.16. . . . . 93

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4.18 Excited state population following a finite-length Gaussian pulse terminated at
±nσ, shown for different values of n. For smaller n the pulse shape is closer to
a square pulse, and the fringes exhibit significant power broadening. As more of
the Gaussian envelope is used the fringe bandwidth is reduced, limiting the power
broadening effect. Here the pulse lengths σ have been adjusted to yield similar
oscillation rates at ∆ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1 Illustration of the plasma-enhanced atomic layer deposition process (PEALD or

ALD for short). After an atomic monolayer of the precursor (TBTDEN) coats
the surface, a nitrogen and hydrogen plasma reacts with the TBTDEN ligand
and incorporates nitrogen into the niobium matrix. . . . . . . . . . . . . . . . . 98
5.2 Film measurements (a) Measured resistivity as a function of temperature show-
ing decreasing resistivity with increasing temperature above the superconducting
transition characteristic for NbN. (b) Thicknesses measured by profilometry as
a function of deposition cycles, with a linear fit overlaid in red. We extract a
growth rate of 0.63 Å per cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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5.3 Superconducting critical temperature (Tc ) and sheet inductance (L□ ) of deposited
NbN as a function of film thickness (t). Bars denote temperatures corresponding
to 90% and 10% reductions from maximum resistivity. The inset shows the
dependence of tTc on R□ with a fit (red) to tTc = AR□
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0.647 ± 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Schematic of millimeter-wave measurement setup for single and two-tone con-
figurations. Colored tabs show temperature stages inside the 4 He adsorption
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refrigerator. A photograph on the right highlights relevant hardware inside the
fridge. The bottom left shows a photograph of the sample with top waveguide
removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Device characterization and design. (a) Six-fold frequency multipliers convert mi-
crowave to millimeter-wave signals (green), which are demodulated with a cryo-
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genic mixer. A cutaway shows copper WR-10 rectangular waveguides coupling

the signal in and out of a Nb coated slot, into which we mount a chip patterned
with 6 resonators. (b) Top down composite micrograph showing a mounted chip
with the top waveguide removed. (c) Scanning Electron Micrograph of a typical
resonator used in this chapter, with wire width w = 4 µm and film thickness
t = 27.8 nm (NbN false colored yellow). Dipole coupling antennas extend on the
left of the quarter-wave resonator. Measurements can be described with input
and loss couplings Qe and Qi using the circuit model in (d), which takes into
account the impedance mismatch between waveguide Z0 and slot with sapphire
chip Z ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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5.6 (a) Power dependence of the internal quality factor for a resonator with Q∗e =
2.759 × 104 patterned on a 27.9nm thick NbN film, measured at 1 K. White
triangles are fits to a nonlinear response model near and above the bifurcation
power (dashed line). The red line is a fit to a model including power-dependent
loss from two-level systems, and power-independent loss. Insets show lineshape
and fits at average photon occupation n̄ph ≈ 1.2. (b) Internal quality factors for
resonators in this chapter, grouped by film thickness. The top and bottom of
the colored bars correspond to fitted low and high power saturation values, while
points correspond to two-level system induced Qi with high power loss subtracted. 105
5.7 Complex transmission along with fits for a nonlinear resonator at powers near and
above the bifurcation power (blue line) demonstrating how the quality factors
can still be extracted from a bifurcated response, albeit with increasingly less
certainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.8 Temperature dependence of BCS conductivity. (a) High power Qi as a function
of normalized temperature for four resonators of different film thickness. Solid
lines correspond to a BCS model with Tc and kinetic inductance fraction α as fit

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parameters. (b) Extracted Tc from fitting to BCS model (red dots), compared
to Tc from DC resistivity measurements. (c) Normalized frequency shift of the
same resonators as a function of temperature, with overlaid predictions from the
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Mattis-Bardeen equations for σ2 /σn with parameters taken from fits in (a). . . 107
5.9 Measuring Kerr nonlinearity (a) Frequency shift versus average resonator photon
number n̄ph in linear and log-scale (n̄ph accurate within a factor of ∼ 10). (b)
Extracted self-Kerr coefficients versus wire width w for resonators fabricated from
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a 29 nm thick film. Predicted w−2 dependence is shown in red. We find no
significant impact of w on Qi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.10 Transmission of a typical resonance at a range of powers near and above bifur-
cation showing good agreement with a Kerr nonlinear response. The inset shows
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overlaid data and fits in the complex plane just below and above the bifurcation
point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.11 (a) Self-Kerr constant |K00 | as a function of parameters in Eq. 5.7, with a linear
fit overlaid as a solid line. Solid bars correspond to value ranges for groups
of similar film thicknesses and wire widths, with error bars marking systematic
uncertainty. (b) Qi as a function of ω02 /|K00 | which corresponds to the loss Q3
associated with kinetic inductance. Points correspond to low and high power
limits of Qi . Note that devices with varying wire width (empty circles) do not
appear to be correlated with Q3 . (c) Transmission as a function of frequency
for a 18.7 nm thick device at 95.15 GHz taken at increasing powers ξ, with the
inset highlighting decreasing Qi near the bifurcation power ξcrit (dashed blue
line) deviating from two level system loss model (red line). Triangles correspond
to nonlinear model fits, with traces shown in main panel marked in blue. . . . . 111

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5.12 Four-wave mixing. (b-c) Parametric conversion gain with a 95.1 GHz device with
the same film thickness as 5.10 as a function of reduced pump frequency δ for
a fixed signal detuning ∆ of +450 kHz, taken at increasing pump powers. Solid
lines correspond to theoretical response. The initial forward de-amplification is
better understood when the response is viewed in the complex plane (c), where
we observe smooth parametric deformation from the single tone response. . . . 115
5.13 Nonlinearity (K) plotted as a function of total linewidth (γT ) for all of the
millimeter-wave CPS resonators made with NbN, grouped by wire width. . . . 117

6.1 Nonlinear resonator consisting of a nonlinear inductor shunted by a single ca-

pacitance C0 . The parasitic inductance of the capacitor in this case reduces the
self-Kerr strength so cannot be neglected. . . . . . . . . . . . . . . . . . . . . . 119
6.2 Illustration of a finger capacitor with increasing number of fingers N , along with
the equivalent parasitic circuit network (the circuit is symmetric, so is shown for
a single finger pair). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Subdivision of a finger capacitor. (a) We begin with a simple two-finger capacitor

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and split up the fingers into squares, each with sheet inductance L□ and a fraction
of capacitance on their edges. This circuit is drawn out explicitly in (b), showing
the parasitic network. The capacitances all still add up to the original capacitance
C. (c) The parasitic network is drawn out for a four-finger capacitor. Assuming
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both ends of the capacitor A and B are well connected, we can use a symmetry
argument to locate equivalent points on the finger capacitor and fold it up into
the simplified circuit shown in (c). This process can be repeated for any number
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of fingers, which will only change the coefficients, but the equivalent circuit will
look stay the same. This circuit only changes depending on how many squares
we divide each finger into. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Simplified parasitic circuit for fingers divided into two squares. Since the circuit
has loops we must use Kirchoff’s theorem and solve for the effective impedance
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given an applied current I0 and voltage VA − VB . . . . . . . . . . . . . . . . . . 125

6.5 Effective reactance of a large (2 pF) finger capacitor, assuming a sheet inductance
of 30 pH/□. For square aspect ratios (n = m shown on left), the parasitics are
largely described by adding a series inductor. Higher order subdivisions add small
corrections to this, which are largely captured by m = 8 subdivisions. Since the
added inductance depends on the aspect ratio of the capacitor multiplied by the
sheet inductance, increasing finger number for a fixed finger length reduces the
parasitic inductance as shown on the right. . . . . . . . . . . . . . . . . . . . . . 126
6.6 Parasitic networks for capacitors whose edges are defined by a Sierpinski fractal
curve, shown for the first three fractal iterations. . . . . . . . . . . . . . . . . . 128
6.7 Effective impedance of a fractal capacitor with parasitic inductance taken into
account, for the first three fractal iterations. Ideal capacitor model (dashed red)
and equivalent footprint square finger capacitor (dashed gray) are also shown for
comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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6.8 Generator and first three iterations of the square Sierpinski curve, on which our
fractal is based, along with effective curve length relative to the overall width
L = Lcurve /W for fractals along with an equivalent square finger capacitor with
n fingers. The curve length grows faster per iteration compared to the curve
length of meander in a finger capacitor. . . . . . . . . . . . . . . . . . . . . . . . 131
6.9 Scanning electron micrograph of a titanium nitride fractal capacitor (second or-
der) shunted by a 50 nm wide nanowire. The circuit is patterned in one step on
a crystalline silicon substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.10 Transmission of a nonlinear resonator plotted for increasing powers. The fre-
quency of the transmission dip shifts down as the circulating power in the res-
onator increases. By fitting the frequency of the transmission minimum can be
used to extract the frequency shift, which is linear in resonator photon num-
ber, and can used to extract the Kerr nonlinearity. Above the bifurcation point,
the transmission minima are determined by the bifurcation point instead of the
resonant frequency (green dashed line). . . . . . . . . . . . . . . . . . . . . . . . 133
6.11 Left: Measured nonlinearity strength K for the titanium nitride devices in this

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chapter plotted as a function of total device linewidth γTot , which is the combined
internal and coupling linewidth. Right: Measured nonlinearity strength K for the
titanium nitride devices in this chapter plotted as a function of device impedance,
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grouped by wire geometry. Dashed lines are the expected scaling from Equation
6.6, which takes into account wire geometry. . . . . . . . . . . . . . . . . . . . . 135
6.12 Top: Direct measurement of the ringdown chirp emitted by a decaying Kerr non-
linear resonator prepared in an initial coherent state α. The microwave ringdown
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was measured directly using an oscilloscope. Short incremental sections of the
ringdown response can be fitted to a sinusoid function, which lets us track output
frequency with respect to the instantaneous power of the emitted signal. For a
weakly nonlinear system measured here, the resulting frequency is proportional
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to the emitted power, as expected for a Kerr nonlinearity. . . . . . . . . . . . . 136

6.13 Heterodyne measurement of the ringdown chirp emitted by a decaying Kerr non-
linear resonator prepared in a coherent state α. . . . . . . . . . . . . . . . . . . 138
6.14 Ringdown Measurement of a highly nonlinear resonator prepared in a coherent
state α. The amplitude decays much faster than expected from classical theory! 139
6.15 a) Regular millimeter-wave CPS resonator made from NbN on sapphire, with
a standard 4 µm wire width. b) A similar CPS resonator with a 500 nm wide
wire. The rest of the geometry is largely similar, so the impedance is not signif-
icantly changed from the standard design. c) A low-impedance millimeter-wave
resonator, inspired by the titanium nitride fractal resonators. The required ca-
pacitance at 90 GHz is significantly smaller so only a zeroth-order fractal (in
other words not a fractal at all) is required. Similar to the microwave design, a
relatively thin wire shorts the capacitor to ground. However since this design is
fabricated with optical lithography, the nanowires are between 1–2 µm wide. . 140

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6.16 Nonlinearity (K) plotted as a function of total linewidth (γT ) for the microwave ti-
tanium nitride resonators, the millimeter-wave NbN coplanar stripline resonators
from last chapter, as well as millimeter-wave resonators with reduced impedance
(highlighted by the dashed circle), all grouped by wire width. Notably the re-
duced impedance does help increase the self-nonlinearity of the millimeter-wave
resonators. However this increase is less than one order of magnitude, likely since
the original millimeter-wave resonators already had relatively low impedance. . 141

7.1 Junction fabrication process. (a) Trilayer is deposited and oxidized in-situ. (b)
First layer is etched with a chlorine RIE. (c) SiO2 is grown isotropically. (d) Side-
wall spacer is formed by anisotropic etching with fluorine chemistry. (e) Surface
oxides are cleaned in vacuum and wiring layer (purple) is deposited. (f) Sec-
ond junction finger (and other circuit elements) are defined by a fluorine plasma
etch selective against Al. (g) Final devices undergo a wet etch to further remove
SiO2 , exposed Al and some NbOx . (h) Color-enhanced electron micrograph of a
finished trilayer junction with an area of ∼ (500 nm)2 . . . . . . . . . . . . . . . 144

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7.2 Atomic-force microscopy measurements of the C-plane sapphire surface. As re-
ceived (left), the surface is still fairly rough after the epitaxial surface polish,
however after a 1.5 hour anneal at 1250 ◦ C in air (right) atomic terraces are visi-
ble, meaning the surface is extremely flat. Annealing in nitrogen instead results
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in oxygen depletion on the sapphire surface, creating a coral-like surface. Data
courtesy of F. Zhao. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3 Spacer Growth, imaged in cleaved samples from HDPCVD growth methods and
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PECVD growth methods. Note the superior conformality of PECVD, while HD-
PCVD has two phases and a breadloaf cross section. . . . . . . . . . . . . . . . 147
7.4 Sidewall-passivating spacer after etching, imaged in cleaved samples from HD-
PCVD growth methods and PECVD growth methods. The smooth conformal
PECVD growth translates to a smooth spacer profile, while the HDPCVD re-
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sults in a chair-like structure, with potentially lossy discontinuities in the seat

area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5 When cleaving a wafer I got lucky and a fault line ran right through a junction,
allowing us to see what the trilayer stack looks like after the wiring layer is
deposited. The spacer is still intact here, and you can see no interface between
the wiring and counterelectrode indicating good contact. . . . . . . . . . . . . . 149
7.6 Junction properties. (a) Current-voltage relations for an un-shunted junction at
860 mK with Ic = 38 µA and an energy gap 2∆ = 2.89 meV. Bulk resistivity
measurements (inset) give a critical temperature of Tc = 9.28 K. Above 4 mV,
a linear fit (red dashed line) gives Rn = 39 Ω, and a fit to the sub-gap region
(blue dashed line), estimates sub-gap resistance Rs > 8 kΩ. (b) Critical current
density Jc (found by fitting room-temperature junction resistance as a function
of junction area) as a function of oxygen exposure E measured for various wafers
made with two deposition processes. The expected empirical E −1/2 relationships
are plotted as guides to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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7.7 Superconductor material quality. (a) Niobium superconducting critical tempera-
ture TC extracted from resistivity measurements as a function of metal deposition
rate. At rates above 0.6 nm/s, TC approaches bulk value (dashed line). The inset
shows deviations from bulk ∆TC = TCbulk − TC are correlated with the residual
resistivity ratio, implying high deposition rates result in high-quality films. (b)
Sheet kinetic inductance Lk and observed London penetration depth λL plot-
ted as a function of deposition rate suggesting that films deposited at higher
rates are closer to the clean superconductor limit. (c) Specific junction resistance
RJ = R/N obtained by measuring the resistance R of a chain of N = 12 junctions
as a function of temperature. A sharp drop in resistance is observed above 9 K
as the niobium electrodes begin to superconduct. As the temperature decreases,
the junction critical currents increase above the excitation current (10 µA), and
below 5 K the measured resistance drops to zero as the excitation is confined
to the superconducting branch, indicating proximitization of the aluminum and
superconducting contact between the counterelectrode and wiring layers. . . . . 153
7.8 Superconductor grain size. (a) In a tilted scanning electron microscope image of

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a junction, microscopic grains are observed on the metal surface. In regions of
the wiring layer that lie directly on the sapphire substrate, the columnar grain
growth is uninterrupted, and the grain pattern is transferred to the top surface of
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the metal. (b) A top down high-resolution scanning electron micrograph reveals
the hexagonal arrangement of the grains. The grain size can be estimated by
measuring the narrow dimension of a grain, marked d. (c) A histogram of repeated
measurements of grain width are fitted to a normal distribution which suggests
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an average grain width of 16.386 nm. . . . . . . . . . . . . . . . . . . . . . . . 155
7.9 Etch residue chemical analysis. (a) Scanning electron micrograph of a plasma
etch residue located on the wiring layer near a junction. (b) Composite Energy
Dispersive Spectroscopy (EDS) image overlaid on the image in (a) showing nor-
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malized element density regions for F, Nb, Al, and O, with individual element
density maps shown in their respective color on the right. Along with clear Nb
and sapphire (Al2 O3 ) regions, a high concentration of fluorine relative to the
background is found in the residue region, suggesting the residue is composed of
fluorinated polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.10 Lossy silicon oxide spacer residue, which is insoluble in HF of NH4 F. . . . . . . 158
7.11 Etch residue NaK reactivity. (a) Scanning electron micrograph of a plasma etch
residue on the edges of the wiring layer. A closer inspection of the bottom left
reveals that the residue extends to cover the sides of the metal, even where the top
crust has been mechanically removed. (b-c) The wiring layer and a junction from
the same wafer imaged after a 15 min exposure to sodium-potassium amalgam
(NaK) showing nearly complete removal of the etch residue. . . . . . . . . . . . 159

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7.12 Residue treatment with potassium napthalenide K[Nap]. (a) Scanning electron
micrograph of a wiring layer residue after immersion in a room temperature
K[Nap] solution for 15 minutes. The residue remains, however is thinner and
slightly damaged. (b) A junction from the same sample shows residue damage
visible as vertical striations especially near the junction. While the residues are
partially attacked by the K[Nap] solution, this treatment is not sufficient for full
lossy residue removal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.13 Residue treatment with EKC. (a-b) Finished junction treated with the EKC
mixture nearly 20 degrees above the target etching temperature, showing signif-
icant metal attack (nearly 60 nm). Notably no sign of the fluorocarbon residues
on the edges of the Nb wiring layer remain. Traces of material remains on the
spacer-niobium interface, which warrants further study. (c-d) When treated be-
tween 70–75 ◦ C, the metal etch rate is reduced to a reasonable level, while the
organometallic residue is still efficiently removed. This leaves incredibly smooth
and virtually residue-free surfaces on the junction. . . . . . . . . . . . . . . . . 162
7.14 (a) Room temperature junction resistance and junction inductance plotted as

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a function of junction area (corrected for lithographic reduction). Original un-
treated junction resistances are shown in red, and etched junctions in teal, with
fits to an inverse relationship to area (dashed lines) yielding the original critical
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current density Jc and an etch dimension reduction of approximately 160 nm.
(b) Junction resistances as a function of the final junction area with a inverse fit
(dashed line) which gives the critical current density. For illustrative purposes
we have shown PECVD junctions in (a) and HDPCVD junctions in (b). (c) To
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estimate reproducibility, spectroscopically measured qubit frequencies are plotted
as a function of design junction area, labelled by wafer and cooldown. Expected
values for the two different qubit capacitor designs (120 and 160 fF) are shown
with dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
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7.15 (a) Average junction critical current density on an individual chip measured across
several chips across a wafer, with deviations from nominal values (2.088 kA/cm2 )
highlighted with color. (b) Junction area measured with optical microscopy rel-
ative to the expected design area, highlighting the distribution of deviations re-
sulting from lithography. (c) Long term stability of junctions measured by the
relative change in Josephson inductance for 5 month old junctions relative to
their original values. Notably the change in high temperature PECVD junctions
is much lower than HDPCVD junctions. . . . . . . . . . . . . . . . . . . . . . . 165
7.16 (a) HDPCVD Junction critical current density reduction after annealing for 5
min plotted as a function of anneal temperature showing activation at 250 ◦ C.
(b) Critical current density reduction as a function of anneal time at 300 ◦ C,
which approaches the factor of 50 reduction observed in the main text (red lines).
The purple line represents an exponential fit saturating at the observed reduction
factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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7.17 Low temperature PECVD junctions (primarily used in Chapter 8). (a) Junction
critical density as a function of oxygen exposure (same as 7.6b) with the addi-
tion of low temperature PECVD junctions, which still have high critical current
density. (b) Critical current density reduction as a function of anneal tempera-
ture, with the addition of the low-temperature PECVD junctions, which are only
mildly annealed. The HDPCVD junctions were annealed for 5 minutes, while
the junctions that went through PECVD spent approximately 15 minutes at the
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.18 Schematic of the microwave measurement setup used for qubit characterization.
Colored tabs show temperature stages inside the dilution refrigerator. A compos-
ite microscope image (top right) shows a single qubit and its readout resonator,
coupled to a waveguide for measurement. A photograph (bottom right) shows
the chip containing 6 qubits mounted in its copper circuit board. . . . . . . . . 169
7.19 Qubit Properties. (a) Average qubit decay time T1 extracted by fitting the ex-
ponential decay of excited state population in (b) plotted as a function of qubit
frequency, grouped by wafer. Lines indicate qubit quality factor Q1 = ωq T1 . We

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find an overall mean Q1 of 2.57 × 105 with some wafer to wafer variation. (c)
Ramsey dephasing time T2∗ (filled points) and Hahn-echo dephasing time T2 (hol-
low points) extracted by fitting the exponential decay of oscillations in (d) as a
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function of qubit frequency. We find an overall average T2∗ and T2 of 6.643 µs and
12.916 µs respectively. (e) Qubit quality factors as a function of their junction
participation ratio plotted for devices in this chapter (reds) and in literature (blue,
black, green). Lines and shaded confidence regions show Q−1
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1 = pJ /QJ + p0 /Q0
as a guide to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.20 Qubit quality factors as a function of their junction participation ratio plotted
for our trilayer qubits (reds) aluminum junction qubits from our lab (purple) and
selected qubits in literature (blue, black, green). Lines and shaded confidence
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regions show Q−1 1 = pJ /QJ + p0 /Q0 as a guide to the eye. . . . . . . . . . . . 173

7.21 (a) Power dependence of the internal quality factor for a readout resonator (Qe =
2.6 × 105 ) with no qubit present. The red line is a fit to a model including
loss from two-level systems (TLS). The insets show the lineshape and fits at an
average photon occupation n̄ph ≈ 0.96. (b) Internal quality factor of resonators
without qubits measured as a function of temperature. Solid lines are fits to a
model including TLS loss and quasiparticle loss. The three red resonators are
formed from the wiring layer, and the blue resonators from the base electrode.
Measurements are taken at n̄ph ≈ 104 so some TLS loss is saturated. (c) Qubit
quality factors Q1 plotted as a function of their readout resonator quality factors
Qi (measured at nph < 1). A grey line indicates a 1:1 relationship. . . . . . . . 174

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7.22 Junction loss regions (a) Cartoon showing regions defined for a resonator made
with the first layer, with dimensions exaggerated. Niobium oxide (metal-air inter-
face) is separated into top oxide (Tox) and side oxide (Sox) regions. For a wiring
layer resonator, the dirty substrate region (DMS) is merged with the substrate
layer. (b) Cartoon showing regions for a junction, which adds the junction barrier
region (Jox) and the spacer region (SiOx). (c) Three dimensional rendering of
the junction with realistic dimensions. Simulated regions are colored in the same
way as in parts (a-b). (d) Transparent rendering of the junction visualizing the
spacer remaining percentage PS relative to the junction width jw . . . . . . . . 176
7.23 Junction losses by region. (a) Participation ratios of the primary lossy materials
in the junction, plotted as a function of niobium oxide thickness tNbOx . As
expected the niobium oxide participation ratio increases as the layer gets thicker.
(b) Junction loss tangent expressed as visual sum of losses from various materials
in the junction with assumed loss tangents, plotted as a function of niobium
oxide thickness. For thicker oxide layers (eg. those used in anodization processes)
niobium oxide loss dominates the junction loss. The junction loss calculated from

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Fig. 7.19c is shown in black dashed lines. (c) We can also solve for the barrier
quality factor based on the junction quality factor and the calculated participation
ratios for varying material quality factors. Solid and dashed lines correspond to
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a SiO2 loss tangent of tan δ = 2.7 × 10−3 and 2.9 × 10−3 respectively. In (d-f) we
repeat parts (a-c) but measure the effect of partially un-removed spacer material
expressed as a fraction PS of the junction width. We find that residual spacer
material contributes a significant amount of loss. For both sets of simulations,
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the unswept variable is set to nominal values of tNbOx = 2 nm and PS = 0.2. . 177
7.24 Qubit quality factors from wafers B, D as a function of temperature. A mild
decrease is observed at higher temperatures consistent with the system bath tem-
perature Qbath , however lifetimes are virtually unaffected by quasiparticles Qqp
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(red lines). We also plot quality factors of an Al junction qubit, whose perfor-
mance is noticeably limited by quasiparticles above 160 mK (green lines), whereas
the Nb junction wouldn’t see an effect until 1.6 K. . . . . . . . . . . . . . . . . 181

8.1 Qubit geometry. a) A scanning electron micrograph of a low-loss niobium trilayer

junction at the core of the qubit. b) A micrograph of the qubit and readout
resonator geometry, with the junction location marked at the top. c) Equivalent
circuit of the qubit and readout resonator coupled inductively to a transmission
line. d) Photograph of a chip containing six qubits mounted in a low-loss K band
circuit board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.2 Cross section view of the K band packaging, showing the chip (blue) secured to
the high-frequency circuit board (gold) by the copper enclosure. . . . . . . . . 186
8.3 Mode-free K Band packaging. Three copper pieces align the chip containing
qubits with the low-loss printed circuit board. . . . . . . . . . . . . . . . . . . 187

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 8.4 Optimized launcher geometry and wirebond configuration which achieves max-
imal transmission up to 27 GHz. Using a manual wirebonder, we attempt to
replicate this wirebond shape, however in practice the circuit board dimensions
requires slightly longer bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.5 Qubit dynamics. a) Deflection of the readout resonator transmission signal as a
function of applied qubit pulse frequency, shown for increasing qubit pulse power.
At low powers (blue) a single peak is observed when the pulse is resonant with
the qubit frequency (fge = 18.474 GHz). As power increases, the linewidth of
this transition increases, and additional peaks appear from excitations into higher
qubit levels through many-photon excitations (fgf /2 etc). These features have
a spacing of α/2 = (fge − fef )/2. b) Measured excited state probability shows
Rabi oscillations when a fixed-length pulse with varying amplitude is applied at
the qubit frequency. The red line is a fit to the expected sinusoidal behavior.
c) Rabi oscillations are measured for frequencies near fge , with brighter colors
corresponding to higher excited state probabilities. Away from the transition
frequency, the Rabi frequency increases while the oscillation amplitude decreases

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and becomes power dependent. d) Rabi oscillations as a function of pulse am-
plitude and length σ, with brighter colors corresponding to higher excited state
probabilities. Dashed red lines mark contours of integer π pulses where σΩ = mπ.
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8.6 Number splitting. a) Deflection of the readout resonator transmission signal as
a function of applied qubit probe frequency shown for increasing powers. We ob-
serve the level transitions separated by anharmonicity α/2 similar to Figure 8.5a,
however on closer inspection each peak is split into smaller features. b) With
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non-negligible readout resonator population, the sub-peaks for each transition
are split by the dispersive shift 2χ. . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.7 Qubit Properties. a) Average qubit decay time T1 extracted by fitting the expo-
nential decay of excited state population. b) T1 plotted as a function of qubit
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frequency, grouped by wafer. Lines indicate qubit quality factor Q1 = ωq T1 .

We find an overall mean Q1 of 0.792 × 105 with some wafer to wafer variation.
c) Ramsey dephasing time T2∗ (filled points) and Hahn-echo dephasing time T2
(hollow points) extracted by fitting the exponential decay of oscillations in (d) as
a function of qubit frequency. Lines indicate dephasing quality factor. We find
an average T2∗ and T2 of 1.124 µs and 1.357 µs respectively. . . . . . . . . . . . 192
8.8 Thermal decoherence and dephasing. a) Decoherence time T1 of three represen-
tative qubits measured as a function of temperature. A mild decrease is observed
at higher temperature, consistent with a model including loss from increased sys-
tem bath temperature (solid lines). b) Ramsey dephasing time T2∗ as a function
of temperature. The behavior is largely captured by a parameter-free thermal
dephasing model assuming a fixed T1 (solid lines). c) Pure dephasing rate Γϕ
which has dephasing from relaxation subtracted, resulting in better agreement
with the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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