Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck EquationsAs a graduate student working in quantum optics I encountered the question that might be taken as the theme of this book. The question definitely arose at that time though it was not yet very clearly defined; there was simply some deep irritation caused by the work I was doing, something quite fundamental I did not understand. Of course, so many things are not understood when one is a graduate student. However, my nagging question was not a technical issue, not merely a mathematical concept that was difficult to grasp. It was a sense that certain elementary notions that are accepted as starting points for work in quantum optics somehow had no fundamental foundation, no identifiable root. My inclination was to mine physics vertically, and here was a subject whose tunnels were dug horizontally. There were branches, certainly, going up and going down. Nonetheless, something major in the downwards direction was missing-at least in my understanding; no doubt others understood the connections downwards very well. In retrospect I can identify the irritation. Quantum optics deals primarily with dynamics, quantum dynamics, and in doing so makes extensive use of words like "quantum fluctuations" and "quantum noise. " The words seem harmless enough. Surely the ideas behind them are quite clear; after all, quantum mechanics is a statistical theory, and in its dynamical aspects it is therefore a theory of fluctuations. But there was my problem. Nothing in Schrodinger's equation fluctuates. |
Contents
The Master Equation Approach | 1 |
12 Inadequacy of an Ad Hoc Approach | 2 |
13 System Plus Reservoir Approach | 3 |
131 The Schrodinger Equation in IntegroDifferential Form | 5 |
132 Born and Markov Approximations | 6 |
133 The Markov Approximation and Reservoir Correlations | 7 |
14 The Damped Harmonic Oscillator | 9 |
142 Some Limitations | 17 |
521 The Green Function | 166 |
522 Moments of MultiDimensional Gaussians | 169 |
523 Formal Solution for TimeDependent Averages | 171 |
524 Equation of Motion for the Covariance Matrix | 174 |
525 SteadyState Spectrum of Fluctuations | 176 |
53 Stochastic Differential Equations | 178 |
531 A Comment on Notation | 179 |
532 The Wiener Process | 180 |
143 Expectation Values and Commutation Relations | 18 |
15 TwoTime Averages and the Quantum Regression Formula | 19 |
151 Formal Results | 22 |
152 Quantum Regression for a Complete Set of Operators | 25 |
153 Correlation Functions for the Damped Harmonic Oscillator | 27 |
2 TwoLevel Atoms and Spontaneous Emission | 29 |
in the Master Equation Approach | 32 |
222 The Einstein A Coefficient | 35 |
223 Matrix Element Equations Correlation Functions and Spontaneous Emission Spectrum | 36 |
224 Phase Destroying Processes | 39 |
23 Resonance Fluorescence | 43 |
231 The Scattered Field | 45 |
232 Master Equation for a TwoLevel Atom Driven by a Classical Field | 48 |
233 Optical Bloch Equations and Dressed States | 51 |
234 The Fluorescence Spectrum | 56 |
235 SecondOrder Coherence | 60 |
236 Photon Antibunching and Squeezing | 65 |
The GlauberSudarshan P Representation | 75 |
31 The GlauberSudarshan P Representation | 76 |
311 Coherent States | 77 |
312 Diagonal Representation for the Density Operator Using Coherent States | 81 |
Coherent States Thermal States and Fock States | 83 |
314 FokkerPlanck Equation for the Damped Harmonic Oscillator | 89 |
315 Solution of the Fokker Planck Equation | 91 |
32 The Characteristic Function for NormalOrdered Averages | 94 |
321 Operator Averages and the Characteristic Function | 95 |
322 Derivation of the FokkerPlanck Equation Using the Characteristic Function | 96 |
P Q and Wigner Representations | 101 |
41 The Q and Wigner Representations | 102 |
412 The Damped Harmonic Oscillator in the Q Representation | 105 |
413 AntinormalOrdered Averages Using the P Representation | 108 |
414 The Wigner Representation | 110 |
42 Fun with Fock States | 114 |
422 Damped Fock State in the P Representation | 117 |
423 Damped Fock State in the Q and Wigner Representations | 120 |
43 TwoTime Averages | 123 |
431 QuantumClassical Correspondence for General Operators | 124 |
432 Associated Functions and the Master Equation | 129 |
433 NormalOrdered TimeOrdered Averages in the P Representation | 131 |
434 More General TwoTime Averages Using the P Representation | 133 |
435 TwoTime Averages Using the Q and Wigner Representations | 137 |
5 FokkerPlanck Equations and Stochastic Differential Equations | 147 |
51 OneDimensional FokkerPlanck Equations | 148 |
511 Drift and Diffusion | 149 |
512 SteadyState Solution | 153 |
513 Linearization and the System Size Expansion | 155 |
514 Limitations of the Linearized Treatment of Fluctuations | 160 |
515 The Truncated KramersMoyal Expansion | 164 |
52 Linear FokkerPlanck Equations | 165 |
533 Stochastic Differential Equations | 183 |
534 Ito and Stratonovich Integrals | 186 |
535 Fokker Planck Equations and Equivalent Stochastic Differential Equations | 190 |
536 MultiDimensional OrnsteinUhlenbeck Process | 192 |
6 QuantumClassical Correspondence for TwoLevel Atoms | 195 |
611 The Characteristic Function and Associated Distribution | 196 |
612 Some Operator Algebra | 197 |
613 PhaseSpace Equation of Motion for the Damped TwoLevel Atom | 199 |
614 A Singular Solution to the PhaseSpace Equation of Motion | 205 |
62 NormalOrdered Representation for a Collection of TwoLevel Atoms | 211 |
621 Collective Atomic Operators | 212 |
622 Direct Product States Dicke States and Atomic Coherent States | 216 |
623 The Characteristic Function and Associated Distribution | 222 |
624 Nonsingular Approximation for the P Distribution | 223 |
625 TwoTime Averages | 226 |
626 Other Representations | 232 |
63 FokkerPlanck Equation for a Radiatively Damped TwoLevel Medium | 233 |
632 Closed Dynamics for NormallyOrdered Averages of Collective Operators | 236 |
633 Operator Averages Without Quantum Fluctuations | 241 |
634 PhaseSpace Equation of Motion for Independently Damped TwoLevel Atoms | 245 |
FirstOrder Treatment of Quantum Fluctuations | 248 |
636 SteadyState Distribution of Inversion | 252 |
Preliminaries | 257 |
71 Laser Theory from Einstein Rate Equations | 258 |
712 Spontaneous Emission and Thermal Photons | 263 |
A Stochastic Model | 268 |
714 TwoLevel Model and Laser Parameters | 276 |
72 PhaseSpace Formulation in the NormalOrdered Representation | 280 |
722 Master Equation for the SingleMode Homogeneously Broadened Laser | 284 |
723 The Characteristic Function and Associated Distribution | 286 |
724 PhaseSpace Equation of Motion for the SingleMode Homogeneously Broadened Laser | 287 |
73 The Laser Output Field | 289 |
732 Coherently Driven Cavities | 293 |
733 Correlations Between the Free Field and Source Field for Thermal Reservoirs | 295 |
734 Spectrum of the Free Field plus Source Field for the Laser Below Threshold | 302 |
PhaseSpace Analysis | 305 |
812 Laser Equations Without Fluctuations | 312 |
813 Linearized Treatment of Quantum Fluctuations Below Threshold | 316 |
814 Adiabatic Elimination of the Polarization and Laser Linewidth | 320 |
82 Laser FokkerPlanck Equation at Threshold | 325 |
821 System Size Expansion and Adiabatic Elimination of Atomic Variables | 326 |
822 SteadyState Solution and Threshold Photon Number | 329 |
Laser FokkerPlanck Equation Above Threshold | 331 |
831 System Size Expansion Above Threshold | 333 |
832 Adiabatic Elimination | 340 |
833 Quantum Fluctuations Above Threshold | 345 |
References | 349 |
357 | |
Other editions - View all
Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck ... Howard J. Carmichael No preview available - 2002 |
Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck ... Howard Carmichael No preview available - 1999 |
Common terms and phrases
8-function adiabatic elimination amplitude antibunching antinormal-ordered approximation calculation cavity characteristic function classical coherent correlation function damped harmonic oscillator damped two-level atom defined density operator derivatives described distribution dynamics e−xt electromagnetic field equation of motion Fock Fokker-Planck equation formal Fourier transform frequency Gaussian given gives Hamiltonian integral interaction inversion J₂ laser field laser mode master equation matrix elements ñ+nspon nonlinear normal-ordered nsat Nspon obtain operator averages phase phase-space equation photon number photopulse polarization quantum fluctuations quantum mechanics quantum optics quantum regression formula quantum-classical correspondence rate equation relationship reservoir resonance fluorescence result right-hand side Sect single-atom solution spectrum spontaneous emission statistical steady-state stochastic differential equations system size expansion theory thermal threshold tion two-level atom two-time averages Wiener process Wigner representation write θα ӘР მ მ მა
Popular passages
Page 354 - FT Arecchi, E. Courtens, R. Gilmore. and H. Thomas, Phys. Rev. A 6, 2211 (1972).