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A guide to experiments in quantum optics / Hans-A. Bachor and Timothy C. Ralph.

By: Bachor, H.-A., (Hans-Albert) [author.].
Contributor(s): Ralph, Timothy C | IEEE Xplore (Online Service) [distributor.] | Wiley [publisher.].
Material type: materialTypeLabelBookPublisher: Weinheim : Wiley-VCH, 2019Distributor: [Piscataqay, New Jersey] : IEEE Xplore, [2019]Edition: Third edition.Description: 1 PDF (588 pages).Content type: text Media type: electronic Carrier type: online resourceISBN: 9783527695805.Subject(s): Quantum optics -- ExperimentsGenre/Form: Electronic books.Additional physical formats: Print version:: No titleDDC classification: 535.15078 Online resources: Abstract with links to resource Also available in print.
Contents:
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 Poincaré 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.
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Includes bibliographical references and index.

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 Poincaré 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.

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