Quantum Theory of Optical Coherence: Selected Papers and Lectures

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 One-Atom Photon Detector 38

2.5 The n-Atom 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 First-Order 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 Field-Theoretical 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 Phase-Space Distributions for the Field 121

2.11.1 The P Representation and the Moment Problem 123

2.11.2 A Positive-Definite “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 Space-Time 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 Positive-Definite 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 ρA(t) 238

6.7 Solutions for an Initially Chaotic B Mode 241

6.8 Solution for Initial n-Quantum 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(α,t) 258

Appendix 260

References 261

7 Quantum Theory of Parametric Amplification II 263

7.1 Introduction 263

7.2 The Two-Mode 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 Power-Series Expansions 345

9.5 s-Ordered Power-Series 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 <α|ρ|α> 399

10.6 Ensemble Averages and s Ordering 402

10.7 Examples of the General Quasiprobability Function W(α,s) 408

10.8 Analogy with Heat Diffusion 416

10.9 Time-Reversed Heat Diffusion and W(α,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 Quantum-Mechanical 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 Schrödinger’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 𝛿 Functions and Fourier Transforms 608

16.7 Operator Expansions 610

16.8 Characteristic Functions 612

16.8.1 The s-Ordered Characteristic Function 613

16.9 s-Ordered 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 Fermion-Counting Experiments 626

16.17 Some Elementary Examples 628

16.17.1 The Vacuum State 628

16.17.2 A Physical Two-Mode Density Operator 629

References 631

Index 633