Circuit Quantum Electrodynamics - Yale University

1 AbstractCircuit Quantum ElectrodynamicsDavid Isaac Schuster2007 This 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 toa 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 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.

2 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 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 2007c 2007 by David rights 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.

3 I will always remember our late night brainstorming sessions, and his willingness tosuspend more critical demands to provide advice. Most of allI thank him for creating RSL andgiving me the opportunity to 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 thejoint 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 , , and , related to the Transmon was performed with my eviltwin, Andrew Houck.

4 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 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 throughoutmy time here, but also for helpingto build such an amazing community on the lab mates have made graduate school the best of times, providing help, camaraderie, and close12friendship.

5 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 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 Quantum Computation .. Cavity Quantum Electrodynamics .. Quantum Circuits .. Circuit Quantum Electrodynamics .. Thesis Overview .. 322 Cavity Quantum Dispersive Limit .. Strong Dispersive Interactions .. 413 Cavity QED with superconducting Transmission Line Cavities .. The LCR Oscillator .. Transmission Line as Series of LC Circuits.

6 Capacitively Coupled LCR Resonator .. Capacitively Coupled Transmission Line Resonator .. Coplanar Waveguide Cavities .. Kinetic Inductance .. Intrinsic Resonator Losses .. Quantization of the LC Oscillator .. Cooper Pair Box .. Charge Basis .. Phase Basis .. Split CPB .. Coupling CPB to Cavity .. Comparison with Traditional Cavity QED .. Measurement Theory .. Quantum Non-Demolition Measurements .. Mapping Qubit State onto Cavity State .. Distinguishing Cavity States .. Small Phase Shift Limit .. Optimizing SNR .. 784 Decoherence in the Cooper Pair Relaxation and Heating .. Voltage Noise .. Voltage Noise Inside a Cavity.

7 Material Loss .. Dipole Radiation .. Dephasing .. Charge Noise .. Flux Noise .. Critical Current/Josephson Energy 1/fNoise .. Summary of Cooper pair box decoherence .. Transmon .. Charge Dispersion .. Anharmonicity .. Transmon as a Josephson Oscillator .. Transmon for Circuit QED .. Other Sources of Decoherence .. Transmon Summary .. 1175 Design and Cavity .. Design Considerations .. Optical Lithography .. Deposition and Liftoff .. Substrates .. Cooper Pair Box .. Josephson Energy .. Charging Energy and Voltage Division .. Electron Beam Lithography .. Veil of Death .. Transmon .. Printed Circuit Boards and Sample Holders.

8 1376 Measurement Cryogenics and Filtering .. Pulse Synthesis .. Demodulation .. Digital Homodyne .. 1487 Characterization of Cavity .. Temperature Dependence .. Magnetic Field Dependence .. Cooper pair box .. Charge Noise .. Measured CPB properties .. Transmon .. 1668 Cavity QED Experiments with Resonant Limit .. Vacuum Rabi Mode Splitting with CPB .. Vacuum Rabi Mode Splitting With Transmon .. Dispersive Weak Limit .. AC Stark Effect .. Off-Resonant AC Stark Effect .. Sideband Experiments .. Dispersive Strong Limit .. Photon Number Splitting .. Anharmonic Strong Dispersive Limit .. 1979 Time Domain Single Qubit Gates .. Single Shot Readout.

9 Single Photon Source .. 21110 Future Evolution of Circuit QED .. New Cavity and Qubit Designs .. Scaling Circuit QED .. Other Quantum circuits .. Hybrid Circuit QED .. 22311 Conclusions225 Appendices226A Operators and Commutation Harmonic Oscillators .. Spin 1/2 .. Jaynes-Cummings Operators .. Interaction with Harmonic oscillator operators .. Interaction with Spin 1/2 Operators .. 228B Derivation of Dressed State Atom Picture229C Mathematica Cooper Pair Box .. 233 CONTENTS7D Recipes234 List of Relaxation and dephasing of qubits leads to decoherence.. Cavity QED setup with alkali atoms at optical frequencies .. Cavity QED setup with Rydberg atoms at microwave frequencies.

10 Cavity QED setup with Quantum dots in semiconductors .. Gallery of superconducting qubits .. Cooper pair box as tunable atom .. Cavity QED setup with superconducting circuits .. Circuit QED sample .. Illustration of atomic cavity QED system .. Energy level diagrams of Jaynes-Cummings Hamiltonian .. Exact calculation of vacuum Rabi avoided crossing and indirect decay rates .. A phase diagram for cavity QED .. Spectra of cavity and atom in strong dispersive limit .. LCR oscillator .. Transmission line as series of LC oscillator .. Impedance of transmission line resonator .. Capacitive coupling to an LCR resonator .. Transmission of asymmetric cavity.