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Nonlinearities in the Quantum Measurement Process of Superconducting Qubits

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Nonlinearities in the Quantum Measurement Process of Superconducting Qubits Nonlinearities in the quantum measurement process of superconducting qubits Ioana S¸erban M¨unchen, 2008 Nonlinearities in the quantum measurement process of superconducting qubits Ioana S¸erban Dissertation an der Fakult¨atf¨urPhysik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Ioana S¸erban aus Bukarest, Rum¨anien M¨unchen, Mai 2008 Erstgutachter: Prof. Dr. Jan von Delft Zweitgutachter: Prof. Dr. Frank K. Wilhelm Tag der m¨undlichen Pr¨ufung:09.07.2008 Contents Abstract 1 I Introduction 3 1 Quantum measurement 5 1.1 Quantum decoherence . 7 1.1.1 Open quantum systems . 8 1.2 Quantum optical methods for the solid state . 9 2 Quantum computing with superconducting devices 11 2.1 The Josephson effect . 13 2.2 Superconducting qubits . 16 2.2.1 Flux qubit readout . 19 II Flux qubit readout and control in the presence of decoher- ence 21 3 Dispersive readout protocols 23 3.1 Introduction . 24 3.2 From circuit to Hamiltonian . 26 3.3 Method . 30 3.4 Results . 33 3.4.1 Measurement . 33 3.4.2 Back-action on the qubit . 40 3.4.3 Comparison of dephasing and measurement times . 43 3.5 Conclusion . 45 3.6 Additional information . 46 3.6.1 Parameter conversion . 46 3.6.2 Parameters for the generalized Fokker-Planck equation . 47 vi CONTENTS 4 Qubit dephasing for dispersive readout 49 4.1 Introduction . 50 4.2 Method . 51 4.3 Weak qubit-oscillator coupling . 53 4.4 Strong qubit-oscillator coupling . 54 4.5 Conclusion . 56 4.6 Additional information . 56 5 QND-like, fast qubit measurement 59 5.1 Introduction . 60 5.2 Model and method . 61 5.2.1 The interaction phase . 63 5.2.2 The post-interaction phase . 65 5.3 Results . 66 5.3.1 Qubit decoherence . 66 5.3.2 Detector dynamics . 68 5.4 Practical implementation . 71 5.5 Conclusion . 73 5.6 Additional information . 74 5.6.1 Solution for the Wigner characteristic functions . 74 5.6.2 Conversion to circuit parameters . 76 6 Macroscopic dynamical tunneling in a dissipative system 77 6.1 Introduction . 78 6.2 The Duffing oscillator . 78 6.3 Transformation to the rotating frame . 80 6.4 Thermal activation . 84 6.5 Conclusion . 87 6.6 Additional information . 88 6.6.1 WKB quantisation . 88 6.6.2 Analytic formulas for the action . 89 7 Qubit relaxation from the Duffing oscillator 91 7.1 Introduction . 92 7.2 The model . 92 7.3 Relaxation from detector noise . 93 7.3.1 The rotating frame . 94 7.3.2 The initial state . 95 7.3.3 Using the quantum regression theorem . 97 7.3.4 Results . 98 7.4 Outlook . 99 Table of contents vii 8 Optimal control of a qubit coupled to a two-level fluctuator 101 8.1 Introduction . 102 8.2 Decoherence model . 103 8.3 Optimal control and results . 105 8.4 Conclusion . 106 Conclusion 106 A System-bath model 111 A.1 The quantum regression theorem . 115 B Wigner representation 117 B.1 Useful relations . 118 C Floquet states 121 Bibliography 123 List of publications 138 Acknowledgment 140 Zusammenfassung 143 Curriculum vitae 144 viii Table of contents List of Figures 2.1 Schematic representation of a Josephson junction . 14 2.2 Tilted washboard potential of a Josephson junction . 16 2.3 Schematic representation of a superconducting flux qubit . 18 2.4 Schematic representation of a superconducting quantum interference device 19 3.1 Simplified circuit for dispersive flux qubit readout using a SQUID . 26 3.2 Detector output: probability distribution of momentum . 37 3.3 Discrimination time . 38 3.4 Dephasing rate dependence on driving . 42 3.5 Comparison of dephasing and measurement rates . 44 3.6 Comparison of dephasing and discrimination rates (short time measurement) 45 4.1 Illustration of the phase Purcell effect . 53 4.2 The qubit dephasing rate overview . 54 4.3 Qubit dephasing rate in the strong coupling regime . 55 4.4 Illustration of the dephasing in the weak coupling regime . 57 5.1 Simplified circuit for fast QND-like flux qubit readout . 62 5.2 Qubit dephasing and relaxation for the QND-like readout . 67 5.3 Detector dynamics in the phase-space during and after the qubit interaction 70 5.4 Detector output: voltage probability distribution . 71 5.5 Circuit diagram for the practical implementation of the QND-like readout . 72 5.6 Voltage probability distribution with additional reference SQUID . 73 6.1 Behavior of the classical Duffing oscillator: a double well parallel . 79 6.2 The Hamiltonian in the phase-space and the classical trajectories . 82 6.3 Quantized energies of the system in the rotating frame . 83 6.4 Estimation of the effective temperature . 85 6.5 Tunneling and activation rates . 86 6.6 Effective double well potential in the rotating frame . 88 7.1 Relaxation rate for the qubit coupled to the Duffing oscillator . 99 7.2 Effective mass and frequency of the Duffing oscillator . 100 x List of figures 8.1 Illustration of the coupled system qubit-fluctuator-bath . 103 8.2 Typical behavior of relaxation in the coupled qubit-TLF system . 104 8.3 Gate error overview . 107 8.4 Comparison of Rabi and optimized pulses . 107 8.5 Gate error dependence on the bath parameters . 108 Abstract As condensed matter systems become smaller, and ultimately reach the nanoscale, quan- tum electrodynamical effects become more important. Mesoscopic systems such as quan- tum dots and ultra-small Josephson junctions can, e. g., be used as artificial atoms or molecules. Compared to natural atoms, they present the significant advantage that their optical and electronic properties can be designed and controlled with high flexibility. Such artificial quantum systems offer an attractive combination of high engineering potential and quantum properties, thus being an ideal test-bed for fundamental physics. These properties make solid state devices promising candidates in the realization of a scalable quantum computer. It has been shown that solid-state quantum bits (qubits) present quantum-coherent features such as Rabi oscillations and Ramsey fringes. Nevertheless, the preservation of such a coherent behavior in the presence of the many degrees of freedom in the environment of such mesoscopic devices remains an important issue, as well as the detection of the quantum state of the qubit. The work described in this thesis focuses on the investigation of decoherence and mea- surement backaction, on the theoretical description of measurement schemes and their improvement. It is motivated by the importance of improving detection schemes for the field of quantum computing, but also by the fundamental interest in the physics of quantum measurement and the implications of nonlinearities for this process. The study presented here is centered around quantum computing implementations using superconducting devices and most important, the Josephson effect. The measured system is invariantly a qubit, i. e. a two-level system. The objective is to study detectors with increasing nonlinearity, e. g. coupling of the qubit to the frequency a driven oscillator, or to the bifurcation amplifier, to determine the performance and backaction of the detector on the measured system and to investigate the importance of a strong qubit-detector coupling for the achievement of a quantum non-demolition type of detection. The first part gives a very basic introduction to quantum information, briefly reviews some of the most promising physical implementations of a quantum computer before fo- cusing on the superconducting devices. It also gives a short summary for the theory of decoherence, which offers powerful tools for the description of open quantum systems, thus providing an insight into the physics of quantum measurement. The second part presents a series of studies of different qubit measurements, describing the backaction of the measurement onto the measured system and the internal dynamics 2 Abstract of the detector. The chapters follow closely publications in peer reviewed journals, or still under review, and have a self contained structure, with separate introduction and conclusion. Methodology adapted from quantum optics and chemical physics (master equations, phase-space analysis etc.) combined with the representation of a complex environment yielded a tool capable of describing a nonlinear, non-Markovian environment, which couples arbitrarily strongly to the measured system. This is described in chapter 3 along with a theoretical description of the the dispersive readout protocol of Ref. [1]. The developed tool allows the investigation of the non-linear coupling between qubit and its detector beyond the weak measurement limit. Chapter 4 focuses on the backaction on the qubit and presents novel insights into the qubit dephasing in the strong coupling regime. Chapter 5 uses basically the same system and technical tools to explore the potential of a fast, strong, indirect measurement, and determine how close such a detection would ideally come to the quantum non-demolition regime. Chapter 6 focuses on the internal dynamics of a strongly driven Josephson junction. The nonlinearity of this device (or rather of the very similar Josephson bifurcation amplifier, JBA) has been already used as a significant ingredient for a qubit measurement, achiev- ing remarkable resolution. However, this feature may be a source of novel macroscopic quantum features such as macroscopic dynamical tunneling, and thus worth attention by itself. Dynamical tunneling refers to tunneling between two patterns of motion rather than the conventional tunneling between two points in space. The possibility for such an phe- nomenon to be detected, in the presence of noise, with present day technological means is investigated. The analytical results are based on phase-space methods, a modified version of the WKB approximation and the Caldeira-Leggett approach. Returning to the application of the JBA as a qubit detector, chapter 7 describes the relaxation of the qubit in contact with its detector.
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