Preface xv
Acknowledgments xix
1 Introduction 1
1.1 Optics in Modern Life 1
1.2 The Origin and Progress of Quantum Optics 3
1.3 Motivation Through Simple and Direct Teaching Experiments 7
1.4 Consequences of Photon Correlations 12
1.5 How to Use This Guide 14
References 16
2 Classical Models of Light 19
2.1 Classical Waves 20
2.1.1 Mathematical Description of Waves 20
2.1.2 The Gaussian Beam 21
2.1.3 Quadrature Amplitudes 24
2.1.4 Field Energy, Intensity, and Power 25
2.1.5 A Classical Mode of Light 26
2.1.6 Light Carries Information 28
2.1.7 Modulations 30
2.2 Optical Modes and Degrees of Freedom 32
2.2.1 Lasers with Single and Multiple Modes 32
2.2.2 Polarization 33
2.2.2.1 Poincare Sphere and Stokes Vectors 35
2.2.3 Multimode Systems 36
2.3 Statistical Properties of Classical Light 37
2.3.1 The Origin of Fluctuations 37
2.3.1.1 Gaussian Noise Approximation 38
2.3.2 Noise Spectra 39
2.3.3 Coherence 40
2.3.3.1 Correlation Functions 44
2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case 46
2.5 Spatial Information and Imaging 50
2.5.1 State-of-the-Art Imaging 50
2.5.2 Classical Imaging 52
2.5.3 Image Detection 55
2.5.4 Scanning 56
2.5.5 Quantifying Noise and Contrast 58
2.5.6 Coincidence Imaging 59
2.5.7 Imaging with Coherent Light 60
2.5.8 Image Reconstruction with Structured Illumination 60
2.5.9 Image Analysis and Modes 61
2.5.10 Detection Modes and Displacement 61
2.6 Summary 62
References 63
Further Reading 64
3 Photons: The Motivation to Go Beyond Classical Optics 65
3.1 Detecting Light 65
3.2 The Concept of Photons 68
3.3 Light from a Thermal Source 70
3.4 Interference Experiments 73
3.5 Modelling Single-Photon Experiments 78
3.5.1 Polarization of a Single Photon 79
3.5.1.1 Some Mathematics 80
3.5.2 Polarization States 81
3.5.3 The Single-Photon Interferometer 83
3.6 Intensity Correlation, Bunching, and Anti-bunching 84
3.7 Observing Photons in Cavities 88
3.8 Summary 90
References 90
Further Reading 92
4 Quantum Models of Light 93
4.1 Quantization of Light 93
4.1.1 Some General Comments on Quantum Mechanics 93
4.1.2 Quantization of Cavity Modes 94
4.1.3 Quantized Energy 95
4.1.4 The Creation and Annihilation Operators 97
4.2 Quantum States of Light 97
4.2.1 Number or Fock States 97
4.2.2 Coherent States 99
4.2.3 Mixed States 101
4.3 Quantum Optical Representations 102
4.3.1 Quadrature Amplitude Operators 102
4.3.2 Probability and Quasi-probability Distributions 104
4.3.3 Photon Number Distributions 108
4.3.4 Covariance Matrix 111
4.3.4.1 Summary of Different Representations of Quantum States and Quantum Noise 112
4.4 Propagation and Detection of Quantum Optical Fields 113
4.4.1 Quantum Optical Modes in Free Space 114
4.4.2 Propagation in Quantum Optics 115
4.4.3 Detection in Quantum Optics 117
4.4.4 An Example: The Beamsplitter 118
4.5 Quantum Transfer Functions 120
4.5.1 A Linearized Quantum Noise Description 121
4.5.2 An Example: The Propagating Coherent State 123
4.5.3 Real Laser Beams 123
4.5.4 The Transfer of Operators, Signals, and Noise 124
4.5.5 Sideband Modes as Quantum States 126
4.5.6 Another Example: A Coherent State Pulse Through a Frequency Filter 129
4.5.7 Transformation of the Covariance Matrix 130
4.6 Quantum Correlations 131
4.6.1 Photon Correlations 131
4.6.2 Quadrature Correlations 132
4.6.3 Two-Mode Covariance Matrix 133
4.7 Summary 134
4.7.1 The Photon Number Basis 134
4.7.2 Quadrature Representations 135
4.7.3 Quantum Operators 135
4.7.4 The Quantum Noise Limit 136
References 136
Further Reading 137
5 Basic Optical Components 139
5.1 Beamsplitters 140
5.1.1 Classical Description of a Beamsplitter 140
5.1.1.1 Polarization Properties of Beamsplitters 142
5.1.2 The Beamsplitter in the Quantum Operator Model 143
5.1.3 The Beamsplitter with Single Photons 144
5.1.4 The Beamsplitter and the Photon Statistics 146
5.1.5 The Beamsplitter with Coherent States 149
5.1.5.1 Transfer Function for a Beamsplitter 149
5.1.6 Comparison Between a Beamsplitter and a Classical Current Junction 151
5.1.7 The Beamsplitter as a Model of Loss 152
5.2 Interferometers 153
5.2.1 Classical Description of an Interferometer 154
5.2.2 Quantum Model of the Interferometer 155
5.2.3 The Single-Photon Interferometer 156
5.2.4 Transfer of Intensity Noise Through the Interferometer 156
5.2.5 Sensitivity Limit of an Interferometer 157
5.2.6 Effect of Mode Mismatch on an Interferometer 160
5.3 Optical Cavities 162
5.3.1 Classical Description of a Linear Cavity 164
5.3.2 The Special Case of High Reflectivities 169
5.3.3 The Phase Response 170
5.3.4 Spatial Properties of Cavities 172
5.3.4.1 Mode Matching 172
5.3.4.2 Polarization 174
5.3.4.3 Tunable Mirrors 175
5.3.5 Equations of Motion for the Cavity Mode 175
5.3.6 The Quantum Equations of Motion for a Cavity 176
5.3.7 The Propagation of Fluctuations Through the Cavity 177
5.3.8 Single Photons Through a Cavity 180
5.3.9 Multimode Cavities 181
5.3.10 Engineering Beamsplitters, Interferometers, and Resonators 182
5.4 Other Optical Components 184
5.4.1 Lenses 184
5.4.2 Holograms and Metasurfaces 185
5.4.3 Crystals and Polarizers 187
5.4.4 Optical Fibres and Waveguides 188
5.4.5 Modulators 189
5.4.5.1 Phase and Amplitude Modulators 191
5.4.6 Spatial Light Modulators 193
5.4.7 Optical Noise Sources 195
5.4.8 Non-linear Processes 195
References 196
6 Lasers and Amplifiers 199
6.1 The Laser Concept 199
6.1.1 Technical Specifications of a Laser 201
6.1.2 Rate Equations 203
6.1.3 Quantum Model of a Laser 207
6.1.4 Examples of Lasers 209
6.1.4.1 Classes of Lasers 209
6.1.4.2 Dye Lasers and Argon Ion Lasers 209
6.1.4.3 The CW Nd: YAG Laser 210
6.1.4.4 Diode Lasers 213
6.1.4.5 Limits of the Single-Mode Approximation in Diode Lasers 213
6.1.5 Laser Phase Noise 214
6.1.6 Pulsed Lasers 215
6.2 Amplification of Optical Signals 215
6.3 Parametric Amplifiers and Oscillators 218
6.3.1 The Second-Order Non-linearity 219
6.3.2 Parametric Amplification 220
6.3.3 Optical Parametric Oscillator 221
6.3.3.1 Noise Spectrum of the Parametric Oscillator 222
6.3.4 Pair Production 223
6.4 Measurement-Based Amplifiers 224
6.4.1 Deterministic Measurement-Based Amplifiers 225
6.4.2 Heralded Measurement-Based Amplifiers 228
6.5 Summary 230
References 231
7 Photon Generation and Detection 233
7.1 Photon Sources 236
7.1.1 Deterministic Photon Sources 239
7.2 Photon Detection 240
7.2.1 Detecting Individual Photons 240
7.2.1.1 Photochemical Detectors 241
7.2.1.2 Photoelectric Detectors 241
7.2.1.3 Photo-thermal Detectors 243
7.2.1.4 Multipixel and Imaging Devices 243
7.2.2 Recording Electrical Signals from Individual Photons 245
7.3 Generating, Detecting, and Analysing Photocurrents 247
7.3.1 Properties of Photocurrents 247
7.3.1.1 Beat Measurements 247
7.3.1.2 Intensity Noise and the Shot Noise Level 248
7.3.1.3 Quantum Efficiency 249
7.3.1.4 Photodetector Materials 250
7.3.2 Generating Photocurrents 251
7.3.2.1 Photodiodes and Detector Circuit 251
7.3.2.2 Amplifiers and Electronic Noise 252
7.3.2.3 Detector Saturation 254
7.3.3 Recording of Photocurrents 255
7.3.4 Spectral Analysis of Photocurrents 257
7.3.4.1 Digital Fourier Transform 257
7.3.4.2 Analogue Fourier Transform 258
7.3.4.3 From Optical Sidebands to the Current Spectrum 258
7.3.4.4 The Operation of an Electronic Spectrum Analyser 259
7.3.4.5 Detecting Signal and Noise Independently 260
7.3.4.6 The Decibel Scale 261
7.3.4.7 Adding Electronic AC Signals 262
7.4 Imaging with Photons 263
References 264
Further Reading 267
8 Quantum Noise: Basic Measurements and Techniques 269
8.1 Detection and Calibration of Quantum Noise 269
8.1.1 Direct Detection and Calibration 269
8.1.1.1 White Light Calibration 273
8.1.2 Balanced Detection 273
8.1.3 Detection of Intensity Modulation and SNR 275
8.1.4 Homodyne Detection 275
8.1.4.1 The Homodyne Detector for Classical Waves 275
8.1.5 Heterodyne Detection 279
8.1.5.1 Measuring Other Properties 280
8.2 Intensity Noise 281
8.2.1 Laser Noise 281
8.3 The Intensity Noise Eater 282
8.3.1 Classical Intensity Control 282
8.3.2 Quantum Noise Control 285
8.3.2.1 Practical Consequences 289
8.4 Frequency Stabilization and Locking of Cavities 290
8.4.1 Pound-Drever-Hall Locking 292
8.4.2 Tilt Locking 293
8.4.3 The PID Controller 294
8.4.4 How to Mount a Mirror 295
8.4.5 The Extremes of Mirror Suspension: GW Detectors 296
8.5 Injection Locking 296
References 299
9 Squeezed Light 303
9.1 The Concept of Squeezing 303
9.1.1 Tools for Squeezing: Two Simple Examples 303
9.1.1.1 The Kerr Effect 304
9.1.1.2 Four-Wave Mixing 307
9.1.2 Properties of Squeezed States 310
9.1.2.1 What Are the Uses of These Various Types of Squeezed Light? 312
9.2 Quantum Model of Squeezed States 314
9.2.1 The Formal Definition of a Squeezed State 314
9.2.2 The Generation of Squeezed States 317
9.2.3 Squeezing as Correlations Between Noise Sidebands 319
9.3 Detecting Squeezed Light 322
9.3.1 Detecting Amplitude Squeezed Light 322
9.3.2 Detecting Quadrature Squeezed Light 322
9.3.3 Using a Cavity to Measure Quadrature Squeezing 324
9.3.4 Summary of Different Representations of Squeezed States 325
9.3.5 Propagation of Squeezed Light 325
9.4 Early Demonstrations of Squeezed Light 330
9.4.1 Four Wave Mixing 330
9.4.2 Optical Parametric Processes 333
9.4.3 Second Harmonic Generation 339
9.4.4 The Kerr Effect 343
9.4.4.1 The Response of the Kerr Medium 343
9.4.4.2 Optimizing the Kerr Effect 345
9.4.4.3 Fibre Kerr Squeezing 346
9.4.4.4 Atomic Kerr Squeezing 348
9.4.4.5 Atomic Polarization Self-Rotation 349
9.5 Pulsed Squeezing 349
9.5.1 Quantum Noise of Optical Pulses 349
9.5.2 Pulsed Squeezing Experiments with Kerr Media 352
9.5.3 Pulsed SHG and OPO Experiments 353
9.5.4 Soliton Squeezing 354
9.5.5 Spectral Filtering 355
9.5.6 Non-linear Interferometers 356
9.6 Amplitude Squeezed Light from Diode Lasers 358
9.7 Quantum State Tomography 360
9.8 State of the Art of CW Squeezing 363
9.9 Squeezing of Multiple Modes 365
9.9.1 Twin-Photon Beams 365
9.9.2 Polarization Squeezing 367
9.9.3 Degenerate Multimode Squeezers 368
9.10 Summary: Quantum Limits and Enhancement 370
References 371
Further Reading 376
10 Applications of Quantum Light 377
10.1 Quantum Enhanced Sensors 377
10.1.1 Coherent Sensors and Sensitivity Scaling 377
10.1.2 Practical Examples of Sensors 380
10.1.3 Ultimate Sensing Limits 382
10.1.4 Adaptive Phase Estimation 384
10.2 Optical Communication 384
10.3 Gravitational Wave Detection 389
10.3.1 The Origin and Properties of GW 389
10.3.1.1 Concept and Design of an Optical GW Detector 392
10.3.2 Quantum Properties of the Ideal Interferometer 393
10.3.2.1 Configurations of Interferometers 396
10.3.2.2 Recycling 397
10.3.2.3 Modulation Techniques 398
10.3.3 The Sensitivity of GW Observatories 400
10.3.3.1 Enhancement Below the SQL 402
10.3.4 Interferometry with Squeezed Light 405
10.3.4.1 Quantum Enhancement Beyond the SQL 410
10.4 Quantum Enhanced Imaging 411
10.4.1 Imaging with Photons on Demand 411
10.4.2 Quantum Enhanced Coincidence Imaging 412
10.5 Multimode Squeezing Enhancing Sensors 414
10.5.1 Spatial Multimode Squeezing 414
10.6 Summary and Outlook 419
References 419
11 QND 425
11.1 QND Measurements of Quadrature Amplitudes 425
11.2 Classification of QND Measurements 427
11.3 Experimental Results 430
11.4 Single-Photon QND 432
11.4.1 Measurement-Based QND 434
References 437
12 Fundamental Tests of Quantum Mechanics 441
12.1 Wave-Particle Duality 441
12.2 Indistinguishability 446
12.3 Non-locality 453
12.3.1 Einstein-Podolsky-Rosen Paradox 453
12.3.2 Characterization of Entangled Beams via Homodyne Detection 458
12.3.2.1 Logarithmic Negativity and Two-Mode Squeezing 459
12.3.2.2 Entanglement of Formation 460
12.3.3 Bell Inequalities 461
12.3.3.1 Long-Distance Bell Inequality Violations 466
12.3.3.2 Loophole-Free Bell Inequality Violations 466
12.4 Summary 468
References 468
13 Quantum Information 473
13.1 Photons as Qubits 473
13.1.1 Other Quantum Encodings 475
13.2 Post-selection and Coincidence Counting 475
13.3 True Single-Photon Sources 477
13.3.1 Heralded Single Photons 477
13.3.2 Single Photons on Demand 480
13.4 Characterizing Photonic Qubits 482
13.5 Quantum Key Distribution 484
13.5.1 QKD Using Single Photons 485
13.5.2 QKD Using Continuous Variables 489
13.5.3 No Cloning 492
13.6 Teleportation 492
13.6.1 Teleportation of Photon Qubits 493
13.6.2 Continuous Variable Teleportation 495
13.6.3 Entanglement Swapping 502
13.6.4 Entanglement Distillation 502
13.7 Quantum Computation 505
13.7.1 Dual-Rail Quantum Computing 506
13.7.1.1 Quantum Circuits with Linear Optics 507
13.7.1.2 Cluster States 511
13.7.1.3 Quantum Gates with Non-linear Optics 513
13.7.2 Single-Rail Quantum Computation 514
13.7.2.1 Quantum Random Walks 515
13.7.2.2 Boson Sampling 516
13.7.3 Continuous Variable Quantum Computation 518
13.7.3.1 Cat State Quantum Computing 519
13.7.3.2 Continuous Variable Cluster States 521
13.7.4 Large-Scale Quantum Computation 522
13.8 Summary 525
References 526
Further Reading 531
14 The Future: From Q-demonstrations to Q-technologies 533
14.1 Demonstrating Quantum Effects 533
14.2 Matter Waves and Atoms 535
14.3 Q-Technology Based on Optics 537
14.3.1 Applications of Squeezed Light 537
14.3.2 Quantum Communication and Logic with Photons 539
14.3.3 Cavity QED 542
14.3.4 Extending to Other Wavelengths: Microwaves and Cryogenic Circuits 542
14.3.5 Quantum Optomechanics 542
14.3.6 Transfer of Quantum Information Between Different Physical Systems 543
14.3.7 Transferring and Storing Quantum States 544
14.4 Outlook 544
References 545
Further Reading 547
Appendices 549
Appendix A: List of Quantum Operators, States, and Functions 549
Appendix B: Calculation of the Quantum Properties of a Feedback Loop 551
Appendix C: Detection of Signal and Noise with an ESA 552
Reference 554
Appendix D: An Example of Analogue Processing of Photocurrents 554
Appendix E: Symbols and Abbreviations 556
Index 559
