C.W. Gardiner. P. Zoller. Quantum Nois e. A Handbook of Markovian and Non-Markovia n Quantum Stochastic Method s with Applications to Quantum Optics
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1 C.W. Gardiner P. Zoller Quantum Nois e A Handbook of Markovian and Non-Markovia n Quantum Stochastic Method s with Applications to Quantum Optics
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3 1. A Historical Introduction Heisenberg's Uncertainty Principle The Equation of Motion and Repeated Measurements The Spectrum of Quantum Noise Emission and Absorption of Light Consistency Requirements for Quantum Noise Theory Consistency with Statistical Mechanics Consistency with Quantum Mechanics Quantum Stochastic Processes and the Master Equation The Two Level Atom in a Thermal Radiation Field Relationship to the Pauli Master Equation Quantum Statistics The Density Operator Density Operator Properties Von Neumann's Equation Quantum Theory of Measurement Precise Measurements Imprecise Measurements The Quantum Bayes Theorem More General Kinds of Measurements Measurements and the Density Operator Multitime Measurements Sequences of Measurements Expression as a Correlation Function General Correlation Functions Quantum Statistical Mechanics Entropy Thermodynamic Equilibrium The Bose Einstein Distribution System and Heat Bath Density Operators for "System" and "Heat Bath" Mutual Influence of "System" and "Bath" Quantum Langevin Equations The Harmonic Oscillator Heat Bath Derivation of the Langevin Equation Commutation Relations for Noise Sources 49
4 3.2 The Field Interpretation-Noise Inputs and Outputs Input and Output Fields Equations of Motion for System Operators The Noise Interpretation Thermal Statistics The Classical Limit Behaviour of the Langevin Correlation Function as a Function of Time Other Noise Statistics Examples and Applications A Particle Moving in a Potential The Brownian Particle Langevin Equation The Harmonic Oscillator The Two Level Atom The Rotating Wave Approximation The Adjoint Equation Derivation of the Adjoint Equation Comments on the Adjoint Equation Summary of the Adjoint Equation The Master Equation The Quantum Brownian Motion Master Equation Quantum Brownian Motion of a Particle in a Potential The Quantum Optical Case Phase Space Methods The Harmonic Oscillator in One Variable Equations of Motion-Classical Equations of Motion-Quantum The Schrödinger Picture: Energy Eigenvalue s and Number States The Heisenberg Picture Coherent States and the Classical Limit Coherent States as Quasi-Classical States Coherent State Solution for the Harmonic Oscillator Coherent States Properties of the Coherent States Coherent States are Driven Oscillator Wavefunctions Phase Space Representation s of the Harmonic Oscillator Density Operator The Q-Representation The Quantum Characteristic Function The P-Representation The Wigner Function Gaussian Density Operators Operator Correspondences and Equations of Motion Application to the Driven Harmonic Oscillator 124
5 4.5.2 The Wigner Functio n and the Quasiclassical Langevin Equation 12 6 Appendix 4A. The Baker-Hausdorff Formula A.1 Corollaries Quantum Markov Processes The Physical Basis of the Master Equation Derivation of the Quantum Optical Master Equation. 13 I A Derivation Based on Projection Operators Relationship to the Quantum Optical Master Equation Quantum Optical Master Equation with Arbitrary Bath Relationship to the Quantum Brownian Motion Master Equation Notational Matters Multitime Structure of Quantum Markov Processes Computation of Multitime Averages The Markov Interpretation Quantum Regression Theorem Inputs, Outputs and Quantum Stochastic Differential Equations Idealized Hamiltonian Derivation of the Langevin Equations Inputs and Outputs, and Causality Several Inputs and Outputs Formulation of Quantum Stochastic Differential Equations Quantum Ito Stochastic Integration Ito Quantum Stochastic Differential Equation The Quantum Stratonovich Integral Connection between the Ito and Stratonovich Integral Stratonovich Quantum Stochastic Differential Equation Comparison of the Two Forms of QSDE Noise Sources of Several Frequencies The Master Equation Description of the Density Operator Derivation of the Master Equation Comparison with Previous Results Master Equation with Several Frequencies Equivalence of QSDE and Master Equation Correlation Functions of Inputs, System, and Outputs Applying the Master Equation Using the Number State Basis The Damped Harmonic Oscillator-Quantum Optical Case The Phase Damped Oscillator Quantum Classical Correspondence Use of the P-Representation Time Correlation Functions in the P-Representation.. 174
6 6.2.3 Application to the Damped Harmonic Oscillator General Form for Time Correlation Functions in the P-Representation Some Amplifier Models A Simple Amplifier Comparison of P-, Q- and Wigner Function Methods The Degenerate Parametric Amplifier Generalized P-Representations The R-Representation Existence Theorems Definition of the Positive P-Representation by Mean s of the Quantum Characteristic Function Operator Identities Time-Development Equations Complex P-Representation Positive P-Representation Applications of the Generalized P-Representations Complex P-Representation Applications of the Positive P-Representation Linear Systems and Linearization Stochastic Simulation The Single Mode Laser Analytic Treatment via the Deterministic Equation Full Stochastic Case Numerical Signatures The Anharmonic Oscillator Numerical Signatures Theoretical Framework of the Problem The Power-Law Tails The Earliest "Spike" Time Conclusions Guidelines for Simulations Example-Quantum Noise in the Parametric Oscillator Amplifiers and Measurement Input-Output Theory of Amplifiers and Attenuators Amplifiers The Inverted Oscillator Heatbath The Amplifier Model Added Noise Signal to Noise Ratio "Noise Temperature" of an Amplifier QSDEs in the Case of a Negative Temperature Bath Ito QSDEs for Positive and Negative Temperature Phase Conjugating Amplifier The Degenerate Parametric Amplifier 220
7 7.3 The Macroscopic Limit in Open Quantum Systems Example-Quantum Brownian Motion Example-The Quantum Optical Situation Application to a Model of Quantum Measurement Photon Counting Quantization of the Electromagnetic Field Maxwell's Equations Expansion in Mode Functions Quantization by Commutation Relations Quantization in an Infinite Volume Optical Electromagnetic Fields The Photon Beams of Light Photodetection and Photon Counting The Physical Basis of the Detection Formulae Coherence and Correlation Functions Normalized Correlation Functions Photon Counting Formulae Development of the Formulae Master Equation and Quantum Stochastic Differential Equations Photon Counting Probabilities Intensity Correlation s and the Hanbury-Brown Twiss Experiment Mandel's Counting Formula Applications to Particular States Model for Efficiency Less Than 100% Homodyne and Heterodyne Detection Schematic Setup of Homodyne and Heterodyne Detection General Formulae Coherent Signal Detection Balanced Homodyne/Heterodyne Detection Input-Output Formulations of Photodetection A One Atom Model A Model Using Fermi Electrons A Spatially Distributed Detector Model Interaction of Light with Atoms Two Level Systems Pauli Matrix Description Pauli Matrix Properties Atoms with More Than Two Levels Two Level Atom in the Electromagnetic Field Lamb and Stark Shifts Rotating Wave Approximation 280
8 9.2.3 Master Equation and QSDE Two Level Atom in a Coherent Driving Field The Laser Quantum Langevin Equations for the Laser Derivation of Laser Equations Solutions of the Laser Equations The Nature of Laser Light Optical Bistability Other Phase Space Methods The Method of Haken, Risken and Weidlich A More Direct Phase Space Method Squeezing Squeezed States of the Harmonic Oscillator Definition of an Ideal Squeezed State The Degenerate Parametric Amplifier Squeezing in the Degenerate Parametric Amplifier Squeezed White Noise Squeezed Light on a Single Atom Simulation Methods Based on the Adjoint Equation Adjoint Equation for Squeezed Light Solution of the Equations of Motion Numerical Methods The Stochastic Schrödinger Equation Quantum Stochastic Schrödinger Equation The Model Validity of the Model QSDE for Time Evolution Operators and State Vectors QSDE in Stratonovich Form Conversion from Stratonovich to Ito Form Formal Solution QSDE for the State Vector QSDE for the Stochastic Density Operator QSDE for System Operators Non-vacuum Initial States Number Processes and Photon Counting Number Processes and Quantum Stochastic Calculus Input and Output Photon Counting as a Measurement of the A(t) Operator Photon Counting and Exclusive Probability Densities Mandel's Counting Formula The Characteristic Functional and System Averages Conditional Dynamics and a Posteriori States Stochastic Schrödinger Equation for Counting Processes Wave Function Simulation : Procedure 367
9 Simulation of Correlation Functions and Spectra Diffusion Processes and Homodyne Detection Homodyne Detection The Characteristic Functional and System Averages Stochastic Schrödinger Equation Applications and Illustrations Resonance Fluorescenc e of Strongly Driven Two-Level Systems Quantum Jumps in Three-Level Atoms Mechanical Light Effects Quantized Atomic Motion in Optical Molasses Localization by Spontaneous Emission Cascaded Quantum Systems Coupling Equations Relation to Input-Output Formalism Conversion to Quantum Ito Equations Master Equation The Lindblad Form Stochastic Schrödinger Equation Imperfect Coupling Application to Harmonic Oscillator Systems Driving by Squeezed Light Two Level Atom Driven by Squeezed Light Two Level Atom Driven by Antibunched Light Coherent Excitation of the Source Atom Incoherent Excitation of the Source Atom Characterizing Non-Classical Light Transmission of Quantum Informatio n Through a Quantum Network Quantum Information Physical Implementation of a Quantum Network with Atoms and Photons Physical Idea behind Ideal Transmission Quantum Transmission in a Quantum Trajectory Picture References 41 9 Bibliography 42 4 Author Index 42 6 Subject Index 427
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