Circuit Quantum Electrodynamics
AbstractCircuit Quantum ElectrodynamicsDavid Isaac Schuster2007This thesis describes the development of circuit quantum electrodynamics (QED), architecturefor studying quantum information and quantum optics. In circuit QED a superconducting qubitacting as an artificial atom is electrostatically coupled to a 1D transmission line resonator. Thelarge effective dipole moment of the qubit and high energy density of the resonator allowed thissystem 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 physicssuch as the vacuum Rabi mode splitting, and the AC Stark effect. These cavity QED effects areused to control and measure the qubit state, while protecting it from radiative decay. The qubitcan also be used to measure and control the cavity state, as shown by experiments detecting andgenerating single photons. This thesis will describe the theoretical framework, implementation, andmeasurements of the circuit QED system.
Circuit Quantum ElectrodynamicsA DissertationPresented to the Faculty of the Graduate SchoolofYale Universityin Candidacy for the Degree ofDoctor of PhilosophybyDavid Isaac SchusterDissertation Director: Professor Robert J. SchoelkopfMay 2007
c 2007by David Schuster.All rights reserved.
AcknowledgementsI would first like to thank my advisor Rob Schoelkopf. He gave me the freedom to explore anamazing world of quantum physics, while his guidance prevented me from ever feeling lost in all ofits complexity. He taught me what it means to be a scientist, and the importance of eliminatingground loops. I will always remember our late night brainstorming sessions, and his willingness tosuspend more critical demands to provide advice. Most of all I thank him for creating RSL andgiving me the opportunity to participate.I have been fortunate to work with many amazing people in the course of this research. Inparticular, most of the work presented in this thesis was the joint effort of Andreas Wallraff andmyself. His influence on me runs deep extending from small things like my (near fanatical) use ofmathematica and my much improved graphics design skills to the way I now approach experimentalquestions. 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 eviltwin, Andrew Houck. His scientific abilities are matched only by his unbounded enthusiasm, andhopefully the former is as contagious as his excitement. I thank Alexandre Blais and Jay Gambettafor teaching me everything I know about cavity QED and quantum measurement. Their patienceexceeds even Andrew’s optimism. I also owe a great debt to Luigi Frunzio for teaching me everythingI 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 MichelDevoret’s office with new understanding of a question much deeper than the one with which I hadentered. Dan Prober is to be thanked not only for his direct role in helping to convince me to cometo Yale, serving on my committee, and advising me throughout my time here, but also for helpingto build such an amazing community on the 4th floor.My lab mates have made graduate school the best of times, providing help, camaraderie, and close1
2friendship. All of my friends from graduate, undergraduate, and high school have provided invaluablesupport. I would especially like to thank my roommate(s) Matt and Sam for their tolerance andfriendship.Most importantly I would like to thank my family, who have encouraged my curiosity andprovided me with unending love. Finally, I thank Carol who helped me to grow as a person as wellas a scientist.
Contents1 Introduction181.1Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181.2Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . .211.3Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261.4Circuit Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . .291.5Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322 Cavity Quantum Electrodynamics342.1Dispersive Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372.2Strong Dispersive Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413 Cavity QED with Superconducting Circuits3.13.245Transmission Line Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .453.1.1The LCR Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463.1.2Transmission Line as Series of LC Circuits . . . . . . . . . . . . . . . . . . . .473.1.3Capacitively Coupled LCR Resonator . . . . . . . . . . . . . . . . . . . . . .503.1.4Capacitively Coupled Transmission Line Resonator . . . . . . . . . . . . . . .513.1.5Coplanar Waveguide Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . .533.1.6Kinetic Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553.1.7Intrinsic Resonator Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563.1.8Quantization of the LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . .60Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .613.2.1Charge Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .613.2.2Phase Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .653
CONTENTS3.2.33.33.44Split CPB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66Coupling CPB to Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .683.3.1Comparison with Traditional Cavity QED . . . . . . . . . . . . . . . . . . . .71Measurement Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .733.4.1Quantum Non-Demolition Measurements . . . . . . . . . . . . . . . . . . . .733.4.2Mapping Qubit State onto Cavity State . . . . . . . . . . . . . . . . . . . . .733.4.3Distinguishing Cavity States . . . . . . . . . . . . . . . . . . . . . . . . . . .753.4.4Small Phase Shift Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773.4.5Optimizing SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .784 Decoherence in the Cooper Pair Box4.14.24.3Relaxation and Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .824.1.1Voltage Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .834.1.2Voltage Noise Inside a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . .854.1.3Material Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .894.1.4Dipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .924.2.1Charge Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .954.2.2Flux Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.3Critical Current/Josephson Energy 1/f Noise . . . . . . . . . . . . . . . . . . 1014.2.4EC Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2.5Summary of Cooper pair box decoherence . . . . . . . . . . . . . . . . . . . . 102Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.3.1Charge Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.3.2Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.3Transmon as a Josephson Oscillator . . . . . . . . . . . . . . . . . . . . . . . 1084.3.4Transmon for Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.5Other Sources of Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3.6Transmon Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 Design and Fabrication5.182119Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
CONTENTS55.1.1Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.1.2Optical Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.1.3Deposition and Liftoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.4Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2.1Josephson Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2.2Charging Energy and Voltage Division . . . . . . . . . . . . . . . . . . . . . . 1305.2.3Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2.4Veil of Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4Printed Circuit Boards and Sample Holders . . . . . . . . . . . . . . . . . . . . . . . 1376 Measurement Setup1416.1Cryogenics and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.2Pulse Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.3Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.3.1Digital Homodyne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 Characterization of CQED7.17.27.3Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.1.1Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.1.2Magnetic Field Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.2.1Charge Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.2.2Measured CPB properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668 Cavity QED Experiments with Circuits8.18.2154169Resonant Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.1.1Vacuum Rabi Mode Splitting with CPB . . . . . . . . . . . . . . . . . . . . . 1718.1.2Vacuum Rabi Mode Splitting With Transmon . . . . . . . . . . . . . . . . . . 173Dispersive Weak Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
CONTENTS8.368.2.1AC Stark Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.2.2Off-Resonant AC Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.2.3Sideband Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Dispersive Strong Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.3.1Photon Number Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.3.2Anharmonic Strong Dispersive Limit . . . . . . . . . . . . . . . . . . . . . . . 1979 Time Domain Measurements2009.1Single Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009.2Single Shot Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.3Single Photon Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110 Future work22010.1 Evolution of Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22010.1.1 New Cavity and Qubit Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 22010.1.2 Scaling Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.1.3 Other quantum circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22210.1.4 Hybrid Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22311 Conclusions225Appendices226A Operators and Commutation Relations227A.1 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.2 Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.3 Jaynes-Cummings Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.3.1 Interaction with Harmonic oscillator operators . . . . . . . . . . . . . . . . . 228A.3.2 Interaction with Spin 1/2 Operators . . . . . . . . . . . . . . . . . . . . . . . 228B Derivation of Dressed State Atom Picture229C Mathematica Notebooks233C.1 Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
CONTENTS7D Recipes234
List of Figures1.1Relaxation and dephasing of qubits leads to decoherence. . . . . . . . . . . . . . .211.2Cavity QED setup with alkali atoms at optical frequencies . . . . . . . . . . . . . . .241.3Cavity QED setup with Rydberg atoms at microwave frequencies . . . . . . . . . . .241.4Cavity QED setup with quantum dots in semiconductors . . . . . . . . . . . . . . . .251.5Gallery of superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281.6Cooper pair box as tunable atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291.7Cavity QED setup with superconducting circuits . . . . . . . . . . . . . . . . . . . .301.8Circuit QED sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312.1Illustration of atomic cavity QED system . . . . . . . . . . . . . . . . . . . . . . . .352.2Energy level diagrams of Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . .362.3Exact calculation of vacuum Rabi avoided crossing and indirect decay rates . . . . .382.4A phase diagram for cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . .422.5Spectra of cavity and atom in strong dispersive limit . . . . . . . . . . . . . . . . . .433.1LCR oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463.2Transmission line as series of LC oscillator . . . . . . . . . . . . . . . . . . . . . . . .483.3Impedance of transmission line resonator . . . . . . . . . . . . . . . . . . . . . . . . .503.4Capacitive coupling to an LCR resonator . . . . . . . . . . . . . . . . . . . . . . . .513.5Transmission of asymmetric cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . .533.6Coplanar waveguide cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543.7Dependence of characteristic impedance, Z0 , on the CPW geometry. . . . . . . . . .553.8Dependence of kinetic inductance prefactor on geometry. . . . . . . . . . . . . . . . .573.9Dependence of kinetic inductance on penetration depth and center-pin width. . . . .578
LIST OF FIGURES93.10 CPB circuit diagram / junction diagram . . . . . . . . . . . . . . . . . . . . . . . . .623.11 CPB energy levels and charge staircase . . . . . . . . . . . . . . . . . . . . . . . . . .633.12 Energy in two state approximation and fictitious field figure . . . . . . . . . . . . . .643.13 CPB Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .653.14 Split CPB sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .673.15 Dipole moment of the Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . .703.16 Measurement schematic and derivatives of CPB energy levels . . . . . . . . . . . . .743.17 State dependent transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .753.18 Q function of coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763.19 State dependent transmission with small phase shift . . . . . . . . . . . . . . . . . .773.20 Selecting Δr for optimal SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .793.21 Selecting κ/2χ for optimal SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .804.1Decoherence through gate voltage coupling circuit diagram and SV (ω) at different temperatures 844.2Quality factor of qubits coupled to transmission line and cavity . . . . . . . . . . . .864.3CPB coupling to slotline mode in resonator . . . . . . . . . . . . . . . . . . . . . . .874.4Flux noise circuit diagram and flux transition matrix elements . . . . . . . . . . . .884.5Electric Field distribution in CPB . . . . . . . . . . . . . . . . . . . . . . . . . . . .904.6Derivatives of energy with respect to gate charge . . . . . . . . . . . . . . . . . . . .964.7Thermal Dephasing of the qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . .974.8Dephasing of qubit due to 1/f charge noise . . . . . . . . . . . . . . . . . . . . . . .984.9Dephasing times due to flux and critical current noise . . . . . . . . . . . . . . . . . 1014.10 CPB energy bands at different EJ /EC ratios . . . . . . . . . . . . . . . . . . . . . . 1044.11 Anharmonicity vs. EJ /EC ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.12 Anharmonic barrier and dimensionless dephasing rates . . . . . . . . . . . . . . . . . 1074.13 Sketch and circuit diagram of the transmon . . . . . . . . . . . . . . . . . . . . . . . 1084.14 Analogy of transmon as quantum rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.15 Transmon matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.16 Transmon dispersive couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.1Resonator sample and gap capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2Optical lithography equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
LIST OF FIGURES105.3Resist Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.4Optical image of gap capacitor in resist . . . . . . . . . . . . . . . . . . . . . . . . . 1255.5Pictures of sputtered Nb resonators showing the “apron” and “flagging” . . . . . . . 1275.6Diagram of rotating angle evaporation process . . . . . . . . . . . . . . . . . . . . . . 1285.7Edge profiles of deposited Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.8CPB sample and equivalent circuit including parasitic capacitances . . . . . . . . . . 1315.9Dolan bridge process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.10 SEM images of tunnel junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.11 Photograph of scanning electron microscope and electron beam evaporator . . . . . . 1365.12 Transmon pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.13 Sample mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.14 “Coffin” style printed circuit board . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.15 Next generation PCB schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.1Measurement setup for cQED experiments . . . . . . . . . . . . . . . . . . . . . . . . 1426.2Annotated images of the cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.1Transmission of a high Q resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.2Quality factor as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . 1567.3a) Comparison of Q’s between under and over coupled resonators and their harmonics 1577.4Resonance frequency shift with temperature due to kinetic inductance . . . . . . . . 1587.5Resonator quality factor dependence on magnetic field . . . . . . . . . . . . . . . . . 1587.6Resonance frequency shift due to magnetic field . . . . . . . . . . . . . . . . . . . . . 1597.7“Footballs” showing phase shift of cQED system as function of gate voltage and magnetic field.1617.83D images of qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.9Spectroscopic determination of qubit energy . . . . . . . . . . . . . . . . . . . . . . . 1647.10 Saturation and power broadening of the qubit . . . . . . . . . . . . . . . . . . . . . . 1657.11 “Footballs” showing parity effects and the presence of charge “switchers” . . . . . . 1667.12 Spectroscopic characterization of the transmon . . . . . . . . . . . . . . . . . . . . . 1688.1A phase diagram for cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.2Vacuum Rabi avoided crossing as function of bias charge . . . . . . . . . . . . . . . . 172
LIST OF FIGURES118.3Transmission spectra of cQED system at and away from cavity-qubit degeneracy . . 1728.4Vacuum Rabi avoided crossing as function of bias flux . . . . . . . . . . . . . . . . . 1748.5Level separation and linewidth near flux avoided crossing . . . . . . . . . . . . . . . 1768.6Vacuum Rabi mode splitting at different drive powers . . . . . . . . . . . . . . . . . 1768.7Density plot showing the AC Stark shift and slices at different drive powers . . . . . 1788.8AC Stark shift and dephasing rate vs. input power . . . . . . . . . . . . . . . . . . . 1798.9Non-linear corrections to the AC Stark effect . . . . . . . . . . . . . . . . . . . . . . 1808.10 Dephasing due to measurement photons showing higher powers and a more comprehensive theory1838.11 Measurement setup for off-resonant AC Stark effect and sideband experiments. . . 1848.12 Lorentzian cavity transmission spectrum and experimental signal frequencies . . . . 1858.13 AC Stark shift using tone detuned from cavity frequency . . . . . . . . . . . . . . . . 1868.14 Plots of AC Stark shifted qubit transition frequency and linewidth with and off-resonant tone1868.15 Qubit-cavity energy levels illustrating sideband transitions. . . . . . . . . . . . . . . 1878.16 Sideband spectroscopy density plot and spectrum . . . . . . . . . . . . . . . . . . . . 1888.17 Tracking sidebands as function of spectroscopy and AC Stark drive tone parameters1908.18 Strong dispersive spectral features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.19 Direct spectroscopic observation of quantized cavity photon number . . . . . . . . . 1958.20 Density and waterfall plots of the dispersive cavity’s inherited non-linearity from its coupling to the qubit18.21 Anharmonic cavity shifts and linewidths . . . . . . . . . . . . . . . . . . . . . . . . . 1999.1Time resolved measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.2Rabi oscillation experiment and individual time slices . . . . . . . . . . . . . . . . . 2049.3Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.4Ramsey fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.5Histograms of single shot measurements . . . . . . . . . . . . . . . . . . . . . . . . . 2109.6Single photon source protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2129.7Mapping of qubit state onto photon states . . . . . . . . . . . . . . . . . . . . . . . . 2159.8Spontaneous emission from qubit Rabi oscillations in cavity . . . . . . . . . . . . . . 2179.9Single photon fluorescence tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.1 Sketch of two cavities and two qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.2 Sketch of hybrid circuit QED with molecules . . . . . . . . . . . . . . . . . . . . . . 223
LIST OF FIGURES12B.1 Radiative decay in the presence of coupling . . . . . . . . . . . . . . . . . . . . . . . 231
List of Tables1.1Types of qubits and effects of different types of noise . . . . . . . . . . . . . . . . . .273.1Table of superconductor properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .593.2Key rates and parameters for different CQED systems . . . . . . . . . . . . . . . . .724.1Comparison of representative relaxation and dephasing times for various superconducting qubit designs. T7.1Sample Info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167D.1 Optical Lithography Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235D.2 Electron beam resist spinning recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . 235D.3 Electron beam resist development recipe . . . . . . . . . . . . . . . . . . . . . . . . . 236D.4 Nb sputtering recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237D.5 Deposition and Liftoff recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23813
List of Symbols and Abbreviations † a ,aPhoton creation and annihilation operators † b ,bTransmon excitation creation and annihilation operators(Cg )Gate capacitance Cin/outResonator input/output coupling capacitor(Cn )Effective resonator capacitance for the nth cavity mode(Cj )Junction capacitance(Cs )Stray capacitance between islands(CΣ )Total capacitance between islands(Cs )Shunting capacitance used to increase CΣ and lower EC and g(CP B)Cooper Pair Box - consisting of two small superconducting islands connected by aJosephson junction(CN OT )Conditional Not, a two qubit gate in which one qubit is flipped depending on thestate of the other(E ,n )Energies with n excitations from exact diagonalization of Jaynes-Cummings Hamiltonian( g/e, n )State vector with atom in ground/excited state and n-photons , n Exact eigenstates of Jaynes-Cummings Hamiltonian, n is total number of excitations(not photon number)(EC )Cooper pair box charging energy (in single e units)(EJ )Josephson energy(g)Interaction rate between qubit and photon(gij )Cavity-qubit coupling between qubit levels i and j14
List of Symbols and Abbreviations15(Ic )Josephson critical current(Jc )Josephson critical current density (per unit area)(LJ )Josephson inductance(LJ 0)Josephson inductance in zero current limit(LK )Kinetic inductance per unit length of the (superconducting) CPW(Lm )Magnetic inductance per unit length of the CPW(Ln )Effective resonator inductance for the nth cavity mode(n)Average photon number in the cavity(ncrit )Number of photons before dispersive approximation fails to converge, ncrit Δ2 /4g 2(nκ )Number of photons before second order term in dispersive Taylor expansion of3energy is comparable to cavity linewidth, nκ κΔ2g4(ng )Gate polarization charge(ns )Density of Cooper pairs(nn )Density of quasiparticles(Q)Cavity quality factor (Q ωr /κ) Qint/extQuality factor due to internal/external losses(Qop )Operations quality factor, defined as number of oscillations in one decay time Top /T2(Qφ )Pure dephasing quality factor(Qres )Quality factor due to resistivity of CPW(Qrad )Quality factor due to radiative losses qin/out2External inverse q factor, the external resonator quality factor is Qint 1/qin/out(QED)quantum electrodynamics(qubit)A quantum mechanical bit, referring to any quantum system with two levels(RK )The resistance quantum (Von Klitzing constant) RK h/e2 25.8 kΩ(Rn )Resistance of tunnel junction in normal state
List of Symbols and Abbreviations16(RSA)Rivest-Shamir-Adelman public key cryptosystem, which bases its security on thecomputational difficulty of factoring products of large primes(SI )Spectral density of current noise in units of A2 /Hz1/2(SΦ )Spectral density of flux noise in units of Φ20 /Hz1/2(SV )Spectral density of voltage noise in units of V 2 /Hz1/2(SQ )Spectral density of charge noise in units of e2 /Hz1/2(1/T )Inverse transit time for atom to leave the cavity(T1 )Qubit state relaxation time(T2 )Total dephasing time(T2 )Inhomogenous dephasing time, due to ensembles of different qubit frequencies (usually due to 1/f noise)(Top )Operation time(Tφ )Pure dephasing time(Vg )Cooper pair box gate voltage(Vj )Voltage across the junction when a gate voltage is applied elsewhere(ZL )Load impedance(ZC )Characteristic impedance(α)Fine structure constant or anharmonicity (depending on context)(αr )Relative anharmonicity α/ωa(Δ)Atom-cavity detuning Δ ωa ωr( m )Charge dispersion of mth CPB energy level( eff )Effective dielectric constant of CPW mode re/imReal and imaginary parts of the substrate dielectric constant( r )Relative dielectric constant(γ )Decay rate of atom into modes other than the cavity(Γeff )Decay rate of atom including all channels of decay(γκ )Radiative decay of the atom by radiating through the cavity
List of Symbols and Abbreviations17(κ)Decay rate of photon out of the cavity(κγ )Decay of photons by decaying through the atom via non-radiative means κin/outPhoton decay rate due to the input/output coupling(λL )The London penetration depth, describing the depth within which currents flow(μr )Relative magnetic permeability(σ )Qubit excitation creation and annihilation operators(σx,y,z )Pauli matrices for a spin 1/2 particle used to measure x,y,z components of qubit(σn )Conductivity of the normal fluid(τn )Scattering time of the normal fluid (Drude model), which is a measure of the realpart of the conductivity(χ)Cavity shift or Stark shift per photon, χ g 2 /Δ for a two level atom(χij )Dispersive shift between cavity and qubit levels i and j(χeff )Effective dispersive shift due to interactions of all qubit levels with the cavity(ωa )Ground-excited state transition (angular) frequency(ωr )Cavity resonance (angular) frequency(ωs )Spectroscopy (angular) frequency
Chapter 1IntroductionProgress in science is often a result of exchanging principles and techniques between seeminglydisparate disciplines. Often this interaction leads both fields in new directions. The subject of thisdissertation, “circuit quantum electrodynamics” is an excellent example of such cross-pollination.In this work, techniques imported from atomic cavity quantum electrodynamics (QED), a
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
Quantum electrodynamics From Wikipedia, the free encyclopedia Quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved.
Circuit quantum electrodynamics with a nonlinear resonator One of the most studied model systems in quantum optics is a two-level atom strongly coupled to a single mode of the electromagnetic eld stored in a cavity, a re-search eld named cavity quantum electrodynamics or CQED (Haroche and Raimond, 2006).
6. Quantum Electrodynamics In this section we finally get to quantum electrodynamics (QED), the theory of light interacting with charged matter. Our path to quantization will be as before: we start with the free theory of the electromagnetic field and see how the quantum th
atomic physics, and astrophysics. Also, quantum field theory has led to new bridges between physics and mathematics. Since this thesis is about scattering cross-section in quantum electrodynamics, so it is instructive to start with an introduction of quantum electrodynamics (QED). It is a field theory of interaction between light and matter.
Circuit Quantum Electrodynamics with Electrons on Helium Andreas Arnold Fragner 2013 This thesis describes the theory, design and implementation of a circuit quantum electro-dynamics (QED) architecture with electrons floating above the surface of superfluid helium.
Testing a Conjecture On Quantum Electrodynamics Christoph Schiller* Abstract A geometric Planck-scale model of quantum electrodynamics is tested against obser-vations. Based on Dirac’s proposal to describe spin 1/2 particles as tethered objects, elementary fermions are conjectured to be fluctuating rational tangles with unobserv-able tethers.
Quantum electrodynamics in a piece of rock John McGreevy Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA Gauge theories, such as quantum electrodynamics, are a cornerstone of high energy particle physics. They may also describe the physics of certain unassuming materials.
ANSI A300 Purpose: To provide performance standards for developing written specifications for tree management. Currently nine individual parts . ANSI Z60 American Nurseryman and Landscape Association began developing this standard back in 1929 Became an ANSI document in 1949 Current version is 2004 . ANSI A300 The Tree Care Industry Association convened a consensus body .
