Quantum Theory of Optical Coherence: Selected Papers and LecturesISBN: 9783527406876
656 pages
April 2007

Written by the Nobel Laureate himself, the concepts described in this book have formed the basis for three further Nobel Prizes in Physics within the last decade.
1 The Quantum Theory of Optical Coherence 1
1.1 Introduction 1
1.2 Elements of Field Theory 2
1.3 Field Correlations 7
1.4 Coherence 10
1.5 Coherence and Polarization 15
Appendix 18
References 20
2 Optical Coherence and Photon Statistics 23
2.1 Introduction 23
2.1.1 Classical Theory 27
2.2 Interference Experiments 30
2.3 Introduction of Quantum Theory 35
2.4 The OneAtom Photon Detector 38
2.5 The nAtom Photon Detector 46
2.6 Properties of the Correlation Functions 51
2.6.1 Space and Time Dependence of the Correlation Functions 54
2.7 Diffraction and Interference 56
2.7.1 Some General Remarks on Interference 58
2.7.2 FirstOrder Coherence 59
2.7.3 Fringe Contrast and Factorization 64
2.8 Interpretation of Intensity Interferometer Experiments 66
2.8.1 Higher Order Coherence and Photon Coincidences 67
2.8.2 Further Discussion of Higher Order Coherence 70
2.8.3 Treatment of Arbitrary Polarizations 71
2.9 Coherent and Incoherent States of the Radiation Field 75
2.9.1 Introduction 75
2.9.2 FieldTheoretical Background 77
2.9.3 Coherent States of a Single Mode 80
2.9.4 Expansion of Arbitrary States in Terms of Coherent States 86
2.9.5 Expansion of Operators in Terms of Coherent State Vectors 89
2.9.6 General Properties of the Density Operator 92
2.9.7 The P Representation of the Density Operator 94
2.9.8 The Gaussian Density Operator 100
2.9.9 Density Operators for the Field 104
2.9.10 Correlation and Coherence Properties of the Field 109
2.10 Radiation by a Predetermined Charge–Current Distribution 117
2.11 PhaseSpace Distributions for the Field 121
2.11.1 The P Representation and the Moment Problem 123
2.11.2 A PositiveDefinite “Phase Space Density” 124
2.11.3 Wigner’s “Phase Space Density” 127
2.12 Correlation Functions and Quasiprobability Distributions 132
2.12.1 First Order Correlation Functions for Stationary Fields 134
2.12.2 Correlation Functions for Chaotic Fields 136
2.12.3 Quasiprobability Distribution for the Field Amplitude 139
2.12.4 Quasiprobability Distribution for the Field Amplitudes at Two SpaceTime Points 145
2.13 Elementary Models of Light Beams 148
2.13.1 Model for Ideal Laser Fields 153
2.13.2 Model of a Laser Field With Finite Bandwidth 156
2.14 Interference of Independent Light Beams 164
2.15 Photon Counting Experiments 170
References 181
3 Correlation Functions for Coherent Fields 183
3.1 Introduction 183
3.2 Correlation Functions and Coherence Conditions 184
3.3 Correlation Functions as Scalar Products 186
3.4 Application to Higher Order Correlation Functions 189
3.5 Fields With PositiveDefinite P Functions 191
References 195
4 Density Operators for Coherent Fields 197
4.1 Introduction 197
4.2 Evaluation of the Density Operator 199
4.3 Fully Coherent Fields 205
4.4 Unique Properties of the Annihilation Operator Eigenstates 209
References 216
5 Classical Behavior of Systems of Quantum Oscillators 217
References 220
6 Quantum Theory of Parametric Amplification I 221
6.1 Introduction 221
6.2 The Coherent States and the P Representation 223
6.3 Model of the Parametric Amplifier 227
6.4 Reduced Density Operator for the A Mode 233
6.5 Initially Coherent State: P Representation for the A Mode 234
6.6 Initially Coherent State; Moments, Matrix Elements, and Explicit Representation for _A(t) 238
6.7 Solutions for an Initially Chaotic B Mode 241
6.8 Solution for Initial nQuantum State of A Mode; B Mode Chaotic 244
6.9 General Discussion of Amplification With B Mode Initially Chaotic 249
6.10 Discussion of P Representation: Characteristic Functions Initially Gaussian 252
6.11 Some General Properties of P(_,t) 258
Appendix 260
References 261
7 Quantum Theory of Parametric Amplification II 263
7.1 Introduction 263
7.2 The TwoMode Characteristic Function 265
7.3 The Wigner Function 267
7.4 Decoupled Equations of Motion 271
7.5 Characteristic Functions Expressed in Terms of Decoupled Variables 273
7.6 W and P Expressed in Terms of Decoupled Variables 275
7.7 Results for Chaotic Initial States 278
7.8 Correlations of the Mode Amplitudes 283
References 286
8 Photon Statistics 287
8.1 Introduction 287
8.2 Classical Theory 288
8.3 Quantum Theory: Introduction 290
8.4 Intensity and Coincidence Measurements 293
8.5 First and Higher Order Coherence 297
8.6 The Coherent States 300
8.7 Expansions in Terms of the Coherent States 307
8.8 Characteristic Functions and Quasiprobability Densities 313
8.9 Some Examples 319
8.10 Photon Counting Distributions 322
References 329
9 Ordered Expansions in Boson Amplitude Operators 331
9.1 Introduction 331
9.2 Coherent States and Displacement Operators 333
9.3 Completeness of Displacement Operators 337
9.4 Ordered PowerSeries Expansions 345
9.5 sOrdered PowerSeries Expansions 353
9.6 Integral Expansions for Operators 358
9.7 Correspondences Between Operators and Functions 366
9.8 Illustration of Operator–Function Correspondences 375
Appendix A 377
Appendix B 378
Appendix C 379
Appendix D 380
References 380
10 Density Operators and Quasiprobability Distributions 383
10.1 Introduction 383
10.2 Ordered Operator Expansions 385
10.3 The P Representation 389
10.4 Wigner Distribution 393
10.5 The Function h___i 399
10.6 Ensemble Averages and s Ordering 402
10.7 Examples of the General Quasiprobability Function W(_,s) 408
10.8 Analogy with Heat Diffusion 416
10.9 TimeReversed Heat Diffusion and W(_,s) 418
10.10 Properties Common to all Quasiprobability Distributions 420
References 423
11 Coherence and Quantum Detection 425
11.1 Introduction 425
11.2 The Statistical Properties of the Electromagnetic Field 426
11.3 The Ideal Photon Detector 428
11.4 Correlation Functions and Coherence 429
11.5 Other Correlation Functions 432
11.6 The Coherent States 434
11.7 Expansions in Terms of Coherent States 437
11.8 A Few General Observations 439
11.9 The Damped Harmonic Oscillator 440
11.10 The Density Operator for the Damped Oscillator 444
11.11 Irreversibility and Damping 447
11.12 The Fokker–Planck and Bloch Equations 449
11.13 Theory of Photodetection. The Photon Counter Viewed as a Harmonic Oscillator 453
11.14 The Density Operator for the Photon Counter 459
References 462
12 Quantum Theory of Coherence 463
12.1 Introduction 463
12.2 Classical Theory 468
12.3 Quantum Theory 471
12.4 Intensity and Coincidence Measurements 474
12.5 Coherence 487
12.6 Coherent States 495
12.7 The P Representation 499
12.8 Chaotic States 514
12.9 Wavepacket Structure of Chaotic Field 521
References 530
13 The Initiation of Superfluorescence 531
13.1 Introduction 531
13.2 Basic Equations for a Simple Model 532
13.3 Onset of Superfluorescence 534
References 536
14 Amplifiers, Attenuators and Schrödingers Cat 537
14.1 Introduction: Two Paradoxes 537
14.2 A QuantumMechanical Attenuator: The Damped Oscillator 542
14.3 A Quantum Mechanical Amplifier 548
14.4 Specification of Photon Polarization States 558
14.5 Measuring Photon Polarizations 561
14.6 Use of the Compound Amplifier 563
14.7 Superluminal Communication? 565
14.8 Interference Experiments and Schr¨odinger’s Cat 569
References 575
15 The Quantum Mechanics of Trapped Wavepackets 577
15.1 Introduction 577
15.2 Equations of Motion and Their Solutions 578
15.3 The Wave Functions 581
15.4 Periodic Fields and Trapping 584
15.5 Interaction With the Radiation Field 587
15.6 Sum Rules 590
15.7 Radiative Equilibrium and Instability 592
References 594
16 Density Operators for Fermions 595
16.1 Introduction 595
16.2 Notation 597
16.3 Coherent States for Fermions 597
16.3.1 Displacement Operators 597
16.3.2 Coherent States 599
16.3.3 Intrinsic Descriptions of Fermionic States 600
16.4 Grassmann Calculus 601
16.4.1 Differentiation 601
16.4.2 Even and Odd Functions 601
16.4.3 Product Rule 602
16.4.4 Integration 602
16.4.5 Integration by Parts 603
16.4.6 Completeness of the Coherent States 604
16.4.7 Completeness of the Displacement Operators 604
16.5 Operators 605
16.5.1 The Identity Operator 605
16.5.2 The Trace 606
16.5.3 Physical States and Operators 606
16.5.4 Physical Density Operators 607
16.6 _ Functions and Fourier Transforms 608
16.7 Operator Expansions 610
16.8 Characteristic Functions 612
16.8.1 The sOrdered Characteristic Function 613
16.9 sOrdered Expansions for Operators 614
16.10 Quasiprobability Distributions 616
16.11 Mean Values of Operators 618
16.12 P Representation 619
16.13 Correlation Functions for Fermions 620
16.14 Chaotic States of the Fermion Field 621
16.15 Correlation Functions for Chaotic Field Excitations 624
16.16 FermionCounting Experiments 626
16.17 Some Elementary Examples 628
16.17.1 The Vacuum State 628
16.17.2 A Physical TwoMode Density Operator 629
References 631
Index 632