MEASUREMENT THEORY QUANTUM AND ITS APPLICATIONS KURT JACOBS. University of Massachusetts at Boston. fg Cambridge WW UNIVERSITY PRESS

Transcription

1 QUANTUM MEASUREMENT THEORY AND ITS APPLICATIONS KURT JACOBS University of Massachusetts at Boston fg Cambridge WW UNIVERSITY PRESS

2 Contents Preface page xi 1 Quantum measurement theory Introduction and overview Classical measurement theory Understanding Bayes'theorem Multiple measurements and Gaussian distributions Prior states-of-knowledge and invariance Quantum measurement theory The measurement postulate Quantum states-of-knowledge: density matrices Quantum measurements Understanding quantum measurements Relationship to classical measurements Measurements of observables and resolving power A measurement of position The polar decomposition: bare measurements and feedback Describing measurements within unitary evolution Inefficient measurements Measurements on ensembles of states 40 2 Useful concepts from information theory Quantifying information The entropy The mutual information Quantifying uncertainty about a quantum system The von Neumann entropy Majorization and density matrices Ensembles corresponding to a density matrix Quantum measurements and information Information-theoretic properties Quantifying disturbance 72 vii

3 viii Contents 2.4 Distinguishing quantum states Fidelity of quantum operations 82 3 Continuous measurement Continuous measurements with Gaussian noise Classical continuous measurements Gaussian quantum continuous measurements When the SME is the classical Kalman-Bucy filter The power spectrum of the measurement record Solving for the evolution: the linear form of the SME The dynamics of measurement: diffusion gradients Quantum jumps Distinguishing quantum from classical Continuous measurements on ensembles of systems Measurements that count events: detecting photons Homodyning: from counting to Gaussian noise Continuous measurements with more exotic noise? The Heisenberg picture: inputs, outputs, and spectra Heisenberg-picture techniques for linear systems Equations of motion for Gaussian states Calculating the power spectrum of the measurement record Parameter estimation: the hybrid master equation An example: distinguishing two quantum states Statistical mechanics, open systems, and measurement Statistical mechanics Thermodynamic entropy and the Boltzmann distribution Entropy and information: Landauer's erasure principle Thermodynamics with measurements: Maxwell's demon Thermalization I: the origin of irreversibility A new insight: the Boltzmann distribution from typicality Hamiltonian typicality Thermalization II: useful models Weak damping: the Redfield master equation Redfield equation for time-dependent or interacting systems Baths and continuous measurements Wavefunction "Monte Carlo" simulation methods Strong damping: master equations and beyond The quantum-to-classical transition Irreversibility and the quantum measurement problem Quantum feedback control 5.1 Introduction Measurements versus coherent interactions Explicit implementations of continuous-time feedback 239

4 Contents ix Feedback via continuous measurements Coherent feedback via unitary interactions Coherent feedback via one-way fields Mixing one-way fields with unitary interactions: a coherent version of Markovian feedback Feedback control via continuous measurements Rapid purification protocols Control via measurement back-action Near-optimal feedback control for a single qubit? Summary Optimization Bellman's equation and the HJB equation Optimal control for linear quantum systems Optimal control for nonlinear quantum systems Metrology Metrology of single quantities The Cramer-Rao bound Optimizing the Cramer-Rao bound Resources and limits to precision Adaptive measurements Metrology of signals Quantum-mechanics-free subsystems Oscillator-mediated force detection Quantum mesoscopic systems I: circuits and measurements Superconducting circuits Procedure for obtaining the circuit Lagrangian (short method) Resonance and the rotating-wave approximation Superconducting harmonic oscillators Superconducting nonlinear oscillators and qubits The Josephson junction The Cooper-pair box and the transmon Coupling qubits to resonators The RF-SQUID and flux qubits Electromechanical systems Optomechanical systems Measuring mesoscopic systems Amplifiers and continuous measurements Translating between experiment and theory Implementing a continuous measurement Quantum transducers and nonlinear measurements Quantum mesoscopic systems II: measurement and control Open-loop control 383

5 X Contents Fast state-swapping for oscillators Preparing non-classical states Measurement-based feedback control Cooling using linear feedback control Squeezing using linear feedback control Coherent feedback control The "resolved-sideband" cooling method Resolved-sideband cooling via one-way fields Optimal cooling and state-preparation 416 Appendix A The tensor product and partial trace 432 Appendix B A fast-track introduction for experimentalists 441 Appendix C A quick introduction to Ito calculus 448 Appendix D Operators for qubits and modes 451 Appendix E Dictionary ofmeasurements 456 Appendix F Input-output theory 458 FA A mode of an optical or electrical cavity 458 F.2 The traveling-wave fields atx = 0: the input and output signals 462 F.3 The Heisenberg equations of motion for the system 463 FA A weakly damped oscillator 467 F.5 Sign conventionsfor input-output theory 467 F.6 The quantum noise equations for the system: Ito calculus 468 F. 7 Obtaining the Redfield master equation 469 F.8 Spectrum of the measurement signal 470 Appendix G Various formulae and techniques 475 G. I The relationship between Hz and s~], and writing decay rates in Hz 475 G.2 Position representation of a pure Gaussian state 475 G.3 The multivariate Gaussian distribution 476 G.4 The rotating-wave approximation (RWA) 476 G.5 Suppression of off-resonant transitions Ml G.6 Recursion relationsfor time-independent perturbation theory 478 G. 7 Finding operator transformation, reordering, and splitting relations 479 G.8 The Haar measure 484 G.9 Generalform of the Kushner-Stratonovich equation 485 G. 10 Obtaining steady states for linear open systems 486 Appendix H Some proofs and derivations 490 //./ The Schumacher-Westmoreland-Wootters theorem 490 H.2 The operator-sum representation for quantum evolution 492 H.3 Derivation of the Wiseman-Milburn Markovian feedback SME 494 References 498 Index 539