This book grew out of our work in the field of the quantum theory of nonlinear optics. Some of this work we have done together and some following our own paths. One major emphasis of this work has been the quantization of electrodynamics in the presence of dielectric media. This is a subject that is often given short shrift in many treatments, and we felt that a book in which it receives a more extensive discussion was warranted.
M.H. would like to thank his thesis advisor, Eyvind Wichmann, for an excellent education in quantum mechanics and quantum field theory, and M. Suhail Zubairy for introducing him to the field of nonlinear optics with quantized fields. Others who played a major role are Leonard Mlodinow, with whom the initial work on quantization in nonlinear media was done, and Janos Bergou and Vladimir Buzek, long-time collaborators with whom it has been a pleasure to work. He also thanks Carol Hutchins for many things.
P.D.D. wishes to acknowledge his parents and family for their invaluable support. The many colleagues who helped form his approach include Crispin Gardiner and the late Dan Walls, who pioneered quantum optics in New Zealand. Subhash Chaturvedi, Howard Carmichael, Steve Carter, Paul Kinsler, Joel Corney, Piotr Deuar and Kaled Dechoum have contributed greatly to this field. He also thanks Margaret Reid, who has played a leading role in some of the developments outlined in the quantum information section, and Qiongyi He, who provided illustrations.

Preface page ix

Introduction 1

1 Classical nonlinear optics 4

1.1 Linear polarizability 4

1.2 Nonlinear polarizability 7

1.3 Frequency dependence and dispersion 10

1.4 Power and energy 12

1.5 Order-of-magnitude estimates 16

1.6 The two-level atom 17

1.7 Local-field corrections 24

1.8 Propagation in a nonlinear medium 29

1.9 Raman processes 32

Additional reading 34

Problems 35

2 Field quantization 37

2.1 Quantum theory 37

2.2 Fock space for bosons 41

2.3 Many-body operators 44

2.4 Fock space for fermions 48

2.5 Canonical quantization 50

2.6 One-dimensional string 53

2.7 Scattering matrix 57

2.8 Quantized free electromagnetic field 66

2.9 Constrained quantization 73

2.10 Exponential complexity 78

Additional reading 81

Problems 81

3 Quantized fields in dielectric media 83

3.1 Dispersionless linear quantization 83

3.2 Scattering in linear media 89

3.3 Quantizing a nonlinear dielectric 92

3.4 Homogeneous nonlinear dielectric 94

3.5 Inhomogeneous nonlinear dielectric 96

3.6 Dispersion 103

3.7 One-dimensional waveguide 108

Additional reading 113

Problems 113

4 Microscopic description of media 116

4.1 The Coulomb gauge 116

4.2 The multipolar gauge 119

4.3 Hamiltonian for a polarizable medium 123

4.4 Dipole-coupling approximation 127

4.5 Linear medium 129

4.6 Quantization of the linear model 134

4.7 Two-level atomic medium 136

4.8 Polaritonic limit 139

Additional reading 142

Problems 142

5 Coherence and quantum dynamics in simple systems 144

5.1 Photon counting and quantum coherence 144

5.2 Quadratures and beam-splitters 149

5.3 Coherent states and P-representations 151

5.4 Nonclassical states 154

5.5 Two-mode states 157

5.6 Mode entanglement 160

5.7 Parametric interactions 163

5.8 Anharmonic oscillator and Schrodinger’s cat 170 ¨

5.9 Jaynes–Cummings dynamics 175

5.10 Parametric approximation 177

Additional reading 181

Problems 182

6 Decoherence and reservoirs 184

6.1 Reservoir Hamiltonians 184

6.2 Absorption 187

6.3 Gain 191

6.4 Phase decoherence 195

6.5 Input–output relations 198

6.6 Photon flux and density 201

6.7 Two-time correlation functions 206

6.8 Master equations 207

6.9 Gain and damping rates 211

6.10 Driven linear cavity example 212

Additional reading 214

Problems 215

7 Phase-space distributions 217

7.1 Diffusion processes 217

7.2 Fokker–Planck equations 220

7.3 Stochastic differential equations 223

7.4 Phase-space representations 228

7.5 Wigner and Q-representations 231

7.6 Nonclassical representations 235

7.7 Operator identities and quantum dynamics 241

7.8 Quasi-probability Fokker–Planck equation 245

7.9 Linearized fluctuation theory 250

7.10 Functional phase-space representations 251

Additional reading 253

Problems 254

8 Single-mode devices 256

8.1 Linear cavity 256

8.2 Phase-space representation methods 257

8.3 Driven nonlinear absorber 262

8.4 Squeezing and photon anti-bunching 264

8.5 High-Q laser 268

8.6 Laser linewidth 272

8.7 Laser quantum state: number or coherent? 274

8.8 Open nonlinear interferometer 276

Additional reading 280

Problems 280

9 Degenerate parametric oscillator 283

9.1 Hamiltonian and stochastic equations 283

9.2 Classical results 285

9.3 Fokker–Planck and stochastic equations 287

9.4 Adiabatic approximation 290

9.5 Multi-mode treatment of parametric down-conversion in a cavity 291

Additional reading 296

Problems 296

10 Quantum field dynamics 298

10.1 Kerr medium 299

10.2 Quantum solitons 301

10.3 Time-dependent Hartree approximation 304

10.4 Quantum solitons in phase space 307

10.5 Parametric down-conversion 308

10.6 Maxwell–Bloch equations 312

Additional reading 317

Problems 318

11 Quantum propagation in fibers and waveguides 319

11.1 Order-of-magnitude estimates 319

11.2 Waveguide modes 321

11.3 Dispersive energy 323

11.4 Nonlinear Hamiltonian 326

11.5 Fiber optic Hamiltonian 328

11.6 Raman Hamiltonian 330

11.7 Gain and absorption 333

11.8 Combined Heisenberg equations 336

11.9 Phase-space methods 337

11.10 Polarization squeezing 341

Additional reading 344

Problems 345

12 Quantum information 346

12.1 The Einstein–Podolsky–Rosen paradox 347

12.2 Bell inequality 350

12.3 Schrödinger cat paradoxes 355

12.4 Probabilistic simulations of Bell violations 358

12.5 Quantum cloning 360

12.6 Teleportation 363

Additional reading 366

Problems 367

List of symbols 369

Index 371

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