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 ..............