Abstract
Circuit Quantum Electrodynamics
David Isaac Schuster 2007
This thesis describes the development of circuit quantum electrodynamics (QED), architecture for studying quantum information and quantum optics. In circuit QED a superconducting qubit acting as an artificial atom is electrostatically coupled to a 1D transmission line resonator. The large effective dipole moment of the qubit and high energy density of the resonator allowed this system to reach the strong coupling limit of cavity QED for the first time in a solid-state system. Spectroscopic investigations explore effects of different regimes of cavity QED observing physics such as the vacuum Rabi mode splitting, and the AC Stark effect. These cavity QED effects are used to control and measure the qubit state, while protecting it from radiative decay. The qubit can also be used to measure and control the cavity state, as shown by experiments detecting and generating single photons. This thesis will describe the theoretical framework, implementation, and measurements of the circuit QED system. Circuit Quantum Electrodynamics
A Dissertation
Presented to the Faculty of the Graduate School of Yale University
in Candidacy for the Degree of Doctor of Philosophy
by
David Isaac Schuster
Dissertation Director: Professor Robert J. Schoelkopf
May 2007 c 2007 by David Schuster.
All rights reserved. Acknowledgements
I would first like to thank my advisor Rob Schoelkopf. He gave me the freedom to explore an amazing world of quantum physics, while his guidance prevented me from ever feeling lost in all of its complexity. He taught me what it means to be a scientist, and the importance of eliminating ground loops. I will always remember our late night brainstorming sessions, and his willingness to suspend more critical demands to provide advice. Most of all I thank him for creating RSL and giving me the opportunity to participate. I have been fortunate to work with many amazing people in the course of this research. In particular, most of the work presented in this thesis was the joint effort of Andreas Wallraff and myself. His influence on me runs deep extending from small things like my (near fanatical) use of mathematica and my much improved graphics design skills to the way I now approach experimental questions. More importantly, Andreas has become one of my closest friends. More recent work, presented in sections 4.3, 8.3.1, and 9.3, related to the “Transmon” was performed with my evil twin, Andrew Houck. His scientific abilities are matched only by his unbounded enthusiasm, and hopefully the former is as contagious as his excitement. I thank Alexandre Blais and Jay Gambetta for teaching me everything I know about cavity QED and quantum measurement. Their patience exceeds even Andrew’s optimism. I also owe a great debt to Luigi Frunzio for teaching me everything
I know about the dark arts of fabrication. Steve Girvin has an uncanny ability to make tangible connections between theory and experiment, while simultaneously telling a hilarious story. Similarly, I often found myself emerging from Michel Devoret’s office with new understanding of a question much deeper than the one with which I had entered. Dan Prober is to be thanked not only for his direct role in helping to convince me to come to Yale, serving on my committee, and advising me throughout my time here, but also for helping to build such an amazing community on the 4th floor. My lab mates have made graduate school the best of times, providing help, camaraderie, and close
1 2 friendship. All of my friends from graduate, undergraduate, and high school have provided invaluable support. I would especially like to thank my roommate(s) Matt and Sam for their tolerance and friendship. Most importantly I would like to thank my family, who have encouraged my curiosity and provided me with unending love. Finally, I thank Carol who helped me to grow as a person as well as a scientist. Contents
1 Introduction 18 1.1 QuantumComputation ...... 18
1.2 CavityQuantumElectrodynamics ...... 21 1.3 QuantumCircuits ...... 26 1.4 CircuitQuantumElectrodynamics ...... 29
1.5 ThesisOverview ...... 32
2 Cavity Quantum Electrodynamics 34
2.1 DispersiveLimit ...... 37 2.2 StrongDispersiveInteractions...... 41
3 Cavity QED with Superconducting Circuits 45
3.1 TransmissionLineCavities ...... 45 3.1.1 TheLCROscillator ...... 46 3.1.2 Transmission Line as Series of LC Circuits...... 47
3.1.3 CapacitivelyCoupledLCRResonator ...... 49 3.1.4 Capacitively Coupled Transmission Line Resonator ...... 51
3.1.5 CoplanarWaveguideCavities ...... 53 3.1.6 KineticInductance...... 55 3.1.7 IntrinsicResonatorLosses ...... 56
3.1.8 QuantizationoftheLCOscillator ...... 60 3.2 CooperPairBox ...... 61 3.2.1 ChargeBasis ...... 61
3.2.2 PhaseBasis...... 65
3 CONTENTS 4
3.2.3 SplitCPB...... 66
3.3 CouplingCPBtoCavity...... 68 3.3.1 Comparisonwith Traditional CavityQED ...... 71 3.4 MeasurementTheory...... 73
3.4.1 Quantum Non-Demolition Measurements ...... 73 3.4.2 Mapping Qubit State onto Cavity State ...... 73 3.4.3 Distinguishing Cavity States ...... 75
3.4.4 SmallPhaseShiftLimit ...... 77 3.4.5 OptimizingSNR ...... 78
4 Decoherence in the Cooper Pair Box 82 4.1 RelaxationandHeating ...... 82 4.1.1 VoltageNoise...... 83
4.1.2 VoltageNoiseInsideaCavity ...... 85 4.1.3 MaterialLoss...... 89 4.1.4 DipoleRadiation ...... 91
4.2 Dephasing...... 92 4.2.1 ChargeNoise ...... 95
4.2.2 FluxNoise ...... 100 4.2.3 Critical Current/Josephson Energy 1/f Noise...... 101
4.2.4 EC Noise ...... 102 4.2.5 SummaryofCooperpairboxdecoherence ...... 102 4.3 Transmon ...... 103 4.3.1 ChargeDispersion ...... 103
4.3.2 Anharmonicity ...... 105 4.3.3 TransmonasaJosephsonOscillator ...... 108
4.3.4 TransmonforCircuitQED ...... 111 4.3.5 OtherSourcesofDecoherence...... 115 4.3.6 TransmonSummary ...... 117
5 Design and Fabrication 119 5.1 Cavity...... 119 CONTENTS 5
5.1.1 DesignConsiderations ...... 119
5.1.2 OpticalLithography ...... 123 5.1.3 DepositionandLiftoff ...... 126 5.1.4 Substrates...... 127
5.2 CooperPairBox ...... 129 5.2.1 JosephsonEnergy ...... 129 5.2.2 ChargingEnergyandVoltageDivision ...... 130
5.2.3 ElectronBeamLithography ...... 133 5.2.4 VeilofDeath ...... 135
5.3 Transmon ...... 137 5.4 PrintedCircuit BoardsandSample Holders ...... 137
6 Measurement Setup 141
6.1 CryogenicsandFiltering...... 143 6.2 PulseSynthesis ...... 146 6.3 Demodulation...... 147
6.3.1 DigitalHomodyne ...... 148
7 Characterization of CQED 154
7.1 Cavity...... 154 7.1.1 TemperatureDependence ...... 155 7.1.2 MagneticFieldDependence ...... 157
7.2 Cooperpairbox ...... 159 7.2.1 ChargeNoise ...... 164
7.2.2 MeasuredCPBproperties ...... 166 7.3 Transmon ...... 166
8 Cavity QED Experiments with Circuits 169
8.1 ResonantLimit...... 171 8.1.1 Vacuum Rabi Mode Splitting with CPB ...... 171 8.1.2 Vacuum Rabi Mode Splitting With Transmon ...... 173
8.2 DispersiveWeakLimit...... 177 CONTENTS 6
8.2.1 ACStarkEffect ...... 177
8.2.2 Off-ResonantACStarkEffect...... 182 8.2.3 SidebandExperiments ...... 185 8.3 DispersiveStrongLimit ...... 191
8.3.1 PhotonNumberSplitting ...... 191 8.3.2 Anharmonic Strong Dispersive Limit ...... 197
9 Time Domain Measurements 200
9.1 SingleQubitGates ...... 200 9.2 SingleShotReadout ...... 207
9.3 SinglePhotonSource...... 211
10 Future work 220 10.1 EvolutionofCircuitQED ...... 220
10.1.1 NewCavityandQubitDesigns ...... 220 10.1.2 ScalingCircuitQED ...... 221
10.1.3 Otherquantumcircuits ...... 222 10.1.4 HybridCircuitQED ...... 223
11 Conclusions 225
Appendices 226
A Operators and Commutation Relations 227 A.1 HarmonicOscillators...... 227
A.2 Spin1/2...... 227 A.3 Jaynes-CummingsOperators ...... 227
A.3.1 Interaction with Harmonic oscillator operators ...... 228 A.3.2 InteractionwithSpin1/2Operators ...... 228
B Derivation of Dressed State Atom Picture 229
C Mathematica Notebooks 233 C.1 CooperPairBox ...... 233 CONTENTS 7
D Recipes 234 List of Figures
1.1 Relaxation and dephasing of qubits leads to decoherence ...... 21 1.2 Cavity QED setup with alkali atoms at optical frequencies...... 24
1.3 Cavity QED setup with Rydberg atoms at microwave frequencies...... 24 1.4 Cavity QED setup with quantum dots in semiconductors ...... 25 1.5 Galleryofsuperconductingqubits...... 28
1.6 Cooperpairboxastunableatom ...... 29 1.7 Cavity QED setup with superconducting circuits ...... 30 1.8 CircuitQEDsample ...... 31
2.1 IllustrationofatomiccavityQEDsystem ...... 35 2.2 Energy level diagrams of Jaynes-Cummings Hamiltonian ...... 36
2.3 Exact calculation of vacuum Rabi avoided crossing and indirectdecayrates . . . . . 38 2.4 AphasediagramforcavityQED ...... 42 2.5 Spectra of cavity and atom in strong dispersive limit ...... 43
3.1 LCRoscillator ...... 46 3.2 Transmission line as series of LC oscillator ...... 48
3.3 Impedance of transmissionline resonator...... 50 3.4 CapacitivecouplingtoanLCRresonator ...... 51 3.5 Transmissionofasymmetriccavity ...... 53
3.6 Coplanarwaveguidecavity...... 54
3.7 Dependence of characteristic impedance, Z0,ontheCPWgeometry...... 55 3.8 Dependence of kinetic inductance prefactor on geometry...... 57
3.9 Dependence of kinetic inductance on penetration depth and center-pin width. . . . . 57
8 LIST OF FIGURES 9
3.10 CPBcircuitdiagram/junctiondiagram ...... 62
3.11 CPBenergylevelsandchargestaircase...... 63 3.12 Energy in two state approximation and fictitious field figure...... 64 3.13CPBMatrixelements ...... 65
3.14SplitCPBsketch ...... 67 3.15 DipolemomentoftheCooperPairBox ...... 70 3.16 Measurement schematic and derivatives of CPB energy levels ...... 74
3.17 Statedependenttransmission ...... 75 3.18 Qfunctionofcoherentstates ...... 76
3.19 State dependent transmission with small phase shift ...... 77
3.20 Selecting ∆r foroptimalSNR ...... 79 3.21 Selecting κ/2χ foroptimalSNR...... 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 andcavity...... 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 ElectricFielddistributioninCPB ...... 90
4.6 Derivatives of energy with respect to gate charge ...... 96 4.7 ThermalDephasingofthequbit ...... 97 4.8 Dephasing of qubit due to 1/f chargenoise ...... 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 diagramof the transmon ...... 108
4.14 Analogyoftransmonasquantumrotor...... 110 4.15 Transmonmatrixelements...... 112 4.16 Transmondispersivecouplings ...... 114
5.1 Resonatorsampleandgapcapacitors...... 120 5.2 Opticallithographyequipment ...... 124 LIST OF FIGURES 10
5.3 ResistProfile ...... 124
5.4 Opticalimageofgapcapacitorinresist ...... 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 EdgeprofilesofdepositedAluminum ...... 128 5.8 CPB sample and equivalent circuit including parasitic capacitances ...... 131 5.9 Dolanbridgeprocess...... 134
5.10 SEMimagesoftunneljunctions...... 135 5.11 Photograph of scanning electron microscope and electronbeamevaporator...... 136
5.12Transmonpictures ...... 136 5.13Samplemounts ...... 138 5.14 “Coffin”styleprintedcircuitboard ...... 139
5.15 NextgenerationPCBschematics ...... 140
6.1 MeasurementsetupforcQEDexperiments ...... 142 6.2 Annotatedimagesofthecryostat ...... 144
7.1 Transmission of a high Q resonator...... 155 7.2 Qualityfactorasafunction oftemperature ...... 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 kineticinductance ...... 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 3Dimagesofqubitspectroscopy ...... 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 AphasediagramforcavityQED ...... 170
8.2 Vacuum Rabi avoided crossing as function of bias charge ...... 172 LIST OF FIGURES 11
8.3 Transmission spectra of cQED system at and away from cavity-qubit degeneracy . . 172
8.4 Vacuum Rabi avoided crossing as function of bias flux ...... 174 8.5 Level separation and linewidth near flux avoided crossing ...... 176 8.6 Vacuum Rabi mode splitting at different drive powers ...... 176
8.7 Density plot showing the AC Stark shift and slices at differentdrivepowers . . . . . 178 8.8 AC Stark shift and dephasing rate vs. input power ...... 179 8.9 Non-linear correctionsto the AC Stark effect ...... 180
8.10 Dephasing due to measurement photons showing higher powers and a more comprehensive theory183 8.11 Measurement setup for off-resonant AC Stark effect and sideband experiments . . . 184
8.12 Lorentzian cavity transmission spectrum and experimental signal frequencies . . . . 185 8.13 AC Stark shift using tone detuned from cavity frequency ...... 186 8.14 Plots of AC Stark shifted qubit transition frequency and linewidth with and off-resonant tone186
8.15 Qubit-cavity energy levels illustrating sideband transitions...... 187 8.16 Sideband spectroscopy density plot and spectrum ...... 188 8.17 Tracking sidebands as function of spectroscopy and AC Stark drive tone parameters 190
8.18 Strongdispersivespectralfeatures ...... 193 8.19 Direct spectroscopic observation of quantized cavity photonnumber ...... 195
8.20 Density and waterfall plots of the dispersive cavity’s inherited non-linearity from its coupling to the qubit198 8.21 Anharmonic cavity shifts and linewidths ...... 199
9.1 Timeresolvedmeasurementsetup ...... 202
9.2 Rabi oscillation experiment and individual time slices ...... 204 9.3 Rabioscillations ...... 206 9.4 Ramseyfringes ...... 206
9.5 Histogramsofsingleshotmeasurements ...... 210 9.6 Singlephotonsourceprotocol ...... 212
9.7 Mapping of qubit state onto photon states ...... 215 9.8 Spontaneous emission from qubit Rabi oscillations in cavity ...... 217 9.9 Singlephotonfluorescencetomography ...... 218
10.1 Sketch of two cavities and two qubits ...... 221 10.2 Sketch of hybrid circuit QED with molecules ...... 223 LIST OF FIGURES 12
B.1 Radiative decay in the presence of coupling ...... 231 List of Tables
1.1 Types of qubits and effects of different types of noise ...... 27
3.1 Tableofsuperconductorproperties ...... 59
3.2 Key rates and parameters for different CQED systems ...... 72
4.1 Comparison of representative relaxation and dephasing times for various superconducting qubit designs. The
7.1 SampleInfo...... 167
D.1 OpticalLithographyRecipe ...... 235 D.2 Electronbeamresistspinningrecipe ...... 235
D.3 Electronbeam resistdevelopmentrecipe ...... 236 D.4 Nbsputteringrecipe ...... 237 D.5 DepositionandLiftoffrecipe ...... 238
13 List of Symbols and Abbreviations
a†,a Photon creation and annihilation operators b†,b Transmon excitation creation and annihilation operators (Cg) Gate capacitance
Cin/out Resonator input/output coupling capacitor