Circuit Quantum Electrodynamics - Yale School of Engineering ...

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LIST OF FIGURES 9 3.10 CPB circuit diagram / junction diagram . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.11 CPB energy levels and charge staircase . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12 Energy in two state approximation and fictitious field figure . . . . . . . . . . . . . . 64 3.13 CPB Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.14 Split CPB sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.15 Dipole moment of the Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.16 Measurement schematic and derivatives of CPB energy levels . . . . . . . . . . . . . 74 3.17 State dependent transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.18 Q function of coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.19 State dependent transmission with small phase shift . . . . . . . . . . . . . . . . . . 77 3.20 Selecting ∆r for optimal SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.21 Selecting κ/2χ for optimal SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 Decoherence through gate voltage coupling circuit diagram and SV(ω) at different temperatures 84 4.2 Quality factor of qubits coupled to transmission line and cavity . . . . . . . . . . . . 86 4.3 CPB coupling to slotline mode in resonator . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Flux noise circuit diagram and flux transition matrix elements . . . . . . . . . . . . 88 4.5 Electric Field distribution in CPB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Derivatives of energy with respect to gate charge . . . . . . . . . . . . . . . . . . . . 96 4.7 Thermal Dephasing of the qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.8 Dephasing of qubit due to 1/f charge noise . . . . . . . . . . . . . . . . . . . . . . . 98 4.9 Dephasing times due to flux and critical current noise . . . . . . . . . . . . . . . . . 101 4.10 CPB energy bands at different EJ/EC ratios . . . . . . . . . . . . . . . . . . . . . . 104 4.11 Anharmonicity vs. EJ/EC ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.12 Anharmonic barrier and dimensionless dephasing rates . . . . . . . . . . . . . . . . . 107 4.13 Sketch and circuit diagram of the transmon . . . . . . . . . . . . . . . . . . . . . . . 108 4.14 Analogy of transmon as quantum rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.15 Transmon matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.16 Transmon dispersive couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.1 Resonator sample and gap capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2 Optical lithography equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
LIST OF FIGURES 10 5.3 Resist Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.4 Optical image of gap capacitor in resist . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Pictures of sputtered Nb resonators showing the “apron” and “flagging” . . . . . . . 127 5.6 Diagram of rotating angle evaporation process . . . . . . . . . . . . . . . . . . . . . . 128 5.7 Edge profiles of deposited Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.8 CPB sample and equivalent circuit including parasitic capacitances . . . . . . . . . . 131 5.9 Dolan bridge process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.10 SEM images of tunnel junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.11 Photograph of scanning electron microscope and electron beam evaporator . . . . . . 136 5.12 Transmon pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.13 Sample mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.14 “Coffin” style printed circuit board . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.15 Next generation PCB schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.1 Measurement setup for cQED experiments . . . . . . . . . . . . . . . . . . . . . . . . 142 6.2 Annotated images of the cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.1 Transmission of a high Q resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Quality factor as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . 156 7.3 a) Comparison of Q’s between under and over coupled resonators and their harmonics 157 7.4 Resonance frequency shift with temperature due to kinetic inductance . . . . . . . . 158 7.5 Resonator quality factor dependence on magnetic field . . . . . . . . . . . . . . . . . 158 7.6 Resonance frequency shift due to magnetic field . . . . . . . . . . . . . . . . . . . . . 159 7.7 “Footballs” showing phase shift of cQED system as function of gate voltage and magnetic field.161 7.8 3D images of qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.9 Spectroscopic determination of qubit energy . . . . . . . . . . . . . . . . . . . . . . . 164 7.10 Saturation and power broadening of the qubit . . . . . . . . . . . . . . . . . . . . . . 165 7.11 “Footballs” showing parity effects and the presence of charge “switchers” . . . . . . 166 7.12 Spectroscopic characterization of the transmon . . . . . . . . . . . . . . . . . . . . . 168 8.1 A phase diagram for cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.2 Vacuum Rabi avoided crossing as function of bias charge . . . . . . . . . . . . . . . . 172
Page 1 and 2: Abstract Circuit Quantum Electrodyn
Page 3 and 4: c○2007 by David Schuster. All rig
Page 5 and 6: friendship. All of my friends from
Page 7 and 8: CONTENTS 4 3.2.3 Split CPB . . . .
Page 9 and 10: CONTENTS 6 8.2.1 AC Stark Effect .
Page 11: List of Figures 1.1 Relaxation and
Page 15 and 16: LIST OF FIGURES 12 B.1 Radiative de
Page 17 and 18: List of Symbols and Abbreviations
Page 19 and 20: List of Symbols and Abbreviations 1
Page 21 and 22: Chapter 1 Introduction Progress in
Page 23 and 24: CHAPTER 1. INTRODUCTION 20 tum phys
Page 25 and 26: CHAPTER 1. INTRODUCTION 22 with lig
Page 27 and 28: CHAPTER 1. INTRODUCTION 24 Probe La
Page 29 and 30: CHAPTER 1. INTRODUCTION 26 used [Re
Page 31 and 32: CHAPTER 1. INTRODUCTION 28 a) Charg
Page 33 and 34: CHAPTER 1. INTRODUCTION 30 5 µm L
Page 35 and 36: CHAPTER 1. INTRODUCTION 32 to contr
Page 37 and 38: Chapter 2 Cavity Quantum Electrodyn
Page 39 and 40: CHAPTER 2. CAVITY QUANTUM ELECTRODY
Page 49 and 50: CHAPTER 3. CAVITY QED WITH SUPERCON
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CHAPTER 3. CAVITY QED WITH SUPERCON
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Chapter 4 Decoherence in the Cooper
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CHAPTER 4. DECOHERENCE IN THE COOPE
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CHAPTER 5. DESIGN AND FABRICATION 1
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CHAPTER 6. MEASUREMENT SETUP 142 ω
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CHAPTER 6. MEASUREMENT SETUP 144 4
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CHAPTER 6. MEASUREMENT SETUP 146 ac
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CHAPTER 6. MEASUREMENT SETUP 148 me
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CHAPTER 6. MEASUREMENT SETUP 150 Q(
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CHAPTER 6. MEASUREMENT SETUP 152 Ad
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Chapter 7 Characterization of CQED
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CHAPTER 7. CHARACTERIZATION OF CQED
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CHAPTER 8. CAVITY QED EXPERIMENTS W
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Chapter 9 Time Domain Measurements
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CHAPTER 9. TIME DOMAIN MEASUREMENTS
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Chapter 10 Future work The biggest
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CHAPTER 10. FUTURE WORK 222 possibl
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CHAPTER 10. FUTURE WORK 224 frequen
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CHAPTER 11. CONCLUSIONS 226 There i
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APPENDIX A. OPERATORS AND COMMUTATI
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APPENDIX B. DERIVATION OF DRESSED S
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APPENDIX B. DERIVATION OF DRESSED S
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Appendix D Recipes This Appendix wi
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APPENDIX D. RECIPES 236 PMMA develo
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APPENDIX D. RECIPES 238 Electron Be
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BIBLIOGRAPHY 240 [Bennett1984] C. H
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BIBLIOGRAPHY 242 [Clauser1974] John
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BIBLIOGRAPHY 244 frequency noise in
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BIBLIOGRAPHY 246 [Kuhn2002] Axel Ku
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BIBLIOGRAPHY 248 controlled by exte
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BIBLIOGRAPHY 250 [Rivest1978] R. L.
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BIBLIOGRAPHY 252 [Walls2006] D. F.
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