ABSTRACT
 Title of dissertation: A STUDY OF TWO-LEVEL SYSTEM
 DEFECTS IN DIELECTRIC FILMS
 USING SUPERCONDUCING RESONATORS
 Moe Khalil, Doctor of Philosophy, 2013
 Directed by: Dr. Kevin D. Osborn
 Laboratory for Physical Sciences
 Professor Frederick C. Wellstood
 Department of Physics
 In this dissertation I describe measurements of dielectric loss at microwave
 frequencies due to two level systems (TLS) using superconducting resonators. Most
 measurements were performed in a dilution refrigerator at temperatures between 30
 and 200 mK and all resonators discussed were fabricated with thin-film supercon-
 ducting aluminum.
 I derive the transmission through a non-ideal (mismatched) resonant circuit
 and find that in general the resonance line-shape is asymmetric. I describe an anal-
 ysis method for extracting the internal quality factor (Qi), the diameter correction
 method (DCM), and compare it to a commonly used phenomenological method,
 the ? rotation method (?RM). I analytically find that the ?RM deterministically
 overestimates Qi when the asymmetry of the resonance line-shape is high.
 Four coplanar resonator geometries were studied, with frequencies spanning
 5-7 GHz. They were all superconducting aluminum fabricated on sapphire and
silicon substrates. These include a quasi-lumped element resonator, a coplanar strip
 transmission line resonator, and two hybrid designs that contain both a coplanar
 strip and a quasi-lumped element. Measured Qi?s were as high as 2 ? 105 for single
 photon excitations and there was no systematic variation in loss between quasi-
 lumped and coplanar strip resonance modes.
 I also measured the microwave loss tangent of several atomic layer deposition
 (ALD) grown dielectrics and obtained secondary ion mass spectrometry (SIMS)
 measurements of the same films. I found that hydrogen defect concentrations were
 correlated with low temperature microwave loss. In amorphous films that showed
 excess hydrogen defects on the surface, two independent TLS distributions were
 required to fit the loss tangent, one for the surface and one for the bulk. In crystalline
 dielectrics where hydrogen contamination was uniform throughout the bulk, a single
 bulk TLS distribution was sufficient.
 Finally, I measured the TLS loss in 250 nm thick HD-PECVD deposited silicon
 nitride (SiNx) while sweeping an independent applied bias electric field across the
 capacitor. With a strong microwave field and an increasing bias rate, the loss tangent
 changed from a low value, where saturation occurs on resonance near the steady
 state, to a larger value approximately equal to the linear-response loss tangent,
 where saturation appears to be avoided. This increase was explained with a new
 theory in which TLSs can experience Landau-Zener transitions as they?re swept,
 where the maximum excitation probability is 1/2 at resonance. Data is found to
 scale if plotted as a function of the dimensionless sweep rate. The functional form
 of this loss tangent agrees well with the theory, and is predicted to hold for any
amorphous dielectric. By fitting the measured loss tangent as a function of bias
 sweep rate to the theory, I was able to extract an average TLS dipole moment of
 7.9 D and a TLS spectral spatial density of P0 = 4.9 ? 1043J?1m?3.
A STUDY OF TWO-LEVEL SYSTEM DEFECTS IN
 DIELECTRIC FILMS USING SUPERCONDUCING
 RESONATORS
 by
 Moe Shwan Khalil
 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
 2013
 Advisory Committee:
 Professor Frederick C. Wellstood, Advisor/Chair
 Dr. Kevin D. Osborn, Advisor/Principal Investigator
 Professor Christopher J. Lobb
 Dr. Ben Palmer
 Professor Ichiro Takeuchi
Dedication
 To my parents who worked so hard to give me opportunities that they never had.
 ii
Acknowledgments
 I was very lucky to have not one but two great advisors in Dr. Kevin Osborn
 at the Laboratory for Physical Sciences (LPS) and Prof. Fred Wellstood at the
 University of Maryland. I cannot thank either of them enough. Kevin was always
 there in the lab happy to get his hands dirty and advise me early in my tenure but
 he also gave me the freedom to pursue my own research interests as I continued my
 research. The strength I gained from knowing that I always had his support cannot
 be overstated.
 Fred?s insight when making sense of puzzling phenomena was invaluable. The
 depth of his knowledge and the incredible intuition he possesses meant there wasn?t
 a phenomenon that he couldn?t come up with a plausible explanation for. I always
 knew that no matter how lost I felt I could go to him to have a sense of direction.
 I?d also like to thank Prof. Chris Lobb, discussions with whom were always as
 entertaining as they were scientifically valuable. Chris brings a sense of joy to his
 work that he simply cannot hide.
 I also have to thank Dr. Ben Palmer who was like a second advisor at LPS.
 Ben was always ready to discuss various experimental and analysis techniques. I
 was so impressed with his love experimental physics which drove him to be in the
 lab more than any other PI I?ve known.
 I should also thank Prof. Ichiro Takeuchi who agreed to serve as the Dean?s
 representative at my thesis defense.
 I?d also like to thank my fellow group members, who aided me in more ways
 iii
than I can possibly enumerate: Bahman Sarabi, Dr. Aruna Ramanayaka, Dr. Yaniv
 Rosen, Dr. Sergey Gladchenko, and Dr. Micah Stoutimore. I would be remiss if I
 didn?t extend a special thanks to Micah. The countless discussions with him were
 enlightening when I was stuck and uplifting when I was frustrated, not to mention
 entertaining when I was not in the mood to do physics. I always knew when arguing
 with Micah that if I could convince him (rare), I was certain to be right.
 I?m grateful to all of the postdocs and graduate students at LPS with which
 I had numerous interactions both scientific and otherwise: Sergey Novikov, Dr.
 Zachary Keane, Dr. Vitaley Zaretskey, Dr. Jennifer Robinson, Baladitya Suri, John
 Bavier, Anita Roychowdhury, Pavel Nagornykh, Dr. Mishkatul Bhattacharya, Dr.
 Nathan Siwak, Dr. Tomek Kott, and Dr. Zaeill Kim. My graduate experience
 would not have been nearly as rewarding or successful without each and every one
 of them. I should also thank Dr. Hanhee Paik who was Kevin?s first postdoc.
 We overlapped in the group only briefly, but she taught me so much about low
 temperature experimental physics and processes fabrication.
 I?m also very grateful to the cleanroom, machine shop, and LPS staff without
 whom my research wouldn?t have been possible: Toby Olver, Steve Brown, Sean
 Flannery, Curt Walsh, John Sugrue, Don Crouse, Bill Donaldson, Paul Hannah,
 and Greg Latini.
 I must also thank my friends how have made this experience so much sweeter.
 I can?t name them all and will not try but you guys know who you are. You were
 there to celebrate with me during my accomplishments and to mourn with me during
 my failures. I can?t imagine what the last six years would have been like without
 iv
each and every one of you.
 Most of all I?d like to thank my family, in particular my brother Salwan Ab-
 dullah and my parents Bangean Abdullah and Zena Khalil. You were always there
 to support me during my trials and I will never be able to adequately express the
 strength I gained from knowing that I always had your full backing. You?ve been
 my safety net, giving me the courage to jump higher knowing that you?d always be
 there if I fall.
 I?m grateful to so many and so I?m sure to have missed several of you. To
 anyone I may have missed, I?m sorry and thank you for all of your support.
 v
Table of Contents
 List of Figures ix
 1 Introduction 1
 1.1 Quantum computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 1.1.2 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . 4
 1.1.3 Superconducting qubit readout . . . . . . . . . . . . . . . . . 7
 1.2 Microwave kinetic inductance detectors . . . . . . . . . . . . . . . . . 9
 1.3 Other superconducting resonator applications . . . . . . . . . . . . . 10
 1.4 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 2 Two-level systems and dissipation 12
 2.1 Background for undriven Two-level systems . . . . . . . . . . . . . . 12
 2.2 Two-level system loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 2.3 Loss in a qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 2.4 TLS micromechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 3 Experimental setup and measurement technique 25
 3.1 Dilution refrigerator setup . . . . . . . . . . . . . . . . . . . . . . . . 25
 3.2 Signal calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 3.3 Resonance derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
 3.4 Determining the voltage across the resonator . . . . . . . . . . . . . . 34
 3.5 Asymmetric resonance line shapes . . . . . . . . . . . . . . . . . . . . 36
 3.5.1 Analysis of asymmetric resonance lineshape . . . . . . . . . . 38
 3.5.2 Analysis of S21 Simulations and Data . . . . . . . . . . . . . . 45
 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
 4 Coplanar resonators 57
 4.1 Resonator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
 4.1.1 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 57
 4.1.2 Designing and simulating resonators to analyze geometry de-
 pendence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
 4.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
 4.2.1 Dicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
 4.2.2 Packaging and wire-bonding . . . . . . . . . . . . . . . . . . . 68
 4.3 Solving for resonance voltage waveform . . . . . . . . . . . . . . . . . 69
 4.3.1 QL resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
 4.3.2 CPS resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
 4.3.3 Quasi-lumped C-CPS resonator . . . . . . . . . . . . . . . . . 78
 4.3.4 Quasi-lumped L- CPS resonator . . . . . . . . . . . . . . . . . 81
 vi
5 Coplanar resonator measurements 85
 5.1 Coplanar resonator loss . . . . . . . . . . . . . . . . . . . . . . . . . . 85
 5.1.1 Sputtered aluminum resonators on sapphire . . . . . . . . . . 85
 5.1.2 Coplanar resonator loss analysis . . . . . . . . . . . . . . . . . 89
 5.1.3 Evaporated aluminum resonators on sapphire . . . . . . . . . 91
 5.1.4 Sputtered aluminum resonators on silicon . . . . . . . . . . . . 94
 5.2 External quality factor . . . . . . . . . . . . . . . . . . . . . . . . . . 95
 5.3 Discussion of likely loss mechanisms . . . . . . . . . . . . . . . . . . . 97
 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
 6 Atomic layer deposition 101
 6.1 Atomic layer deposition and its use in trilayer junctions and capacitors101
 6.2 Microwave resonator fabrication on ALD materials and field simula-
 tions of coplanar geometry . . . . . . . . . . . . . . . . . . . . . . . . 104
 7 ALD grown dielectric measurements 108
 7.1 Deuterated AlOx ALD films . . . . . . . . . . . . . . . . . . . . . . . 108
 7.1.1 OH defect model . . . . . . . . . . . . . . . . . . . . . . . . . 108
 7.1.2 SIMS measurements of AlOx films . . . . . . . . . . . . . . . . 109
 7.1.3 Resonator measurements . . . . . . . . . . . . . . . . . . . . . 110
 7.2 Varying ALD oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
 7.2.1 CPS Qi measurements and field simulations . . . . . . . . . . 113
 7.2.2 SIMS analysis for ALD dielectrics . . . . . . . . . . . . . . . . 119
 7.2.3 Fitting to CPS Qi measurements using TLS loss model . . . . 119
 7.3 Surface Loss effects in ALD grown AlOx . . . . . . . . . . . . . . . . 124
 7.4 Hydrogen vs Carbon TLS analysis . . . . . . . . . . . . . . . . . . . . 124
 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
 8 Biased resonator 132
 8.1 Biasing TLSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
 8.2 Previous bias measurements . . . . . . . . . . . . . . . . . . . . . . . 133
 8.3 Swept bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
 8.4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . 142
 8.4.1 Theory for Monte Carlo simulations . . . . . . . . . . . . . . . 142
 8.4.2 Monte Carlo steady state-results . . . . . . . . . . . . . . . . . 147
 8.4.3 A closer look at the response of the TLS distribution . . . . . 151
 8.4.4 Individual TLS response . . . . . . . . . . . . . . . . . . . . . 154
 8.4.5 Monte Carlo simulation with a bias field . . . . . . . . . . . . 155
 8.4.6 Monte Carlo simulation of TLS dynamics and adiabatic rapid
 passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
 9 Biased resonator results 164
 9.1 Biased resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
 9.1.1 The resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
 vii
9.1.2 Resonator simulations . . . . . . . . . . . . . . . . . . . . . . 165
 9.1.3 Resonator fab . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
 9.2 Dependence on loss on a static (DC) bias voltage . . . . . . . . . . . 168
 9.3 Loss with swept bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
 9.3.1 Pulsed bias effect . . . . . . . . . . . . . . . . . . . . . . . . . 170
 9.3.2 Biased and steady state losses . . . . . . . . . . . . . . . . . . 172
 9.4 Varying bias rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 9.4.1 Methods for varying bias rate . . . . . . . . . . . . . . . . . . 176
 9.4.2 Fitting S curve for different microwave (ac) powers and ex-
 tracting the TLS dipole moment . . . . . . . . . . . . . . . . . 182
 9.5 S curves at 100 mK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
 10 Conclusion 189
 10.1 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . 189
 10.1.1 Diameter correction method for extracting Qi . . . . . . . . . 189
 10.1.2 ALD dielectric results . . . . . . . . . . . . . . . . . . . . . . 189
 10.1.3 Biased resonator results . . . . . . . . . . . . . . . . . . . . . 190
 10.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
 10.2.1 Trilayer resonator fabrication . . . . . . . . . . . . . . . . . . 191
 10.2.2 Biasing TLSs in coplanar resonator . . . . . . . . . . . . . . . 192
 10.2.3 Lasing and loss reduction with biased resonator . . . . . . . . 192
 10.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
 Bibliography 194
 viii
List of Figures
 1.1 Rabi oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
 1.2 Loss tangent and qubit Rabi oscillations. . . . . . . . . . . . . . . . . 8
 2.1 Two-level system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
 2.2 Splitting Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 3.1 Dilution refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 3.2 Input line calibration with frequency. . . . . . . . . . . . . . . . . . . 28
 3.3 Resonator measurement schematic . . . . . . . . . . . . . . . . . . . . 30
 3.4 Measurement schematic redrawn . . . . . . . . . . . . . . . . . . . . . 31
 3.5 Norton equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . 32
 3.6 Asymmetric line shape data . . . . . . . . . . . . . . . . . . . . . . . 37
 3.7 Measurement schematic with mismatched lines . . . . . . . . . . . . . 39
 3.8 Resonance circle plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 3.9 Microwave Office Schematic . . . . . . . . . . . . . . . . . . . . . . . 47
 3.10 Fitting simulated asymmetric resonances . . . . . . . . . . . . . . . . 48
 3.11 Extracted Qi and asymmetry vs. calculated asymmetry . . . . . . . . 49
 3.12 Extracted vs. actual Qis . . . . . . . . . . . . . . . . . . . . . . . . . 51
 3.13 Fitting an asymmetric line shape . . . . . . . . . . . . . . . . . . . . 52
 3.14 Extracted Qis from asymmetric line shapes . . . . . . . . . . . . . . . 53
 3.15 Extracted Qi vs. Qe . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
 4.1 Microwave Office layout and simulation . . . . . . . . . . . . . . . . . 58
 4.2 Optical picture of coplanar resonators . . . . . . . . . . . . . . . . . . 61
 4.3 Resonator schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
 4.4 Current and E-field simulations . . . . . . . . . . . . . . . . . . . . . 64
 4.5 Higher order mode simulations . . . . . . . . . . . . . . . . . . . . . . 66
 4.6 Sample box picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
 4.7 QL schematic and Thevenin equivalent . . . . . . . . . . . . . . . . . 70
 4.8 CPS schematic and Thevenin equivalent . . . . . . . . . . . . . . . . 73
 4.9 CPS transmission line transformation . . . . . . . . . . . . . . . . . . 75
 4.10 QLC-CPS transmission line transformation . . . . . . . . . . . . . . . 79
 4.11 QLL-CPS transmission line transformation . . . . . . . . . . . . . . . 82
 5.1 Sputtered coplanar resonators . . . . . . . . . . . . . . . . . . . . . . 86
 5.2 Compare two sputtered films . . . . . . . . . . . . . . . . . . . . . . . 87
 5.3 Compare resonators from the same film . . . . . . . . . . . . . . . . . 88
 5.4 Calculated internal quality factors from simulated fields . . . . . . . . 90
 5.5 Compare sputtered to evaporated devices . . . . . . . . . . . . . . . . 92
 5.6 AFM of sputtered and evaporated films . . . . . . . . . . . . . . . . . 93
 5.7 Comparing sapphire and silicon substrates . . . . . . . . . . . . . . . 94
 5.8 Compare simulated to measured Qe?s of two sputtered films . . . . . 95
 5.9 Compare simulated to measured Qe?s of sputtered and evaporated films 96
 ix
5.10 Loss in semi-log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
 6.1 ALD process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
 6.2 ALD thickness control . . . . . . . . . . . . . . . . . . . . . . . . . . 103
 6.3 ALD Conformality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
 6.4 COMSOL field simulation . . . . . . . . . . . . . . . . . . . . . . . . 106
 7.1 SIMS measurements of deuterated vs non-deuterated films . . . . . . 111
 7.2 Loss measurements of deuterated vs non-deuterated films . . . . . . . 114
 7.3 Loss measurements of varying ALD films . . . . . . . . . . . . . . . . 116
 7.4 COMSOL field simulation of ALD AlOx film . . . . . . . . . . . . . . 117
 7.5 COMSOL field simulation of ALD LaAlOx film . . . . . . . . . . . . 118
 7.6 SIMS measurements of varying ALD films . . . . . . . . . . . . . . . 120
 7.7 Loss measurements of varying ALD films with fits . . . . . . . . . . . 122
 7.8 Loss measurements of ALD films with varying thickness . . . . . . . . 125
 7.9 SIMS measurements of ALD AlOx films . . . . . . . . . . . . . . . . . 126
 7.10 Filling Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
 7.11 Loss measurement in terms of H and C impurity concentration . . . . 130
 8.1 Landau-Zener transition . . . . . . . . . . . . . . . . . . . . . . . . . 137
 8.2 Loss tangent vs. dimensionless bias rate . . . . . . . . . . . . . . . . 141
 8.3 TLS energy cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
 8.4 TLS Monte Carlo distribution . . . . . . . . . . . . . . . . . . . . . . 148
 8.5 Loss tangent from Monte Carlo simulation . . . . . . . . . . . . . . . 150
 8.6 Transients of loss from simulation . . . . . . . . . . . . . . . . . . . . 152
 8.7 TLS excited state probability (?11) . . . . . . . . . . . . . . . . . . . 153
 8.8 Distribution of TLS loss . . . . . . . . . . . . . . . . . . . . . . . . . 154
 8.9 Loss from individual TLSs . . . . . . . . . . . . . . . . . . . . . . . . 156
 8.10 ?11 for individual TLS with varying microwave drive field . . . . . . . 157
 8.11 Transient loss with slow bias . . . . . . . . . . . . . . . . . . . . . . . 159
 8.12 Transient loss with fast bias . . . . . . . . . . . . . . . . . . . . . . . 160
 8.13 TLS biased quickly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 8.14 TLS biased slowly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
 9.1 Picture and schematic of DC bias device . . . . . . . . . . . . . . . . 165
 9.2 Microwave Office device picture and transmission simulation . . . . . 167
 9.3 Loss tangent of resonator with bias fields . . . . . . . . . . . . . . . . 169
 9.4 Resolving statistical fluctuations in TLS distribution with bias field . 171
 9.5 Increase of loss tangent with swept bias field . . . . . . . . . . . . . . 173
 9.6 Loss tangent increase to intrinsic loss value at several temperatures . 175
 9.7 Loss tangent increase as a function of bias pulse amplitude . . . . . . 177
 9.8 Loss tangent increase with low pass filtered bias . . . . . . . . . . . . 179
 9.9 Loss tangent vs. bias rate with different techniques . . . . . . . . . . 181
 9.10 Loss tangent vs. bias rate for three ac drive fields . . . . . . . . . . . 183
 9.11 Loss tangent vs. bias rate with fits . . . . . . . . . . . . . . . . . . . 184
 9.12 Rescaled loss tangent vs. bias rate . . . . . . . . . . . . . . . . . . . . 186
 x
9.13 Loss tangent vs. bias rate for three ac drive fields at 100 mK . . . . . 187
 xi
Chapter 1
 Introduction
 1.1 Quantum computing
 1.1.1 Background
 Quantum computing is a profound and potentially groundbreaking result of
 quantum mechanics. Since the genesis of the idea in 1982 [1] by Richard Feynman,
 physicists, engineers, and computer scientists have worked towards achieving this
 ambitious goal. This pursuit has led to new devices, new concepts, and a better and
 perhaps more intuitive understanding of some parts of quantum mechanics.
 In 1982 [1] Feynman noted that there appear to be fundamental difficulties in
 simulating quantum mechanical systems with classical computers. This realization
 led him to the idea of a quantum computer for the purpose of simulating such
 systems. This is a very open ended and arguably the most rewarding ends for a
 quantum computer, with the potential to affect many fields of science. For example
 protein folding is ultimately a quantum mechanical process, the understanding of
 which could revolutionize medicine [2]. In addition, a quantum computer with an
 effective quantum simulator would have the potential to solve much of the problems
 facing quantum computing today and could make the design and construction of
 a quantum computer much simpler?it seems the universe is not without a sense of
 1
irony.
 The Church-Turing thesis [3, 4] states that any algorithmic process can be
 simulated efficiently, meaning in polynomial time, with a probabilistic Turing ma-
 chine. In other words one cannot write an algorithm that solves a problem more
 efficiently than a probabilistic Turing machine. In 1985 David Deutsch [5] showed
 that a quantum computer was not limited by the Church-Turing thesis and could
 efficiently solve a problem that has no efficient solution by a probabilistic Turing
 machine. In 1992 David Deutsch along with Richard Jozsa demonstrated an exam-
 ple of this with the Deutsch-Josza algorithm [6] which solved a proposed problem
 where one is attempting to determine the nature of a black box function. The idea
 is that you are given a function that acts on an n-bit number and is known to either
 be constant, meaning that it returns 1 (or 0) for all possible inputs, or balanced,
 meaning that it returns 0 for exactly half of all inputs and 1 for the other half.
 You are tasked with querying an oracle that evaluates the function until you de-
 termine whether the function is constant or balanced. In the worst, case a classical
 computer will require 2n?1 + 1 queries to solve the problem but the Deutsch-Josza
 algorithm on a quantum computer can solve the problem with a single query by
 using a superposition of all 2n possible inputs.
 In 1994 Peter Shor discovered a quantum algorithm (Shor?s algorithm) [7]
 that can factor numbers exponentially faster than classical algorithms. This again
 utilizes the principle of superposition to evaluate a function for what is classically
 multiple inputs. This is considered by many to be the most practical use for a
 quantum computer since modern RSA encryption is based on the inability to factor
 2
sufficiently large numbers in polynomial time. Then in 1995 Lov Grover proposed a
 quantum algorithm that can search an unsorted list with a quadratic speedup over
 classical algorithms. [8] While this did not give as dramatic a speedup as Shor?s
 algorithm, the myriad of potential uses have many excited about its prospects.
 While the number of distinct quantum algorithms has been thus far limited, it
 is the opinion of many, myself included, that this is not a fundamental limit on
 quantum algorithms but rather an indication of the unintuitive nature of quantum
 computing and the relative novelty of the field. There is little doubt in my mind
 that given enough time, we will discover quantum algorithms that will revolutionize
 many aspects of life.
 Another potential technique that could be used on a quantum computer is
 quantum annealing or adiabatic quantum computing. [9] Quantum annealing could
 be applied to the classically intractable problem of function optimization. While
 often finding a local minimum is relatively easy, there is no known classically efficient
 technique for finding a global minimum. Quantum annealing could effectively allow
 the solution to tunnel into its global minimum. Qualitatively this could be done by
 encoding a function into a potential that?s turned on adiabatically. If the system is
 initialized in its ground state, it will remain there adiabatically as the potential is
 turned on.
 The power of quantum computing comes from two fundamental properties of
 quantum mechanics. The first, which has already been mentioned, is the principal
 of quantum superposition. From linearity, a quantum mechanical system with two
 states can in general exist in a quantum superposition of both states. Thus a qubit
 3
unlike a classical bit can exists in a superposition of both the 0 state and the 1 state
 with some complex amplitude. The second unique property of quantum mechanics
 necessary for quantum computing is entanglement. Entanglement is a nonlocal
 property that allows the state of two quantum systems to be strongly correlated in
 a way that is not possible in classical mechanics.
 1.1.2 Superconducting qubits
 In principal any two-level-quantum-system, or more generally any quantum
 system where two levels can be isolated, can act as a qubit. The first realization
 of a qubit in a logic gate was in 1995 with a trapped beryllium ion, [10] where the
 internal energy levels of the atom were used as the qubit. In the late 1990?s some
 attempted to utilize NMR technology to use nuclear magnetic spins as qubits [11]
 but this was met with limited success. The problem with these early techniques
 is that of scalability. Coupling to an atom can be difficult and the infrastructure
 required in scaling up to many qubits and eventually a quantum computer is not
 trivial.
 At first sight, one might instead consider using a harmonic oscillator that has
 been cooled down to its quantum mechanical ground state and use single photon
 excitations to exchange quantum information. Superconducting LC microwave res-
 onators are easy to build and can be cooled below their resonant frequency with
 existing dilution refrigerator technology. However, a harmonic oscillator has equally
 spaced energy levels and as a result an external drive field can?t isolate individ-
 4
ual energy levels. What we need is not a quantum spring (harmonic) but rather a
 quantum pendulum (anharmonic). This can be achieved with a Josephson junction
 [12], a thin insulating layer separating two superconducting metals which allows
 Cooper pairs to tunnel across. A Josephson junction is ideally lossless and gives
 superconducting circuits the non-linearity needed to isolate two energy levels. The
 first superconducting qubit was realized in 1999 [13] with a Cooper pair box, where
 the quantum states used for the qubit are zero/one excess Cooper pairs on a super-
 conducting island.
 The advantage of superconducting qubits is that it is easier to couple to them
 and for them to couple to each other. The disadvantage is that it is easier for the
 universe to couple to them as well, and in so doing decohere their quantum state.
 Figure 1.1(a) shows a simulation of a qubit?s excited state probability while the
 qubit is being driven at the transition frequency. It is oscillating between the ground
 and the excited state due to absorption and stimulated emission of the microwave
 photons from the input drive. These are called Rabi oscillations after Isidor Isaac
 Rabi who first measured them in nuclear magnetic spins [14]. Figure 1.1(b) shows
 what happens when the drive is stopped. Rather than remaining in its current state
 the qubit decays into its grounds state exponentially with characteristic time T1.
 This decay to the ground state as well as the decay in the Rabi oscillations is caused
 by coupling of the qubit to other degrees of freedom.
 It is a major goal of research on superconducting quantum computing to in-
 crease these qubit lifetimes. Figure 1.2(a) [15] shows a loss tangent measurement
 of a linear superconducting resonator as a function of the microwave voltage across
 5
0
 0.2
 0.4
 0.6
 0.8
 1
 P 1
 0 200 400 600 800 1000
 0
 0.2
 0.4
 0.6
 0.8
 1
 t (ns)
 P 1
 (a)
 (b)
 Figure 1.1: (a) Simulated Rabi oscillations for a qubit with T1 = 300ns
 and a T2 = 200ns. (b) The drive is turned off at t=500 ns at which point
 the qubit decays into its ground state with the spontaneous emission time
 constant T1
 6
the capacitor for different dielectric materials. The functional form of the loss will
 be discussed in Chapter 2 but for now we note that silicon nitride has more than an
 order of magnitude less dielectric loss than silicon dioxide. Figure 1.2(b) (from Ref.
 [15]) shows Rabi oscillations for superconducting phase qubits where silicon nitride
 and silicon dioxide were used as wiring dielectrics; the qubits where silicon nitride
 was used clearly show longer coherence times. This result is evidence that coher-
 ence times in these superconducting qubits were limited by material losses. This
 work also shows that one can probe these losses not only with qubits which can be
 complicated to fabricate and measure, but also with simple linear resonators which
 are by comparison much easier to deal with. Thus understanding and improving
 dielectric loss in superconducting microwave resonators is a pathway to improving
 the performance of superconducting qubits.
 1.1.3 Superconducting qubit readout
 In addition to being useful in diagnosing sources of loss and decoherence,
 superconducting linear resonators are also useful in reading out qubits. This idea
 was the genesis of circuit cavity quantum electrodynamics in 2004 by the Schoelkopf
 group at Yale [16]. They demonstrated that by coupling a qubit to a linear resonator,
 the state of the qubit creates a dispersive shift in the resonance frequency of the
 resonator. Thus by measuring the resonance frequency of the linear resonator the
 state of the qubit can be determined.
 When coupling a linear resonator to a qubit, the loss of the resonator can
 7
(a) 
(b) 
Figure 1.2: From Ref. [15] (a) Superconducting resonator loss tangent
 measurement for silicon nitride and silicon dioxide. Silicon nitride shows
 significantly lower loss (b) Rabi oscillations of phase qubit where the
 same silicon nitride and silicon dioxide are used as wiring dielectrics.
 Qubit made with the lower loss silicon nitride show longer coherence
 times indicating that qubit lifetimes are likely limited by dielectric loss.
 8
limit the lifetime of the qubit. This was observed recently when the Yale group and
 eventually much of the community started using 3 dimensional cavities to measure
 their qubits [17]. The electromagnetic fields inside the 3-D cavities mostly reside in
 empty space. With little lossy dielectric, they exhibit much lower internal loss than
 planar LC resonators. Due to this improvement in the loss of the linear resonator,
 state of the art decay times were observed [17]. This presents another clear example
 where improvement in fabrication and understanding of linear resonators led to
 breakthroughs for non-linear superconducting qubits.
 1.2 Microwave kinetic inductance detectors
 While superconducting quantum computing is our underlying motivation for
 studying superconducting resonator loss, improved telescopes to better view the
 cosmos is no less romantic a pursuit. Superconducting microwave resonators are
 now used in some large array infrared telescopes; the resonators are in the form of
 microwave kinetic inductance detectors (MKIDs) that act as far-infrared photon de-
 tectors. [18] While visible light telescopes can give us beautiful pictures, seeing the
 universe in the far-infrared gives us information about ?cold? objects such as plan-
 ets, events inside interstellar gases and dust clouds where visible light is scattered,
 and distant galaxies with a high red shift. MKIDs are sensitive to far-infrared pho-
 tons because an absorbed photon will break Cooper pairs and increase the kinetic
 inductance of the resonator. This decreases the resonance frequency and quality
 factor which can then be detected using standard microwave measurements. By
 9
understanding and reducing loss in superconducting resonators, we can potentially
 improve the sensitivity of MKIDs.
 1.3 Other superconducting resonator applications
 Superconducting resonators have several other applications that I have not
 mentioned. Niobium superconducting resonators have been used in particle accel-
 erators [19] for years and have thus aided in understanding our universe at a more
 fundamental level. High-temperature superconducting microwave resonators have
 also been used for telecommunication purposes [20]. Superconducting resonators
 have recently even been used to read out nanomechanical resonators [21]. These
 applications as well as the yet unknown potential applications make the study of
 superconducting resonators, and the dielectric films that often cause their loss, an
 important one.
 1.4 Overview of thesis
 In this thesis, I discuss my research on thin-film superconducting resonators.
 My goal was to understand and improve their loss, usually by understanding the
 dielectric films that limit their loss. In Chapter 2, I discuss the two-level system
 model that has been used to explain dielectric loss and acoustic loss in amorphous
 films. In Chapter 3, I describe the experimental apparatus and details of the mea-
 surements performed as well as intricacies in the analysis of these measurements. In
 Chapter 4, I discuss the design and fabrication of four coplanar resonators on sap-
 10
phire and silicon wafers. In Chapter 5, I discuss my results from the measurement
 of these four resonators. In Chapter 6, I discuss Atomic Layer Deposition (ALD)
 and its potential for growing thin dielectric films useful for superconducting quan-
 tum computing. In Chapter 6, I also propose using coplanar resonators as a test
 bed for ALD grown dielectrics. In Chapter 7, I describe measurements of several
 ALD grown dielectric films with coplanar resonators and analysis of their loss. In
 Chapter 8, I discuss a new experiment designed to electrically bias the defects in
 dielectric films in superconducting resonators. In Chapter 9, I discuss measurements
 of this experiment and analyze the results. Finally, in Chapter 10, I conclude with
 a summary of key results obtained and a discussion of possible future research.
 11
Chapter 2
 Two-level systems and dissipation
 2.1 Background for undriven Two-level systems
 In the early 1970?s measurements of specific heat and thermal conductivity of
 different amorphous solids taken between 70 mK and 1 K [22] highlighted that there
 was a fundamental lack of understanding of low temperature thermal properties of
 amorphous solids. With this motivation, in 1972 Phillips [23] and Anderson [24]
 independently developed the standard tunneling two-level system (TLS) model to
 explain the experimental results. In this model each defect atoms can tunnel between
 two distinct local minima in a two-well potential. Needless to say this assumption
 of two wells is made to simplify the math and is in general not exactly true.
 The microscopic identity of actual TLSs in real samples is typically unknown,
 but they can be modeled as ions hopping between two local minima (see Fig. 2.1(a))
 i.e. by a particle of mass m in a double well potential (see Fig. 2.1(b)). For simplic-
 ity we assume that these are identical harmonic wells, with minima separated by
 distance d and shifted in energy by an asymmetry energy ?. Since we are consid-
 ering low-temperature properties, we restrict ourselves to the ground states of the
 two wells and solve the Schrodinger equation H? = E?. We assume the state ?
 obeys the ansatz ? = a?l + b?r, where ?l and ?r are the normalized ground state
 wave functions for the left and right wells respectively, and the coefficients a and b
 12
 4 
.? 
4? 8 
h ' @ M 
(a) 
(b) 
@ 
Figure 2.1: (a) Spatial representation of a TLS as a charged defect hop-
 ping between two positions a distance d apart that make angle ? with
 respect to the applied electric field. (b) Spectral representation (poten-
 tial energy description) of a TLS as a particle in a double well potential
 with asymmetry energy ?, tunneling rate ?0, and barrier height V ,
 where (V >> ?,?0).
 13
are assumed to be real. Multiplying Schrodinger?s equation by ?? and solving for
 the energy eigenvalues we have
 E =
 ? ??H?d3x? ???d3x = a2Hll + b2Hrr + 2abHlra2 + b2 + 2abS , (2.1)
 where Hll and Hrr are the eigenvalues of the left and right well states, Hlr =? ??l H?rd3x is the exchange energy between the two states, and S = ? ??l ?rd3x
 is the overlap between them. We know that the true wave function ? in the ground
 state will minimize E . Thus the true eigenvalue is always smaller than or equal to
 the one calculated from the ansatz. Therefore we minimize E with respect to the
 coefficients a and b by solving for ?E/?a = 0 and ?E/?b = 0, resulting in
 a
 b =
 Hlr ? ES
 Hll ? E
 , (2.2)
 and
 a
 b =
 E ?Hrr
 Hlr ? ES
 . (2.3)
 From Eqs. 2.2 and 2.3 we then obtain:
 (Hll ? E)(Hrr ? E) ? (Hlr ? ES)2 ? E2 + (Hll ?Hrr)E +HllHrr ?H2lr = 0, (2.4)
 where we?ve assumed that the overlap between the left and right wavefunctions is
 small and have neglected the term ES in the second step. Choosing the average of the
 left and right ground states to be the zero energy gives Hrr,ll = ~?/2 ? ?/2, where
 ? is the harmonic frequency of the left and right wells (defined by the curvature of
 the wells). Solving for the energy eigenvalues then gives
 E? =
 1
 2
 (~? ?
 ?
 ?2 + 4H2lr). (2.5)
 14
The energy splitting between the two states is then
 E = E+ ? E? =
 ?
 ?2 + 4H2lr =
 ?
 ?2 + ?20, (2.6)
 where we have defined ?0 = 2Hlr for the exchange energy; ?0/~ is the tunneling
 rate. Thus the Hamiltonian in the left right basis is simply
 H0 =
 1
 2
 ?
 ??? ? ??0
 ??0 ??
 ?
 ??? , (2.7)
 and diagonalizing it into the eigenbasis gives
 H0 =
 1
 2
 ?
 ??? E 0
 0 ?E
 ?
 ??? . (2.8)
 It can be shown using the WKB approximation that the tunneling energy is ?0 ?
 ~?e??, where ? ?
 ?
 2mV d2/2~ [25]. Assuming a uniform distribution of barrier
 heights V it is easily shown that one expects uniform distribution in log ?0. That
 along with the assumption of a uniform distribution of ? gives the standard distri-
 bution for the TLS barrier height and asymmetry
 d2N
 d?d?0
 =
 P0
 ?0
 , (2.9)
 where P0 is the TLS spectral and spatial density found to be of order 1044 J?1m?3
 [25] for most amorphous dielectrics.
 2.2 Two-level system loss
 Charge TLS defects can couple to electric fields or strain fields. In this dis-
 cussion we?ll focus on electric field perturbation of TLSs but the results can be
 generalized to include strain fields.
 15
An applied electric field ~E alters ? and ?0 of a TLS via a perturbation
 Hamiltonian
 HS =
 1
 2
 ?
 ??? ?? ???0
 ???0 ???
 ?
 ??? . (2.10)
 Thus the total Hamiltonian in the left right basis is H = H0 + HS. Here we will
 make the assumption that the change in asymmetry ?? is much bigger than the
 change in tunneling rate ??0. This is plausible because the electric field wavelength
 is in general much longer than the TLS length scales and thus we don?t expect
 the barrier height and the separation of the wells to be affected by the application
 of a field, but we do expect the field to effect the EM environment of the TLSs,
 thus shifting their asymmetry energy, ?. For small perturbations, we expect the
 asymmetry energy shift ?? to vary linearly with the applied field,
 ?? = 2~p ? ~E, (2.11)
 where ~p is the dipole moment of the TLS and ~E is the applied electric field. Trans-
 forming the perturbation Hamiltonian into the eigenbasis of H0 gives [26]
 HS =
 1
 E
 ?
 ??? ? ?0
 ?0 ??
 ?
 ??? (~p ? ~E). (2.12)
 We note that this Hamiltonian for a TLS in an electric field is analogous to that of
 a spin 1/2 system in a magnetic field [27, 28],
 HM = HM0 + HMS = ?~? ~B ? ~S = ?~?( ~B0 ? ~S) ? ~?( ~B? ? ~S), (2.13)
 where ? is the gyromagnetic ratio, ~B0 is a static magnetic field representing the
 unperturbed Hamiltonian, ~B? is an oscillating magnetic field representing the per-
 16
turbation Hamiltonian, and ~S = ~?/2, where ~? represents the Pauli matrices:
 ?x =
 ?
 ??? 0 1
 1 0
 ?
 ??? , ?y =
 ?
 ??? 0 ?i
 i 0
 ?
 ??? , ?z =
 ?
 ??? 1 0
 0 ?1
 ?
 ??? . (2.14)
 The two systems are identical once the following substitutions are made
 ?~? ~B0 = (0, 0, E) and ? ~? ~B? =
 (
 2
 ?0
 E ~p ?
 ~E, 0, 2?E ~p ?
 ~E
 )
 . (2.15)
 Without relaxation processes (infinite spin lifetimes), the dynamics of a free spin in
 a magnetic field are simply represented by
 d
 dt
 ~S(t) = ?~S ? ~B, (2.16)
 where ~S(t) is the spin operator.
 In 1946 Felix Bloch derived the Bloch equations describing a spin 1/2 system
 with relaxation in the context of nuclear magnetic resonance (NMR) [29]
 d
 dt?Sx(t)? = ? (?Sy(t)?Bz(t) ? ?Sz(t)?By(t)) ?
 ?Sx(t)?
 T2
 d
 dt?Sy(t)? = ? (?Sz(t)?Bx(t) ? ?Sx(t)?Bz(t)) ?
 ?Sy(t)?
 T2
 d
 dt?Sz(t)? = ? (?Sx(t)?By(t) ? ?Sy(t)?Bx(t)) ?
 ?Sz(t)??S0z [Bz(t)]
 T1 , (2.17)
 where T1 and T2 are the longitudinal and transverse relaxation times, respectively,
 and S0z [Bz(t)] is the equilibrium population difference between the two levels with
 field Bz(t),
 S0z [Bz(t)] = 12 tanh
 (
 ~?Bz(t)
 2kT
 )
 . (2.18)
 The Bloch equations are for an ensemble of spins, so ?~S(t)? represents the average
 spin of the system. Thus to solve for the behavior of a TLS in an electric field, we
 need only solve the analogous Bloch equations for a spin in a magnetic field.
 17
In the Bloch equations, the relaxation times, added in an ad hoc manner, are
 essential in understanding the loss in our system so let?s briefly discuss their role.
 First, the longitudinal relaxation time T1 is simply the characteristic timescale for
 a TLS to fall from the excited state to the ground state. It can be defined by the
 rate at which an ensemble of TLSs approaches their equilibrium state
 dSz
 dt = ?
 Sz ? S0z
 T1
 , (2.19)
 where one can substitute polarization P for spin S when considering a TLS instead
 of a magnetic spin. It can be shown [27] that for an ensemble of TLSs
 1
 T1
 =
 (
 ?0
 E
 )2 1
 T1,min
 , (2.20)
 where T1,min is a characteristic of the material. This inverse relationship between T1
 and the tunneling energy ?0 can be understood qualitatively. For lower tunneling
 rates, transitions between the two states are of course less common, naturally leading
 to a higher state lifetime T1.
 The transverse relaxation time T2 is more subtle. Quantitatively it?s the rate
 of decay of the off-diagonal terms of the TLS density matrix in the master equation
 (discussed in more detail in Chapter 8), or it?s the rate at which a pure state ap-
 proaches a mixed state. One can also think of it as the loss of quantum coherence,
 which includes both loss of phase information and loss of energy; this is why T2 is
 also called the coherence time. Loss of phase information occurs on a time scale T?,
 the dephasing time. Thus T2 incorporates T1 relaxation as well as dephasing,
 1
 T2
 =
 1
 2T1
 +
 1
 T?
 . (2.21)
 18
The correspondence between TLS behavior and the Bloch equations was rec-
 ognized and used to solve the acoustic loss problem of a TLS in a strain field [28].
 A year later this result was generalized to solve the electric loss problem [30]
 tan (?) = piP0|~p|
 2
 3
 tanh (~?/2kBT )?
 1 + ?2R0? 2
 , (2.22)
 where f = ?/2pi is the measurement frequency, ? is a characteristic TLS lifetime
 that depends on the decoherence limit, and ?R0 is the characteristic Rabi frequency
 of the TLS ensemble,
 ?R0 =
 ~p ? ~E
 ~ . (2.23)
 Note that ? =
 ?
 T1,minT2/3 for constant coherence time T2 and that the Rabi
 frequency for an individual TLS is
 ?R =
 ~p ? ~E
 ~
 ?0
 E . (2.24)
 To simplify the expression for the loss tangent we define the intrinsic loss
 tangent of a material
 tan ?0 =
 piP0|~p|2
 3 , (2.25)
 and the characteristic field
 Ec =
 ~
 |~p|? . (2.26)
 Thus the Eq. 2.22 becomes
 tan (?) = tan(?0)tanh (~?/2kBT )?
 1 + (| ~E|/Ec)2
 . (2.27)
 A qualitative way to understand the functional form of the TLS loss is to
 think of the TLSs as a bath that absorbs and reemits photons through absorption
 19
and stimulated emission. For low excitation fields | ~E|  Ec the TLS bath is in
 its ground state and thus is able to absorb whatever energy is available to it, thus
 there?s a constant loss tangent, tan ?0 tanh (~?/2kBT ), with no field dependence. In
 contrast when the field increases such that the Rabi frequency of the TLSs is faster
 than the decay rate, the TLSs become more and more likely to be in the excited
 state, and are just as likely to reemit a photon through stimulated emission as they
 are to absorb a photon. This leads to a reduction in loss tangent as ? 1/| ~E| for
 high fields | ~E|  Ec.
 2.3 Loss in a qubit
 Individual TLSs were first noticed as splittings in the spectrum of phase qubits
 [31] (see Fig. 2.2) and were initially believed to be TLSs causing critical current
 noise in the junction, but a year later were understood as TLSs coupling to the
 electric field of the qubit [15, 32]. I helped take a related resonator measurement of
 individual TLS with a device called a Josephson Junction Defect Spectrometer [33].
 By solving for the Hamiltonian of a TLS in the junction dielectric of a phase qubit
 [15], it can be shown that the splitting size is given by
 S = Smax cos ? cos ?, (2.28)
 where ? is the angle between the TLS dipole and the field in the junction, ? =
 arctan ?/?0, and Smax is the maximum splitting size given by
 Smax = 2
 |~p|
 x
 ?
 E
 2C , (2.29)
 20
Figure 2.2: Splitting in the spectrum of phase qubit from Ref. [31] due
 to coupling to TLSs in the junction. The dotted lines mark the location
 of prominent splittings on the current bias axis.
 21
where C is the capacitance of the qubit and x is the thickness of the junction. If we
 assume a surface density of TLSs for the junction, ?, as was done in Ref. [15] and
 the same uniform distribution of asymmetry energies ? and logarithmic distribution
 of tunneling rates ?0 we arrive at this expression for the TLS density of states
 d2N
 dEd? =
 ?A
 cos ? , (2.30)
 where A is the junction area and we?ve changed the distribution variables from ?
 and ?0 to E and ?. It can be shown that switching from ? to S and averaging over
 ? yields the density
 d2N
 dEdS = ?A
 ?
 1 ? S2/S2max
 S , (2.31)
 for S < Smax and 0 otherwise [15].
 Assuming that the qubit couples to a continuum of many TLSs, which is
 reasonable for large area junctions, we calculate the decay rate of the qubit from
 Fermi?s golden rule
 ?1 =
 2pi
 ~
 ? Smax
 0
 d2N
 dEdS (S/2)
 2dS (2.32)
 = (pi/6)?AS2max/~ (2.33)
 =
 pi(?/x)|~p|2
 3
 E
 ~ (2.34)
 where we?ve substituted in Eq. 2.31 for the density of states and used the qubit-TLS
 interaction Hamiltonian,
 Hint = i(S/2)(|1?|g??0|?e| ? |0?|e??1|?g|) (2.35)
 where, |g? and |e? are the ground and excited states of the TLS and |0? and |1? are
 the ground and excited states of the qubit.
 22
By comparing the qubit decay rate in Eq. 2.34 to the intrinsic loss tangent
 from TLSs in Eq. 2.25 it can be seen that the qubit decay rate is proportional to
 the low field limit of the dielectric loss tangent. Thus we need to minimize dielectric
 loss to optimize qubit lifetimes.
 2.4 TLS micromechanism
 TLSs have been germane to the study of low temperature properties of amor-
 phous solids for over 40 years, however rarely has a TLS micromechanism been
 identified. One problem is that TLSs are so ubiquitous in low temperature solids
 that their effect can be due to many different types of structures. The amorphous
 nature of the materials inevitably creates variations in the environment, presenting a
 vast parameter space with which to form TLS defects, making the task of identifying
 such a defect very difficult.
 Despite the difficulties there have been some successes in using density func-
 tional theory to identify potential TLS candidates. Holder et al. [34] found that in
 amorphous aluminum oxide (AlOx), hydrogen impurities are likely to fill aluminum
 cation vacancies and create an OH?1 rotors that can act as tunneling TLSs. Such
 defects were originally proposed by Phillips [26] as a leading TLS candidate. Some
 experiments have correlated low temperature TLS dielectric loss with room temper-
 ature defect spectroscopy of the material. Paik et al. [35] found that nitrogen rich
 silicon nitride (SiNx) films had a relatively high density of N-H bonds which were
 correlated with low temperature loss.
 23
TLSs can also be found in crystals, and because crystals are relatively sim-
 ple systems, it has been possible to identify the defects responsible for them. For
 example, it?s been shown that lithium ion (Li+) defects can replace potassium ions
 (K+) in potassium chloride (KCl) and form tunneling TLSs [25]. It?s also been found
 through thermal conductivity measurements that doping sodium fluoride (NaF) with
 hydroxide (OH?) ions can create TLSs, suggesting an OH rotor defect model. Com-
 pared to crystalline systems, it seems likely that amorphous systems will tend to
 have many more types of TLSs. Thus a good starting point for research into loss in
 amorphous dielectrics is to find the leading TLS contributors to loss.
 24
Chapter 3
 Experimental setup and measurement technique
 3.1 Dilution refrigerator setup
 All measurements in this thesis were performed in a dilution refrigerator. A
 dilution refrigerator uses a mixture of He-3 and He-4 to reach millikelvin tempera-
 tures. The specific type of refrigerator I used was an Oxford Instrument Kelvinox
 model 400 dilution refrigerator which has a cooling power of 400 ?W at 100 mK
 (see Fig. 3.1(a)). I typically reached a base temperature of about 30 mK, although
 there were instances when either vacuum leaks or insufficient heat sinking produced
 a higher base temperature.
 For all of my measurements only two microwave lines were required (in mea-
 surements discussed in Chapters 8 and 9 an additional dc bias line was required).
 A schematic of the wiring setup in the refrigerator is shown in Fig. 3.1(b). Stainless
 steel UT85 coax cables (0.085? outer diameter) were used for the input lines for ther-
 mal isolation and copper nickel UT85 coax cables were used for the output lines for
 thermal isolation and reduced attenuation of the measurement signal. In addition
 to the attenuation in the lines themselves, 20 dB Midwest Microwave attenuators
 were used at the 1.5 K stage and the mixing chamber (base temperature) to elimi-
 nate thermal noise from higher temperatures. The output line had two PAMTECH
 circulators (CTH1409KS) terminated by 50 ? at the mixing chamber and one at the
 25
Figure 3.1: (a) Photograph of the inner vacuum chamber of the dilution
 refrigerator with the sample box mounted at the bottom. (b) Schematic
 of microwave wiring in the refrigerator.
 1.5 K stage for microwave isolation between the sample and the amplifiers. Each
 provides 18 dB of isolation in the 4-8 GHz range. After the circulators there was a
 low noise, cryogenic HEMT amplifier, purchased from the Weinreb group in Caltech
 [36], in the 4 K helium bath followed by two additional room temperature amplifiers
 (see Fig. 3.1(b)).
 3.2 Signal calibration
 As shown in Fig. 3.1(b), an Agilent N5242A PNA-X network analyzer was
 used to perform device spectroscopy. Here I?ll discuss the calibration of the system
 26
to convert from the applied network analyzer voltage to the voltage at the sample.
 This calibration is critical because as discussed in Chapter 2 TLS dielectric loss is
 power-dependent. Since stainless steel has little change in electrical conductivity
 with temperature, I performed the calibration of the input microwave lines at room
 temperature and assumed this was accurate at low temperatures.
 First, to determine the attenuation of the input lines at room temperature, I
 used the network analyzer to measure the throughput S21 from the network analyzer
 to the input connection at the bottom of the mixing chamber where the device will
 be mounted. This was done for three input powers (see Fig. 3.2(a)). Three relatively
 high powers were used (-20, -10, and 0 dBm) to minimize the required averaging for
 sufficient signal to noise. I fit the three measurements of S21 to a line on a semi log
 plot (see Fig. 3.2(a)). I then averaged the slopes, and plotted the intercepts as a
 function of the input power (see Fig. 3.2(b)). The slope in Fig. 3.2(b) should ideally
 be unity but I fit to the slope in order to capture any nonlinearities in the system; the
 slope obtained was 0.981. The average slope with frequency from Fig. 3.2(a) (-3.53
 dBm/GHz) was then combined with the slope with input power from Fig. 3.2(b) to
 obtain a calibration function for the power at the mixing chamber as a function of
 the frequency and the input power,
 Pdevice = ?3.53(dBm/GHz) ? freq(GHz) + 0.981 ? PNA ? 57.04 dBm. (3.1)
 Note that since the slopes with frequency from Fig. 3.2(a) are generally very close
 and the slope with power from Fig. 3.2(b) is generally close to 1, in principle a single
 power could be used to calibrate the lines. Three was chosen somewhat arbitrarily
 27
4 5 6 7
 ?110
 ?100
 ?90
 ?80
 ?70
 f (GHz)
 Throughput (dBm
 )
 ?3.48[f(GHz)]?57.07
 ?3.53[f(GHz)]?66.78
 ?3.57[f(GHz)]?76.68
 (a)
 ?20 ?10 0
 ?80
 ?75
 ?70
 ?65
 ?60
 ?55
 PNA (dBm)
 P devic
 e (dBm
 )
 (b)
 Figure 3.2: (a) Throughput calibration measurements of input mi-
 crowave lines performed at room temperature with -20 (green), -10
 (blue), and 0 (red) dBm at the Network analyzer. (b) Plot of the inter-
 cepts from the fits in (a).
 28
to check and marginally improve the calibration.
 From the 50 ? characteristic impedance of the input lines and the input power
 at the device Pdevice we can calculate the input RMS voltage at the device
 Vdevice,RMS =
 ?
 Pdevice ? 50?. (3.2)
 This input voltage will be used later to calculate the voltage across the resonators.
 The setup has two sets of input and output lines and I?ve shown here the calibration
 for only one. The calibration for the other input line is performed in an identical
 manner. The calibration equation for the other line analogous to Eq. 3.1 is
 Pdevice = ?3.51(dBm/GHz) ? freq(GHz) + 1.02 ? PNA ? 58.03 dBm. (3.3)
 3.3 Resonance derivation
 I measure a resonator?s loss by coupling it to a coplanar waveguide (CPW)
 transmission line and measure the waveguide?s throughput S21 ? Vout/Vin. I next
 present a derivation of S21 that is similar to the one originally derived for this project
 by Kevin Osborn. The throughput can be modeled by the circuit schematic shown
 in Fig. 3.3, where the resonator, composed of the inductor L and the capacitor ?C,
 is coupled to a CPW. Both inductive and capacitive coupling is included through
 M and CC respectively, and ?C is complex to account for dielectric losses. Vin and
 Vout are the input and output voltage amplitudes, and V is the voltage across the
 capacitor ?C. For this model, I assume the transmission lines are perfectly coupled
 with the characteristic impedance Z0 and L1 is small (L1 << CCZ20).
 29
Figure 3.3: a schematic of the resonator measurement setup with both
 inductive and capacitive coupling.
 Figure 3.3 can be redrawn as Fig. 3.4(a) and assuming low loss
 (
 Im
 {
 ?C
 }
 << Re
 {
 ?C
 })
 ,
 the complex capacitance can be split into a capacitor and a resistor as in Fig. 3.4(b),
 where C = Re
 {
 ?C
 }
 and
 R = 1
 ?Im
 {
 ?C
 } . (3.4)
 To solve for the transmission through the CPW I write the following Kirchhoff?s
 equations for the circuit in Fig. 3.4(b),
 2Vin ? Vout = I1(Z0 + i?L1) ? i?MIL
 V ? Vout =
 I2
 i?CC
 Vout = (I1 + I2)Z0
 V = i?LIL ? i?MI1
 V = ?I2 + IL
 i? ?C
 . (3.5)
 Solving these equations we find an expression for the transmission S21 as a function
 30
Z0 
Z0 
CC M 
C R ~ 2Vin 
V 
(b) 
Z0 
Z0 
CC M 
~ 2Vin V 
(a) 
Vout 
Vout 
IL 
I1 
I1 
IL 
I2 
I2 
Figure 3.4: (a) Equivalent circuit to that shown in Fig. 3.3. (b) Equiv-
 alent circuit to (a) where ?C has been separated into its real (C) and
 imaginary (1/?R) parts.
 31
NI C?L
  
CT RT V 
Figure 3.5: Norton equivalent circuit for resonator measurement where
 V is the voltage across the capacitor.
 of the voltage across the capacitor:
 S21 =
 Vout
 Vin
 = 1 +
 V
 2Vin
 (
 M
 L
 + Z0i?CC
 )
 , (3.6)
 where some approximations have been used (?C << 1/Z0, {?L1, ?M2/L} << Z0).
 Using Eqs. 3.5 again, this time to eliminate Vout we get the following independent
 equation:
 ?V
 (
 1
 i?L
 +i? ?C+ i?CC
 i?CCZ0 + 1
 +
 1
 2Z0
 (
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 )2)
 =
 Vin
 Z0
 (
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 )
 .
 (3.7)
 To proceed, we note that Eq. 3.7 is of the form
 IN = V
 (
 1
 i?L
 + i? ?C +GN
 )
 , (3.8)
 where GN and IN are the Norton equivalent conductance and current respectively.
 The circuit in Fig. 3.4 thus has the Norton equivalent circuit shown in Fig. 3.5.
 From Eq. (3.7) we see that
 IN = ?
 Vin
 Z0
 (
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 )
 , (3.9)
 and
 GN =
 i?CC
 i?CCZ0 + 1
 +
 1
 2Z0
 (
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 )2
 . (3.10)
 32
Since the real part of GN loads the resonator measurement and the imaginary
 part shifts the resonance frequency, it is useful to separate GN into its real (con-
 ductive) and imaginary (susceptive) components. We can now define Re{GN} ?
 1/RT , Im{GN} ? ?CT , R?1eff ? R?1 + R?1T , Q ? Reff?0(C + CT ), and ?0 =
 1/?L(C + CT ). Expanding Eq. 3.10 around the small values M/L and ?CCZ0 and
 keeping only the lowest order non-vanishing terms we find
 CT = CC , (3.11)
 and
 R?1T =
 1
 2Z0
 ((
 M
 L
 )2
 + ?2C2CZ20
 )
 . (3.12)
 Equation (3.6) can then be rewritten as
 S21 = 1 ?
 Reff/RT
 1 + 2iQ???0
 ?0
 , (3.13)
 where we?ve assumed that ? ? ?0. Next it is convenient to define the external
 quality factor Qe = RT?0(C + CT ), which effectively gives the resonator?s coupling
 loss, allowing us to rewrite Eq. 3.13 as
 S21 = 1 ?
 Q/Qe
 1 + 2iQ???0
 ?0
 . (3.14)
 Equation 3.14 has the functional form of a Lorentzian with three free param-
 eters (Q, Qe, and ?0) and can now in principle be used to fit transmission measure-
 ments to extract the resonance frequency as well as the total and external quality
 factors. Note that what we?re generally most interested in is the loss tangent of the
 resonator tan ? = Im
 {
 ?C
 }
 /Re
 {
 ?C
 }
 . We can now use Eq. 3.4 to define the internal
 33
quality factor Qi = R?C = 1/ tan ? such that
 1
 Qi =
 1
 Q ?
 1
 Qe . (3.15)
 This relationship allows us to use extracted values of Q and Qe to determine Qi and
 thus the loss tangent of the resonator.
 Note that the internal quality factor Qi of the resonator is modeled here as
 the loss tangent of the dielectric, but in principle the loss can come from any inter-
 nal loss mechanism in the device. When multiple loss mechanisms are active they
 add reciprocally as the internal and external quality factors do. For example if a
 resonator exhibited resistive loss (Qres), radioactive loss (Qrad), and dielectric loss
 (Qtan ?), the total internal quality factor would be
 1
 Qi =
 1
 Qrad +
 1
 Qres +
 1
 Qtan ? . (3.16)
 3.4 Determining the voltage across the resonator
 In section 3.2 I discussed how I calibrated the microwave input lines to know
 what voltage I?m applying at the CPW. In section 3.3 I derived the output transmis-
 sion through the CPW as a function of the resonator parameters. Here I combine
 these two results in order to calculate the voltage V across the resonator as a func-
 tion of the resonator parameters. From Eq. 3.8 we can write the following expression
 for the voltage across the resonator,
 V = IN1
 i?L + i? ?C +GN
 . (3.17)
 34
The denominator can be split into its real and imaginary components using i? ?C =
 i?C + R?1 and GN = i?CT + R?1T . On resonance, the imaginary components of
 the denominator cancel leaving only real components, thus the voltage on resonance
 can be rewritten as
 V (? = ?0) = INR?1 +R?1T
 = INReff . (3.18)
 Substituting Eq. 3.9 in for IN , Eq. 3.18 becomes
 V (? = ?0) = ?VinReffZ0
 (
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 )
 , (3.19)
 and recognizing that we?re interested in the RMS voltage across the resonator on
 resonance, we can rewrite Eq. 3.19 as
 VRMS(? = ?0) = ReffZ0
 ????
 M
 L
 ?
 i?CCZ0
 1 + i?CCZ0
 ????Vin,RMS . (3.20)
 Then using Eq. 4.5, this can be rewritten as
 VRMS =
 Reff
 Z0
 ?
 2Z0
 RT
 Vin,RMS. (3.21)
 Finally using the definitions of Q and Qe, we find the voltage on resonance is
 VRMS(? = ?0) =
 ?
 2
 Z0?0CQeQVin,RMS, (3.22)
 where we?ve assumed CC << C.
 We now have an expression for the voltage across the resonator in terms of ex-
 tracted parameters (Q, Qe, and ?0) from throughput measurement fits to Eq. 3.14,
 the calibrated input voltage Vin,RMS of the device from Eq. 3.2, and known design
 parameters of the resonator and the circuit (Z0 and C). Note that a lumped el-
 ement resonator was assumed for this derivation, and Chapter 4 contains further
 35
analysis for the situation where the resonator length is comparable to the resonance
 wavelength.
 3.5 Asymmetric resonance line shapes
 In section 3.3 we derived the Lorentzian line shape for a matched transmission
 line coupled to a resonator. However, as Fig. 3.6 shows non-ideal experimental
 setups can lead to asymmetry in the resonance line shape [37, 27, 35, 38, 39, 40, 41],
 corresponding to, as will be shown later in this section, a rotation of the resonance
 circle. This complicates the analysis and the interpretation of internal and external
 quality factors (Qi and Qe). Several methods for extracting the Qi of a resonator
 have been used for different experimental setups [42, 43, 44, 45, 40]. However, these
 methods either require a single port reflective measurement [43, 44] (incompatible
 with most qubit measurements), full two-port characterization [42, 45] (typically
 unavailable for millikelvin measurements), or identifying and fitting to a second
 coupled mode [40] (a special case). The most widely used technique for analyzing
 millikelvin resonator measurements, prior to our publication of this work, the ?
 rotation method (?RM), simply adds an empirical rotation of the resonance circle
 before extracting the quality factor [35, 27, 38].
 In this section I show how asymmetry in the resonance line shape can arise
 from coupling the resonator to mismatched input and output transmission lines as
 well as non-negligible transmission line series inductance (L1), mutual inductance
 (M) or coupling capacitance (CC). Based on this understanding of the origin of
 36
5.7647 5.7648 5.7649 5.765
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 f (GHz)
 |S 2
 1
 |
 Figure 3.6: Resonance line shape of a coplanar superconducting alu-
 minum QLL-CPS resonator taken at 30 mK (see Chapter 4). The line
 shape is clearly asymmetric and does not fit to Eq. 3.14.
 37
the asymmetry, I?ll derive the Diameter Correction Method (DCM) and use it to
 extract Qi. I?ll compare this to the conventional analysis method, the ? rotation
 method (?RM) [35, 27, 38], and show that there is a one-to-one mapping between the
 two methods but that the ?RM systematically overestimates Qi by an analytically
 quantifiable amount. Note, I do not address fitting techniques here but interested
 readers can find a comprehensive quantitative comparison of fitting techniques in
 Ref. [46].
 3.5.1 Analysis of asymmetric resonance lineshape
 Here I derive an expression for the resonance line shape of a resonator cou-
 pled to a transmission line. In contrast to section 3.3, here I assume mismatched
 transmission lines and non-negligible coupling components (see Fig. 3.7). Solving
 Kirchhoff?s equations for this circuit yields a more general form of Eq. 3.6
 S21 = (1 + ?)
 (
 1 +
 V
 2Vin
 (
 M
 L
 + Z ?ini?CC
 ))
 , (3.23)
 where 1 + ? ? 2/(1 + (i?CC + /Zout)Z ?in), Z ?in ? Zin + i?L1 ? i?M2L , and |?| << 1.
 Solving Kirchhoff?s equations and eliminating Vout, we get the independent equation:
 ?V
 (
 1
 i?L
 +i? ?C + i?CCi?CCZout+1 +
 (
 M
 L ?
 i?CCZout
 1+i?CCZout
 )2
 Z?out+Zin+i?
 (
 L1? M
 2
 L
 )
 )
 = 2Vin
 (
 M
 L ?
 i?CCZout
 1+i?CCZout
 Z?out+Zin+i?
 (
 L1? M
 2
 L
 )
 )
 , (3.24)
 where Z ?out ? Zout/(1 + i?CCZout). Again we note that Eq. (3.24) is of the form
 V
 (
 1/(i?L) + i? ?C +GN
 )
 = IN , where
 IN = ?2Vin
 (
 M
 L ?
 i?CCZout
 1+i?CCZout
 Z ?out + Zin + i?
 (
 L1 ? M
 2
 L
 )) , (3.25)
 38
Figure 3.7: (a) Schematic of resonator measurement setup with inductive
 and capacitive coupling and mismatched input and output transmission
 lines. (b) equivalent circuit to (a), where ?C has been separated into its
 capacitive part (C) and its resistive parts (1/?R). (c) Norton equiva-
 lent circuit for resonator measurement where V is the voltage across the
 capacitor.
 39
and,
 GN =
 i?CC
 i?CCZout + 1
 +
 (
 M
 L ?
 i?CCZout
 1+i?CCZout
 )2
 Z ?out + Zin + i?
 (
 L1 ? M
 2
 L
 ). (3.26)
 Since the real part of GN loads the resonator measurement and the imaginary part
 shifts the resonance frequency, it is useful to separate GN into its real (conductive)
 and imaginary (susceptive) components. We can now make the same definitions as
 in section 3.3:
 Re{GN} ? 1/RT , (3.27)
 Im{GN} ? ?CT , (3.28)
 R?1eff ? R?1 +R?1T , (3.29)
 Q ? Reff?0(C + CT ), (3.30)
 ?0 = 1/
 ?
 L(C + CT ), (3.31)
 Qi ? R?0C = Re{
 ?C}
 Im{ ?C} =
 1
 tan ? , (3.32)
 and we add the definition
 G? ? ? IN
 2Vin
 (
 M
 L
 + Z ?ini?CC
 )
 . (3.33)
 Then, Eq. (3.23) can be rewritten as
 S21 = 1 ?
 G?Reff
 1 + 2iQ???0
 ?0
 , (3.34)
 where I?ve dropped the 1 + ? multiplicative factor because it can be accounted for
 by assuming an arbitrary attenuation and phase shift. Note that a fundamental dif-
 ference between section 3.3?s derivation of Eq. 3.13 and this more general derivation
 40
of Eq. 3.34 is that here if higher order terms in the small parameters (M/L) and
 ?CCZout are kept, G? 6= R?1T . Proceeding, I now rewrite Eq. (3.34) as
 S21 = 1 ?
 (
 GD +R?1T
 )
 Reff
 1 + 2iQ???0
 ?0
 , (3.35)
 where I?ve defined
 GD ? G? ?R?1T . (3.36)
 Expanding to third-order in M/L we find:
 GD = i?CC
 M
 L
 (
 Zin ? Zout
 Zin + Zout
 )
 +
 i
 (Zin + Zout)2
 (
 (?CC)2 Z2out
 (
 L1 ? CCZ2in
 )
 ?
 (
 M
 L
 )2(
 L1 ? CCZ2out
 ))
 ,
 (3.37)
 and note that GD is purely imaginary.
 At this point in the analysis it is useful to generalize the definition of the
 external quality factor
 ?Q?1e ?
 R?1T +GD
 ?0 (C + CT ) , (3.38)
 and recognize that sinceGD is purely imaginary, combining Eqs. 3.38, 3.35, 3.29, 3.30,
 and 3.32 gives
 Q?1i = Q?1 ? Re
 {
 ?Q?1e
 }
 , (3.39)
 where I?ve assumed CT  C. We can now rewrite Eq. 3.35 as
 S21 = 1 ?
 Q ?Q?1e
 1 + 2iQ???0
 ?0
 , (3.40)
 which adds another free parameter to the ideal throughput transmission of Eq. 3.14.
 This parameter accounts for asymmetry in the line shape. Note that the prefactor
 41
1 + ? has been dropped from Eq. 3.40 because it only adds a constant phase and
 contributes a constant multiplicative factor to the amplitude which is convoluted
 with the amplification and attenuation of the signal.
 I emphasize here that GD is the term that creates the asymmetry in the line
 shape or equivalently a rotation of the resonance circle around the off resonance
 point. If GD were zero then ?Q?1e would be real, reducing Eq. (3.40) to a symmetric
 Lorentzian line shape (Eq. 3.14). Equation (3.40) can also be written as
 S21 = 1 ?
 Q
 ??? ?Q?1e
 ??? ei?
 1 + 2iQ???0
 ?0
 , (3.41)
 where ?Q?1e is represented in terms of its magnitude and phase ?. Another equivalent
 representation is
 S21 = 1 ?
 Q
 Qe
 (
 1 + 2iQ ??
 ?0
 )
 1 + 2iQ???0
 ?0
 , (3.42)
 where we?ve defined
 1
 Qe ? Re
 {
 ?Q?1e
 }
 , (3.43)
 and ?? is the difference between the resonance frequency and the new rotated in-
 phase point on the resonance circle, ?1 (see Fig. 3.8). The form of Eq. (3.42) can
 be understood by noting that S21 is real when ? = ?? + ?0.
 Here I stress that Eqs. (3.40-3.42) are equivalent representations of the asym-
 metric line shape, each highlighting a different interpretation of the asymmetry.
 In Eq. (3.40) the asymmetry is quantified by Im
 {
 ?Q?1e
 }
 and one can think of the
 asymmetry as coming from a complex loading of the resonator. In Eq. (3.41) the
 asymmetry is quantified by ?, where ? is the rotation angle of the resonance circle
 around the off-resonance point (see Fig. 3.8). And finally in Eq. (3.42) the asym-
 42
?0.2
 0
 0.2
 0.4
 0.6
 Im
 {S
 21
 }
 -?
 ?0
 ?1
 ?0.2 0 0.2
 ?0.4
 ?0.2
 0
 0.2
 0.4
 ?0 ?0
 ?RM
 DCM
 Re{S21}
 Im
 {S
 21
 }
  
 
(b)
 (a)
 Q/Qe
 Q/| ?Qe|
 Q/Qe
 Figure 3.8: (a) Simulated transmission through mismatched coupling
 lines (Zin = 24.5?, Zout = 84.5?) plotted as Im{S21} vs. Re{S21}
 with a fit to a circle. The asymmetry is represented as a rotation of
 the resonance circle by the angle ? away from the real (in-phase) axis
 or equivalently as ?? = ?1 ? ?0, the frequency shift of the in-phase
 point on the resonance circle. (b) Shows the simulated transmission with
 asymmetry removed using both analysis techniques. The ?RM (4) only
 rotates the circle to the real axis while the DCM () both rotates the
 circle and removes the factor of 1/ cos(?) increase to the diameter. The
 DCM shows that the invariant quantity is not, as the ?RM assumes,
 the diameter of the circle (equal to Q/| ?Qe| and Q/Qe before and after
 the DCM transformation respectively) but rather the length of the real
 axis segment intersecting the circle (shown in bold and equal to Q/Qe),
 where 1/Qe ? Re
 {
 1/ ?Qe
 }
 .
 43
metry is quantified by ??, where ?? = ?1 ??0 is the frequency shift of the in-phase
 point on the resonance circle from ?0 to ?1 (see Fig. 3.8). Since the three rep-
 resentations are equivalent, there exists a one-to-one mapping between the three
 notations:
 ? = arctan
 (
 Im{ ?Q?1e }
 Re{ ?Q?1e }
 )
 = arctan
 (
 2Q??
 ?0
 )
 . (3.44)
 One method the, ? rotation method (?RM), has been used in the literature
 [35, 27, 38] to extract internal quality factors accounts for the asymmetry by simply
 adding an empirical ? rotation. Unfortunately this method incorrectly substitutes
 ??? ?Q?1e
 ??? for Q?1e and defines
 1
 Qi, ?RM =
 1
 Q ?
 ????
 1
 ?Qe
 ???? . (3.45)
 In effect the ?RM accounts for the asymmetric line shape phenomenologically by
 adding the rotation, ?, without accounting for its origin and its impact on the
 interpretation of Qi. It corresponds to rotating the resonance circle back an angle
 ? in the complex S21 plane, thereby putting ?0 on the in-phase axis. The rotation
 of the ?RM can best be seen by examining the difference between Fig. 3.8(a) and
 Fig. 3.8(b)(4).
 The problem with the ?RM is that simply rotating the resonance circle by
 angle ? does not take into account the fact that the asymmetry has also caused the
 circle to grow by a factor of 1/ cos(?), assuming the circle has been normalized to
 full transmission off resonance (S21(? << ?0) = S21(? >> ?0) = 1). I have shown
 here that instead of Eq. 3.45 one has
 1
 Qi, DCM =
 1
 Q ?
 1
 Qe . (3.46)
 44
I call this the Diameter Correction Method (DCM) because in addition to rotating
 the circle by the asymmetry angle ? it also corrects the diameter by accounting
 for the complex Qe. This can be seen by examining the transformation between
 Fig. 3.8(a) and Fig. 3.8(b)().
 Another interpretation of the DCM is that the quantity that remains constant
 in the asymmetry transformation is not the diameter of the resonance circle, as
 the ?RM assumes, but rather the distance between the in-phase axis intercepts,
 shown in bold in Fig. 3.8. That invariant length is the diameter of the circle for
 a symmetric resonance and becomes a chord of the circle as asymmetry is added,
 but remains equal to Q/Qe, while the diameter grows as Q/| ?Qe|. The analytical
 discrepancy between the two methods can be determined by subtracting Eq. (3.46)
 from Eq. (3.45),
 1
 Qi, DCM ?
 1
 Qi, ?RM =
 ????
 1
 ?Qe
 ???? (cos(?) ? 1) . (3.47)
 From Eq. (3.47) we see that the error in the ?RM diverges for high asymmetry
 angle, ? ? ?pi, and for low Qe (high coupling). Note that for ? = 0, ?Q?1e is real
 and Eq. (3.46) reduces to Eq. (3.45) and therefore Eq. (3.47) goes to zero.
 3.5.2 Analysis of S21 Simulations and Data
 To test the ?RM and the DCM, I simulated the transmission through reso-
 nant circuits using the numerical linear solver, Microwave Office. Simulations were
 run varying a range of parameters: Qi, impedance mismatches, strength of both
 inductive and capacitive coupling and inductance L1. The resonator capacitance
 45
and inductance were held at 0.3 pF and 2.5 nH, respectively, producing a resonance
 frequency that ranged from 5.717-5.802 GHz (resonance frequency varies with cou-
 pling capacitance). The simulated S21 data was then fit and analyzed using both
 methods. Figure 3.9 shows an example of a Microwave Office schematic from such a
 simulation. Microwave Office doesn?t allow for variation of the port impedance (set
 to 50 ?), so to vary Zin and Zout I used the built in coax transmission line element
 and calculated the effective input and output impedances by using the well-known
 equation for impedance of a loaded transmission line [47]
 Z = ZC
 ZL + iZ0 tan ?l
 Z0 + iZL tan ?l , (3.48)
 where Z is the effective input impedance, ZL is the load impedance, ZC is the
 characteristic transmission line impedance, l is the length of the transmission line,
 and ? = 2pi/? is the wavenumber. For an electrical length (?l) of 90?, the tangent
 term in Eq. 3.48 dominates, and the equation reduces to Z = Z2C/ZL. The input and
 output impedances for the example shown in Fig. 3.9 are Zin = (30 ?)2/(50 ?) =
 18 ? and Zout = (70 ?)2/(50 ?) = 98 ?.
 I typically modeled asymmetry by varying Zin/Zout. Asymmetry can also be
 created by increasing L1. However, L1 values in the nanohenries are required to
 create significant asymmetry, which is far too large to be physical in my setup.
 The impedance mismatches used are sometimes large, but after this work was pub-
 lished [41] a related study found that these large effective impedance mismatches
 are consistent with wire bonds made to the coplanar waveguides on the chip [48].
 Figure 3.10 shows results from simulations and fits with the same simulated
 46
Figure 3.9: Schematic from the numerical linear solver Microwave Office
 showing a typical simulated resonance circuit.
 quality factor, Qi = 105 and a range of Zin/Zout values. The coupling line mismatch
 creates a clear asymmetry in the line shape which I quantified by extracting the
 asymmetry angle ? (see Fig. 3.10). In addition to the value of ? extracted from the
 fit, I also analytically determined the expected asymmetry using
 ? = arctan
 (
 Im{GD}
 R?1T
 )
 , (3.49)
 which can be obtained from Eqs. 3.26, 3.27, and 3.37.
 Two internal quality factors were extracted from these fits, one using the ?RM
 and the other using the DCM. Figure 3.11 shows both extracted quality factors as
 well as the fit extracted asymmetry angle ?, plotted against the predicted ? from
 Eq. (3.49). Examination of Fig. 3.11 reveals that the DCM is more accurate than the
 ?RM for large asymmetry ?, and that the two methods agree for small asymmetry.
 I also compared both analysis techniques when asymmetry is held constant but
 Qi is varied, which I will show later in this chapter corresponds to some experimen-
 47
5.8017 5.8018 5.8019 5.802
 ?5
 0
 5
 10
 f(GHz)
 |S 2
 1
 |(d
 B
 )
 Zin = 50 ?
 Zout = 50 ?
 Zin = 40.5 ?
 Zout = 60.5 ?
 Zin = 32 ?
 Zout = 72 ?
 Zin = 24.5 ?
 Zout = 84.5 ?
 Zin = 18 ?
 Zout = 98 ?
 ? = 0 rad
 ? = ?0.2 rad
 ? = ?0.39 rad
 ? = ?0.57 rad
 ? = ?0.74 rad
 Figure 3.10: Plot of simulated S21 vs. freq f fit to symmetric and asym-
 metric resonance line shapes. Here the asymmetry was created using
 mismatched coupling lines (Zin and Zout). Asymmetry angles ? are ex-
 tracted from the fits.
 48
?0.8 ?0.6 ?0.4 ?0.2 0
 ?0.8
 ?0.6
 ?0.4
 ?0.2
 0
 ?pred.(radians)
 ? s
 im
 .
 (ra
 di
 a
 n
 s)
 90
 100
 110
 120
 130
 Q i
 ?
 10
 ?
 3
  
 
DCM
 ?RM
 (a)
 (b)
 Figure 3.11: (a) Qi extracted from fits to circuit simulations using ?RM
 (?) and DCM () analysis techniques, as a function of predicted asym-
 metry angle ?pred. calculated using Eq. (3.49). The dashed line indicates
 actual simulation Qi. At low asymmetry the two methods agree. As
 asymmetry is increased, the ?RM extracted Qi deviates from the actual
 Qi. (b) The fit extracted asymmetry angle, ?sim. (), as a function of
 predicted asymmetry angle, ?pred.. The solid line is the ?pred. = ?sim.
 line. Good agreement of that line with the results () indicates that this
 method is accurate at predicting the asymmetry.
 49
tal data sets. Figure 3.12 shows Qi extracted using both analysis techniques plotted
 as a function of the actual (simulation set) Qi for simulations with low and high
 asymmetry. For low asymmetry (matched ports) both analysis techniques yield the
 correct Qi within the expected first-order error (CC/C). However, for high asym-
 metry (mismatched ports), the ?RM yields quality factors that are systematically
 too high. In addition, for sufficiently high Qis, the ?RM yields negative Qis (this
 is why the asymmetric data analyzed by the ?RM appears to stop at large Qi in
 Fig. 3.12).
 The occurrence of these unphysical, negative, Qis in the ?RM can be under-
 stood by examining the circle plots in Fig. 3.8. As discussed earlier when there is a
 large asymmetry, in addition to being rotated, the resonance circle grows by a factor
 of 1/ cos(?) (assuming full transmission off resonance). Since the ?RM only rotates
 the circle back, it does not account for the increase in size, shown in Fig. 3.8(b).
 So if Q ? Qe (Qi >> Qe), the circle diameter is larger than 1, almost crossing
 the y-axis. Rotating the circle using the ?RM causes the circle to cross the y-axis
 and this yields a negative Qi. In Fig. 3.8(b) the ?RM analyzed simulation almost
 crosses the origin. This corresponds to the Qi = 8 ? 105 simulation in Fig. 3.12; it
 is an example of a simulation data set with a Qi and asymmetry not large enough
 to create a negative Qi but still large enough to create a considerable discrepancy
 between Qi, ?RM and Qi, DCM .
 To further evaluate both methods, I also analyzed the asymmetric data shown
 in Fig. 3.6. Figure 3.13 shows the fit used for that resonance line shape. In Fig. 3.14 I
 show theQi extracted using both techniques and the asymmetry angle ?. Figure 3.14
 50
105 106
 ?0.6
 ?0.4
 ?0.2
 0
 Qi (Actual)
 ? s
 im
 .
 (ra
 d)
  
 
matched
 mismatched
 105
 106
 107
 Q i
  
 
matched (DCM)
 matched (?RM)
 mismatched (DCM)
 mismatched (?RM)
 (a)
 (b)
 Figure 3.12: (a) Qi extracted with both analysis techniques (?RM and
 DCM) as a function of the actual Qi from two sets of simulations. The
 first set of simulations had high asymmetry (mismatched ports: Zin =
 24.5?, Zout = 84.5?) and the second had low asymmetry (matched ports:
 Zin = Zout = 50?). Solid line is actual Qi equal to extracted Qi line and
 dashed line indicates the coupling (Qe). Both analysis techniques work
 well with low asymmetry but only the DCM works with high asymmetry
 at large Qi. At low simulation internal quality factors (Qi = 105) the
 DCM extracted internal quality factors (Qi = 1.002?105) with less than
 1% deviation from the actual value in both low and high asymmetry
 simulations and at high simulation internal quality factors (Qi = 4 ?
 106) the DCM extracted internal quality factors (Qi = 3.85 ? 106) with
 less than 4% deviation from the actual value for both low and high
 asymmetry simulations. The deviation at high internal quality factors
 is limited numerically by the fit and is not a limit of the method. (b)
 The fit extracted asymmetry angle ?sim. for both low (?) and high ()
 asymmetry simulations plotted against the actual simulation Qi.
 51
5.7647 5.7648 5.7649 5.765
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 f (GHz)
 |S 2
 1
 |
 Figure 3.13: The resonance line shape from Fig. 3.6 fit with the derived
 asymmetric resonance line shape (Eq. 3.40).
 is similar to the high asymmetry simulations shown in Fig. 3.12. As expected from
 Eq. (3.37), the asymmetry ? is independent of Qi for both the real device measure-
 ments and the simulated data. Note also the last two data points for the ?RM in
 Fig. 3.14 are negative (and off the plot) in the same manner that the last points in
 the simulated data of Fig. 3.12 are negative.
 Another way to test the analysis techniques is by varying Qe while keeping
 Qi constant. In Fig. 3.15 I show the results form a set of simulations where I
 increased Qe by lowering the capacitive coupling. As expected, for low asymmetry
 (matched ports) both analysis techniques do a good job of extracting Qi = 105.
 However, for high asymmetry (mismatched ports) the ?RM overestimates Qi by a
 decreasing amount as Qe/Qi increases. Interestingly, in the ?RM, as Qe increases,
 52
10?6 10?5 10?4 10?3
 ?1
 ?0.8
 ?0.6
 ?0.4
 VRMS(V)
 ? e
 x
 p.
 (ra
 di
 a
 n
 s)
 105
 106
 107
 108
 Q i
  
 
DCM
 ?RM
 (b)
 (a)
 Figure 3.14: Extracted results from the same resonator that produced
 the data shown in Fig. 3.6 and Fig. 3.13. (a) Qi, extracted using two
 analysis techniques, ?RM (N) and DCM (), as a function of voltage
 across the resonator, VRMS. As with simulated results in Fig. 3.12, the
 ?RM systematically extracts higher Qis or unphysical negative Qis. The
 dashed line indicates the fit extracted external quality factor Qe. (b) The
 fit extracted asymmetry angle ?exp. () plotted against VRMS. Again
 similar to the simulated results in Fig. 3.12, ?exp. is independent of Qi,
 which is varying with VRMS.
 53
104 105
 ?0.6
 ?0.4
 ?0.2
 0
 0.2
 Qe
 ? s
 im
 .
 (ra
 di
 a
 n
 s)
  
 
matched
 mismatched
 80
 100
 120
 140
 160
 180
 200
 Q i
 (in
 th
 o
 u
 sa
 n
 ds
 )
  
 
matched (DCM)
 matched (?RM)
 mismatched (DCM)
 mismatched (?RM)
 (a)
 (b)
 Figure 3.15: (a) Qi extracted using DCM and ?RM techniques from two
 sets of simulations plotted as a function of Qe. One set has high asym-
 metry (mismatched ports: Zin = 24.5 ?, Zout = 84.5 ?) and one has low
 asymmetry (matched ports: Zin = Zout = 50?). Qe is varied by varying
 the coupling capacitance (1-10 fF), with a constant mutual inductance
 (5 pH). The dashed line indicates the actual Qi of the simulations. With
 increasing Qe the inaccuracy of the ?RM is diluted due to the decreasing
 weight of Qe in the analysis. (b) Plot of extracted asymmetry angles vs.
 Qe for the two simulations with low (?) and high () asymmetry.
 54
the extracted Qi approaches the real value, although the asymmetry ? is increasing.
 This is because as Qe increases the ?RM is less sensitive to the asymmetry. This
 behavior is captured in Eq. (3.47), which shows that as Qe increases, the difference
 between the two analysis methods vanishes. In fact for Qe >> Qi, the asymmetry
 becomes completely irrelevant and the two techniques converge.
 3.6 Conclusion
 In summary, in this chapter I first described the dilution refrigerator I used
 for cooling samples to 30 mK. I next discussed the input microwave lines and how I
 calibrated the voltage at the sample and the voltage across the resonance capacitor,
 based on the resonance line shape. Some resonance line shapes were found to be
 asymmetric so I derived an analytical resonance line shape based on circuit parame-
 ters and found that for non-ideal conditions the line shape is asymmetric. Based on
 this analysis I (with the help of my advisor Kevin Osborn) developed a technique
 (DCM) for extracting accurate values for internal quality factors Qi from asymmet-
 ric resonator measurements using only transmission data. By analyzing simulated
 resonator measurements, I found that the DCM was superior at extracting accurate
 internal quality factors in contrast to the conventional ?RM technique found in the
 literature. I found that in the limit where the asymmetry is low, the two methods
 agreed, but when the asymmetry is high, particularly when Qi >> Qe, the DCM
 accurately determines Qi while the ?RM systematically overestimates it. I also
 found that sufficiently high asymmetry and coupling, the ?RM gives an unphysical
 55
negative Qi. Finally, I showed that the two methods can produce different results
 on real data taken on a coplanar superconducting aluminum resonator with high
 asymmetry.
 56
Chapter 4
 Coplanar resonators
 4.1 Resonator Design
 4.1.1 Design Procedure
 In this section I discuss how I designed, simulated and fabricated thin-film
 coplanar aluminum resonators on sapphire and silicon substrates. All resonators
 were designed based on simulations done with Microwave Office. Microwave Office
 is a finite element numerical E&M solver that accepts material parameters (e.g.
 conductivity, dielectric constant, loss tangent, ...) and boundary conditions on the
 enclosing box. Figure 9.2(a) shows a typical view of a resonator design (details of
 the design to be discussed later in this section) in Microwave Office. The metal
 (green in Fig. 9.2(a)) is modeled as a perfect conductor and the dielectric constant
 of the substrate is set to 10.7 since it?s been found that this closely approximates
 the effective dielectric constant of the sapphire wafers we use.
 As seen in Fig. 9.2(a), two ports are used to measure throughput of the copla-
 nar waveguide (CPW) transmission line. Figure 9.2(b) shows the simulated through-
 put S21 of the resonant circuit of Fig. 9.2(a). From fits to S21, I can extract the
 design value for Qe, the coupling of the CPW to the resonator. Another method
 to determine Qe for a design is to set the loss in the resonator and just record the
 57
Simulated Transmission 
~350?m 
~450?m 
Pin 
Pout 
(a) 
(b) 

5 65

  
(d
 B)
  
Figure 4.1: (a) Microwave Office layout of coplanar resonator. Green
 is prefect conductor and white is dielectric. Line width is 5 ?m. (b)
 Microwave Office simulation of transmission from port 1 to port 2 (S21) as
 a function of frequency. On resonance the CPW couples to the resonator
 and lower transmission is observed.
 58
minimum throughput transmission (on resonance). By setting the loss you know
 the Qi which allows you to determine Qe since on resonance Eq. 3.14 reduces to
 S21 = 1 ? QQe and solving for Qe gives
 Qe = Qi
 (
 S21
 1 ? S21
 )
 . (4.1)
 Perhaps the most useful thing about using Microwave Office to design res-
 onators is that it allows for precise tuning of the resonance frequencies. Using a
 sapphire dielectric constant of 10.7, I found the Microwave Office estimation of the
 resonance frequency of coplanar resonators is off by less than 1% from the measured
 resonance frequency values.
 Once the resonator designs were determined, the next step was to transfer all
 of the resonator designs onto one layout in the design software, Cadence. Since I
 use a notch type measurement that only affects the transmission of the CPW near
 resonance, I could multiplex several resonators (in this case 4) on a single CPW.
 4.1.2 Designing and simulating resonators to analyze geometry de-
 pendence
 Coplanar resonators have been ubiquitous in superconducting quantum com-
 puting, but most previous research has focused on quasi-one-dimensional cavity
 resonators, such as 1/2-wavelength or 1/4 wavelength CPW transmission line res-
 onators [16, 49, 50]. Lumped element devices (where dimensions are much shorter
 than a wavelength) are less popular but have been used as part of qubits [51, 52],
 and in Josephson junction resonators [53, 33]. The interest in coupling quasi-lumped
 59
element resonators to qubits is that the lack harmonic modes in lumped element res-
 onators reduces loss from the Purcell effect [54]. When coupling to a qubit, it?s also
 been found that the symmetry of the qubit must be considered, since the type of
 coupling may affect coherence [55].
 With this in mind I designed four resonators with frequencies between 5 and
 7 GHz. The resonators had both quasi-lumped and quasi-one-dimensional cavity
 (transmission line) elements, in order to study the transmission between the two
 regimes. All the resonators were designed with a symmetric shape to induce induc-
 tive, rather than capacitive, coupling.
 Figure 4.2 shows optical images of the four resonators. Device QL is a quasi-
 lumped element resonator, composed of a meandering quasi-lumped inductor (QLL)
 and a quasi-lumped interdigital capacitor (IDC) (see Fig. 4.2(a)) with an inductance
 and a capacitance of approximately 2 nH and 0.3 pF respectively. Device CPS is
 a 4.5 mm long shorted ?/4 coplanar strip (CPS) resonator, shown in Fig. 4.2(b).
 Unlike in the QL resonator the electric field in the CPS resonator is distributed
 along the entire length of the resonator rather than being confined to an IDC. The
 other two resonators have both a lumped and distributed element (see Fig. 4.2(c,
 d)). The four resonators were embedded in the ground plane of the same 50 ? CPW
 and were inductively coupled to the waveguide (see Fig. 4.3).
 To understand the behavior of the resonators when they are driven, note that
 the point connecting the two nominally symmetric halves of each resonator is a
 current anti-node and a voltage node, and the sides of each resonator far from the
 coplanar waveguide are voltage anti-nodes. The fundamental resonance frequency
 60
(a) 
(b) 
(d) 
(c) 
~350?m 
~450?m 
l/4=4.5mm 
Figure 4.2: Optical images of four resonators (green is aluminum metal
 and black is sapphire substrate). (a) Quasi-lumped resonator (QL) at
 5.46 GHz. (b) Coplanar strip resonator (CPS) at 6.44 GHz. (c) Quasi-
 lumped inductor with a CPS resonator (QLL-CPS) at 5.76 GHz. (d)
 Quasi-lumped capacitor with a coplanar strip resonator (QLC-CPS) at
 6.01 GHz.
 61
Figure 4.3: (a) Schematic drawing of an arbitrary resonator inductively
 coupled to a coplanar waveguide. (b-e) Schematic drawing of each of the
 four types of resonators: (b) QL, (c) CPS, (d) QLL-CPS, and (e) QLC-
 CPS. (f) Circuit schematic of capacitive coupling to a resonator. Due
 to the symmetry of these resonators C1 = C2, C3 = C4, C5 = C6, and
 C7 = C8, capacitive coupling cannot excite an antisymmetric resonance
 and only inductive coupling remains.
 62
is voltage antisymmetric in all four resonators. The effective capacitive coupling
 between the CPW and the resonators for this mode is weak, due to the design and
 layout of the structures.
 The fact that the capacitive coupling is weak can be understood from the
 nearly symmetric shape of these structures, which implies the capacitances to either
 side of the resonators are approximately equal. Since the lowest frequency mode is
 antisymmetric, we expect only inductive coupling between the resonators and the
 CPW. As another example, Fig. 4.3(f) shows a circuit schematic, where a capacitive
 network couples voltage V to a resonator. With a symmetric design, C1 = C2,
 C3 = C4, C5 = C6, and C7 = C8, the voltage V cannot excite antisymmetric modes.
 In order to confirm that the QL resonator showed lumped element behavior, I
 used Microwave Office to simulate the current amplitude (see Fig. 4.4(a)) and electric
 field amplitude (see Fig. 4.4(b)) on resonance. As can clearly be seen in Fig. 4.4, the
 simulations show nearly uniform current amplitude in the meandering inductor and
 nearly uniform electric field amplitude in the IDC, while the meandering inductor
 shows nearly no electric field and the IDC shows nearly no current. This is indicative
 of quasi-lumped behavior in this device with the meandering inductor giving nearly
 no capacitance and the IDC giving nearly no inductance.
 Since the CPS resonator is effectively a one-dimensional cavity, we should
 expect to see higher order modes at each harmonic of the resonator. However, if the
 QL resonator is truly lumped, we wouldn?t expect to see any higher order modes.
 To test this, I used Microwave Office to simulate the spectrum of the CPS and the
 QL resonators from 2 to 40 GHz (see Fig. 4.5). As shown in Fig. 4.5(a), the CPS
 63
High 
Current 
Low 
Current 
(a) 
.=NCA '  
5I=HH '  
(b) 
Figure 4.4: (a) Current simulation for the QL resonator on resonance.
 As expected, most of the current is in the inductor. (b) Electric field
 simulation for the QL resonator on resonance. As expected, most of the
 E-field is in the IDC.
 64
device shows a higher order mode at each harmonic while the QL device doesn?t
 show any higher order modes until over 30 GHz, indicating that the device behaves
 like a lumped element resonator for a large bandwidth.
 4.2 Fabrication
 I fabricated all of the resonators with 100 nm thick aluminum films (both sput-
 tered and evaporated) on either c-plane sapphire wafers [56] or <1-0-0> undoped 20
 k?-cm silicon wafers [57]. The aluminum was patterned with positive photoresist
 and wet etched in an acidic solution.
 For the sputtered aluminum the procedure was to first transfer the 3? wafer
 into the CMS-18 Kurt J. Lesker sputtering chamber and pump to a pressure of
 2 ? 10?8 torr with a CTI Cryo Torr 8 cryo-pump. I then used a 3 cm argon ion
 beam source at 300 V beam voltage with 0.3 mT of argon gas to ion mill the surface
 of the wafer for 30 seconds. This both cleaned the surface and roughened it on an
 atomic scale to allow for better adhesion of the aluminum. Next I sputtered 100
 nm of aluminum with 400 W DC power from a 3? diameter target. The rate of
 deposition was 4.8 nm/min and was calibrated with a profilometer measurement. I
 then used a programmable spinner at 3000 RPM for 60 seconds to spin OIR 906-10
 photoresist [58] on the aluminum. Next I baked the wafer on a 90 ?C hot plate for
 60 seconds and then exposed it to the device pattern on a mask in the stepper for
 0.3 seconds (exposure time depends on UV power output which can vary). With a
 3? wafer and a sample size of 6.35 ? 6.35 mm, the stepper independently exposed 88
 65
Higher order modes 1st Mode 
Higher order modes 1st Mode 
(a) 
(b) 
Figure 4.5: (a) Microwave Office simulation for CPS resonator. Higher
 order harmonics are present every quarter wavelength, as expected for
 a transmission line resonator. (b) Microwave Office simulation for QL
 resonator. No higher order modes are present close to the first mode,
 indicating a quasi-lumped element resonator. The second mode occurs
 at about 33 GHz.
 66
samples. After exposing the resist, I baked the wafer at 120 ?C for 60 seconds. Next
 I developed the exposed photoresist with OPD-4262 developer [58] for 40 seconds.
 After rinsing with di water, I etched the now exposed aluminum in an aluminum etch
 solution (Aluminum Etch 80-15-3-2) composed mainly of phosphoric acid. Finally
 the photoresist was cleaned off with acetone, methanol, and isopropanol.
 For the evaporated aluminum I used an e-beam evaporator (CHA Mark 40)
 and didn?t ion mill the wafer before deposition because there was no ion source
 in the system. After the deposition the procedure was identical to the sputtered
 aluminum samples.
 4.2.1 Dicing
 After the aluminum was patterned and the photoresist was removed, I used
 the spinner to coat the wafer with a layer of FSC-M resist to protect the resonators
 while dicing the wafer. I then applied tape to the back of the wafer to insure that
 the samples stay in place after the wafer was diced. After performing a calibration
 cut on a test wafer (hairline alignment), I mounted the wafer in the dicing saw.
 After aligning the dicing saw with the pattern on the wafer, I began dicing the
 wafer first in one direction and then in the other producing 88 6.35 mm ? 6.35 mm
 samples. I used a Disco-DAD 321 dicing saw with a coarse grit resinoid diamond
 blade (type CX-010-325-080-4) with a rotation speed of 22,000 RPM. The cutting
 speed for sapphire wafers was 0.75 mm/s and for silicon wafers was 2.0 mm/s. I set
 the blade height to 0.2 mm to insure that the blade cut through the wafer but not
 67
Figure 4.6: Photograph of a sample mounted on a PC board attached
 to a Cu sample box.
 all the way through the tape.
 Once the wafer was diced I went back into the clean room to find a good sample
 (without scratches). I used the microscope to first find a few good candidates. I
 then cleaned them by first spraying with acetone, then letting them sit in acetone for
 10 minutes, and finally rinsing them with methanol and isopropanol. After cleaning
 I examined the samples in the microscope to confirm that they were indeed clean
 and selected the best one to mount in the sample box.
 4.2.2 Packaging and wire-bonding
 The sample chip was mounted to a PC board with GE varnish and the PC
 board was attached to a copper box. Figure 4.6 shows a photograph of the sample
 box with a sample mounted in it and Fig. 3.1(a) shows the sample box mounted
 to the dilution refrigerator. The PC board had wire bond pads that lined up with
 the CPW launchers of the samples. I connected these pads to the samples? CPW
 launchers using a West-Bond 7476E-79 wire bonder and aluminum wire.
 68
Transmission through the sample box was tested at room temperatures to
 ensure that it has no resonance box modes for our group?s frequencies of interest
 (4-8 GHz). The sample box seal is a copper metal-metal seal screwed tight with
 2-56 brass screws.
 4.3 Solving for resonance voltage waveform
 I this section, I discuss solving for the resonance frequency of the four resonator
 types and in so doing solving for the voltage waveform of the devices on resonance.
 This allows me extract the voltage across each device despite not being able to use
 the lumped element approximation Eq. 3.22.
 4.3.1 QL resonator
 In this part, I solve for the resonance of the QL resonator. This is actually
 just a specific case of the circuit solved in Chapter 3. Since this device only has
 inductive coupling, I can model the measurement setup with Fig. 4.7(a), where
 I?ve assumed balanced impedances for simplicity and put the loss in series with the
 LC circuit rather than in parallel because it simplifies the calculation when only
 inductive coupling is considered.
 The input impedance of the resonant circuit is given by
 Zin = R + i?L?
 i
 ?C . (4.2)
 Resonance occurs when the impedance |Zin| is minimized, when the inductive and
 69
Figure 4.7: (a) Circuit schematic for a QL resonator measurement. A
 series RLC resonance circuit is assumed because it simplifies the system
 when only inductive coupling is present. (b) Thevenin equivalent circuit
 for measurement schematic shown in (a).
 70
capacitive impedances cancel we have
 ?0 =
 1?
 LC
 . (4.3)
 Assuming the system is driven close to resonance (|? ? ?0|/?0 << 1), the input
 impedance can be rewritten as
 Zin ' R + 2iL?, (4.4)
 where ? = ? ? ?0 is the detuning.
 Similar to the derivation shown in Chapter 3, by solving Kirchhoff?s equations
 for the circuit in Fig. 4.7(a), it can be shown that it?s equivalent to the Thevenin
 equivalent circuit in Fig. 4.7(b), where
 RT =
 (?0M)2
 2Z0
 (4.5)
 and
 VT =
 2i?MVin
 2Z0
 , (4.6)
 depend on the coupling and we?ve made the approximation that 2Z0 >> ?L1. Since
 we know that the impedances of the inductor and the capacitor in Fig. 4.7(b) will
 cancel on resonance, the current through the circuit on resonance will be simply
 Ires =
 VT
 Reff
 , (4.7)
 where we?ve defined
 Reff = R +RT . (4.8)
 Solving the circuit in Fig. 4.7(a) for Vout, as done in Chapter 3, gives
 Vout
 Vin
 =
 R
 Reff
 1 + 2i?L/R
 1 + 2i?L/Reff . (4.9)
 71
The quality factors can now be defined in the usual way
 Q ? 1
 ?0ReffC
 , (4.10)
 Qi ? 1
 ?0RC
 , (4.11)
 and
 Qe ? 1
 ?0RTC
 , (4.12)
 This allows us to rewrite Eq. 4.9 as
 S21 =
 Vout
 Vin
 = 1 ?
 Q/Qe
 1 + 2Qi?/?0 . (4.13)
 From Eqs. 4.6 and 4.7 we can calculate the voltage across the capacitor,
 VC =
 Ires
 i?C =
 MVin
 Z0CReff
 . (4.14)
 Writing this in terms of the quality factors recovers Eq. 3.22,
 VC =
 ?
 2
 Z0?0CQeQVin. (4.15)
 This is of course to be expected because the QL resonator is simply a lumped element
 resonator with only inductive coupling (a specific case of the problem we solved in
 Chapter 3).
 4.3.2 CPS resonator
 To illustrate the parallels between the QL resonator and the CPS resonator
 I have redrawn the measurement schematic for the CPS resonator in Fig. 4.8(a).
 Here ZCPS represents the input impedance of the CPS as the series RLC circuit did
 72
Figure 4.8: (a) Measurement schematic for CPS resonator. (b) Thevenin
 equivalent circuit for the CPS resonator.
 73
in Fig. 4.7(a). To solve for the voltage across the CPS resonator, we first examine
 the geometry of the resonance mode. Figure 4.9 shows a representation of the CPS
 resonator enclosed in a ground plane. Because of the symmetry of the layout, we
 can assume that the capacitances of either finger to the ground plane are equal i.e.
 C11 = C22.
 This problem would be easier to solve if we could turn the resonator into a
 simple transmission line with a single voltage electrode and ground. This is achieved
 by recognizing that the resonance will excite the odd (antisymmetric) mode of the
 CPS, which allows us to assume that the middle of the CPS is grounded. To turn
 this circuit into a simple transmission line we just need to find the total capacitance
 of one line to ground. That?s simply equal to
 C = C11 + 2C12, (4.16)
 where the factor of 2 in front of the C12 comes from the fact that the capacitance
 between one of the lines and the effective ground plane in the middle is twice the
 capacitance C12 between the two lines. Equation 4.16 can be rewritten as
 C ? = C ?11 + 2C ?12, (4.17)
 where the primes indicate per unit length, thus C ? = C/l. And the effective char-
 acteristic impedance of this transmission line is
 ZC =
 1
 vC ? , (4.18)
 where v is the speed of light on the substrate
 v =
 c?
 re
 , (4.19)
 74
%55 
%66 
%56 V 
L r V 
L H 
V 
L r V 
L H 
8 

F8 
8 %55
Et%56 
(a) 
(b) 
Figure 4.9: (a) Drawing of the CPS resonator inside a ground plane that
 emphasizes the symmetry of the layout. Note that due to the symmetry,
 an odd electric mode can be assumed to be grounded down the middle
 of the resonator. (b) A transformed transmission line resonator created
 by taking the capacitances between one of the fingers of the CPS to
 ground, where a ground plane (actually a ground line) is assumed down
 the middle of the resonator in (a).
 75
and re is the effective r for a coplanar geometry
 re =
 r + 1
 2
 , (4.20)
 assuming the substrate has a relative permittivity r and the vacuum above has a
 relative permittivity 1. With the dimensions of the CPS (CPS separation s = 15 ?m,
 CPS-ground separation s2 = 10 ?m, conductor width w = 5 ?m, substrate height
 h = 430 ?m) I used Ref. [59] to calculate the characteristic odd mode impedance of
 the device ZC = 62 ?.
 The next step is to calculate the input impedance of the CPS transmission
 line and solve for the resonance. The input impedance Zin of a transmission line
 open at the end with attenuation constant ? and wavenumber ? = 2pi/? is [47]:
 Zin = ZC coth[(? + i?)l] = ZC cot ?l + i?l
 ?l cot ?l + i
 = ZC
 (?l cot2 ?l + ?l) + ((?l)2 cot ?l ? cot ?l)i
 1 + (?l)2 cot ?l . (4.21)
 To find the resonance we set the imaginary part of the input impedance to zero
 giving cot ?l = 0. This essentially means that on resonance the length l of the
 transmission line is ?/4 (i.e. l = ?/4 = (2pi/?0)/4 ? ?0l = pi/2). Assuming we?re
 close to resonance (cot ?l << 1) and have low attenuation (?l << 1) Eq. 4.21
 becomes
 Zin ? ZC(?l ? i cot ?l). (4.22)
 It is more convenient in the experiment to work with frequency than with
 wavenumber. Using the substitutions ? = v? and ? = ? ? ?0 we have
 ?l = ?l
 v
 =
 ?0l
 v
 +
 ?l
 v
 =
 pi
 2
 +
 ?
 v
 pi
 2?0 =
 pi
 2
 +
 pi
 2
 ?
 ?0
 , (4.23)
 76
thus
 cot ?l = cot( pi
 2
 (1 + ?
 ?0
 )) ? ? pi
 2
 ?
 ?0
 , (4.24)
 where I have expanded the cot term to first order in ?/?0. Now combining Eq. 4.22
 and Eq. 4.24 gives
 Zin = ZC(?l + pi2
 ?
 ?0
 i). (4.25)
 Now we can see that the input impedance for the CPS, Eq. 4.25, is of the same form
 as the series RLC input impedance, Eq. 4.4, with the substitutions
 ?R = ZC?l, (4.26)
 ?L = pi
 4
 ZC
 ?0
 , (4.27)
 and
 ?C = 4
 piZC?0
 . (4.28)
 We can now use the input impedance transformation to create the Thevenin
 equivalent schematic for the CPS geometry (see Fig. 4.8(b)), and as was the case for
 the QL resonator RT and VT are given by Eq. 4.5 and Eq. 4.6 respectively since they
 only depend on the coupling. Again we set the imaginary part of the impedance to
 zero to calculate the current on resonance and find
 I0 =
 ?0MVin
 Z0 ?Reff
 , (4.29)
 where ?Reff ? ?R + RT . I?ve defined the maximum current as I0 here because it?s
 the current maximum amplitude on the transmission line (at the current anti-node).
 The maximum voltage amplitude is then:
 V0 = ZCI0 = Zc
 ?MVin
 Z0 ?Reff
 =
 ?
 8ZC
 piZ0QeQVin, (4.30)
 77
where Q and Qe have been defined the same way as they were in the QL case (with
 ?C and ?Reff substituted in for C and Reff ). Note however that V0 in Eq. 4.30 is the
 voltage amplitude on the transformed transmission lines, which is just one of our
 CPS fingers to ground. To get the actual voltage from one finger of the CPS to the
 other we have to double V0 i.e. VCPS = 2V0.
 4.3.3 Quasi-lumped C-CPS resonator
 The case of the CPS terminated by an IDC capacitor is slightly more com-
 plicated than the previous two cases but the approach used for the CPS can still
 be used here. Again we can think of the resonator as having an input impedance
 that?s coupled to the CPW. Figure 4.10(a) shows a representation of the resonator
 enclosed in a ground plane. Again recognizing that we?re exciting the odd mode
 we can assume that the middle of the resonator is grounded and recalculate the
 capacitances from one of the fingers to ground (see Fig. 4.10(b)). Since the CPS has
 the same geometry we know from the last section that ZC = 62 ? and from Ref.
 [59] we use the dimensions of the IDC (5 ?m separation and width, 22 fingers, and
 320 ?m finger length) to its capacitance CIDC = 0.30pF .
 Next we find the input impedance of a 62 ? transmission line of length l
 (l = 1.6 mm for this device) terminated by a 2CIDC load. It can be shown that the
 input impedance is given by
 ZCPS,IDC = ZC
 ?l((ZC + tan ?l2?CIDC )2 + (ZC tan ?l ? 12?CIDC )2)
 (ZC + tan ?l2?CIDC )2
 +ZC
 (ZC + tan ?l2?CIDC )(ZC tan ?l ? 12?CIDC )i
 (ZC + tan ?l2?CIDC )2
 . (4.31)
 78
%55 
%66 
%56 V 
L r V 
L H 
V 
L r V 
L H 
8 

F8 
8 %55
Et%56 
(a) 
(b) 
%??? 
t%??? 
Figure 4.10: (a) Drawing of QLC-CPS resonator inside a ground plane.
 Note that due to the symmetry, an odd electric mode can be assumed to
 be grounded down the middle of the resonator. (b) Transformed resonant
 circuit created by taking the capacitances between one of the fingers of
 the QLC-CPS resonator to ground, where a ground plane (actually a
 ground line) is assumed down the middle of the resonator in (a).
 79
To solve for the resonance condition, we make the imaginary part vanish
 ZC tan ?0l ? 12?0CIDC = 0. (4.32)
 Using the substitution ? = ?/v and expanding around ?0, it can be shown that the
 input impedance of the resonator close to resonance is
 ZCPS,IDC ' ZC
 ?l((ZC + tan ?l2?0CIDC )2 + (ZC tan ?l ? 12?0CIDC )2)
 (ZC + tan ?l2?0CIDC )2
 +ZC
 (1 + tan2(?0l
 v
 )ZC l
 v
 ) + 12CIDC?2
 ZC +
 tan(?0l
 v
 )
 2?0C
 ?i. (4.33)
 Examination of Eq. 4.33 reveals that the input impedance for this resonator can
 again be put in the same form as a series RLC resonator (Eq. 4.4) with the substi-
 tutions
 ?L = ZC
 (1 + tan2(?0l
 v
 )ZC l
 v
 ) + 12CIDC?2
 2(ZC + tan(
 ?0l
 v
 )
 2?0C )
 , (4.34)
 and
 ?C = 1
 ?20
 ?L
 , (4.35)
 where ?0 can be solved numerically from Eq. 4.32. These transformations allow
 us to model this resonator as another Thevenin equivalent circuit. Solving for the
 current amplitude on resonance we find
 I0 =
 ?0MVin
 Z0 ?Reff
 i, (4.36)
 where once again Q = 1/(?0 ?C ?Reff ).
 Note that this case is slightly more complicated than a simple transmission
 line so to solve for the voltage across the capacitor on resonance we first write the
 voltage waveform as generally as possible
 V (z) = V+ei?z + V?e?i?z, (4.37)
 80
where z is distance along the CPS, the boundary value V (0) = 0 is known and we?re
 attempting to solve for V (l) = V ?C , where V ?C is the voltage across the capacitor in
 this transformed transmission line resonator system. From our boundary condition
 we know that V
 ?
 = ?V+, so V (l) becomes
 V (l) = V+(ei?l ? e?i?l) = 2iV+ sin ?l = V0 sin ?l = V ?C , (4.38)
 where we have defined the voltage amplitude as V0 = 2iV+. Recognizing that we
 can solve for the voltage amplitude from the current amplitude (V0 = ZCI0) we now
 find an expression for the voltage waveform of the resonator
 V (z) = V0 sin ?z = ZC?0MVinZ0Reff i sin
 (?0
 v
 z
 )
 =
 (
 2 ?C?0
 Z0Qe
 )
 ZCQVin sin
 (?0
 v
 z
 )
 . (4.39)
 To solve for the voltage across the capacitor on resonance, we again recognize that
 this was a transformed system on one of the fingers to ground thus solving for the
 voltage across the actual capacitor requires a factor of two giving,
 VC = 2V ?C = 2V (l) = 2
 (
 2 ?C?0
 Z0Qe
 )
 ZCQVin sin
 (?0
 v
 l
 )
 . (4.40)
 4.3.4 Quasi-lumped L- CPS resonator
 Finally we examine the case of the QLL-CPS resonator. Figure 4.11(a) shows
 a representation of the QLL-CPS resonator enclosed in a ground plane. For this
 device l = 3.355 mm and L = 2.9 nH, obtained from the capacitance and the
 resonance frequency of the QL device. Again, because we?re exciting an odd mode
 we can assume the middle of the resonator is grounded and solve a transformed
 device of one half of the resonator relative to ground (see Fig. 4.11(b)).
 81
V 
L r 
%55 
%66 
%56 V 
L r V 
L H 
V 
L H 
8 

F8 
8 %55
Et%56 
(a) 
(b) 
. 
.t 
Figure 4.11: (a) Schematic picture of QLL-CPS resonator inside a ground
 plane. Note that due to the symmetry, an odd electric mode can be
 assumed to be grounded down the middle of the resonator. (b) Trans-
 formed resonant circuit created by cutting the QLL-CPS resonator down
 the middle and taking the capacitances between one of the fingers to
 ground, where a ground plane (actually a ground line) is assumed down
 the middle of the resonator in (a).
 82
Solving for the input impedance of this resonator, we find
 ZL,CPS =
 i?L
 2
 + ZC coth((? + i?)l) = ZC?l(cot2 ?l + 1) +
 (
 ?L
 2
 ? ZC cot ?l
 )
 i.
 (4.41)
 For the resonance condition, the imaginary part vanishes
 ?0L
 2
 ? ZC cot ?0l = 0. (4.42)
 Solving for the resonance frequency and expanding around it, we find the input
 impedance on resonance
 ZL,CPS =
 (
 L
 2
 + ZC
 (
 1 + cot2
 (?0
 v
 l
 ))l
 v
 )
 ?, (4.43)
 and again note that it is of the same form as the series RLC input impedance
 (Eq. 4.4) with the substitutions
 ?L = 1
 2
 (
 L
 2
 +
 (
 1 + cot2
 (
 ?0l
 v
 ))
 lZC
 v
 )
 , (4.44)
 and
 ?C = 1
 ?20
 ?L
 . (4.45)
 As in the previous geometries, this transformation allows us to model the
 resonator as a Thevenin equivalent circuit and solve for the current amplitude on
 resonance
 IL =
 ?0MVin
 Z0 ?Reff
 i, (4.46)
 where I have defined the current amplitude through the inductor as IL. As with the
 previous case, we can write the general form of the voltage waveform on the CPS
 V (z) = V+e?i?z + V?ei?z. (4.47)
 83
Here the boundary condition is
 V (0) = VL = i?0LIL2 = ?
 ?20LMVin
 2Z0 ?Reff
 . (4.48)
 From the transmission line characteristic impedance, we can also write the current
 waveform
 I(z) = V (z)
 ZC
 =
 1
 ZC
 (V+e?i?z + (V+ ? VL)ei?z) (4.49)
 Since the transmission line is open at the end z = l, we also know that the current
 at the end is zero, i.e.
 I(l) = 1
 ZC
 (V+e?i?l + (V+ ? VL)ei?l) = 0. (4.50)
 Solving for V+ gives
 V+ =
 VL
 e?2i?l + 1
 . (4.51)
 And from Eq. 4.47 we then obtain the voltage at the ends of the transmission line
 V (l) = VLe
 ?i?l
 e?2i?l + 1
 +
 VLei?l
 e2i?l + 1
 =
 VL
 cos ?l =
 (
 ?C?30
 2Z0Qe
 )
 LQVin
 cos
 (
 ?0
 v
 l
 ) . (4.52)
 And as in the previous cases, this is the voltage of one of the fingers to ground and
 it should be doubled to calculate the voltage between the two fingers.
 84
Chapter 5
 Coplanar resonator measurements
 5.1 Coplanar resonator loss
 As discussed in Chapter 2, the loss tangent of a material that is limited by a
 standard distribution of TLS defects can be described by
 tan ? = tan ?0
 tanh (~?/2kBT )?
 1 + (E/Ec)2 , (5.1)
 where tan ?0 is the low-power intrinsic loss tangent, dependent on the frequency
 distribution and density of the TLSs, EC is the critical field dependent on the T1
 and T2 of the TLSs, and E is the applied electric field at the location of the TLSs.
 In general, the electric field and the loss tangent in a resonator my not be uniform.
 I that case, the inverse of the internal quality factor of the resonator can be written
 as a weighted distribution of the loss tangent:
 1
 Qi =
 ?
 LossyMaterial tan ?(~r)( ~E(~r))2d3r?
 All (~r)( ~E(~r))2d3r
 . (5.2)
 Therefore, in the case of a coplanar resonator, one must consider the electric field
 distribution for the entire resonator when interpreting the internal quality factor.
 5.1.1 Sputtered aluminum resonators on sapphire
 I measured coplanar resonators fabricated from several aluminum films, most
 of which were sputtered films on a sapphire substrate. Here I present measurements
 85
10?5 10?4 10?3 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q i
  
 
QL
 CPS
 QLC?CPS
 QLL?CPS
 Figure 5.1: The inverse internal quality factor 1/Qi vs. RMS microwave
 voltage for the four resonators with different geometries from sample sp1.
 of resonators from three such chips: two are different chips from the same sputtered
 film on the same wafer (sp1 and sp1-B) and one is a chip from a different sputtered
 film with nominally the same film and device parameters (sp2). Figure 5.1 shows
 a comparison of the 1/Qi for the four coplanar devices, measured on one chip with
 a single CPW, as a function of the rms excitation voltage. Figure 5.1 indicates
 that the CPS resonator has a lower loss (higher Qi) than the other three devices.
 However, Fig. 5.2 shows that measurements to those on nominally identical devices
 (sputtered using the same technique and components on the same type of wafer
 86
10?5 10?4 10?3 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q i
  
 
Figure 5.2: plot of the internal loss 1/Qi vs. microwave voltage VRMS,
 comparing previous measurements from sp1 (?) to measurements of
 nominally identical resonators from sample sp2 (+), using the same de-
 vice color code as before (QL: red, CPS: blue, QLC-CPS: green, QLL-
 CPS: magenta).
 with nominally the same processing) gives quite different results. In Fig. 5.2 the ?
 symbols represent measurements from chip sample sp1 as shown in Fig. 5.1, and the
 + symbols represent the measurements from sp2. Despite being nominally identical,
 the second set of devices from sp2 have a different distribution of losses with three
 devices clumped together and the QLL-CPS device showing a distinctly higher loss.
 The discrepancy between the two sets of measurements in Fig. 5.2 implies that
 there is at least one uncontrolled factor that is affecting the loss. To check if this
 87
10?5 10?4 10?3 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q i
  
 
CPS:Sample A
 QLL?CPS: Sample A
 CPS:Sample B
 QLL?CPS: Sample B
 Figure 5.3: Comparing plots of the internal loss 1/Qi vs. microwave
 voltage VRMS for the CPS (blue) and the QLL-CPS (magenta) resonators
 from samples sp1 and sp1-B from two different chips from the same film
 discrepancy occurs between different samples from the exact same film, I cooled
 down another chip sp1-B from the same sputtered film as sp1. Figure 5.3 shows
 two sets of measurements for the CPS and QLL-CPS resonators. As can clearly
 be seen in Fig. 5.3, the differences between the different resonator geometries are
 reproducible from sample to sample within the same film. This behavior suggests
 that significant variations in the resonator qualities are coming from uncontrolled
 variations in the fabrication process.
 88
5.1.2 Coplanar resonator loss analysis
 Taken as a whole, these results suggest that there is little device to device vari-
 ation due to geometry and no significant loss-based advantages or disadvantages to
 particular geometry types. However, since the quasi-lumped element resonator does
 not have harmonic cavity modes, it may still provide an advantage when measuring
 the lifetime of a coupled qubit.
 Another significant characteristic of the data is that the power dependence of
 the loss is very weak, nowhere near the ? 1/V dependence predicted by TLS theory
 in Eq. 9.1. Also, the measured power dependence of 1/Qi seems to vary from device
 to device in the range from ? 1/V 0.1 to ? 1/V 0.2, as reported in [39].
 Note Eq. 9.1 gives the expected loss tangent and the measurement is only of
 the device quality factors. It?s been found that the loss in these coplanar devices
 scales with the surfaces [60]. A possible explanation for the weak power dependence
 is that the loss is occurring at the surface. To model the effects of surface loss, I
 assumed that the loss is coming from either the native oxide on the aluminum (blue
 and green in Fig. 5.4), the substrate surface (red in Fig. 5.4), or the substrate metal
 interface (black in Fig. 5.4). I then used the finite element E&M simulator COMSOL
 to calculate the electric field at those surfaces as a function of the voltage across
 the device. Assuming a surface thickness of 5 nm, and an intrinsic loss tangent of
 10?3, I used Eq. 7.3 for each surface to find the effective internal loss as a function
 of the voltage. As Fig. 5.4(b) shows, no matter what lossy surface was assumed,
 1/Qi never decreases slower than 1/V for more than a decade in voltage. One way
 89
sapphire 
gnd. gnd. V -V 
15 m  
100 nm 
2.0
 ~
 ?V
 1
 ~
 ?V
  s r?9 
 s r?: 
 s r?; 
 s r?< 
 s3 ? ?  
8??? :=
? Q
?;  s r?5  s r4  s r5  s r6  s r7 
(a) 
(b) 
Figure 5.4: (a) Cross sectional diagram of CPS resonator. (b) Calculated
 1/Qi for the CPS geometry from simulated fields as a function of the
 voltage across the resonator assuming the loss is dominated by different
 possible surfaces (5 nm thicknesses), color coded from (a). Magenta
 represents the addition of loss from the substrate vacuum interface (red)
 and the top of electrode surface (blue).
 90
to lower the power dependence was to pathologically consider just the loss from the
 substrate vacuum interface (red) and the top of electrode surface (blue). But as the
 magenta curve in Fig. 5.4 shows even that only lowered the power dependence to
 1/V 0.2 for less than a decade in voltage. From these simulations it would appear
 that surface loss is not sufficient to explain the shallow power dependence observed
 in the coplanar resonators.
 5.1.3 Evaporated aluminum resonators on sapphire
 As mentioned earlier, I also tested the four resonator geometries using devices
 made from evaporated aluminum films. Figure 5.5 shows a comparison between
 the evaporated film ev1 measurements (?) and the sputtered film measurements
 sp1 (+) (devices color coded as in the previous figures). We see from Fig. 5.5 that
 the evaporated film tends to have a slightly higher loss than the sputtered film. A
 possible explanation for this is that the evaporated film is rougher and therefore has
 more lossy surface area. AFM measurements were performed on the two films (see
 Fig. 5.6) which confirmed that the evaporated film is in fact rougher with 20-40 nm
 features compared to the sputtered film?s 5-10 nm features. Assuming approximately
 a factor of 4 increased amplitude and density (in both surface dimensions) of ?bumps?
 in the evaporated film would mean more than a factor of 3 increase in the total
 surface area. This of course is not the whole story because surface area topography
 also affects the electric field, but it?s certainly a plausible explanation.
 91
10?6 10?4 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q i
  
 
Figure 5.5: plot of the internal loss 1/Qi vs. microwave voltage VRMS,
 comparing previous measurements from sputtered sample sp1 (+) to
 evaporated sample ev1 (?), using the same device color code as before
 (QL: red, CPS: blue, QLC-CPS: green, QLL-CPS: magenta).
 92
(a) 
(b) 
40 nm 
500 nm 
10 nm 
500 nm 
Figure 5.6: (a) AFM measurement of evaporated Al film showing 20-
 40 nm sized features. (a) AFM measurement of the sputtered Al film
 showing 5-10 nm sized features.
 93
10?6 10?4 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q i
  
 
Figure 5.7: plot of the internal loss 1/Qi vs. microwave voltage VRMS,
 comparing previous measurements from sputtered sample on sapphire
 sp1 (+) to a sputtered sample on a silicon substrate that was cleaned
 with HF sp3-Si (?). The same device color code is used as before (QL:
 red, CPS: blue, QLC-CPS: green, QLL-CPS: magenta).
 5.1.4 Sputtered aluminum resonators on silicon
 I also measured sputtered aluminum films on undoped high resistivity silicon
 wafers sp3-Si. In Fig. 5.7, I compare results from these devices ? to results from sp1 +
 (devices color coded as in the previous figures). The measurements on silicon wafers
 showed lower loss which is likely due to a cleaner silicon surface as compared to the
 sapphire surface; I used an HF treatment of the silicon wafers prior to deposition
 (see Chapter 4).
 94
10?6 10?5 10?4 10?3 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q e
  
 
Figure 5.8: Inverse external quality factors 1/Qi (coupling to CPW) vs.
 RMS microwave voltage for devices from the two sputtered films sp1
 and sp2, using the same device color code as before (QL: red, CPS: blue,
 QLC-CPS: green, QLL-CPS: magenta).
 5.2 External quality factor
 Another parameter that I monitored during resonator measurements is the
 external quality factor Qe. As discussed in Chapter 3, the Qe is a measure of the
 coupling of the CPW to the resonator and can be simulated with Microwave Office.
 Figure 5.8 shows the measured values of 1/Qe for samples sp1 and sp2 along with the
 four simulated values for the parameter. As Shown in Fig. 5.8, the simulations don?t
 predict the accurate values for Qe to better than within a factor of two. Also the
 95
10?6 10?5 10?4 10?3 10?2
 10?6
 10?5
 VRMS (V)
 1/
 Q e
  
 
Figure 5.9: Inverse external quality factors 1/Qi (coupling to CPW) vs.
 RMS microwave voltage for devices from the sputtered sample sp1 (+)
 and the evaporated sample ev1 (?), using the same device color code as
 before (QL: red, CPS: blue, QLC-CPS: green, QLL-CPS: magenta).
 difference between the measurements and simulations is not systematic. For exam-
 ple, for the two QL resonators, the measurement of sp1 shows higher coupling than
 the simulated value and sp2 shows lower. When comparing the Qe?s for the evap-
 orated and sputtered films (see Fig. 5.9) similar deviations are observed, although
 the predictions appear closer for the evaporated film. It?s possible, that while these
 devices are in nominally identical packaging, small uncontrollable deviations in the
 packaging can play a large role on the coupling.
 96
5.3 Discussion of likely loss mechanisms
 As discussed earlier, a likely loss contributor is the surface native oxide and
 other surface contaminants. As shown in Fig. 5.4 and discussed in more detail
 in Ref. [61], the electric field distribution is such that the dominant loss arises
 from the metal-substrate interface and the substrate-vacuum interface. It?s also
 been found through electric field simulations that the corner at the metal-substrate-
 vacuum interface is a significant contributor to loss [61]. So a small undercut of the
 substrate at the corner of the metal would likely go a long way to reducing the loss
 [62]. Even more evidence for substrate surface limited loss is an experiment where
 the measured loss in a coplanar resonator was reduced with an 850 oC O2 anneal of
 the sapphire substrate before deposition [63].
 Barends et al. [64] have found that if the devices aren?t properly shielded,
 stray infrared light can generate quasiparticles that break up Cooper pairs in the
 superconductor and cause a power independent loss. It?s likely that this at least
 partially limits these measurements because in other measurements of these de-
 vices, performed by S. Gladchenko, a steeper power dependence was found when
 sample box shielding was improved and SiC stycast mixture (see Ref. [64]) was
 used. However, the microwave line connectors were also changed in that sample
 box design, so it?s possible that other factors caused the change in saturation power
 dependence.
 Thus far I have only discussed electric loss in the capacitor. But it is also
 possible that these resonators are limited by inductive loss [65]. It?s been shown in
 97
10?6 10?5 10?4 10?3 10?2
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 VRMS (V)
 1/
 Q i
  x 1
 05
  
 
Figure 5.10: Replotted measurements from Fig. 5.5 again with sputtered
 sample sp1 (+) and evaporated sample ev1 (?) and identical device color
 code (QL: red, CPS: blue, QLC-CPS: green, QLL-CPS: magenta).
 SQUIDs that inductance noise is consistent with surface magnetic spin interactions
 [66], likely through RKKY interactions [67]. It?s plausible that such surface spins
 are causing inductive loss in these coplanar devices which is convoluted with the
 capacitive loss resulting in the measured Qi. Although such magnetic losses would
 likely still be small in this regime because a comprehensive study of scaling of a
 quai-lumped resonator based on the QL resonator discussed here found scaling of
 measured Qi with the spacing of the IDC fingers and not with the geometry of the
 meandering inductor [68].
 98
Recently Faoro et al. [69] motivated by our results along with others in the
 community proposed a model where interactions between TLSs confined to res-
 onator surfaces can lead to a shallower loss saturation with electric field. In this
 model they assumed coherent TLSs with relaxation times longer than their Rabi fre-
 quency (saturation time), which they called TLSs, and relatively incoherent TLSs
 with relaxation times shorter than their Rabi frequency, which they called fluctua-
 tors. In their model when a long lived TLS interacts with a short lived fluctuator,
 its frequency drifts moving it in and out of the resonator?s bandwidth. So if a TLS
 is on resonance and is excited by the resonator, it may shift its frequency through
 interaction with a fluctuator and dissipate the photon outside the resonator?s band-
 width. Thus a photon that could have been returned back to the resonator through
 stimulated emission has been lost to the system and through this mechanism ex-
 cess loss can be observed. Faoro et al. found that such interactions can lead to a
 logarithmic TLS loss saturations. In Fig. 5.10 I?ve replotted the measurements of
 films sp1 and ev1 from Fig. 5.5 on a semi-log plot. While certainly not conclusive,
 a logarithmic saturation seems plausible. Note that somewhat coincidently a very
 similar TLS loss mechanism will be addressed in Chapters 8 and 9.
 5.4 Conclusion
 In conclusion I measured four types of coplanar devices and they showed sim-
 ilar loss with no clear geometry dependence. This suggests that the four resonator
 designs are equally appropriate for quantum computing circuits, and there does not
 99
seem to be a large advantage for one type of resonator design over another if only
 considering the fundamental mode. However, since the quasi-lumped element res-
 onator does not have harmonic cavity modes, it may still provide an advantage when
 measuring the lifetime of a coupled qubit.
 I also found that the power dependence of the loss in these resonators is not
 consistent with a conventional surface distribution of identical TLSs. These mea-
 surements, along with other measurements in the quantum computing community
 indicate that the low power loss tangent is limited by TLS material losses, but that
 the shallow power dependence may be associated with quasiparticles caused by in-
 frared light absorption. I also observed that evaporated films tended to have higher
 loss, likely because they were rougher with more native oxide, and that films on HF
 cleaned high-resistivity silicon substrates showed lower loss, most likely because the
 silicon surface was cleaner. Finally, I observed variations in the Qe from simulation
 predicted values in an unsystematic seemingly random manner.
 100
Chapter 6
 Atomic layer deposition
 6.1 Atomic layer deposition and its use in trilayer junctions and ca-
 pacitors
 Atomic layer deposition (ALD) is a film deposition technique which alternates
 the use of two or more gaseous chemical precursors to create a self-limiting layer-
 by-layer deposition process. Figure 6.1 shows a summary of a generic ALD process.
 In addition to the self-limiting nature of the process, it also provides atomic level
 thickness control (see Fig. 6.2) with superb conformality (see Fig. 6.3).
 The level of control that ALD provides makes it a potentially attractive choice
 for Josephson junction preparation. If metal deposition can also be integrated into
 the ALD chamber, it can allow for in situ trilayer fabrication and the potential
 to produce almost defect-free junctions and capacitors. As with conventional tri-
 layer deposition techniques, the advantage of in situ ALD tilayer fabrication is that
 the deposition of the metal/insulator/metal layers can be done without breaking
 vacuum, reducing film contamination.
 Despite its potential ALD is not without its drawbacks. Incomplete reactions
 can implant impurities from precursors into the film. Suitable precursors are often
 limited and for certain materials, research groups have to make their own precursors.
 101
Figure 6.1: ALD cycles exposure and purge of two precursors to grow
 thin films with monolayer control. In this generic example ML4 and
 H2O are cycled to grow MO2 [70].
 102
Figure 6.2: A plot of thickness of an ALD grown ZrO2 as a function of
 the number of precursor cycles. This shows the level of thickness control
 that ALD provides [71].
 103
Figure 6.3: ALD MIM nanocapacitor grown on anodized aluminum oxide
 (AAO) nanopores highlighting the conformality of ALD growth. MIM
 structure has a thickness of TiN 5.6 nm, AlOx 6.6 nm, TiN 12.6 nm [70].
 And finally, unlike the relatively rapid process of the more common CVD growth,
 ALD growth tends to have a very low deposition rate.
 6.2 Microwave resonator fabrication on ALD materials and field sim-
 ulations of coplanar geometry
 Perhaps the most direct method for measuring the loss of an ALD dielectric
 at microwave frequencies is to fabricate it into a parallel-plate capacitor as part of
 a microwave resonator and measure the resonator?s Qi, as described in Chapter 3.
 Since all of the electric energy would be in the ALD dielectric, and that is by far
 the limiting source of loss, 1/Qi would be equal to the loss tangent of the material.
 If not all of the electric energy is in the ALD dielectric, then one needs to find
 the filling factor. The filling factor is just the fractional energy stored in a particular
 104
part of a resonator,
 F =
 ?
 LossyMaterial (~r)( ~E(~r))2d3r?
 All (~r)( ~E(~r))2d3r
 , (6.1)
 Assuming the resonator?s quality is limited entirely by the ALD material?s loss
 tangent, the loss tangent of the ALD material can be calculated from the measured
 Qi with
 tan ? = 1
 FQi . (6.2)
 For a parallel-plate capacitor F = 1, but the loss tangent can be calculated even
 if F 6= 1 as long as it?s known. If the electric field distribution is known, Eq. 7.10
 can be calculated and Eq. 6.2 can be used to find the loss tangent of the material
 after Qi is measured. This method allowed me to measure the loss tangent of ALD
 films using simple fabrication techniques that did not require being able to build a
 parallel-plate capacitor.
 In particular, I fabricated coplanar resonators discussed in Chapters 4 and 5
 atop ALD dielectrics. I simulated the field distribution of the resonators with the
 finite element simulator COMSOL and by knowing the thickness of the predeposited
 ALD dielectric, I was able to calculate Eq. 7.10 for each deposited dielectric.
 Figure 6.4 shows a COMSOL simulation of the cross-section of a CPS resonator
 on top of a thin dielectric film on a substrate. Fig. 6.4(a) shows the electric field in
 the vacuum and dielectric induced by the CPS fingers and Fig. 6.4(b) shows the
 electric energy density from said field. In this simulation a difference of 2 mV was
 placed on the two conductors and the electric field distribution was exported. To
 find the filling factor I numerically integrated the electric energy stored in the top
 105
0 10 20 -20 -10 
x (?m) 
0 10 20 -20 -10 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
10-8 
10-10 
10-12 
10-14 
J/m3 
10-6 
10-4 
10 
1 
V/m 
102 
103 
(a) 
(b) 
Figure 6.4: (a) Simulated electric field distribution for 100 nm thick CPS
 resonator (with 2 mV across the CPS electrodes) on 90 nm of ALD grown
 AlOx atop a sapphire wafer. (b) Electric energy density distribution of
 (a).
 106
ALD dielectric, 90 nm thick in this case, as a percentage of the total electric energy
 of the resonator and assumed that the lossy dielectric was entirely responsible for the
 resonator?s Qi. There were added complications on the analysis due to the voltage
 gradient on the resonator, the field dependence of the loss tangent, and as it turns
 out the non-uniform loss of the ALD material, but these subtleties will be discussed
 when the data is presented in Chapter 7.
 107
Chapter 7
 ALD grown dielectric measurements
 In this chapter I summarize my measurements on ALD dielectric films and
 discuss the implications for future quantum computing studies. As discussed in
 Chapter 6, the low temperature loss measurements were performed by fabricating
 coplanar resonators (discussed in Chapters 4 and 5) on top of the ALD dielectrics
 and measuring their Qi. In this chapter, I will also discuss the OH defect model,
 show secondary ion mass spectrometry (SIMS) measurements of ALD dielectrics
 and discuss their implications.
 7.1 Deuterated AlOx ALD films
 7.1.1 OH defect model
 As discussed in Chapter 2, perhaps the most commonly considered TLS mi-
 cromechanism in oxides is the OH rotor. As early as 1987, Phillips proposed that
 OH? groups could act as TLS defects [26]. It has recently been shown using den-
 sity functional theory that within particularly O-rich AlOx, hydrogen is likely to fill
 cation vacancies and form a significant density of microwave frequency TLSs [34].
 Hydrogen is particularly difficult to remove from ALD AlOx films because it
 exists in both the aluminum precursor, trimethylaluminum (TMA), and the oxygen
 precursor, water. It was not known which precursor contributed most of the hydro-
 108
gen, nor whether OH? was really dominating the loss. It was with this in mind that
 we attempted to substitute deuterium for hydrogen by using deuterated water as a
 precursor [72]. As calculated by Holder et al. [34], the increased mass of deuterium
 would result in a substantially reduced tunneling rate ?0 . Naturally this could
 move TLS defects into a much lower frequency range and improve the loss tangent
 of AlOx at microwave frequencies.
 7.1.2 SIMS measurements of AlOx films
 Two 55 nm thick AlOx films were grown by Alexander Kozen in the Rubloff
 group at the University of Maryland, one with regular water as the oxygen precur-
 sor and the other with 100% deuterated water. To confirm that we were actually
 substituting deuterium for hydrogen when we were using deuterated water, we had
 SIMS measurements performed on them [73]. Figures 7.1(a,c) show SIMS results for
 a non-deuterated film and Figs. 7.1(b,d) show SIMS results for a deuterated film.
 In Figs. 7.1(a,b), the concentration of hydrogen, deuterium, and carbon are shown
 as a function of depth in the film (0 is the top and 55 nm is the interface with
 the sapphire wafer), while in Figs. 7.1(c,d) the integrated concentrations of each
 impurity are shown. The first feature of note is that using deuterated water does in
 fact reduce the amount of excess hydrogen and increase the amount of deuterium
 in the bulk. However, there also appears to be a substantial amount of hydrogen
 surface contamination at the sapphire-AlOx interface that doesn?t depend on the
 precursor used. Comparing Fig. 7.1(c) to Fig. 7.1(d), we see that the deuterated
 109
film has a total of 4 ? 1022 H atoms/cm2 while the non-deuterated film has a total
 of 1023 Hatoms/cm2, there is still about a factor of 2 difference between the amount
 of total hydrogen in the two films. If OH? defects are a dominant loss contributor
 we may then expect to see a difference in the loss tangent of the two films. Note
 that the carbon impurities found in these films come from the aluminum precursor
 TMA and will be discussed later in the chapter.
 7.1.3 Resonator measurements
 As mentioned earlier, I used the coplanar resonators discussed in Chapters
 4 and 5 to measure the loss of the ALD grown dielectrics. Figure 7.2(a) shows
 results from the inverse internal quality factor measurement as a function of the
 RMS microwave voltage across the devices, taken at 80 mK, for two resonators
 (CPS and QLC-CPS) fabricated on 55 nm of ALD grown AlOx. Sample dut1 was
 grown with deuterated water as the oxygen precursor and sample hyd1 was grown
 with non-deuterated water as the oxygen precursor. All other grown parameters
 were identical (growth was performed at 250 oC).
 The first thing we notice in Fig. 7.2(a) is that qualitatively this behavior is
 clearly due to TLS loss, with an intrinsic voltage-independent-loss at low-voltages
 and a decrease in loss at higher-voltages due to TLS saturation. Next note that there
 doesn?t seem to be a difference in low-voltage Qi for the same device types between
 the two films hyd1 and dut1. The non-deuterated film (red symbols) seems to have
 a higher background loss at high voltages, but that may be due to the measurement
 110
0 20 40 60 80
 1016
 1018
 1020
 1022
 Concentration (atoms/cc
 )
  
 
H
 D
 C
 0 20 40 60 80
 1016
 1018
 1020
 1022
  
 
H
 D
 C
 0 20 40 60 80
 1018
 1019
 1020
 1021
 1022
 1023
 1024
 Depth (nm)
 Integrated Concentration (atoms/c
 m2
 )
  
 
H
 D
 C
 0 20 40 60 80
 1018
 1019
 1020
 1021
 1022
 1023
 1024
 Depth (nm)
  
 
H
 D
 C
 (a) (b)
 (d)(c)
 Figure 7.1: (a) SIMS analysis for carbon (black), hydrogen (red), and
 deuterium (blue) impurities of ALD grown AlOx. (b) SIMS analysis for
 carbon, hydrogen, and deuterium impurities of ALD grown AlOx with
 deuterated water precursor. (c) Integration of impurities in (a) through
 the 55 nm film. (d) Integration of impurities in (b) through the 55 nm
 film. Sapphire-ALD interface is at d=55 nm. The vertical dashed lines
 indicate the film depth. Dashed lines aren?t aligned with the defect peaks
 because of limitations in the SIMS depth resolution.
 111
setup rather than the TLS defects. The difference between the films is at such a
 high quality factor that it?s likely similar to the discrepancies observed between the
 coplanar device on bare sapphire in Chapter 5. This result suggests that either the
 TLSs causing the loss are not hydrogen based or perhaps that there?s another factor
 that compensates for the decreased hydrogen in the deuterated film.
 One possibility is that the deuterium substitution worked but that because of
 their lower tunneling rates and because the TLS distribution is inversely proportional
 to tunneling rates ? 1/?0, the deuterium based TLSs ended up having a higher
 density. To understand this consider an example of a distribution of TLSs with
 tunneling rates from 100 to 1000 (in arbitrary units). If their tunneling rate is
 reduced by a factor of 10 then they would be squeezed into having tunneling rates
 from 10 to 100, thus increasing their density by a factor of 10 as well. So if the
 distribution of TLSs is sufficiently broad, it can have a negligible effect on the loss.
 Note that if the TLS well frequency is proportional to the inverse of the square
 root of the defect mass as one would expect (?2 ? 1/m), we would still expect the
 deuterated TLS to have a factor of
 ?
 2 less loss, since of course deuterium has twice
 the mass of hydrogen, but such a difference in loss wasn?t resolved.
 We also note that the CPS resonators show a lower inverse Qi than the QLC-
 CPS resonators. This is because of the electric energy filling factor,
 F =
 ?
 LossyMaterial (~r)(E(~r))2d3r?
 All (~r)(E(~r))2d3r
 , (7.1)
 for the lossy ALD grown film is higher for the IDC than it is for the CPS. Simply put,
 the IDC teeth are closer together than the CPS electrodes are, so more electric energy
 112
is stored in the 55 nm directly below the resonator, where the ALD film is. The
 filling factors for the two devices can be numerically calculated using the COMSOL
 simulations mentioned in Chapter 6. I found that the IDC filling factor for the 55
 nm ALD film is FIDC,55nm = 0.052, while the CPS filling factor is FCPS,55nm = 0.031.
 As seen in Fig. 7.2(b) when the 1/Qi measurements for the four resonators are scaled
 for the different filling factors they lie on top of each other. This is consistent with
 loss coming uniformly from the ALD film.
 7.2 Varying ALD oxides
 7.2.1 CPS Qi measurements and field simulations
 To better understand TLSs in amorphous and crystalline ALD oxides, I next
 measured three different ALD-grown films: crystalline BeO (known to be crystalline
 from XRD measurements [74]), amorphous AlOx [75], and amorphous LaAlOx[76].
 The BeO film from the Banerjee group at UT Austin [74] and was grown at 200 oC
 with the beryllium precursor, dimethyl beryllium Be(CH3)2, and the oxygen pre-
 cursor, water. The AlOx [75] and LaAlOx [76] films from the Gordon group at Har-
 vard were grown at 300 oC with the aluminum precursor TMA and the lanthanum
 precursor tris(N,N?-diisopropylacetamidinato)lanthanum (La(iPrAMD)3). The BeO
 film was grown on high-resistivity silicon and the amorphous films were grown on
 sapphire. The crystalline BeO film is of particular interest because crystalline ma-
 terials tend not to show TLS loss. Bulk single crystal Al2O3 (sapphire) as discussed
 in Chapter 5, has a low, TLS free, loss tangent, which suggests that either the
 113
10?7 10?6 10?5 10?4 10?3 10?2
 10?6
 10?5
 10?4
 10?3
 VRMS (V)
 1/
 Q i
  
 
CPS, H
 CPS, D
 QLC?CPS, H
 QLC?CPS, D
 10?7 10?6 10?5 10?4 10?3 10?2
 10?5
 10?4
 10?3
 10?2
 VRMS (V)
 1/(
 Q iF
 )
  
 
CPS, H
 CPS, D
 QLC?CPS, H
 QLC?CPS, D
 Figure 7.2: (a) Inverse Qi measurement of CPS (?) and QLC-CPS (?)
 resonators on a 55 nm ALD grown AlOx film grown with deuterated
 (blue) and conventional (red) water. (b) Measurements from (a) divided
 by the filling factor of the ALD film for the two resonator geometries.
 At low voltage, this corresponds to the intrinsic voltage-independent loss
 tangent of the film. Deuterated and undeuterated films show the same
 level of loss. Data plotted for devices: hyd1-CPS (? in red), dut1-CPS
 (? in blue), hyd1-QLC-CPS (? in red), dut1-QLL-CPS (? in blue).
 114
amorphous nature or impurities in the films create additional loss. Also, the high
 thermal conductivity at low temperature [77] and the low loss tangent near room
 temperature [78] of crystalline BeO suggests that it [74] may have low loss useful
 for superconducting devices.
 Figure 7.3 shows a plot of the inverse Qi of the CPS resonators on the three
 different films taken at 35 mK as a function of the voltage across the CPS elec-
 trodes. Immediately we notice that even the crystalline BeO film shows behavior
 that is clearly consistent with TLS loss. However as discussed earlier, the inverse
 Qi only corresponds to the loss tangent when the filling factor of the dielectric is
 1, as in a parallel-plate capacitor. Here we must use COMSOL to simulate the
 field distribution of the CPS geometry using the known permittivity of the oxides
 (r,BeO = 6.7, r,AlOx = 8, r,LaAlOx = 16) and the film thicknesses (78 nm of BeO,
 90 nm of AlOx, and 97 nm of LaAlOx). For example, Fig. 7.4 shows a COMSOL
 simulation of the electric field and electric energy distribution for a cross section of
 the CPS geometry with 2 mV across the electrodes atop a 90 nm ALD AlOx film
 grown on a sapphire wafer. As one would expect, the electric field is symmetric
 about the vacuum substrate interface, but electric energy of course scales with the
 permittivity of the substrate. Another factor to consider is that the LaAlOx film
 was also etched by the Aluminum Etch acid used to pattern the aluminum, com-
 pletely etching away the LaAlOx film that was not covered by Al, creating trenches
 in the LaAlOx. After this was realized, I adjusted the COMSOL field simulations
 to account for this (see Fig. 7.5).
 115
10?6 10?4 10?2
 10?6
 10?5
 10?4
 10?3
 VRMS (V)
 1/
 Q i
  
 
90nm a?Al2O3
 97nm a?LaAlO3
 78nm c?BeO
 Figure 7.3: Inverse Qi of coplanar strip resonators on three different ALD
 grown film types. Data plotted for devices: BeO-CPS, AlOx-Harv-CPS,
 and LaAlOx-Harv-CPS.
 116
0 10 20 -20 -10 
x (?m) 
0 10 20 -20 -10 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
10-8 
10-10 
10-12 
10-14 
J/m3 
10-6 
10-4 
10 
1 
V/m 
102 
103 
(a) 
(b) 
Figure 7.4: (a) COMSOL simulation of electric field distribution for 100
 nm thick CPS resonator (with 2 mV across the CPS electrodes) on 90 nm
 of ALD grown AlOx atop a sapphire wafer. (b) Electric energy density
 distribution of (a).
 117
0 10 20 -20 -10 
x (?m) 
0 10 20 -20 -10 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
y 
(?m
 ) 
0 
2 
4 
-4 
-2 
10-8 
10-10 
10-12 
10-14 
J/m3 
10-6 
10-4 
10 
1 
V/m 
102 
103 
(a) 
(b) 
Figure 7.5: (a) COMSOL simulation of electric field distribution for 100
 nm thick CPS resonator (with 2 mV across the CPS electrodes) on 97 nm
 of ALD grown LaAlOx atop a sapphire wafer. The LaAlOx is only under
 the electrodes because elsewhere it was etched away by the aluminum
 etchant. (b) Electric energy density distribution of (a).
 118
7.2.2 SIMS analysis for ALD dielectrics
 Since hydrogen was still the most likely culprit for TLS loss in our ALD films,
 SIMS measurements were performed on these films to determine the hydrogen im-
 purity concentration. The carbon concentration for the BeO film and the hydrogen
 concentration for all three film types is plotted as a function of film depth in Fig. 7.6.
 Examination of the figure shows that there is an 8 times greater concentration in
 the bulk of the crystalline BeO film than in the amorphous AlOx or LaAlOx films.
 In the BeO films the hydrogen is almost uniformly distributed throughout the bulk
 of the film (although some H diffuses into the Si substrate). In contrast, the amor-
 phous AlOx or LaAlOx films have a large peak in hydrogen concentration at the
 surface. This distribution of hydrogen strongly suggests that BeO has incorporated
 hydrogen during growth with its precursors (dimethyl beryllium and water), while
 the other oxides have mostly incorporated hydrogen from ambient exposure rather
 than growth. Figure 7.6 also indicates that there is a high concentration of C impu-
 rities in the c ? BeO film, but unlike the hydrogen, it is concentrated on the surface
 as the impurities in the amorphous films are.
 7.2.3 Fitting to CPS Qi measurements using TLS loss model
 Next I discuss fitting the inverse Qi measurements of the CPS devices using
 the COMSOL simulated field distributions, SIMS hydrogen impurity measurements,
 and the TLS loss tangent model,
 tan ?(~r, E) = tan ?0 tanh(~?/2kBT )?
 1 + (E(~r)/Ec)2 . (7.2)
 119
0 20 40 60 80 100 120
 1019
 1020
 1021
 1022
 1023
 1024
 h (nm)
 N (atoms/cc
 )
  
 
BeO:H
 BeO:C
 LaAlO
 x
 :H
 AlO
 x
 :H
 Figure 7.6: Hydrogen impurity concentration measured by SIMS for
 films with nominally identical parameters to those shown in Fig. 7.3 as
 well as carbon impurity concentration in BeO film. The crystalline BeO
 film (75 nm thick) shows a large hydrogen concentration that is nearly
 uniform throughout the film depth, while the AlOx and LaAlOx films
 (each approximately 48 nm thick) show hydrogen concentrated at the
 surface of the films. The BeO film also shows a large surface distribution
 of carbon impurities. Vertical dotted lines indicate film thicknesses.
 120
To do this I started by taking the COMSOL simulated field distribution for the CPS
 geometry and used the assumption that the field throughout the cross section would
 scale linearly with the voltage across the electrodes. Knowing the field throughout
 the resonator geometry I then assumed a tan ? and an Ec for the ALD dielectric and
 that the rest of the volume was lossless . Numerically integrating over the volume
 for every voltage data point in Fig. 7.3, I could then find:
 1
 Qi(V ) =
 ?
 Lossyfilm tan ?(~r, E)film(E(~r, V ))2d3r?
 All (~r)(E(~r))2d3r
 , (7.3)
 remembering of course to also integrate over the entire length of the CPS (z) with
 Vcps(z) = V0 sin( pi2
 z
 lcps
 ), (7.4)
 where lcps = 4.8 mm.
 Figure 7.7 shows the inverse Qi data from Fig. 7.3 with loss tangent fits. Not
 surprisingly when considering the impurity distribution in Fig. 7.6, for the two amor-
 phous films, a single bulk loss term representing the film thickness was insufficient
 to fit Qi and an independent surface-loss mechanism was added. I assumed the
 thickness of the lossy material at the surface was 5 nm. The fit parameters for the
 curves in Fig. 7.7 are shown in Table 7.1. As expected from the SIMS measurements
 in Fig. 7.6, I found that the dominant loss occurs in the bulk of the crystalline BeO
 film and the surface of the amorphous AlOx or LaAlOx films. A relatively small
 background loss term was also included in the fit and labeled as the floor in Ta-
 ble 7.1. This power-independent background loss is similar to that discussed earlier
 and present in coplanar resonator measurements on bare substrate.
 121
10?6 10?4 10?2
 10?6
 10?5
 10?4
 10?3
 VRMS (V)
 1/
 Q i
  
 
90nm a?Al2O3
 97nm a?LaAlO3
 78nm c?BeO
 Figure 7.7: Data points same 1/Qi data plotted in Fig. 7.3 solid curves
 are fits to the TLS loss expression (Eq. 7.2) for the CPS resonator (6.4
 GHz) geometry with the fit parameters from Table 7.1. Measurements
 were taken at 34 mK.
 122
Table 7.1: Fit parameters for the curves in Fig. 7.7. The bold values correspond
 to the dominant loss terms of the films, in the surface of the amorphous films and
 the bulk of the crystalline film. All films were fit assuming a bulk loss value, a floor
 limited by extrinsic effects, and amorphous AlOx or LaAlOx films were fit with an
 additional 5 nm thickness of lossy material at the top surface. Films were measured
 at 34 mK with 6.4 GHz CPS resonator.
 Fit parameters BeO AlOx LaAlOx
 tan ?0,surface(?10?3) - 13.0 4.5
 Ec,surface(V/m) - 0.75 0.55
 tan ?0,bulk(?10?3) 6.2 0.70 1.1
 Ec,bulk(V/m) 0.70 5.9 60
 tan ?0,f loor(?10?6) 32 4.0 0.68
 thickness (nm) 78 90 97
 The results from these fits also indicate that despite the high carbon impurity
 concentration shown in Fig. 7.6, carbon impurities are unlikely to be the cause of
 TLS loss in the film. This is because unlike the hydrogen impurities in the film, the
 carbon impurities are concentrated at the surface and so would have resulted in a
 measureable surface loss term as the amorphous films did.
 The BeO loss measurements indicate that TLS defects are not limited to amor-
 phous materials and can in fact have an even greater effect in crystalline materials.
 This reduces the parameter space available in searching for a TLS micromechanism,
 perhaps simplifying the task of modeling and identifying it.
 123
7.3 Surface Loss effects in ALD grown AlOx
 To verify that the AlOx loss tangent was dominated by surface defects, I
 measured two films that were nominally identical in every way except for their
 thicknesses. Figure 7.8 shows the inverse Qi measurement of the resonators on two
 films with 48 and 90 nm thicknesses measured at 34 mK. The idea is that if the
 TLS were uniformly distributed in the dielectric, then the Qi of the resonator would
 differ by a factor of 1.7, scaling linearly with the electric energy stored in the film
 thickness. Instead, the quality factor of the resonators matches to within 15% at the
 low-voltage limit, consistent with loss that is primarily at the surface. I note that
 the development of in situ deposition of dielectrics within MIM trilayer structures
 would greatly reduce such surface loss [72].
 7.4 Hydrogen vs Carbon TLS analysis
 One of the most common impurities in ALD films is carbon due to incomplete
 reaction of the organometallic precursors. In the case of AlOx the precursor is
 trimethylaluminum. I therefore had SIMS measurements performed for both carbon
 and hydrogen impurities in AlOx films grown in separate ALD chambers in separate
 labs, but with the same precursors. Figure 7.9 shows the integration of hydrogen and
 carbon impurities over the thickness of the films. The Lab 1 film from the Rubloff
 group at the University of Maryland has a much higher (40 times greater) carbon
 defect concentration than that of Lab 2 from the Gordon group at Harvard. We
 also measured CPS resonators on these films and found them to have very similar
 124
10?8 10?6 10?4 10?2
 10?6
 10?5
 10?4
 1/
 Q i
  
 
VRMS (V)
 90nm
 48nm
 Figure 7.8: plot of 1/Qi vs. voltage of the CPS resonators measured
 on nominally the same AlOx films grown in the same chamber but to
 two different thicknesses. Because the loss is primarily at the surface,
 the quality factor of the two devices is practically identical and does not
 scale with the thickness of the film.
 125
0 20 40 60 80
 1013
 1014
 1015
 1016
 h (nm)
 N (atoms/c
 m2
 )
  
 
C: Lab 1, a?Al2O3
 C: Lab 2, a?Al2O3
 H: Lab 1, a?Al2O3
 H: Lab 2, a?Al2O3
 Figure 7.9: Integrated hydrogen and carbon concentration from the sur-
 face (0 nm) downward in ALD aluminum oxide films from different labs
 (Lab 1=Rubloff Lab 2 = Gordon) as measured by SIMS. They show
 similar hydrogen concentration but the film from Lab 1 shows a much
 greater carbon concentration. Since the two films have similar loss but
 very different carbon concentrations, C-based TLSs can be eliminated
 as a candidate for TLS loss in this regime.
 126
Qi?s, indicating that carbon plays a negligible role in the TLS loss. From Fig. 7.9
 we see that hydrogen is a much more viable TLS candidate.
 To quantitatively analyze H and C impurities as TLS candidates we assume
 that the loss tangent for each type of impurity tangent,
 tan ?0,x(~r) = KxCx(~r), (7.5)
 is proportional to the impurity concentration Cx(~r) of species x, where Kx is a
 TLS-loss proportionality constant. Assuming defect x limits the resonator?s loss,
 the low-voltage-amplitude internal quality factor of the film is then
 1
 Qi(V ? 0) = Kx?Cx?FfilmWx, (7.6)
 where ?Cx? is the average impurity concentration
 ?Cx? =
 ?
 film d
 3rCx(~r)?
 film d3r
 , (7.7)
 Ffilm is the geometric filling factor,
 Ffilm =
 ?
 film d
 3r(~r)|E(~r)|2?
 all d3r(~r)|E(~r)|2
 , (7.8)
 and Wx is impurity weighting coefficient,
 Wx =
 ?
 film d
 3r(~r)|E(~r)|2Cx(~r)
 ?Cx? ?film d3r(~r)|E(~r)|2 . (7.9)
 Note that Ffilm = 1 for a device with a single dielectric and Wx = 1 for a uniform
 impurity distribution. WH,C can be calculated from the SIMS measurements shown
 in Fig. 7.9 and Ffilm can be calculated by numerically integrating the energy density
 of the COMSOL field simulations for the ALD film. Note that instead of defining the
 127
0 20 40 60 80 100
 4
 5
 6
 7
 8
 h (nm)
 dF/dh x 1
 04
 Figure 7.10: Numerically integrated filling factor for each nm depth slice
 of a 90 nm film of AlOx under the CPS resonator geometry as a function
 of the film depth.
 filling factor over the entire film depth, it can instead be defined over any arbitrary
 thickness of the film, so with that in mind I define
 F (h) ?
 ? ?
 dxdy
 ?h
 0 dh(~r)|E(~r)|2?
 all d3r(~r)|E(~r)|2
 , (7.10)
 where h = 0 is defined as the surface of the film and hf is the film thickness, such
 that F (hf ) = Ffilm. In Fig. 7.10 I?ve plotted the filling factor depth derivative as a
 function of the depth below the CPS resonator with 1 nm resolution, as calculated
 by a numerical integration from the COMSOL field simulations.
 In Fig. 7.11 I?ve plotted 1/(QiFWH?CH?) and 1/(QiFWC?CC?) for the two
 films as a function of the microwave voltage using the measured values of Qi (at 80
 128
mK) and CH/CC , as well as the simulated field distributions for WH/WC and F. If we
 assume hydrogen impurities are the dominant cause of TLS-loss, at low field values,
 I?ve found that the two films from different labs have KH = 3? 10?24 cm3 to within
 50%. However, as made clear by Fig. 7.11, if we instead assume carbon impurities
 are the dominant cause of TLS-loss, the two films would produce drastically different
 values for KH , differing by more than a factor of 30, indicating that carbon is not
 the dominant source of TLS loss in this experiment. This provides evidence that
 oxide films can be optimized for low temperature devices by lowering their hydrogen
 concentration.
 7.5 Conclusion
 In conclusion, I measured the loss of several ALD grown dielectric oxide films
 at millikelvin temperatures using the coplanar superconducting resonators discussed
 in chapter 4. By using deuterated water to grow AlOx films, we reduced the amount
 of hydrogen in the film but were not able to correlate that reduction in hydrogen
 concentration with a change in low temperature microwave loss, possibly because
 deuterium also causes TLS loss.
 In addition, I found that the TLS loss of the crystalline film, BeO, was higher
 than that of the amorphous films, AlOx and LaAlOx, despite the fact that BeO is
 crystalline and has been shown to have low-loss at room temperature with high ther-
 mal conductivity at low temperatures. Using SIMS measurements I correlated the
 low temperature loss with excess hydrogen defects on the surface of the amorphous
 129
10?6 10?4 10?2
 10?26
 10?25
 10?24
 10?23
 10?22
 VRMS (V)
 1/
 (Q
 iF
 fi
 lm
 W
 x
 ?C
 x
 ?)
  
 
x=H: Lab 1
 x=H: Lab 2
 x=C: Lab 1
 x=C: Lab 2
 Figure 7.11: Measurements of 1/(QiFWH,C?CH,C?) assuming H or C
 impurities are the main cause of TLS loss. At low fields this reduces
 to the proportionality constant of hydrogen impurities to loss tangent,
 KH,C . Since assuming hydrogen impurities are the dominant cause of
 TLS loss results in consistent KH between the two labs, it?s a much
 more likely TLS candidate than carbon.
 130
films and in the bulk of the crystalline film. The BeO film had a uniform bulk loss
 tangent with no excess surface loss consistent with its uniform H impurity concen-
 tration and inconsistent with its excess surface C impurity concentration, excluding
 C as a TLS candidate in BeO. In AlOx and LaAlOx films, the bulk loss tangent was
 found to be similar, but the surface loss dominated indicating that the loss tangent
 was limited by surface impurities, likely hydrogen. A thickness study of AlOx films
 confirmed that the majority of the loss was on the surface. By testing AlOx with
 different carbon concentrations, I found that carbon has a negligible effect on TLS
 loss. I conclude that various low temperature oxides can be optimized via rapid
 room temperature measurements of hydrogen.
 131
Chapter 8
 Biased resonator
 8.1 Biasing TLSs
 Once a linear resonator is fabricated, its resonance frequency is determined
 and is a physical characteristic of the device. In contrast, a Josephson junction
 acting as a tunable nonlinear inductor allows for tuning of the resonance frequency,
 such as in the Josephson Junction Defect Spectrometer, but for linear resonators,
 this cannot be done. However, it is possible is to tune the TLSs that are couple
 to the resonator, moving them in and out of the resonator?s bandwidth. In this
 chapter, I discuss such an experiment.
 As discussed in Chapter 2, the TLS asymmetry energy, ?, couples to an
 applied electric field ~EBias through the TLS?s dipole p according to
 ? = ?E=0 + 2 ~EBias ? ~p. (8.1)
 Since an applied field?s effect on the tunneling rate, ?0, can generally be neglected
 we can write the TLS?s energy as
 E =
 ?
 ?20 + |?old + 2 ~EBias ? ~p|2. (8.2)
 One implication of Eq. 8.2 is that if we applied an additional bias field to the
 dielectric, independent of the microwave probe field, we could bias TLSs out of the
 resonator?s bandwidth so they can no longer couple to the resonator, and limit its
 132
loss. However, we also know from Chapter 2 that the standard distribution of TLSs
 is white in ?:
 d2N = P0?0
 d?d?0. (8.3)
 Thus on average the same number of TLSs would be biased into the resonator?s
 bandwidth as were biased out resulting in a similar TLS loss tangent within statis-
 tical fluctuations. If a bias field could be applied with a sufficiently high amplitude,
 such an experiment could act as a check on the limits of the uniformity of the ?
 distribution. Similarly, if the distribution is sufficiently narrow, we could polarize
 most of the TLSs out of the resonator?s bandwidth and significantly lower the loss.
 8.2 Previous bias measurements
 Experiments have previously tried to measure the response of a glassy dielectric
 in the low-frequency regime (~?  kBT ) after it was driven out of equilibrium.
 It was found that after the application of either a static electric field or strain
 field, there was a sudden increase in the permittivity of glassy dielectrics, followed
 by a subsequent slow logarithmic relaxation back to the equilibrium permittivity
 [79, 80, 81]. It was also found that there was a persistent change in permittivity
 at a previously applied bias field value [79]. This memory effect was explained by
 TLS-TLS interactions causing a pseudogap which decreased the density of TLSs at
 energies bellow 20 GHz [82, 83]. The sudden increase in permittivity was attributed
 to the biasing of TLSs by the dc field and closing of the pseudogap, thus increasing
 the density of low-frequency TLSs, while the subsequent relaxation was explained
 133
by the reforming of the gap via long-range TLS-TLS interactions. This led to a
 better understanding of TLSs and amorphous dielectrics in general. More recently
 the biasing of individual TLSs with strain has been observed in Josephson junctions
 [84].
 8.3 Swept bias
 After performing the experiments described in the next chapter, Professor A.
 Burin derived the theory I discuss next to explain the data [85].
 As mentioned earlier, if the TLS distribution is completely white in the asym-
 metry energy, then a change in the bias field would have no effect on the resonator?s
 loss because on average the same number of TLSs would remain in the resonator.
 However, even without a change in the TLS density, if a swept bias field is used, a
 change in the measured loss tangent of a material should be expected. Qualitatively
 this is because if the TLSs are biased fast enough, they can take energy away from
 the resonator not only through their own relaxation but also through the bias.
 To understand this quantitatively, we first define the energy of a TLS that is
 being swept through the resonator?s frequency,
 E(t) = ~?0 + ~?(t? t0) (8.4)
 where ? is the energy bias rate of the TLS and t0 is the time when the TLS passes
 the resonator?s frequency ?0. Expanding Eq. 8.2 around small shifts in ? (because
 we?re only interested in TLSs close to resonance) gives the energy bias rate
 ? = 2
 ?
 1 ?
 (
 ?20
 ~?
 )2 p
 ~
 dEbias
 dt cos ? = ?0
 ?
 1 ?
 (
 ?20
 ~?
 )2
 cos ? (8.5)
 134
where ? is the angle between the dipole moment and the field and ?0 is the ensemble
 TLS energy bias rate
 ?0 = 2
 p
 ~
 dEbias
 dt . (8.6)
 Note that ? has dimensions of frequency squared or more intuitively frequency per
 second because it?s effectively the velocity at which the TLS?s energy is biased.
 Schrodinger?s equation for a TLS in a resonant field with states ? = (c1, c2)
 can be written as
 dc1
 dt = ?iE(t)c1/2 ? i?R cos(?t)c2,
 dc2
 dt = iE(t)c2/2 ? i?R cos(?t)c1, (8.7)
 where, ?R is the Rabi frequency of the TLS
 ?R =
 pEac
 ~ cos ?
 ?0
 E = ?R0 cos ?
 ?0
 E , (8.8)
 ?R0 is the ensemble TLS Rabi frequency, and c1 (c2) is the amplitude of being in
 the ground (excited) state of the TLS. If we make a transformation into a new basis
 (a1, a2) = (c1ei?t, c2ei?t), Eq. 8.7 becomes
 da1
 dt = ?i (E(t) ? ~?)
 a1
 2
 ? i
 ?R
 2
 a2 ? i
 ?R
 2
 a2e
 2i?t
 ,
 da2
 dt = i (E(t) ? ~?)
 a2
 2
 ? i
 ?R
 2
 a1 ? i
 ?R
 2
 a1e
 2i?t
 . (8.9)
 Using the rotating wave approximation, we can neglect the last term because it?s
 oscillating much faster than the rest of the expression. Thus Eq. 8.9 becomes
 da1
 dt = ?i
 ?(t? t0)
 2
 a1 ? i
 ?R
 2
 a2,
 da2
 dt = i
 ?(t? t0)
 2
 a2 ? i
 ?R
 2
 a1, (8.10)
 135
where I?ve also substituted in Eq. 8.4.
 Here we can recognize that Eq. 8.10 is identical to the equation for a Landau-
 Zener transition of a two-level quantum system (see Fig. 8.1) [86, 87]. We know from
 the Landau-Zener problem that if at t = ??, only the ground state is populated
 |a1|2 = 1, |a2|2 = 0 then long after the TLS crossing, t = ?, we have
 |a1|2 = exp(? pi?
 2
 R
 2?
 ),
 |a2|2 = 1 ? exp(? pi?
 2
 R
 2?
 ). (8.11)
 Thus for fast bias (?  ?2R) the TLS crosses the resonance while staying it its
 ground state (green to blue in Fig. 8.1) and for slow bias the TLS transitions into
 its excited state (green to red in Fig. 8.1), following the bottom path in Fig. 8.1.
 Using Eq. 8.11 for the probability of making a Landau-Zener transition we
 can derive an expression for the bias rate induced loss. We first define the tunneling
 rate as ?0 = x~?, allowing us to write the Rabi frequency as ?R = xpEac cos(?)/~
 and the bias rate as ? = 2p ?Ebias cos(?)?1 ? x2, where ? is the angle between the
 dipole moment and the external field. We know from the Landau-Zener formula
 that the probability of a TLS excitation is Pex = 1 ? e??, where ? = pi?2R/(2?) is
 the dimensionless Landau-Zener parameter from Eq. 8.11. So the energy lost from
 the bias integrated over the entire volume of the dielectric is
 dE = ~?
 ?
 d3r
 ?
 dNPex, (8.12)
 where the ~? comes from the fact that that?s the amount of energy lost every time
 a TLS is excited when crossing the resonance bandwidth and the integral over dN
 136
Figure 8.1: TLS-resonator energy spectrum as the TLS is biased across
 the resonance frequency of the resonator. If biased quickly it is likely
 to stay in the ground state not coupling to the resonator (green to blue
 TLS). If biased slowly enough it?s likely to couple to the resonator and be
 pumped up to its excited state, taking a photon away from the resonator,
 thus |n? ? |n? 1? (green to red TLS).
 137
is only preformed over the TLSs that cross the resonance frequency. Expanding the
 integral over the TLSs in Eq. 8.12, we have
 dE = ~?P0V
 ? 1
 0
 d cos(?)
 ? ?
 (1 ? e??)d?d?0?0 , (8.13)
 where we are integrating over dipole angles ? in addition to the asymmetry energies
 ? and tunneling rates ?0, and the d3r integral over volume (V) has been performed.
 Next it is convenient to convert the integral of ? into an integral over TLS energy
 E =
 ?
 ?2 + ?20 in order to more easily set a limit on the TLSs crossing the resonance,
 dE = ~?P0V
 ? 1
 0
 d cos(?)
 ? ~?
 0
 d?0
 ? ~?+~?dt
 ~??~?dt
 dE (1 ? e
 ??)E
 ?0
 ?
 E2 ? ?20
 , (8.14)
 where the bias is assumed to be on for a time period dt only TLSs with energies
 ~?dt away from the resonance frequency will cross it. The integral over E can be
 evaluated as a Dirac delta function at ~? with magnitude 2~?dt
 dE = ~?P0V
 ? 1
 0
 d cos(?)
 ? ~?
 0
 2~?dt(1 ? e??) ~?d?0
 ?0
 ?(~?)2 ? ?20 . (8.15)
 Next substituting x~? in for ?0 we have
 dE
 dt = 2~
 2?P0V
 ? 1
 0
 d cos(?)
 ? 1
 0
 ?(1 ? e??) dx
 x
 ?
 1 ? x2
 . (8.16)
 Expressing the bias rate ? = 2p ?Ebias cos(?)?1 ? x2 in terms of the dimensionless
 Landau-Zener parameter ? gives
 dE
 dt = 2~
 2?P0V
 ? 1
 0
 d cos(?)
 ? 1
 0
 pi?2R
 2?
 (1 ? e??) dx
 x
 ?
 1 ? x2
 = pi?E2acP0p2V
 ? 1
 0
 d cos(?) cos2(?)
 ? 1
 0
 (1 ? e??)
 ?
 xdx?
 1 ? x2
 . (8.17)
 138
Next using the definition of loss tangent (or quality factor) I can define the
 loss tangent in terms of dE/dt
 tan ? = Power loss rate
 ?(Energy stored in resonator) =
 dE/dt
 ?E2acV
 . (8.18)
 Combining Eq. 8.17 and Eq. 8.18 gives
 tan ? = piP0p
 2
 
 ? 1
 0
 d cos(?) cos2(?)
 ? 1
 0
 (1 ? e??)
 ?
 xdx?
 1 ? x2
 . (8.19)
 Here we recognize from previous discussions of TLS loss that Eq. 8.19 can be rewrit-
 ten as
 tan ? = 3 tan ?0
 ? 1
 0
 d cos(?) cos2(?)
 ? 1
 0
 (1 ? e??)
 ?
 xdx?
 1 ? x2
 , (8.20)
 where tan ?0 = piP0p2/(3) is the intrinsic TLS loss without electric field or thermal
 saturation. Here it?s useful to also remind ourselves of the electric field saturated
 TLS loss tangent
 tan (?) = tan ?0 tanh (~?/2kBT )?
 1 + (Eac/Ec)2 . (8.21)
 In the limit of fast bias (?  ?2R) we can use the approximation e?? ' 1 ? ?
 which simplifies Eq. 8.20 and makes its evaluation trivial
 tan ? = 3 tan ?0
 ? 1
 0
 d cos(?) cos2(?)
 ? 1
 0
 xdx?
 1 ? x2
 = tan ?0, (8.22)
 thus reducing the loss tangent to its intrinsic low Eac value as given by Eq. 9.1.
 This is obviously not a coincidence. A qualitative way to understand this is that in
 this limit, almost all of the TLSs passing through resonance remain in their ground
 state so the resonator effectively sees a bath of ground state TLSs ready to absorb
 a photon as in the low ac field limit. A slightly more comprehensive explanation
 139
for this phenomena is that as the bias rate increases, proportionally more TLSs
 are being swept through the Landau-Zener region in each unit of time. Since the
 TLSs are in the ground state they are all capable of absorbing power. However, the
 higher the bias rate, the faster a given TLS will cross through the Landau-Zener
 transition region, and the smaller the probability of the TLS absorbing energy from
 the resonator and being left in its excited state. These two effects conspire to yield
 the intrinsic low ac field loss tangent.
 We note here that we?ve neglected TLS relaxation for this derivation (T1 =
 T2 = ?). So the loss calculated thus far comes entirely from energy lost to the
 system through the bias and does not include the possibility that an excited TLS
 can lose energy and be available to absorb more from the resonator. This assumption
 will of course fail for sufficiently fast TLS relaxation. The crossover occurs when a
 TLS is likely to decohere during the time it takes it to pass through the resonance
 i.e. for ? ? ?R/? , where ? ? ?T1,minT2 is a characteristic TLS lifetime. Thus for
 sufficiently slow biases, TLS relaxation dominates and the loss tangent reduces to
 the steady state value given by Eq. 9.1.
 In the regime where the bias is sufficiently fast ?  ?R/? such that the loss
 limited by the bias rate, Eq. 8.20 can be evaluated numerically. The numerical
 solution is plotted in Fig. 8.2 as a function of the dimensionless bias rate
 ? = 2?0
 pi?2R0
 . (8.23)
 As discussed earlier the blue curve in Fig. 8.2 is only applicable at high bias
 rates and will fail at lower bias rates where the loss is limited by TLS relaxation.
 140
10?3 10?2 10?1 100 101 102 103
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 ?
 ta
 n?
 /ta
 n?
 0
 Figure 8.2: Plot of tan ?/ tan ?0 as a function of the dimensionless bias
 rate ? = 2?0/(pi?2R0) found by evaluating Eq. 8.20 numerically.
 141
The crossover at low bias sweep rate is dependent on the Rabi frequency and the
 TLS lifetime ?0 ? ?R0/? . Thus given a Rabi frequency and a TLS lifetime, we can
 calculate the characteristic dimensionless bias rate ?c = 1/(?R0?) below which TLS
 relaxations dominate and the blue curve in Fig. 8.2 is no longer applicable.
 8.4 Monte Carlo simulations
 One way to address the behavior of a random distribution of TLSs coupled to
 a cavity is through a Monte Carlo simulation. It involves choosing a set of TLSs
 with the right distribution, turning on a field similar that which exists in the cavity
 and calculating the coupling of the TLSs to the field. Some advantages of this
 approach are that it allows me to include T1 and T2 effects, and examine statistical
 fluctuations resulting from using a finite number of TLSs.
 8.4.1 Theory for Monte Carlo simulations
 As discussed in Chapter 2, the Hamiltonian of a single TLS in an electric field
 can be split into the Hamiltonian of the TLS in its eigenbasis
 H0 =
 1
 2
 ?
 ??? E 0
 0 ?E
 ?
 ??? , (8.24)
 and the perturbation Hamiltonian due to interaction with the applied electric field
 HS =
 1
 E
 ?
 ??? ? ?0
 ?0 ??
 ?
 ??? (~p ? ~E). (8.25)
 142
The total Hamiltonian is then
 H = H0 + HS. (8.26)
 We know from the Heisenberg equation of motion that
 i~?? = [H, ?], (8.27)
 where ? is the TLS density matrix. By combining Eqs. 8.24-8.27 for a single TLS
 we can write the set of equations
 i~??00 = (?01 ? ?10)?0E ~p ? ~E
 i~??01 = ?00
 ?0
 E ~p ?
 ~E + ?01
 (
 E + 2?E ~p ?
 ~E
 )
 ? ?11
 ?0
 E ~p ?
 ~E
 i~??10 = ??00
 ?0
 E ~p ?
 ~E ? ?10
 (
 E + 2?E ~p ?
 ~E
 )
 + ?11
 ?0
 E ~p ?
 ~E
 i~??11 = (?10 ? ?01)?0E ~p ? ~E (8.28)
 for the TLS density matrix.
 Note that Eq. 8.28 does not include dissipation. We add dissipation and
 decoherence to the equations of the density matrix in an ad hoc manner [88] and
 arrive at
 i~??00 = (?01 ? ?10)?0E ~p ? ~E + i~
 ?11
 T1
 i~??01 = ?00
 ?0
 E ~p ?
 ~E + ?01
 (
 E + 2?E ~p ?
 ~E
 )
 ? ?11
 ?0
 E ~p ?
 ~E ? i~?01
 T2
 i~??10 = ??00
 ?0
 E ~p ?
 ~E ? ?10
 (
 E + 2?E ~p ?
 ~E
 )
 + ?11
 ?0
 E ~p ?
 ~E ? i~?10
 T2
 i~??11 = (?10 ? ?01)?0E ~p ? ~E ? i~
 ?11
 T1
 . (8.29)
 143
Equation 8.29 can be rewritten as a matrix equation?
 ???????????
 ??00
 ??01
 ??10
 ??11
 ?
 ???????????
 = M
 ?
 ???????????
 ?00
 ?01
 ?10
 ?11
 ?
 ???????????
 , (8.30)
 where
 M = 1
 i~
 ?
 ???????????
 0 ?0E ~p ? ~E ?
 ?0
 E ~p ? ~E
 i~
 T1
 ?0
 E ~p ? ~E E + 2
 ?
 E ~p ? ~E ?
 i~
 T2 0 ?
 ?0
 E ~p ? ~E
 ?
 ?0
 E ~p ? ~E 0 ?
 (
 E + 2?E ~p ? ~E ?
 i~
 T2
 )
 ?0
 E ~p ? ~E
 0 ??0E ~p ? ~E
 ?0
 E ~p ? ~E ?
 i~
 T1
 ?
 ???????????
 .
 (8.31)
 To solve Eq. 8.30 we use the ansatz?
 ???????????
 ?00(t)
 ?01(t)
 ?10(t)
 ?11(t)
 ?
 ???????????
 = etM
 ?
 ???????????
 C1
 C2
 C3
 C4
 ?
 ???????????
 , (8.32)
 where we know that M is in general time dependent (M = M(t)) since it?s a
 function of the applied field and we have the initial conditions?
 ???????????
 ?00(0)
 ?01(0)
 ?10(0)
 ?11(0)
 ?
 ???????????
 =
 ?
 ???????????
 1
 0
 0
 0
 ?
 ???????????
 . (8.33)
 144
To solve for ?(t) we can take short iterative time steps where we assume M(t) is a
 constant. This is effectively an induction proof, we know ?(0), so let?s assume we
 know ?(t0) and try to calculate ?(t0 + ?t), thus proving that we can solve for ?(t)
 iteratively. First we know the solution around t = t0 to be?
 ???????????
 ?00(t)
 ?01(t)
 ?10(t)
 ?11(t)
 ?
 ???????????
 = etM(t0)
 ?
 ???????????
 C1(t0)
 C2(t0)
 C3(t0)
 C4(t0)
 ?
 ???????????
 , (8.34)
 since we?re assuming M(t) and by extension C(t) are constant over small time steps.
 We can now calculate the constant matrix C(t0) from our initial conditions?
 ???????????
 ?00(t0)
 ?01(t0)
 ?10(t0)
 ?11(t0)
 ?
 ???????????
 = et0M(t0)
 ?
 ???????????
 C1(t0)
 C2(t0)
 C3(t0)
 C4(t0)
 ?
 ???????????
 ?
 ?
 ???????????
 C1(t0)
 C2(t0)
 C3(t0)
 C4(t0)
 ?
 ???????????
 =
 [
 et0M(t0)
 ]
 ?1
 ?
 ???????????
 ?00(t0)
 ?01(t0)
 ?10(t0)
 ?11(t0)
 ?
 ???????????
 ,
 (8.35)
 and with Eq. 8.36 calculate ?(t0 + ?t)?
 ???????????
 ?00(t0 + ?t)
 ?01(t0 + ?t)
 ?10(t0 + ?t)
 ?11(t0 + ?t)
 ?
 ???????????
 = e(t0+?t)M(t0)
 ?
 ???????????
 C1(t0)
 C2(t0)
 C3(t0)
 C4(t0)
 ?
 ???????????
 . (8.36)
 By using this method we can choose a Monte Carlo distribution of TLSs and
 an arbitrary electric field bias to shift the asymmetry energies ? and propagate
 the density matrix for each of the TLSs in time. What we are interested in is to
 145
7??? 2??  $E=O 2?- 
Figure 8.3: A cartoon illustrating how the TLS energy UTLS is affected
 by the microwave drive PBias, the bias, and the TLS relaxation PT1
 calculate the loss to the microwave field caused by the TLSs. Figure 8.3 illustrates
 that the TLS energy is effected by: the microwave field excitations, the bias field
 biasing TLSs, and T1 relaxation. Writing this as a conservation of energy equation
 gives
 dU
 dt = PMW + PBias ? PT1 . (8.37)
 Since we?re interested in the loss to the microwave field we simply solve for the
 microwave field
 PMW =
 ?
 TLS
 ~?TLS ??11 ?
 ?
 TLS
 ~?TLS
 ?11
 T1
 . (8.38)
 where we?ve evaluated the TLS energy as dU/dt =?TLS ~?TLS ??11+?TLS ~??TLS?11,
 the bias caused energy change as PBias =
 ?
 TLS ~??TLS?11, and the T1 relaxation
 caused energy change as PT1 =
 ?
 TLS ~?TLS?11/T1.
 146
We can now assume that the total energy in the system is
 U = Volume ? ? | ~E|2, (8.39)
 making the loss tangent of the system
 tan ? = PMW
 ?cU
 , (8.40)
 where ?c is the resonance frequency and is the frequency of the applied ac field in
 the simulation.
 Note that if we break up the sums in Eq. 8.38 when solving Eq. 8.40 we can
 get the loss tangent contribution from each individual TLS. Also note that I have
 not quantized the resonator for this result. Rather I?ve treated it as a classical
 externally applied field. This is an approximation that ignores some details of the
 behavior of the solution and treating the Jaynes-Cummings Hamiltonian is required
 to solve the full quantum mechanical problem.
 8.4.2 Monte Carlo steady state-results
 In this section I discuss the results from the Monte Carlo simulation when no
 bias is turned on and show that we can reproduce the well-known steady-state loss
 tangent response. Figure 8.4 shows the TLS distribution used to evaluate the loss.
 A T1,min of 10 ns was assumed for this simulation with the T1 distribution described
 in Chapter 2
 1
 T1
 =
 (
 ?0
 E
 )2 1
 T1,min
 . (8.41)
 I assumed that there was no pure dephasing and set T2 = 2T1. Iterative steps of
 ?t = 0.1 ns were used to solve for the density matrix. All TLSs were assumed
 147
0 5 10 15
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
 ?0/h (GHz)
 ?/h (GHz
 )
 Figure 8.4: The Monte Carlo TLS distribution used for the simulation.
 The black curve represents the resonance frequency of the cavity. The
 limits set on the TLS distribution were 0.01 GHz < ETLS/h < 15 GHz
 and 0 < arctan(?/?0) < 9pi/20.
 148
to have the same dipole moment p = 1 D and the TLS density used was P0 =
 1.28 ? 1042J?1m?3, calculated from the number of TLSs (8000), the distribution
 bandwidth (14.99 GHz), and the maximum angle used for the TLS distribution
 (arctan(?0/?)max = 9pi/20)
 NTLS =
 ?
 d3r
 ?
 P0d?d?0
 ?0
 = V P0
 ? 9pi/20
 0
 d?
 cos ?
 ?
 BW
 dE
 = V ? P0 ? BW ? 2.54 ? P0 = NTLS/(V ? BW ? 2.54). (8.42)
 I also set the volume to V = 100 nm ? 50 ?m ? 50 ?m and the resonance frequency
 to f = 5 GHz. Figure 8.4 shows and example of a distribution of 8000 TLSs and
 the black curve represents the resonance frequency used.
 The loss tangent was calculated using Eq. 8.40 after 100 ns of propagation,
 averaged for 100 time steps (10 ns). Figure 8.5 shows the loss tangent calculated
 for these conditions with several Eac drive fields. The dashed line is a 1/E reference
 slope, the vertical black line at 2.2?103 V/m represents the expected characteristic
 saturation field with these parameters calculated from Ec = ~/(?T1,minT2p) and
 the horizontal black line at 8 ? 10?8 represents the expected intrinsic loss tangent
 from the parameters calculated by P0p2/(3). The value for Ec matches well with
 the simulated loss tangent, reaffirming the integrity of the simulation and the as-
 sumptions used. Figure 8.5 shows a variation of about a factor of two between the
 simulated intrinsic loss and the analytically calculated value (horizontal line) but
 statistical fluctuations in the simulation are on that scale.
 Next I look at the transients of the loss tangent to see how the steady state
 loss in Fig. 8.5 is reached, both at low ac fields for the intrinsic loss (see Fig. 8.6(a))
 149
100 102 104 106 108
 10?11
 10?10
 10?9
 10?8
 10?7
 10?6
 1/E
 ERMS (V/m)
 ta
 n?
 Figure 8.5: Calculated loss tangent vs. applied electric field from the
 Monte Carlo TLS simulation with Eq. 8.40 averaged for the last 100
 time steps (10 ns) of the simulation. The dashed line is a 1/E reference
 slope. The simulation reproduces the expected TLS loss with a field
 independent low field loss and an onset of ? 1/E loss tangent behavior.
 The steepening to a 1/E2 response comes from hitting limits to the
 TLS distribution in the simulation. The vertical black line at 2.2 ?
 103 V/m represents the expected characteristic saturation field Ec =
 ~/(?T1,minT2p) and the horizontal black line at 8 ? 10?8 represents the
 expected intrinsic loss tangent P0p2/(3). The deviation between the
 simulated intrinsic loss tangent and the horizontal black line is due to
 statistical fluctuations in the number of TLSs near resonance in the
 simulation.
 150
and at high ac fields (see Fig. 8.6(b)). As seen in Fig. 8.6, the loss starts at around
 the same value for both high and low ac field drives. This happens because initially
 all the TLSs are in their ground state and will absorb photons proportional to
 the drive field energy. For low drive (Fig. 8.6(a)) the TLSs aren?t excited very
 much and the loss response stays around the same with roughly 10% fluctuations
 in tan ?. In contrast for high drive (Fig. 8.6(b)) the TLSs are saturated and their
 loss contribution decreases after sufficient time (determined by their Rabi frequency
 and relaxation time).
 8.4.3 A closer look at the response of the TLS distribution
 Figure 8.7 shows the TLS excitation response as ac drive is increased. In this
 figure, each point represents a TLS with its color scale representing its excitation
 probability (blue being ground state and red being excited state). For each sim-
 ulation the same TLS distribution is used but with a different electric field drive.
 Figure 8.7(d) with the highest electric field drive shows that this method also sim-
 ulates two and three photon processes (10 and 15 GHz). Figure 8.7(d) also shows
 that at these relatively large ac fields the simulation starts being limited by the TLS
 distribution used. This is why at high fields in Fig. 8.5 the loss response changes
 from what is expected for a broad distribution of, ? 1/E, to the expected response
 from a single TLS, ? 1/E2.
 The loss tangent contribution of individual TLSs with 105 V/m ac field drive
 is plotted in Fig. 8.8. As can be seen in Fig. 8.8 the loss contribution of TLSs is
 151
1
 2
 3
 4
 ta
 n?
  x 1
 07
 0 20 40 60?1
 0
 1
 2
 3
 t (ns)
 ta
 n?
  x 1
 07
 Eac,RMS=10
 3 V/m
 Eac,RMS=10
 5 V/m
 (a)
 (b)
 Figure 8.6: (a) Simulated loss as a function of time with an ac drive
 field of 103 V/m, near the intrinsic loss low field limit. Initially the loss
 is higher because all the TLSs are in their ground state. (b) Simulated
 loss as a function of time with an ac drive field of 105 V/m, above the
 TLS saturation knee in Fig. 8.5. Initially the loss is high because all the
 TLSs are in their ground state but, after a short period determined by
 the ac drive field, the TLSs are saturated and the loss decreases to the
 value found in Fig. 8.5 at ERMS = 105 V/m.
 152
0 5 10 15
 0
 5
 10
 15
  
 
?/h (GHz
 )
 0
 0.2
 0.4
 0.6
 0.8
 1
 0 5 10 15
 0
 5
 10
 15
  
 0
 0.2
 0.4
 0.6
 0.8
 1
 0 5 10 15
 0
 5
 10
 15
  
?0/h (GHz)
  
?/h (GHz
 )
 0
 0.2
 0.4
 0.6
 0.8
 1
 0 5 10 15
 0
 5
 10
 15
  
?0/h (GHz)
  0
 0.2
 0.4
 0.6
 0.8
 1
 (b)(a)
 (c) (d)
 Eac,RMS=10
 3 V/m Eac,RMS=10
 4 V/m
 Eac,RMS=10
 5 V/m Eac,RMS=10
 6 V/m
 ?11?11
 ?11?11
 Figure 8.7: The TLS distribution excited state probability (?) after equi-
 librium was reached with a 5 GHz drive field of 103 V/m (a) 104 V/m
 (b) 105 V/m (c) 106 V/m (d). As the drive field was increased the TLSs
 on resonance (as well as at higher modes of the resonance) began to
 saturate. As shown by Fig. 8.5, at higher fields the simulation became
 less accurate and (d) strongly suggests that this is due to the limits in
 the TLS distribution used.
 153
0 5 10 15
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18  
?0/h (GHz)
  
?/h (GHz
 )
 0
 1
 2
 3
 4
 x 10?11
 tan?
 Figure 8.8: Loss contribution from each TLS after equilibrium was
 reached with 105 V/m ac field drive.
 higher for TLSs with higher tunneling rates. This is both because it?s easier to drive
 transitions with higher tunneling rates (as indicated by the ?0/E terms in Eq. 8.29)
 and because TLSs with higher tunneling rates have shorter lifetimes (Eq. 8.41).
 8.4.4 Individual TLS response
 To gain a better understanding of the TLS loss I examined the loss contribution
 for individual planted TLSs in the simulation as a function of ac field drive. I chose
 nine TLSs at 4, 5, and 6 GHz with arctan(?/?0) = pi/6, pi/4, pi/3 (see Fig. 8.9
 154
inset). The loss from each of the TLSs as a function of the applied ac field is plotted
 in Fig. 8.9. As expected 4 and 6 GHz TLSs (dotted and solid lines respectively)
 contribute much lower loss than the resonant 5 GHz TLSs (dashed lines). And also
 as expected, as the tunneling rate to TLS energy ratio ?0/E is decreased (red to
 green to blue in Fig. 8.9) the TLS loss contribution decreases. As can be seen from
 the field dependence in Fig. 8.9, an individual TLS?s loss response goes as 1/E2.
 It?s the sum of an ensemble of TLSs that gives the 1/E response.
 Figure 8.10 shows a plot of the excited state probability ?11 for TLS number
 2 in Fig. 8.9 with increasing ac drive field. For low applied fields the TLS ?11
 increases to a steady state value (see Fig. 8.10(a,b)). For sufficiently high fields the
 TLS begins to show Rabi oscillations (see Fig. 8.10(c,d)).
 8.4.5 Monte Carlo simulation with a bias field
 Once convinced of the Monte Carlo?s integrity in simulating individual TLS
 loss I returned to the distribution of 8000 TLSs from Fig. 8.4 and turned on a bias
 field sweep that varied the asymmetry of each TLS by 2~p ? ~EBias. I simulated the
 TLS response for 200 ns and, as shown in Fig. 8.11(a), with the bias field increasing
 from 0 to 500 kV/m between 75 and 125 ns resulting in ?EBias = 1013 V/m/s. For
 this bias ramp Fig. 8.11(b) shows the corresponding loss response at 104 V/m ac
 field drive and Fig. 8.11(c) shows the response at 105 V/m. As predicted by the
 theory discussed in section 8.3, the loss tangent increased when the bias is swept.
 In Fig. 8.11(b), with the lower ac field drive, the loss increases to close to the same
 155
100 102 104 106
 10?14
 10?13
 10?12
 10?11
 10?10
 10?9
 10?8
 10?7
 Eac, RMS (V/m)
 ta
 n?
 0 2 4 6
 0
 2
 4
 6
 ?0/h (GHz)
 ?/h (GHz
 )
 1
 34
 6
 7
 9
 2
 5
 8
 2
 8
 5
 1,3
 7,9
 4,6
 Figure 8.9: The loss contribution from each of 9 TLSs that were sim-
 ulated at 4 (dotted line), 5 (dashed line), and 6 (solid line) GHz with
 arctan(?/?0) = pi/6 (red), pi/4 (green), pi/3 (blue).
 156
0 20 40 60 80 100
 0
 2
 4
 6 x 10
 ?4
 ? 1
 1
 0 20 40 60 80 100
 0
 0.02
 0.04
 0.06
 0 20 40 60 80 100
 0
 0.2
 0.4
 0.6
 0.8
 t (ns)
 ? 1
 1
 0 20 40 60 80 100
 0
 0.2
 0.4
 0.6
 0.8
 1
 t (ns)
 Eac,RMS=10
 2 V/m Eac,RMS=10
 3 V/m
 Eac,RMS=10
 5 V/mEac,RMS=10
 4 V/m
 (d)
 (b)(a)
 (c)
 Figure 8.10: Simulated excited state probability ?11 for TLS number 2
 from Fig. 8.9 with an ac drive of (a) 102 V/m (b) 103 V/m (c) 104 V/m
 and (d) 105 V/m.
 157
value it started with (t=0). In Fig. 8.11(c) when the ac field is relatively high and
 therefore the TLS Rabi frequency is high, there?s only a small increase in the loss
 tangent response during the sweep.
 In Fig. 8.12 I increased the field ramp sweep rate by a factor of 10 resulting in
 ?EBias = 1014 V/m/s. Here again Fig. 8.12(b) shows an increase in loss to close to
 the initial (t=0) loss value, but because of the increased bias rate Fig. 8.12(c) shows
 a much greater increase in the loss.
 8.4.6 Monte Carlo simulation of TLS dynamics and adiabatic rapid
 passage
 In this section I examine the excited state probability response of a single TLS
 in a swept bias field. As discussed in section 8.3, a TLS crossing the resonance
 frequency of the cavity can experience a Landau-Zener transition. If the bias is
 sufficiently fast, as in Fig. 8.13(a) where ?EBias = 9.88 ? 1014 V/m/s, then the TLS
 is likely to stay in the ground state, as shown in Fig. 8.13(b). But if the bias is
 sufficiently slow, as in Fig. 8.14(a) where ?EBias = 1.32 ? 1013 V/m/s, then the
 TLS experiences adiabatic rapid passage and is likely to be excited as shown in
 Fig. 8.14(b). The insets in Fig. 8.13 and Fig. 8.14 show the TLS before and after
 it was biased. In both simulations the Rabi frequency for the TLS is higher after
 the bias and that?s because as shown by the insets the TLS is further detuned
 from the resonance frequency after the bias. The lifetime used for this TLS was
 T1 = T2/2 = 83 ns and the ac drive was 105 V/m.
 158
0
 0.2
 0.4
 E Bia
 s MV/
 m
 0
 1
 2
 3
 ta
 n?
  x 1
 07
 0 50 100 150 2000
 0.5
 1
 1.5
 2
 t (ns)
 ta
 n?
  x 1
 07
 (a)
 (b)
 (c)
 Eac,RMS=10
 4 V/m
 Eac,RMS=10
 5 V/m
 Figure 8.11: (a) Bias field used in Monte Carlo simulation of distribution
 in Fig. 8.4 to sweep the asymmetry energy ?. (b) The loss tangent with
 104 V/m ac drive starts high then reaches an equilibrium after TLSs are
 saturated then when the bias is turned on and new TLSs are moved into
 the resonator?s bandwidth, the loss increases again. (c) The loss tangent
 with 105 V/m ac drive doesn?t increase as much as (b) when the bias is
 swept because the Rabi frequency of the TLSs is higher.
 159
0
 2
 4
 6
 E Bia
 s MV/
 m
 ?1
 0
 1
 2
 3
 ta
 n?
  x 1
 07
 0 50 100 150 200?1
 0
 1
 2
 t (ns)
 ta
 n?
  x 1
 07
 (a)
 (c)
 (b)
 Eac,RMS=10
 4 V/m
 Eac,RMS=10
 5 V/m
 Figure 8.12: (a) Bias field used in Monte Carlo simulation with a sweep
 rate 10 times greater than Fig. 8.11. (b) The loss tangent with 104 V/m
 ac drive starts high then reaches an equilibrium after TLSs are saturated
 then the loss increases again when the bias is turned on and new TLSs
 are moved into the resonator?s bandwidth. (c) Loss tangent with 105
 V/m ac drive, as in Fig. 8.11, the loss doesn?t increase as much as in
 (b) when the bias is swept because the Rabi frequency of the TLSs is
 higher but increases more than it did in Fig. 8.11(c) because the bias
 sweep rate is higher.
 160
0 10 20 30 40 50
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 E Bia
 s (MV/m
 )
 0 10 20 30 40 50
 0
 0.2
 0.4
 0.6
 0.8
 1
 t (ns)
 ? 1
 1
 0 5
 0
 2
 4
 6
 ?0/h (GHz)
 ?/h (GHz
 ) t=50 ns
 t=0
 (b)
 (a)
 Figure 8.13: (a) Bias field vs. time for a fast sweep (9.88 ? 1014 V/m/s)
 of a single TLS. (b) Simulation of excited state probability ?11 vs. time
 for the bias in (a). Because the bias was very fast compared to the Rabi
 frequency, the TLS stayed in its ground state. Inset shows asymmetry
 ? vs. tunneling ?0 for the TLS as the start and end of the sweep and
 the black curve represents the resonance frequency.
 161
0 10 20 30 40 50
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 E Bia
 s (MV/m
 )
 0 10 20 30 40 50
 0
 0.2
 0.4
 0.6
 0.8
 1
 t (ns)
 ? 1
 1
 0 5
 0
 2
 4
 6
 ?0/h (GHz)
 ?/h (GHz
 )
 t=0
 t=50 ns
 (a)
 (b)
 Figure 8.14: (a) Bias field vs. time for a slow sweep (1.32?1013 V/m/s)
 of a single TLS. (b) Simulation of excited state probability ?11 vs. time
 for the bias in (a). Because the bias was slow compared to the Rabi
 frequency, The TLS underwent adiabatic rapid passage and was pumped
 to the excited state with a high probability. Inset shows asymmetry ?
 vs. tunneling ?0 for the TLS as the start and end of the sweep and the
 black curve represents the resonance frequency.162
8.5 Conclusion
 In conclusion, I?ve discussed biasing TLSs with a field while probing and ex-
 citing them with an independent microwave (ac) field. It can be shown analytically
 that a TLS biased across the resonance frequency can experience a Landau-Zener
 transition. One consequence of this is that if the bias is sufficiently fast the loss
 tangent response of the TLSs at high ac fields approaches the unsaturated linear
 response at low ac fields [85]. I?ve also demonstrated a method for simulating the
 TLS loss response both with and without a swept bias field. Using this method I
 simulated the steady state TLS loss response, including the onset of TLS saturation
 as the ac field is increased. By simulating the system with a swept bias field, I also
 showed the predicted loss increases as the bias sweep rate increases. I also simulated
 the behavior of a TLS at two bias sweep rates and showed that for the fast bias
 sweep rate a TLS is likely to stay in the ground state as it crosses the resonance
 frequency while for a slower bias sweep rate the TLS experienced adiabatic rapid
 passage and is likely to be excited by the resonator.
 163
Chapter 9
 Biased resonator results
 In this chapter I discuss the experimental setup I used for measuring the effects
 of a swept bias on a resonator and I compare the results from the experiment to the
 theory discussed in Chapter 8.
 9.1 Biased resonators
 9.1.1 The resonator
 For these experiments I designed (most of the design was done with my advi-
 sor Kevin Osborn who suggested this experiment) and fabricated a resonator with
 capacitors that can be biased independently from the microwave (ac) resonant field.
 Figure 9.1(a) shows an optical image of a devices that is nominally identical to the
 one measured (from the same wafer) and Fig. 9.1(b) shows a schematic of the res-
 onator design. The resonator was composed of a meandering inductor and four 250
 nm thick amorphous SiNx-dielectric parallel-plate capacitors. As shown by Fig. 9.1,
 the resonator capacitance was broken up into four nominally equivalent capacitors
 with capacitance C = 1.8 pF in a bridge design, such that the equivalent resonator
 capacitance was C. One side of the capacitance bridge was grounded and the other
 was connected to a bias line with an applied bias voltage Vbias creating a bias of
 Vbias/2 across each capacitor. As with previously discussed resonators, the reso-
 164
C1 
C2 
C3 
C4 
L
  
80?m 
Ground Plate 
Bias Plate 
Al
  
VAC, in VAC, out 
(a) (b) VBias 
C1 
C2 
C3 
C4 
L
  
VBias 
Figure 9.1: (a) Optical image of the thin-film superconducting alu-
 minum resonator (nominally identical to devices bias1-4.7GHz-A and
 bias1-4.7GHz-B) comprised of a meandering inductor and four equal
 amorphous SiNx parallel-plate capacitors. C1 and C3 are connected to
 the ground plane while C2 and C4 are connected to a bias plate. The res-
 onator is inductively coupled to a CPW which is probed at microwave fre-
 quencies. (b) Circuit schematic of the resonator highlighting the bridge
 nature of the capacitors and their bias.
 nance is driven through coupling to a CPW transmission line, creating a notch
 filter. As will be shown in the next section, the bias line is not coupled to the
 fundamental microwave resonance so as not to limit the resonance quality factor.
 The reason the coupling to the bias line is so weak on resonance is because at the
 location on the bias plate where the bias line is connects there is very little voltage
 and current from the fundamental microwave resonance mode.
 9.1.2 Resonator simulations
 Figure 9.2(a) shows the resonator design used for the Microwave Office sim-
 ulation. A third port was added to simulate the bias line. As in the experiment,
 165
the bias line was separated from the ground plane by 250 nm of SiNx dielectric. As
 shown by the transmission simulations in Fig. 9.2(b), port 3 (bias port) has less
 than -40 dB coupling to port 1 even on resonance. This corresponds to limiting the
 internal quality factor of the device to about Qi < 2 ? 106, much higher than the
 expected quality of the dielectric used (SiNx), indicating that port 3 can be assumed
 to be isolated from the resonant circuit and will not affect its quality factor.
 9.1.3 Resonator fab
 Fabrication of this multi-layer resonator is more complicated than the fabri-
 cation of the coplanar resonators discussed in Chapter 4. For the bottom layer?
 comprised of the meandering inductor, the bottom plate of the capacitors and the
 ground plane?aluminum was sputtered, patterned and etched in exactly the same
 way that was discussed for the coplanar resonators in chapter 4. Next 250 nm of
 SiNx was deposited with an Oxford high density plasma enhanced chemical vapor
 deposition (HD-PECVD) system. Nitrogen (N2) and silane (SiH4) precursor gasses
 were used with flow rates of 11 sccm and 10 sccm, respectively. The depositions rate
 was calibrated by first depositing SiH4 on a silicon wafer and then measuring the
 thickness with an n&k diffraction measurement system [89]. Next I spun the same
 OIR 906-10 photoresist [58] that was used for the bottom layer in the same man-
 ner. The photoresist was patterned with a stepper and developed with OPD-4262
 developer [58] again using the same technique discussed for the bottom aluminum
 layer in chapter 4. Next I exposed the bottom aluminum layer by etching vias in
 166
565 575 
B (GHz) 
Tr
 an
 sm
 is
 si
 on
  

?
?
? ?
 ? 
Figure 9.2: (a) Layout of 4-capacitor bridge resonator used for Microwave
 Office simulations. (b) Simulated transmission from port 1 to ports 2
 (magenta) and 3 (blue). The coupling between ports 1 and 3 is very low
 at the resonant frequency (less than -40 dB) thus the bias line will not
 limit the resonator?s loss.
 167
the SiNx were etched using a sulfur hexafluoride (SF6) reactive ion etching (RIE)
 process. After cleaning the photoresist, in the same manner discussed earlier, the
 wafer was again put in the sputtering chamber where it was ion milled with argon
 at 800 V beam voltage for 60 seconds in order to ensure that the native oxide on the
 bottom aluminum layer was entirely etched for good electrical contact. Next, the
 top aluminum layer (200 nm) was sputtered, patterned, and etched leaving only the
 top plates of the capacitors, the vias connecting the capacitors to the inductor, and
 the bias line connecting to the bias plate. Finally the excess Si3N4 was patterned
 and etched away (using the same RIE process) everywhere except where it was re-
 quired to support a top aluminum layer (the capacitors and the bias line). Once the
 fabrication process was complete the wafer was diced (in the same manner discussed
 for the coplanar devices in Chapter 4) and mounted in a copper box similar to that
 shown in Fig. 4.6. The devices was wire bonded to in the same manner discussed
 in Chapter 4 with the simple addition of also wire bonding to the bias line.
 9.2 Dependence on loss on a static (DC) bias voltage
 The first measurement I made on these devices was to bias the resonator with
 a static (DC) electric field and see if the TLS loss was affected. Figure 9.3 shows
 measurements of the loss tangent taken at 30 mK for device bias1-4.7GHz-A. The
 presence of a DC bias field did not have a significant effect on the TLS loss even
 with a bias field of 200 V/m, corresponding to a shift of 2.0 THz for the asymmetry
 energy (?/h) of a TLS assuming a 1 Debye dipole moment aligned with the field.
 168
10?1 100 101 102
 10?4
 10?3
 Eac, RMS (V/m)
 ta
 n?
  
 
0V (0 V/m)
 25V (50 MV/m)
 50V (100 MV/m)
 75V (150 MV/m)
 100V (200 MV/m)
 Figure 9.3: Loss tangent measurements of resonator bias1-4.7GHz-A at
 30 mK as a function of the microwave excitation field taken with various
 DC bias fields. The DC bias fields showed no substantial effect on the
 loss.
 For higher biases the SiNx broke down, and thus I can only conclude that the TLS
 asymmetry energy distribution is flat in the 0-2 THz range.
 As mentioned in Chapter 8, even if a systematic shift in loss tangent as a
 function of bias field is not measured, the measurement of random shifts in loss
 tangent due to the discrete nature of the TLSs is still possible. Figure 9.4 shows
 measurements where such texture in both loss (Fig. 9.4(b)) and dielectric constant
 (Fig. 9.4(c)) is resolved by varying the bias field (Fig. 9.4(a)). As shown in Fig. 9.4
 169
when a bias field is applied, the shifts in loss and dielectric constant (capacitance)
 are random, as expected. It was somewhat surprising however to see the loss and
 permittivity systematically return to the same value when the bias is turned off. This
 suggests that the TLSs are not rearranging themselves in some irreversible, glassy
 manner when the bias field changes. Note that these measurements were taken over a
 period of several hours which seems like a relatively long timescale for the stability of
 atomic systems. We can also use the scale of these TLS induced loss tangent jumps
 to estimate the number of TLSs in the resonator?s bandwidth. The loss tangent
 jumps are on the scale of about ? tan ?/ tan ? ? 0.02, and since the loss tangent
 scales with the number of TLS in the resonator, we can say 1/?N ? 0.02. Thus
 the number of TLSs in the bandwidth of the resonator is approximately N ? 2500,
 which for the capacitor volume of V = 4 ? 80 ?m ? 80 ?m ? 250 nm, a resonator
 bandwidth of about 1 MHz, and a TLS linewidth of about 1 MHz corresponds to a
 TLS spectral spatial density of about P0 ? 1043, which as I will show later in this
 chapter comparable to another measurement I preform in this system.
 9.3 Loss with swept bias
 9.3.1 Pulsed bias effect
 As discussed in the previous section, the changes in TLS loss due to a DC
 bias were typically only a few percent. I next discuss results from measurements
 of the loss tangent response with a rapidly swept bias field. Figure 9.5(a) shows
 a bias pulse of 40 MV/m with a rise and fall time of 8.5 ms. The color scale in
 170
0 2 4 6 8 10
 0
 5
 10
 E Bia
 s (kV/m
 )
 0 2 4 6 8 10
 ?10
 0
 10
 ?C/C x 1
 06
 0 2 4 6 8 10
 ?0.05
 0
 0.05
 ?tan
 (?)/tan
 (?)
 time (s)
 (a)
 (b)
 (c)
 Figure 9.4: (a) Three bias field pulses applied sequentially. (b) The
 corresponding fractional change in loss tangent from the value at zero
 bias field in device bias1-4.7GHz-B. The shifts are due to the random
 and discrete nature of the TLSs. (c) Similar (but smaller) random shifts
 are also observed in the capacitance, also caused by small changes in the
 TLS distribution.
 171
Fig. 9.5(b) shows the transmission through the CPW of device bias1-6.5GHz-A at
 the resonance frequency. Before the bias ramp start up (step up) the resonance line
 shape is constant, but once the bias step occurs, the resonance line shape becomes
 wider and shallower indicating a decrease in the quality factor. After some time at
 the constant bias field of 40 MV/m, the resonance line shape returned to its original
 value. When the bias ramp down occurred, the resonator responded again in the
 same manner, with a lower quality factor that eventually returns back to its original
 value.
 Each of the time slices in Fig. 9.5(b) were fit to a resonance line shape and
 the extracted loss tangents are plotted in Fig. 9.5(c). Examination of Fig. 9.5(c)
 makes it clear that the loss jumps to the same value at both the step up and down
 of the bias field and returns to this same value once the bias field stabilizes. As
 discussed in Chapter 8, this behavior is caused by TLSs being swept through the
 resonance frequency and taking energy from the microwaves not only by relaxing but
 also by moving outside the resonator?s bandwidth. It is also worth remarking that
 these effects were observed experimentally before the phenomenon was explained
 theoretically, and it was this peculiar behavior which motivated the theoretical effort.
 9.3.2 Biased and steady state losses
 Based on the model discussed in chapter 8 [85], for a sufficiently fast bias
 rate, the loss tangent can be expected to increase to its intrinsic unsaturated value.
 Figure 9.6(a) shows the steady state loss tangent measurements for the resonator at
 172
1 
0.8 
0.6 
S21 
B (GHz
 ) 
tan?
 H s r7  
E B
 ia
 s 
(MV/
 m)
  
time (s) 
(a) 
(b) 
(c) 
Figure 9.5: (a) Bias electric field pulse applied with a rise and fall time
 of 8.5 ms resulting in a maximum bias rate of ?EBias = 4.7 GV/m/s. (b)
 Transmission of CPW showing the resonance line shape as the bias is
 applied. Both at the rise and fall of the pulse the resonance line shape
 becomes shallower and broader indicating an increase in the loss. (c)
 Each time slice in (b) is fit to a resonance line shape and the extracted
 loss tangent is plotted. The loss tangent initially increases as the bias
 field is swept up or down and then relaxes back to its steady state value.
 This measurement was performed on device bias1-6.5GHz-A.
 173
33 and 200 mK as a function of the magnitude of the ac resonant field. Figure 9.6(b)
 shows the loss tangent response at those same temperatures while exciting with
 a single ac drive field as the bias is ramped (see inset). Comparing Fig. 9.6(a)
 to Fig. 9.6(b), we see that the loss tangent does in fact increase to the intrinsic
 unsaturated value at both temperatures.
 In Fig. 9.6(c) the intrinsic loss tangent at low ac field with no bias (black
 triangles) is compared to the maximum loss tangent jump height with a ramped
 bias field (blue circles). The loss for a ramped bias field show an approximately
 5% greater loss than the linear-response loss tangent at the lowest temperatures.
 Assuming this non-equilibrium loss should exactly equal the linear response loss, it
 is possible that the small systematic difference may be due to slow TLS interactions,
 which move TLSs towards the global ground state that creates a slightly different loss
 tangent [83]. The solid curve in Fig. 9.6(c) is a fit to tan ?0 tanh(~?/2kBT ), which
 describes the thermal saturation of the loss tangent in the conventional theory. I
 find good agreement with this fit.
 9.4 Varying bias rate
 As discussed in chapter 8, the loss tangent is dependent on the bias ramp rate
 and the Rabi frequency (ac field amplitude). At low bias ramp rates we expect to
 be in the steady state limit, resulting in a loss tangent that is dependent on the ac
 field amplitude according to the conventional tunneling model:
 tan (?) = tan ?0 tanh (~?/2kBT )?
 1 + (Eac/Ec)2 . (9.1)
 174
10?1 100 101 102
 10?4
 10?3
 Eac, RMS (V/m)
 ta
 n?
  
 
0 1 2 3
 time (s)
  
 
20
 40
 33mK
 200mK
 33mK
 200mK
 (a) (b)
 Ebias(MV/m)
 0 50 100 150 200
 3
 3.5
 4
 4.5
 5
 5.5
 6
 6.5
 Temperature (mK)
 ta
 n?
  x 1
 04
  
 
Intrinsic Steady?State
 Biased Jump Height
 (c)
 Figure 9.6: (a) Steady-state loss tangent, measured in device bias1-
 4.7GHz-B as a function of the microwave field at temperatures of 33
 (blue) and 200 (red) mK. (b) Corresponding time-dependent loss tan-
 gent measured (main panel) with an electric field bias that sweeps up
 (down) at 0.53 s (1.78 s) (inset). (c) Comparison of steady state intrin-
 sic unsaturated loss tangent (blue circles) to the maximum loss tangent
 reached with a swept bias (black triangles) as a function of tempera-
 ture. The solid curve shows a fit of the conventional theory of thermally
 saturated TLS, tan ?0 tanh(~?/2kBT ), to the steady state data.
 175
At high bias rates, as seen in Fig. 9.6, the model in Chapter 8 predicts a loss tangent
 equal to the intrinsic unsaturated value for the material. In between, the same model
 predicts a loss tangent that is dependent on the dimensionless bias rate
 ? = 2?0
 pi?2R0
 , (9.2)
 where, as discussed in chapter 8,
 ?0 = 2
 p
 ~
 dEbias
 dt , (9.3)
 and
 ?R0 =
 pEac
 ~ . (9.4)
 Here I?ll discuss measurements of loss tangent as a function of the bias rate and
 show that I get a good fit to the model discussed in Chapter 8 and in Ref. [85].
 9.4.1 Methods for varying bias rate
 To vary the bias rate, I used a voltage supply with a constant rise time and
 varied the amplitude of the bias. I used a Stanford Research Systems SIM928
 Isolated Voltage Source with a characteristic rise/fall time of 8.5 ms. I varied the bias
 pulse amplitude (see Fig. 9.7(a)) which resulted in varying loss tangent responses
 (see Fig. 9.7(b)). Figure 9.7(c) shows the resulting plot of loss tangent as a function
 of bias rate bound by taking the maximum loss tangent jump height from each
 pulse and plotting it against the maximum bias rate. The maximum bias rate was
 calculated by dividing the pulse amplitude by the characteristic rise and fall time
 of the pulse.
 176
100
 102
 104
 E Bia
 s(kV/m
 )
 0 1 2
 0
 2
 4
 6
 time (s)
 ta
 n?
  x 1
 04
 (a)
 (b)
 102 104 106 108
 0
 2
 4
 6
 8
 ta
 n?
  x 1
 04
 ?EBias(V/m/s)
 (c)
 Figure 9.7: (a) Various bias pulse amplitudes with the same rise and fall
 time of 8.5 ms resulting in various bias sweep rates. (b) Loss tangent
 as response for the various bias pulses. (c) Plotting the maximum loss
 tangents reached from (b) against the corresponding maximum bias rate
 from (a).
 177
Another technique used see how the loss tangent depends on the bias rate is
 to add a low pass filter to the voltage supply, slowing the characteristic rise and fall
 time of the voltage bias to 50 ms (see Fig. 9.8(a)). The resulting response of the
 loss tangent (see Fig. 9.8(b)) can be monitored as the time derivative of the voltage
 bias changes in time. Thus in Fig. 9.8(c) I plotted the loss tangents from Fig. 9.8(b)
 against the color coded instantaneous bias field time derivative (bias rates) from
 Fig. 9.8(a). With this technique I was able to get the entire loss tangent vs. bias
 rate S-curve with a single pulse.
 Figure 9.9 shows my results for device bias1-4.7GHz-B from both techniques
 showing that they produce the same loss vs. bias rate S-curve. When plotting the
 instantaneous loss tangent against the instantaneous bias rate I used both the step
 up (blue) and the step down (red). The dashed curve in Fig. 9.9 is the numerical
 evaluation of Eq. 8.18 which was also plotted in Fig. 8.2. In Fig. 8.2 the loss
 tangent is plotted as a function of the dimensionless bias rate
 ? = 2?0
 pi?2R0
 =
 4~ ?EBias
 pE2ac
 , (9.5)
 were the ac field across the resonator on resonance, Eac, is known. However, the
 dipole moment p of the TLSs is unknown. The dashed curve in Fig. 9.9 is a fit of
 the model to the data by using the dipole moment as a free parameter. Using this
 method I extracted a dipole moment of 7.9 ? 0.5 Debye. It is noteworthy that this
 technique allows for determination of the TLS dipole moment.
 Examining the dashed curve in Fig. 9.9, we see however that the fit only
 catches the top of the S-curve. This is because, as discussed in chapter 8, when
 178
0
 20
 40
 E Bia
 s (MV/m
 )
 0 1 2 3 4
 0
 2
 4
 6
 time (s)
 ta
 n?
  x 1
 04
 (b)
 (a)
 104 106 108 1010
 0
 2
 4
 6
 8
 ?EBias(V/m/s)
 ta
 n?
  x 1
 04
 (c)
 Figure 9.8: (a) Applied bias pulse with a rise and fall time of 50 ms.
 (b) Loss tangent response to the applied bias. (c) The instantaneous
 loss tangent response from (b) as a function of the instantaneous time
 derivative of the applied bias from (b) (the bias rate).
 179
deriving Eq. 8.18 we neglected TLS decoherence (T1 = T2 = ?) and assumed all of
 the loss comes from the Landau-Zener process produced by biasing the TLSs. Thus
 it is not surprising that the calculation fails when the bias is so slow that the loss
 due to TLS decoherence cannot be ignored (? ? ?R/?T1,minT2).
 The solid curve in Fig. 9.9 is a fit to the data using a Monte Carlo averaged
 solution of the TLS density matrix similar to that discussed in Chapter 8. Note
 that this Monte Carlo simulation was performed with code written by Professor
 Alex Burin from Tulane University and is more streamlined than the Monte Carlo
 simulations I created and discussed in Chapter 8 (i.e. only TLSs near resonance
 were considered). The solution assumed a T1 limited T2 (T2 = 2T1) and that there
 was a loss tangent background of tan ?floor = 1.8? 10?5, as found from steady-state
 saturation measurements. From this fit we extract the same average TLS dipole
 moment of p = 7.9 Debye, and a TLS relaxation time of T1 = 3.0 ?s. This value of
 p is comparable to that found for TLSs in an amorphous Al2O3 Josephson junction
 tunneling barrier [15] and OH in alkali halides [90]. However, unlike previous mea-
 surements our ramped bias technique does not require fabrication of a Josephson
 junction and allows for TLS dipole measurements in any insulating film thickness.
 From the dipole moment and the intrinsic loss tangent, I can calculate a value of
 P0 = 4.9 ? 1043J?1m?3 for the TLS spectral spatial population density using the
 relation tan ?0 = p2P0/(3).
 180
104 106 108 1010
 0
 0.2
 0.4
 0.6
 0.8
 1
 ?EBias(V/m/s)
 ta
 n?
 /ta
 n?
 0
  
 
Bias step up
 Bias step down
 Varying pulse amplitude
 Landau?Zener calculation
 Monte Carlo simulation
 Figure 9.9: Loss tangent as a function of bias rate in device bias1-
 4.7GHz-B taken with different techniques. The blue (red) symbols show
 the loss tangent during the 50 ms rise (fall) time step up (down) of a
 bias pulse. The black symbols show data taken by varying the bias step
 amplitude and using the maximum loss tangent measured. The dashed
 curve is a fit based on the theory discussed in chapter 8 which assumes
 TLSs with infinite relaxation times experience a Landau-Zener transi-
 tion as they?re biased through the resonator?s bandwidth. It therefore
 fails at low bias rates where the loss becomes limited by TLS relaxation
 instead of the bias. The solid curve is a fit using a Monte Carlo averaged
 solution of the TLS density matrix (see section 9.4.1).
 181
9.4.2 Fitting S curve for different microwave (ac) powers and extract-
 ing the TLS dipole moment
 The next measurement I made on device bias1-4.7GHz-B was a bias field
 measurement with different ac drive strengths (see Fig. 9.10). Note that while the
 amplitude of the input ac field is constant for any given curve, as the loss tangent
 varies it also changes the ac field across the capacitors; the low sweep rate (steady-
 state) ac field is given for each curve in Fig. 3A. The ac field for the middle curve
 (blue) varies from Eac = 4.55 V/m at the highest bias rate to Eac = 15.52 V/m at
 the lowest bias rate (the steady-state regime). To calculate Eac for a given bias, I
 first measure the quality factor of the resonator, as discussed earlier, and then use
 Eq. 3.22 along with the microwave line calibrations. As expected, the different ac
 field measurements approach the same intrinsic loss tangent limit for fast bias rates
 and they saturate at different steady state loss tangents, based on their ac field
 drive.
 A qualitative way to understand the shifts in bias rate between the different ac
 field curves in Fig. 9.10 is that the ac field is proportional to the Rabi frequency of
 the TLSs and, as discussed earlier, the Rabi frequency is what you need to compare
 the bias rate to decide if the bias was fast enough for the TLS to Landau-Zener
 transition and remain in the ground state. The key quantity is the dimensionless
 bias rate ? = 2?0/(pi?2R0).
 In Fig. 9.11(a) the solid curves are fits of the data from Fig. 9.10 for different
 ac fields to the same Monte Carlo solution discussed earlier (see section 9.4.1).
 182
104 106 108 1010
 0
 2
 4
 6
 ta
 n?
  x 1
 04
 ?Ebias(V/m/s)
 E
 ac, ss
 =3.4 V/m
 E
 ac, ss
 =16 V/m
 E
 ac, ss
 =58 V/m
 Figure 9.10: Loss tangent vs. bias field ramp rate in device bias1-
 4.7GHz-B for three different input ac fields.
 The fits show excellent agreement especially considering that a single value for the
 dipole moment was assumed. The curves in Fig. 9.11(a) are rescaled in Fig. 9.11(b)
 by plotting them against the dimensionless bias rate ?, using the extracted dipole
 moment p = 7.9 D. I find a good collapse to a single curve at high bias rates where
 the loss is limited by the bias rate (? > ?R/?T1,minT2), but the curves don?t collapse
 at low bias rates since there the loss is limited by TLS relaxations which the model
 neglected.
 Figure 9.12 shows another version of the data from Fig. 9.11. Here I also
 rescaled y-axis by subtracting the zero bias rate steady-state loss tangent value
 (tan ?ss) for each of the three sets of ac field amplitudes to account for the varying
 steady-state losses. This procedure is ad hoc and not predicted by theory, but is
 183
104 106 108 1010
 0
 0.2
 0.4
 0.6
 0.8
 1
 ta
 n?
 /ta
 n?
 0
 ?Ebias(V/m/s)
 10?4 10?2 100 102 104
 0
 0.2
 0.4
 0.6
 0.8
 1
 E
 ac, ss
 =3.4 V/m
 E
 ac, ss
 =16 V/m
 E
 ac, ss
 =58 V/m
 ta
 n?
 /ta
 n?
 0
 ?
 (a)
 (b)
 Figure 9.11: (a) The loss tangents from Fig. 9.10 normalized by the
 intrinsic zero field loss. The solid curves are fits from a Monte Carlo
 averaged solution of the TLS density matrix where the same dipole mo-
 ment was assumed (p = 7.9 D) for all three fits (i.e. no additional free
 parameters). (b) The bias rate from (a) rescaling by the dimensionless
 bias rate ?. The three curves converge at high bias rates where the loss
 is limited by the bias rate and not by TLS relaxation.
 184
done to emphasize the agreement in the scaling. From Fig. 9.12 it?s even clearer
 that rescaling by the dimensionless bias rate ? collapses the data onto one curve.
 9.5 S curves at 100 mK
 All of the swept bias rate measurements discussed thus far in this chapter have
 been taken at 33 mK on device bias1-4.7GHz-B. In Fig. 9.13 I show similar measure-
 ments on the same device taken at 100 mK. Figure 9.13(b) shows the corresponding
 plot of the loss vs. dimensionless bias rate ?. Again, we see good collapse onto one
 curve at high ramp rates but poor collapse as they approach the steady state regime
 where the loss is limited by TLS relaxation. The solid curve in Figure 9.13(b) shows
 a fit to the numerical evaluation of Eq. 8.18 discussed in chapter 8 and plotted in
 Fig. 8.2. The three horizontal lines indicate the known loss tangent values of the
 three data sets measured in the steady state regime. These fits yield a TLS dipole
 moment of p = 8.0 Debye, close to that extracted from the 33 mK data.
 9.6 Conclusion
 In conclusion, with help from my advisor Kevin Osborn, I designed, fabricated,
 and measured a parallel-plate thin-film microwave (ac) resonator which due to its
 bridge design allowed for electric field biasing of the capacitors independent of the ac
 field drive. This biased the asymmetry energy ? of the TLSs in the SiNx dielectric,
 shifting their energies in and out of the resonator?s bandwidth. Consistent with a
 flat distribution of the TLS asymmetry energies, only small random changes of a
 185
104 106 108 1010
 0
 0.2
 0.4
 0.6
 0.8
 1
 y
 ?Ebias(V/m/s)
 10?4 10?2 100 102 104
 0
 0.2
 0.4
 0.6
 0.8
 1
 ?
 y
 (a)
 (b)
 Figure 9.12: (a) Plot of the quantity y = (tan ??tan ?ss)/(tan ?0?tan ?ss)
 vs. bias rate. This is rescaling Fig. 9.12 by subtracting the zero bias rate
 steady-state loss tangent value (tan ?ss), effectively pinning the data to
 zero at low bias rates. (b) Data from (a) with x-axis rescaled by the
 dimensionless bias rate ?. The rescaling collapses the three curves on
 top of each other.
 186
104 106 108 1010
 0
 0.2
 0.4
 0.6
 0.8
 1
 ta
 n?
 /ta
 n?
 0
 ?Ebias(V/m/s)
 10?2 100 102 104
 0
 0.2
 0.4
 0.6
 0.8
 1
 E
 ac, ss
 =2.0 V/m
 E
 ac, ss
 =10 V/m
 E
 ac, ss
 =53 V/m
 ta
 n?
 /ta
 n?
 0
 ?
 (a)
 (b)
 Figure 9.13: (a) Normalized loss tangent as a function of the bias rate
 sweep rate for three different input ac fields taken at 100 mK. (b) Rescal-
 ing the bias rate from (a) to the dimensionless bias rate ? and collapsing
 the three curves at high bias rates where the loss is limited by the bias
 rate and not by TLS relaxation. Solid curve shows a fit based on the
 theory discussed in chapter 8 assuming a TLS Landau-Zener transition.
 The horizontal lines indicate the low bias limit loss tangent measured in
 the steady state regime.
 187
few percent were observed in the TLS loss tangent.
 In contrast, when the bias was swept, an increase in loss tangent was observed.
 For a sufficiently fast bias rate the loss approached the intrinsic unsaturated loss
 tangent of the material, consistent with the theory discussed in Chapter 8. This be-
 havior was seen for temperatures spanning 30 to 200 mK. The loss tangent increased
 with the bias rate, from its low bias steady state limit to its high bias intrinsic loss
 tangent value. The variation of loss tangent with bias rate was fit to a calculation
 based on the TLSs making Landau-Zener transitions as they crossed the resonance
 field. This model did not capture the loss caused by TLS relaxation, so a Monte
 Carlo simulation of the TLS density matrix was used to fit the data for the full bias
 rate range. This analysis allowed me to deconvolute the TLS dipole moment p from
 the TLS spectral density P0 which cannot be done in the corresponding steady-state
 measurement.
 188
Chapter 10
 Conclusion
 10.1 Summary of key results
 10.1.1 Diameter correction method for extracting Qi
 In chapter 3 I analyzed the transmission through a non-ideal resonant circuit
 and it was found that the empirically measured resonance line shape asymmetry
 can be modeled analytically. I also found that the conventional method used for
 extracting Qi from an asymmetric resonance line shape (?RM) systematically over-
 estimated Qi and proposed a novel method (DCM) for correctly extracting Qi from
 an asymmetric resonance line shape.
 10.1.2 ALD dielectric results
 In chapter 7 I discussed my measurements of loss in several ALD grown di-
 electrics by fabricating and measuring coplanar aluminum resonators on top of the
 ALD grown films. By comparing low temperature microwave measurements to room
 temperature SIMS measurements of the dielectrics, I was able to correlate excess
 hydrogen impurities to low temperature microwave loss. I also found that the amor-
 phous dielectrics (AlOx and LaAlOx) showed excess surface hydrogen impurities and
 this was associated with greater surface loss than bulk loss. While the crystalline
 189
dielectric (BeO) showed excess carbon impurities at the surface it had a uniform
 hydrogen impurity concentration and the data could be explained by a single bulk
 loss tangent. This suggested that hydrogen impurities were the source of TLS loss
 in BeO film.
 I also described measurements in which I compared two ALD grown AlOx
 films deposited in different labs with nominally the same precursors. The films
 showed drastically different quantities of carbon impurities but similar hydrogen
 impurity levels. The two films had similar loss, consistent with hydrogen based
 TLSs. By comparing low temperature microwave loss measurements on AlOx to
 SIMS impurity analysis of the films, I found a low field loss tangent (intrinsic loss
 tangent) per hydrogen concentration of KH = 3 ? 10?24 cm3.
 10.1.3 Biased resonator results
 In Chapter 9, I described my work on electric field biased resonators. I fab-
 ricated and measured a biased bridge superconducting resonator designed to bias
 TLSs in the SiNx parallel-plate capacitors in and out of the resonator?s bandwidth.
 By applying several DC biases, I was able to observe small (a few percent) repro-
 ducible fluctuations in the TLS distribution but due to the wideband white distri-
 bution of TLS asymmetry energies ? the loss did not vary monotonically with the
 bias.
 In contrast, if I swept the bias sufficiently rapidly I observed an increase in loss
 tangent due to the biasing of unsaturated (ground state) TLSs into the resonator?s
 190
bandwidth and the biasing of excited TLSs out of the resonator?s bandwidth. When
 measuring with a high (saturating) microwave field, I found that a sufficiently high
 bias rate increased the loss tangent to the unsaturated intrinsic loss tangent of the
 material. This result motivated work on a theory in which TLSs undergo Landau-
 Zener transitions as they are biased across the resonator?s bandwidth. Using this
 model I fit my bias rate loss tangent data and was able to extract an average dipole
 moment of p = 7.9 Debye and a spectral spatial density of P0 = 4.9 ? 1043 J?1m?3
 for the TLSs in an amorphous film of Si3N4.
 10.2 Future work
 10.2.1 Trilayer resonator fabrication
 In chapter 7, I showed that in some cases dielectric loss can be limited by
 surface impurities that are likely introduced when the surface is exposed to the
 ambient atmosphere during processing. This suggests that the loss can be reduced by
 adopting trilayer fabrication techniques where the dielectric is grown in situ with the
 metal on top of it. Such an approach would be especially useful after optimizing the
 dielectric growth to minimize impurities introduced during the growth process, as
 discussed in chapter 7. Trilayer processes can also be useful for junction fabrication.
 For example such trilayer fabrication has recently been developed to improve phase
 qubits [91] and is currently being developed by B. Sarabi in Kevin Osborn?s group.
 191
10.2.2 Biasing TLSs in coplanar resonator
 As I discussed in Chapters 8 and 9, the ability to bias TLSs in a resonator
 can reveal much about their properties and their energy distribution. A possible
 extension of the bias experiments is to attempt TLS biasing for coplanar resonators
 with interdigital capacitors as discussed in chapters 4 and 5. A similar capacitor
 bridge design can be used but with IDCs instead of parallel-plate capacitors. This
 will allow for biasing of TLSs in resonators fabricated on low-loss wafers which can
 lead to a better understanding of loss mechanisms in such geometries. Work along
 these lines is now being pursued by K. Voigt et al. [92].
 10.2.3 Lasing and loss reduction with biased resonator
 While not discussed in Chapters 8 and 9, one implication of the swept bias
 analysis is that the TLSs biased across a resonator?s bandwidth can undergo pop-
 ulation inversion, and in principle this might allow which can theoretically be used
 for lasing. Attempts at measuring this phenomenon in the same device used for this
 thesis are already underway by Y. Rosen in Kevin Osborn?s group.
 A possible technique for taking advantage of TLS biasing to achieve is to fab-
 ricate a resonator with multiple modes that share the same parallel-plate capacitors.
 This would allow TLSs to be excited in one resonance mode and then biased into
 another resonance mode where they can be measured through lasing. A variation
 on this would be to have three modes in total: Two side modes excited with high
 microwave power to excite TLSs, and a middle mode for the TLSs to emit into by
 192
stimulated emission (lase). The reason for two side modes is to allow TLSs with a
 dipole in one direction to be excited by one side mode and TLSs with a dipole in
 the other direction to be excited by the other side mode.
 If this device with multiple modes is used, in addition to lasing, it can also be
 used to reduce the loss tangent of a material at low microwave fields. The idea is
 to continually saturate TLSs by applying power at the side modes and then biasing
 them into the central mode which is being probed with single photon energies in the
 unsaturated regime.
 10.3 Concluding remarks
 This field has progressed so much in the brief six years that I have been for-
 tunate to observe it. If you had asked me six years ago what I thought the first
 quantum computer (as determined by a reasonable person) would look like, I would
 have said a bunch of atoms with lasers everywhere. Today I believe it will be a su-
 perconducting circuit at the bottom of a dilution refrigerator. Part of that of course
 was my naivety at the time, but much of it is to the credit of the accomplishments of
 the superconducting quantum computing community in such a short time period. If
 we continue the two pronged approach of improving device performance empirically
 while not forgetting to hone a better understanding of decoherence mechanisms, I
 have little doubt that a quantum computer is achievable within my lifetime. Actu-
 ally I think (also hope) my lifetime is a very conservative estimate, but it seems like
 an appropriate timescale to reference.
 193
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