A Guide to Experiments in Quantum Optics, 2nd, Revised and Enlarged Edition

Preface XI

1 Introduction 1

1.1 Historical perspective 1

1.2 Motivation: Practical effects of quantum noise 3

1.3 How to use this guide 8

Bibliography 10

2 Classical models of light 12

2.1 Classical waves 13

2.1.1 Mathematical description of waves 13

2.1.2 The Gaussian beam 14

2.1.3 Quadrature amplitudes 16

2.1.4 Field energy, intensity, power 18

2.1.5 A classical mode of light 19

2.1.6 Classical modulations 20

2.2 Statistical properties of classical light 23

2.2.1 The origin of fluctuations 23

2.2.2 Coherence 23

2.2.3 Correlation functions 27

2.2.4 Noise spectra 29

2.2.5 An idealized classical case: Light from a chaotic source 31

Bibliography 36

3 Photons – the motivation to go beyond classical optics 37

3.1 Detecting light 37

3.2 The concept of photons 40

3.3 Light from a thermal source 41

3.4 Interference experiments 43

3.5 Modelling single photon experiments 46

3.5.1 Polarization of a single photon 47

3.5.2 Some mathematics 49

3.5.3 Polarization states 50

3.5.4 The single photon interferometer 51

3.6 Intensity correlation, bunching, anti-bunching 53

3.7 Single photon Rabi frequencies 56

Bibliography 57

4 Quantum models of light 60

4.1 Quantization of light 60

4.1.1 Some general comments on quantum mechanics 60

4.1.2 Quantization of cavity modes 61

4.1.3 Quantized energy 62

4.1.4 The quantum mechanical harmonic oscillator 63

4.2 Quantum states of light 64

4.2.1 Number or Fock states 64

4.2.2 Coherent states 65

4.2.3 Mixed states 68

4.3 Quantum optical representations 68

4.3.1 Quadrature amplitude operators 68

4.3.2 Probability and quasi-probability distributions 70

4.3.3 Photon number distributions, Fano factor 75

4.4 Propagation and detection of quantum optical fields 77

4.4.1 Propagation in quantum optics 77

4.4.2 Detection in quantum optics 81

4.4.3 An example: The beam splitter 82

4.5 Quantum transfer functions 84

4.5.1 A linearized quantum noise description 84

4.5.2 An example: The propagating coherent state 86

4.5.3 Real laser beams 87

4.5.4 The transfer of operators, signals and noise 88

4.5.5 Side band modes as quantum states 90

4.6 Quantum correlations 92

4.6.1 Photon correlations 92

4.6.2 Quadrature correlations 93

4.7 Summary: The different quantum models 95

Bibliography 97

5 Basic optical components 99

5.1 Beam splitters 99

5.1.1 Classical description of a beam splitter 99

5.1.2 The beam splitter in the quantum operator model 102

5.1.3 The beam splitter with single photons 104

5.1.4 The beam splitter and the photon statistics 106

5.1.5 The beam splitter with coherent states 108

5.1.6 The beam splitter in the noise sideband model 110

5.1.7 Comparison between a beam splitter and a classical current junction 111

5.2 Interferometers 112

5.2.1 Classical description of an interferometer 113

5.2.2 Quantum model of the interferometer 115

5.2.3 The single photon interferometer 115

5.2.4 Transfer of intensity noise through the interferometer 116

5.2.5 Sensitivity limit of an interferometer 117

5.3 Cavities 119

5.3.1 Classical description of a linear cavity 121

5.3.2 The special case of high reflectivities 125

5.3.3 The phase response 126

5.3.4 Spatial properties of cavities 128

5.3.5 Equations of motion for the cavity mode 132

5.3.6 The quantum equations of motion for a cavity 133

5.3.7 The propagation of fluctuations through the cavity 133

5.3.8 Single photons through a cavity 137

5.4 Other optical components 138

5.4.1 Lenses 138

5.4.2 Crystals and polarizers 140

5.4.3 Modulators 141

5.4.4 Optical fibres 143

5.4.5 Optical noise sources 143

5.4.6 Nonlinear processes 144

Bibliography 145

6 Lasers and Amplifiers 147

6.1 The laser concept 147

6.1.1 Technical specifications of a laser 148

6.1.2 Rate equations  150

6.1.3 Quantum model of a laser 154

6.1.4 Examples of lasers 156

6.1.5 Laser phase noise 161

6.2 Amplification of optical signals 162

6.3 Parametric amplifiers and oscillators 164

6.3.1 The second-order non-linearity 165

6.3.2 Parametric amplification 167

6.3.3 Optical parametric oscillator 168

6.3.4 Pair production 169

6.4 Summary 171

Bibliography  171

7 Photo detection techniques 173

7.1 Photodetector characteristics 173

7.2 Detecting single photons 174

7.3 Photon sources and analysis 178

7.4 Detecting photocurrents 180

7.4.1 The detector circuit 184

7.5 Spectral analysis of photocurrents 187

Bibliography 197

8 Quantum noise: Basic measurements and techniques 200

8.1 Detection and calibrationo f quantum noise 200

8.1.1 Direct detection and calibration 200

8.1.2 Balanced detection 204

8.1.3 Detection of intensity modulation and SNR 205

8.1.4 Homodyne detection 206

8.1.5 Heterodyne detection 210

8.2 Intensity noise 211

8.3 The intensity noise eater 212

8.3.1 Classical intensity control .213

8.3.2 Quantum noise control 216

8.4 Frequency stabilization, locking of cavities 221

8.4.1 How to mount a mirror 225

8.5 Injection locking 226

Bibliography 229

9 Squeezing experiments 232

9.1 The concept of squeezing 232

9.1.1 Tools for squeezing, two simple examples 232

9.1.2 Properties of squeezed states 238

9.2 Quantum model of squeezed states 242

9.2.1 The formal definition of a squeezed state 242

9.2.2 The generation of squeezed states 245

9.2.3 Squeezing as correlations between noise sidebands 247

9.3 Detecting squeezed light 250

9.3.1 Reconstructing the squeezing ellipse 253

9.3.2 Summary of different representations of squeezed states 254

9.3.3 Propagation of squeezed light 254

9.4 Four wave mixing 260

9.5 Optical parametric processes 263

9.6 Second harmonic generation 267

9.7 Kerr effect 275

9.7.1 The response of the Kerr medium 275

9.7.2 Fibre Kerr Squeezing 277

9.7.3 Atomic Kerr squeezing 279

9.8 Atom-cavity coupling 280

9.9 Pulsed squeezing 283

9.9.1 Quantum noise of optical pulses 283

9.9.2 Pulsed squeezing experiments with Kerr media 285

9.9.3 Pulsed SHG and OPO experiments 287

9.9.4 Soliton squeezing 288

9.9.5 Spectral filtering 289

9.9.6 Nonlinear interferometers 290

9.10 Amplitude squeezed light from diode lasers 292

9.11 Twin photon beams 294

9.12 Polarization squeezing 295

9.13 Quantum state tomography 298

9.14 Summary of squeezing results 300

9.14.1 Loopholes in the quantum description 303

Bibliography 303

10 Applications of squeezed light 310

10.1 Optical communication 310

10.2 Spatial squeezing and quantum imaging 313

10.3 Optical sensors 315

10.4 Gravitational wave detection 321

10.4.1 The origin and properties of GW 321

10.4.2 Quantum properties of the ideal interferometer 323

10.4.3 The sensitivity of real instruments 328

10.4.4 Interferometry with squeezed light 333

Bibliography 338

11 QND 343

11.1 The concept of QND measurements 343

11.2 Classification of QND measurements 346

11.3 Experimental results 348

11.4 Single photon QND 350

Bibliography 353

12 Fundamental tests of quantum mechanics 355

12.1 Wave-Particle duality 355

12.2 Indistinguishability 358

12.3 Nonlocality 362

12.3.1 Einstein-Podolsky-Rosen Paradox 362

12.3.2 Generation of entangled CW beams 365

12.3.3 Bell inequalities 367

12.4 Summary 371

Bibliography 371

13 Quantum Information 374

13.1 Photons as qubits 374

13.2 Post selection and coincidence counting 376

13.3 True single photon sources 377

13.3.1 Heralded single photons 377

13.3.2 Single photons on demand 379

13.4 Characterizing photonic qubits 381

13.5 Quantum key distribution 382

13.5.1 QKD using single photons 383

13.5.2 QKD using continuous variables 385

13.5.3 No cloning 387

13.6 Teleportation 387

13.6.1 Teleportation of photon qubits 388

13.6.2 Continuous variable teleportation 390

13.7 Quantum computation 395

13.8 Summary 400

Bibliography 401

14 Summary and outlook 404

15 Appendices 407

Appendix A: Gaussian functions 407

Appendix B: List of quantum operators, states and functions 408

Appendix C: The full quantum derivation of quantum states 410

Appendix D: Calculation of the quantum properties of a feedback loop 412

Appendix E: Symbols and abbreviations 414

Index 416