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1 Introduction 1

1.1 Scope and aims of this book 1

1.2 History 2

1.3 The contents of this book 7

References 8

Suggestions for further reading 8

2 Field quantization 10

2.1 Quantization of a single-mode field 10

2.2 Quantum fluctuations of a single-mode field 15

2.3 Quadrature operators for a single-mode field 17

2.4 Multimode fields 18

2.5 Thermal fields 25

2.6 Vacuum fluctuations and the zero-point energy 29

2.7 The quantum phase 33

Problems 40

References 41

Bibliography 42

3 Coherent states 43

3.1 Eigenstates of the annihilation operator and minimum uncertainty states 43

3.2 Displaced vacuum states 48

3.3 Wave packets and time evolution 50

3.4 Generation of coherent states 52

3.5 More on the properties of coherent states 53

3.6 Phase-space pictures of coherent states 56

3.7 Density operators and phase-space probability distributions 59

3.8 Characteristic functions 65

Problems 71

References 72

Bibliography 73

4 Emission and absorption of radiation by atoms 74

4.1 Atom-field interactions 74

4.2 Interaction of an atom with a classical field 76

4.3 Interaction of an atom with a quantized field 82

4.4 The Rabi model 87

4.5 Fully quantum-mechanical model；the Jaynes-Cummings model 90

4.6 The dressed states 99

4.7 Density-operator approach：application to thermal states 102

4.8 The Jaynes-Cummings model with large detuning：a dispersive interaction 105

4.9 Extensions of the Jaynes-Cummings model 107

4.10 Schmidt decomposition and von Neumann entropy for the Jaynes-Cummings model 108

Problems 110

References 113

Bibliography 114

5 Quantum coherence functions 115

5.1 Classical coherence functions 115

5.2 Quantum coherence functions 120

5.3 Young’s interference 124

5.4 Higher-order coherence functions 127

Problems 133

References 133

Bibliography 134

6 Beam splitters and interferometers 135

6.1 Experiments with single photons 135

6.2 Quantum mechanics of beam splitters 137

6.3 Interferometry with a single photon 143

6.4 Interaction-free measurement 144

6.5 Interferometry with coherent states of light 146

Problems 147

References 149

Bibliography 149

7 Nonclassical light 150

7.1 Quadrature squeezing 150

7.2 Generation of quadrature squeezed light 165

7.3 Detection of quadrature squeezed light 167

7.4 Amplitude（or number）squeezed states 169

7.5 Photon antibunching 171

7.6 Schrodinger cat states 174

7.7 Two-mode squeezed vacuum states 182

7.8 Higher-order squeezing 188

7.9 Broadband squeezed light 189

Problems 190

References 192

Bibliography 194

8 Dissipative interactions and decoherence 195

8.1 Introduction 195

8.2 Single realizations or ensembles？ 196

8.3 Individual realizations 200

8.4 Shelving and telegraph dynamics in three-level atoms 204

8.5 Decoherence 207

8.6 Generation of coherent states from decoherence：nonlinear optical balance 208

8.7 Conclusions 210

Problems 211

References 211

Bibliography 212

9 Optical test of quantum mechanics 213

9.1 Photon sources：spontaneous parametric down-conversion 214

9.2 The Hong-Ou-Mandel interferometer 217

9.3 The quantum eraser 219

9.4 Induced coherence 222

9.5 Superluminal tunneling of photons 224

9.6 Optical test of local realistic theories and Bell’s theorem 226

9.7 Franson’s experiment 232

9.8 Applications of down-converted light to metrology without absolute standards 233

Problems 235

References 236

Bibliography 237

10 Experiments in cavity QED and with trapped ions 238

10.1 Rydberg atoms 238

10.2 Rydberg atom interacting with a cavity field 241

10.3 Experimental realization of the Jaynes-Cummings model 246

10.4 Creating entangled atoms in CQED 249

10.5 Formation of Schrodinger cat states with dispersive atom-field interactions and decoherence from the quantum to the classical 250

10.6 Quantum nondemolition measurement of photon number 254

10.7 Realization of the Jaynes-Cummings interaction in the motion of a trapped ion 255

10.8 Concluding remarks 258

Problems 259

References 260

Bibliography 261

11 Applications of entanglement：Heisenberg-limited interferometry and quantum information processing 263

11.1 The entanglement advantage 264

11.2 Entanglement and interferometric measurements 265

11.3 Quantum teleportation 268

11.4 Cryptography 270

11.5 Private key crypto-systems 271

11.6 Public key crypto-systems 273

11.7 The quantum random number generator 274

11.8 Quantum cryptography 275

11.9 Future prospects for quantum communication 281

11.10 Gates for quantum computation 281

11.11 An optical realization of some quantum gates 286

11.12 Decoherence and quantum error correction 289

Problems 290

References 291

Bibliography 293

Appendix A The density operator，entangled states，the Schmidt decomposition，and the von Neumann entropy 294

A.1 The density operator 294

A.2 Two-state system and the Bloch sphere 297

A.3 Entangled states 298

A.4 Schmidt decomposition 299

A.5 von Neumann entropy 301

A.6 Dynamics of the density operator 302

References 303

Bibliography 303

Appendix B Quantum measurement theory in a（very small） nutshell 304

Bibliography 307

Appendix C Derivation of the effective Hamiltonian for dispersive（far off-resonant）interactions 308

References 311

Appendix D Nonlinear optics and spontaneous parametric down-conversion 312

References 313

Index 314