ISBN: 3540548823
TITLE: Statistical Methods in Quantum Optics 1
AUTHOR: Carmichael
TOC:

Volume 1. Master Equations and Fokker-Planck Equations 1
1. Dissipation in Quantum Mechanics: The Master Equation Approach 1
1.1 Introduction 2
1.2 Inadequacy of an Ad Hoc Approach 3
1.3 System Plus Reservoir Approach 5
1.3.1 The Schrdinger Equation in Integro-Differential Form 5
1.3.2 Born and Markov Approximations 6
1.3.3 The Markov Approximation and Reservoir Correlations 7
1.4 The Damped Harmonic Oscillator 9
1.4.1 Master Equation for the Damped Harmonic Oscillator 9
1.4.2 Some Limitations 17
1.4.3 Expectation Values and Commutation Relations 18
1.5 Two-Time Averages and the Quantum Regression Formula 19
1.5.1 Formal Results 22
1.5.2 Quantum Regression for a Complete Set of Operators 25
1.5.3 Correlation Functions for the Damped Harmonic Oscillator 27
2. Two-Level Atoms and Spontaneous Emission 29
2.1 Two-Level Atom as a Pseudo-Spin System 29
2.2 Spontaneous Emission in the Master Equation Approach 32
2.2.1 Master Equation for a Radiatively Damped Two-Level Atom 32
2.2.2 The Einstein A Coefficient 35
2.2.3 Matrix Element Equations, Correlation Functions, and Spontaneous Emission Spectrum 36
2.2.4 Phase Destroying Processes 39
2.3 Resonance Fluorescence 43
2.3.1 The Scattered Field 45
2.3.2 Master Equation for a Two-Level Atom Driven by a Classical Field 48
2.3.3 Optical Bloch Equations and Dressed States 51
2.3.4 The Fluorescence Spectrum 56
2.3.5 Second-Order Coherence 60
2.3.6 Photon Antibunching and Squeezing 65
3. Quantum-Classical Correspondence for the Electromagnetic Field I: The Glauber-Sudarshan P Representation 75
3.1 The Glauber-Sudarshan P Representation 76
3.1.1 Coherent States 77
3.1.2 Diagonal Representation for the Density Operator Using Coherent States 81
3.1.3 Examples: Coherent States, Thermal States, and Fock States 83
3.1.4 Fokker-Planck Equation for the Damped Harmonic Oscillator 89
3.1.5 Solution of the Fokker-Planck Equation 91
3.2 The Characteristic Function for Normal-Ordered Averages 94
3.2.1 Operator Averages and the Characteristic Function 95
3.2.2 Derivation of the Fokker-Planck Equation Using the Characteristic Function 96
4. Quantum-Classical Correspondence for the Electromagnetic Field II: P, Q, and Wigner Representations 101
4.1 The Q and Wigner Representations 102
4.1.1 Antinormal-Ordered Averages and the Q Representation 102
4.1.2 The Damped Harmonic Oscillator in the Q Representation 105
4.1.3 Antinormal-Ordered Averages Using the P Representation 108
4.1.4 The Wigner Representation 110
4.2 Fun with Fock States 114
4.2.1 Wigner Distribution for a Fock State 114
4.2.2 Damped Fock State in the P Representation 117
4.2.3 Damped Fock State in the Q and Wigner Representations 120
4.3 Two-Time Averages 123
4.3.1 Quantum-Classical Correspondence for General Operators 124
4.3.2 Associated Functions and the Master Equation 129
4.3.3 Normal-Ordered Time-Ordered Averages in the P Representation 131
4.3.4 More General Two-Time Averages Using the P Representation 133
4.3.5 Two-Time Averages Using the Q and Wigner Representations 137
5. Fokker-Planck Equations and Stochastic Differential Equations 147
5.1 One-Dimensional Fokker-Planck Equations 148
5.1.1 Drift and Diffusion 149
5.1.2 Steady-State Solution 153
5.1.3 Linearization and the System Size Expansion 155
5.1.4 Limitations of the Linearized Treatment of Fluctuations 160
5.1.5 The Truncated Kramers-Moyal Expansion 164
5.2 Linear Fokker-Planck Equations 165
5.2.1 The Green Function 166
5.2.2 Moments of Multi-Dimensional Gaussians 169
5.2.3 Formal Solution for Time-Dependent Averages 171
5.2.4 Equation of Motion for the Covariance Matrix 174
5.2.5 Steady-State Spectrum of Fluctuations 176
5.3 Stochastic Differential Equations 178
5.3.1 A Comment on Notation 179
5.3.2 The Wiener Process 180
5.3.3 Stochastic Differential Equations 183
5.3.4 Ito and Stratonovich Integrals 186
5.3.5 Fokker-Planck Equations and Equivalent Stochastic Differential Equations 190
5.3.6 Multi-Dimensional Ornstein-Uhlenbeck Process 192
6. Quantum-Classical Correspondence for Two-Level Atoms 195
6.1 Haken's Representation and the Damped Two-Level Atom 195
6.1.1 The Characteristic Function and Associated Distribution 196
6.1.2 Some Operator Algebra 197
6.1.3 Phase-Space Equation of Motion for the Damped Two-Level Atom 199
6.1.4 A Singular Solution to the Phase-Space Equation of Motion 205
6.2 Normal-Ordered Representation for a Collection of Two-Level Atoms 211
6.2.1 Collective Atomic Operators 212
6.2.2 Direct Product States, Dicke States, and Atomic Coherent States 216
6.2.3 The Characteristic Function and Associated Distribution 222
6.2.4 Nonsingular Approximation for the P Distribution 223
6.2.5 Two-Time Averages 226
6.2.6 Other Representations 232
6.3 Fokker-Planck Equation for a Radiatively Damped Two-Level Medium 233
6.3.1 Master Equation for Independently Damped Two-Level Atoms 233
6.3.2 Closed Dynamics for Normally-Ordered Averages of Collective Operators 236
6.3.3 Operator Averages Without Quantum Fluctuations 241
6.3.4 Phase-Space Equation of Motion for Independently Damped Two-Level Atoms 245
6.3.5 Fokker-Planck Equation: First-Order Treatment of Quantum Fluctuations 248
6.3.6 Steady-State Distribution of Inversion 252
7. The Single-Mode Homogeneously Broadened Laser I: Preliminaries 257
7.1 Laser Theory from Einstein Rate Equations 258
7.1.1 Rate Equations and Laser Threshold 258
7.1.2 Spontaneous Emission and Thermal Photons 263
7.1.3 Quantum Fluctuations: A Stochastic Model 268
7.1.4 Two-Level Model and Laser Parameters 276
7.2 Phase-Space Formulation in the Normal-Ordered Representation 280
7.2.1 Model and Hamiltonian 280
7.2.2 Master Equation for the Single-Mode Homogeneously Broadened Laser 284
7.2.3 The Characteristic Function and Associated Distribution 286
7.2.4 Phase-Space Equation of Motion for the Single-Mode Homogeneously Broadened Laser 287
7.3 The Laser Output Field 289
7.3.1 Free Field and Source Field for a Lossy Cavity Mode 289
7.3.2 Coherently Driven Cavities 293
7.3.3 Correlations Between the Free Field and Source Field for Thermal Reservoirs 295
7.3.4 Spectrum of the Free Field plus Source Field for the Laser Below Threshold 302
8. The Single-Mode Homogeneously Broadened Laser II: Phase-Space Analysis 305
8.1 Linearization: Laser Fokker-Planck Equation Below Threshold 305
8.1.1 System Size Expansion Below Threshold 312
8.1.2 Laser Equations Without Fluctuations 305
8.1.3 Linearized Treatment of Quantum Fluctuations Below Threshold 316
8.1.4 Adiabatic Elimination of the Polarization and Laser Linewidth 320
8.2 Laser Fokker-Planck Equation at Threshold 325
8.2.1 System Size Expansion and Adiabatic Elimination of Atomic Variables 326
8.2.2 Steady-State Solution and Threshold Photon Number 329
8.3 Quasi-Linearization: Laser Fokker-Planck Equation Above Threshold 331
8.3.1 System Size Expansion Above Threshold 333
8.3.2 Adiabatic Elimination 340
8.3.3 Quantum Fluctuations Above Threshold 345
References 349
Index 357
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