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Abstract Circuit Quantum Electrodynamics Lev Samuel Bishop 2010 Circuit Quantum Electrodynamics (cQED), the study of the interaction between supercon- ducting circuits behaving as artificial atoms and 1-dimensional transmission-line resonators, has shown much promise for quantum information processing tasks. For the purposes of quantum computing it is usual to approximate the artificial atoms as 2-level qubits, and much effort has been expended on attempts to isolate these qubits from the environment and to invent ever more sophisticated control and measurement schemes. Rather than focussing on these technological aspects of the field, this thesis investigates the opportunities for using these carefully engineered systems for answering questions of fundamental physics. The low dissipation and small mode volume of the circuits allows easy access to the strong-coupling regime of quantum optics, where one can investigate the interaction of light and matter at the level of single atoms and photons. A signature of strong coupling is the splitting of the cavity transmission peak into a pair of resolvable peaks when a single resonant atom is placed inside the cavity—an effect known as vacuum Rabi splitting. The cQED architecture is ideally suited for going beyond this linear response effect. This thesis shows that increasing the drive power results in two unique nonlinear features in the transmitted heterodyne signal: the supersplitting of each vacuum Rabi peak into a doublet, and the appearance of additional peaks with the characteristic √n spacing of the Jaynes–Cummings ladder. These constitute direct evidence for the coupling between the quantized microwave field and the anharmonic spectrum of a superconducting qubit acting as an artificial atom. This thesis also addresses the idea of Bell tests, which are experiments that aim to disprove certain types of classical theories, presenting a proposed method for preparing maximally entangled 3-qubit states via a ‘preparation by measurement’ scheme using an optimized filter on the time-dependent signal obtained via homodyne monitoring of the transmitted microwave field. 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 Lev Samuel Bishop Dissertation Director: Professor Steven M. Girvin May 2010 © 2010 by Lev Samuel Bishop All rights reserved. Contents Contents iv List of Figures vii Acknowledgementsx Publication list xii Nomenclature xiii 1 Introduction 18 1.1 Outline of thesis.................................... 19 2 Circuit QED 23 2.1 Circuit quantization.................................. 23 2.2 Quantum LC oscillator................................ 26 2.3 Transmission line resonator............................. 27 2.4 Qubits and artificial atoms: the need for anharmonicity............. 29 2.5 Charge qubit Hamiltonian.............................. 30 2.5.1 Charge dispersion.............................. 32 2.5.2 Anharmonicity................................ 34 2.5.3 Matrix elements and selection rules.................... 35 2.5.4 Flux tuning: the split transmon...................... 37 2.6 Coupling a transmon to a resonator........................ 37 2.7 Jaynes–Cummings Physics.............................. 40 2.8 Driving......................................... 41 2.8.1 Introducing the drive............................ 41 iv CONTENTS v 2.8.2 The rotating frame of the drive...................... 42 2.9 Displacement transformation............................ 42 2.10 Dispersive limit.................................... 43 2.10.1 One-qubit gates............................... 45 2.10.2 Readout.................................... 45 3 Master equation 46 3.1 Quantum operations................................. 47 3.1.1 Positive maps................................. 47 3.1.2 Complete positivity............................. 47 3.1.3 Reduced dynamics.............................. 48 3.1.4 Aside: Not completely positive maps?.................. 49 3.2 Markovian dynamics................................. 49 3.3 Weak coupling..................................... 51 3.3.1 Heat bath................................... 55 3.4 Damped harmonic oscillator............................ 56 3.5 Bloch equations.................................... 56 3.6 Master equation for the transmon-cavity system................. 58 3.6.1 Possible microscopic mechanisms of decoherence for transmons.. 60 3.6.2 Putting the pieces together......................... 63 4 Nonlinear response of the vacuum Rabi splitting 65 4.1 Strong coupling: the fine structure limit...................... 68 4.2 Experimental setup.................................. 70 4.2.1 Details of the Sample............................ 70 4.2.2 Measurement Details............................ 72 4.3 Input-output theory.................................. 73 4.4 Heterodyne detection................................. 75 4.5 Two-level behavior: Supersplitting......................... 75 4.5.1 Simple model of supersplitting....................... 81 4.6 Multi-photon transitions: Climbing the Jaynes–Cummings ladder...... 83 4.6.1 Solving the master equation........................ 87 4.6.2 Fitting the experimental data....................... 89 4.6.3 Parameter analysis.............................. 91 4.6.4 Final thoughts................................ 95 5 Generating and detecting Greenberger–Horne–Zeilinger states 96 5.1 Bell tests........................................ 99 5.1.1 Idealized Bell test.............................. 99 5.1.2 How many repetitions of the protocol are needed?........... 101 5.1.3 Loopholes................................... 102 5.1.4 So what are we trying to do?........................ 103 CONTENTS vi 5.2 Quantum trajectories................................. 104 5.3 Idealized preparation and detection of GHZ states................ 105 5.3.1 Preparation scheme............................. 106 5.3.2 Detection scheme.............................. 107 5.4 Model.......................................... 110 5.5 Preparation of the GHZ state under realistic conditions............. 112 5.6 GHZ state detection under realistic conditions.................. 115 5.7 Conclusions...................................... 119 6 Conclusions and outlook 121 6.1 Vacuum Rabi splitting................................ 121 6.2 Future trends...................................... 122 6.3 New qubit designs................................... 123 6.4 A metaphor....................................... 126 Bibliography 127 Appendices 140 A Mathematica code for strongly-driven vacuum Rabi 140 Copyright Permissions 168 List of Figures 2 Circuit QED 2.1 The LC oscillator................................... 26 2.2 The transmission line................................. 28 2.3 The charge qubit.................................... 30 2.4 Transmon wavefunctions............................... 32 2.5 Charge dispersion................................... 33 2.6 Charge dependence of the matrix elements.................... 35 2.7 Coupling a transmon to a resonator........................ 38 2.8 The capacitance network for the transmon in a CPW resonator........ 39 3 Master equation 3.1 Multimode Purcell effect............................... 61 4 Nonlinear response of the vacuum Rabi splitting 4.1 Jaynes–Cummings level diagram of the resonator–qubit system........ 67 4.2 Two-transmon circuit QED sample......................... 71 4.3 Schematic of measurement setup.......................... 72 4.4 Transmission versus magnetic field and drive frequency............ 73 4.5 Extended Jaynes–Cummings level diagram of the resonator–transmon system 76 4.6 Supersplitting of the vacuum Rabi resonance................... 77 4.7 Comparison of heterodyne detection and photon counting........... 80 4.8 Quadratures of the vacuum Rabi signal...................... 80 4.9 Supersplitting: Heterodyne vs photon counting vs linear response...... 81 4.10 Bloch sphere picture for the qubit–photon 2-level system............ 82 4.11 Supersplitting of the vacuum Rabi peak in experiment and theory...... 83 4.12 Emergence of √n peaks under strong driving of the vacuum Rabi transition 84 vii LIST OF FIGURES viii 4.13 Quadratures of √n peaks under strong driving of the vacuum Rabi transition 85 4.14 Qubit–cavity avoided crossing at high drive power................ 86 4.15 Strongly driven vacuum Rabi at elevated temperature.............. 92 4.16 Vacuum Rabi splitting at elevated temperature.................. 93 4.17 Strong driving of √n peaks in the limit of low dissipation........... 94 5 Generating and detecting Greenberger–Horne–Zeilinger states 5.1 Sketch of the circuit QED architecture....................... 97 5.2 Dispersive measurements for generating and detecting GHZ state...... 108 5.3 GHZ state preparation in presence of decay.................... 113 5.4 Time traces of the signal J(t) for individual quantum trajectories....... 114 5.5 Expectation value of the Mermin operator ⟨M⟩ as a function of acceptance probability....................................... 116 5.6 False positive and false negative rates versus threshold............. 119 6 Conclusions and outlook 6.1 An artificial atom with a quadrupole transition.................. 125 for Cleo Acknowledgements his thesis would never have been possible without the ideas, friendship, support and Tassistance of numerous people. Foremost among them is of course Steve Girvin, who is infinitely more patient than the stereotypical infinitely-patient
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