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Introduction to Experimental Quantum Measurement With Introduction to Experimental Quantum Measurement with Superconducting Qubits Mahdi Naghiloo, Murch Lab Washington University in St. Louis, April 2019 Abstract|Quantum technology has been rapidly growing due to its potential revolutionary applications. In particular, superconducting qubits provide a strong light-matter interaction as required for quantum computation and in principle can be scaled up to a high level of complexity. However, obtaining the full benefit of quantum mechanics in superconducting circuits requires a deep understanding of quantum physics in such systems in all aspects. One of the most crucial aspects is the concept of measurement and the dynamics of the quantum systems under the measurement process. This document is intended to be a pedagogical introduction to the concept of quantum measurement from an experimental perspective. We study the dynamics of a single superconducting qubit under continuous monitoring. We demonstrate that weak measurement is a versatile tool to investigate fundamental questions in quantum dynamics and quantum thermodynamics for open quantum systems. arXiv:1904.09291v1 [quant-ph] 19 Apr 2019 Contents Abstract i List of Figures v 1 Introduction 1 1.1 Overview . .3 2 The Light-Matter Interaction 6 2.1 One-dimensional cavity modes . .6 2.1.1 How to visualize the state of light . .9 2.2 Qubit . 16 2.2.1 Josephson junctions . 17 2.2.2 Transmon qubit . 18 2.3 Qubit-cavity interaction . 21 2.3.1 Jaynes-Cummings model . 23 2.3.2 Dispersive approximation . 27 2.4 Dynamics of a driven qubit . 29 2.4.1 Rabi oscillations: The semi-classical approach . 29 2.4.2 Dynamics in the presence of dissipation . 34 3 Superconducting Quantum Circuits 38 3.1 Cavity . 38 3.2 Qubit . 42 3.2.1 Transmon fabrication: . 42 3.2.2 JJ characterization . 45 3.3 Qubit-cavity system characterization . 47 3.3.1 One-tone spectroscopy: \punch-out" . 48 3.3.2 Two-tone spectroscopy . 50 3.3.3 Time domain measurement: basics . 52 3.3.4 Rabi measurements . 56 3.3.5 T1 Measurement . 58 ∗ 3.3.6 Ramsey Measurement (T2 )........................ 58 3.3.7 Full state tomography . 59 3.4 Josephson Parametric Amplifier . 60 ii Contents 3.4.1 Classical nonlinear oscillators . 60 3.4.2 Paramp operation . 64 3.4.3 Phase-sensitive amplification: phase vs amplitude . 67 4 Quantum Measurement 69 4.1 Projective measurement . 69 4.2 Generalized measurement in the σz basis . 72 4.2.1 Simple Model . 72 4.2.2 POVM . 75 4.2.3 POVM in terms of physical parameters . 76 4.3 Continuous measurement in σz basis . 79 4.3.1 Stochastic Schr¨odingerequation . 80 4.3.2 Stochastic master equation . 81 4.3.3 Inefficient measurement . 82 4.4 Bayesian update . 85 4.4.1 Bayesian update in terms of the Bloch components . 87 4.5 Bayesian vs SME . 88 4.6 Generalized measurement in the σx basis . 89 4.6.1 POVM . 89 4.6.2 SME . 92 4.7 z-measurement procedure . 95 4.7.1 Basic characterization . 95 4.7.2 Paramp calibration . 95 4.7.3 Quantum efficiency calibration . 96 4.7.4 Tomography pulse calibration . 101 4.7.5 Data acquisition . 101 4.7.6 Post-processing: Quantum trajectory update . 102 4.8 σx measurement procedure . 106 4.8.1 Quantum efficiency calibration . 106 4.8.2 State update and quantum trajectory . 106 5 Monitoring Spontaneous Emission of a Quantum Emitter 108 5.1 Spontaneous emission . 108 5.2 Photon Detection . 109 5.3 Homodyne detection of spontaneous emission . 111 6 Quantum Thermodynamics: Quantum Maxwell's Demon 117 6.1 Fluctuation theorems: thermodynamics at the microscope scale . 117 6.2 Maxwell's demon and the 2nd law . 119 6.3 Continuous monitoring: a quantum Maxwell's demon . 120 6.3.1 Examining the Jarzynski equality . 121 6.3.2 The demon's information . 123 6.3.3 Test of the generalized Jarzynski equality . 125 iii Contents 6.4 Information gain and loss . 125 Bibliography 137 iv List of Figures 1.1 Photon detection vs. homodyne detection . .4 1.2 Quantum Maxwell's demon . .5 2.1 One dimensional cavity . .7 2.2 Wigner distribution for photon-number states . 12 2.3 Photon number distributions for coherent states . 13 2.4 Winger function for a coherent state . 15 2.5 Phase shifts for coherent state in the rotating frame . 16 2.6 Circuit QED toolbox . 16 2.7 Josephson junction . 17 2.8 Transmon circuit . 19 2.9 Transmon energy levels . 21 2.10 The qubit-cavity interaction . 22 2.11 Dressed states vs bare states . 26 2.12 Avoided crossing: . 27 2.13 Rabi oscillations . 31 2.14 Eigenstates on the Bloch sphere for a driven qubit . 32 2.15 Driven qubit evolution in the Bloch sphere . 33 2.16 Relaxation and dephasing of a qubit. ......................... 35 2.17 Dephasing and relaxation for the qubit . 37 3.1 TE101 mode in rectangular 3D cavity . 39 3.2 HFSS simulation for cavity transmission . 40 3.3 The cavity linewidth characterization . 41 3.4 The cavity phase shift across the resonance . 41 3.5 Double stack e-beam resist . 43 3.6 A simple design for transmon qubit . 44 3.7 e-beam resist development recipe . 44 3.8 Double-angle evaporation and Josephson junction fabrication . 45 3.9 Qubit pattern SEM . 46 3.10 The HFSS simulation for the transmon shunting capacitor . 46 3.11 The minimum experimental setup for basic qubit characterization . 49 3.12 The \punch-out" measurement . 50 3.13 Two-tone spectroscopy . 51 v List of Figures 3.14 I=Q mixer . 53 3.15 Qubit rotation pulses . 54 3.16 Single sideband modulation (SSB) . 55 3.17 Qubit state readout, homophone detection . 55 3.18 Readout in phase space representation . 56 3.19 Rabi measurement . 57 3.20 Chevron plot . 58 3.21 T1 measurement . 59 3.22 Ramsey measurement . 60 3.23 Full state tomography readout pulses . 61 3.24 JPA Schematic . 61 3.25 Duffing resonator response . 63 3.26 JPA transfer function . 64 3.27 The minimum experimental setup with paramp . 65 3.28 The paramp single pump operation . 66 3.29 The paramp double pump operation . 67 3.30 Phase sensitive amplification . 68 4.1 Bloch sphere . ..
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