A Group-Theoretical Approach to Quantum Optics

A Group-Theoretical Approach to Quantum Optics......Page 5
Contents......Page 7
Preface......Page 11
1.1 Kinematics of an Atom with Two Energy Levels......Page 13
1.2 Dicke States......Page 17
1.3 Atomic Coherent States......Page 19
1.4 Squeezed Atomic States......Page 24
1.5.1 Systems with n Energy Levels......Page 29
1.5.2 Systems with Three Energy Levels......Page 32
1.6 Problems......Page 33
2.1 Spin in a Constant Magnetic Field......Page 35
2.2.1 The Rotating Wave Approximation......Page 36
2.3 A Two-level Atom in a Circularly Polarized Field......Page 38
2.4 Evolution of the Bloch Vector......Page 40
2.5 Dynamics of the Two-level Atom without the RWA......Page 41
2.6 Collective Atomic Systems......Page 45
2.7 Atomic System in a Field of a Single Pulse......Page 51
2.8 Problems......Page 55
3.1 Quantization of the Electromagnetic Field......Page 57
3.2 Coherent States......Page 59
3.3 Properties of the Coherent States......Page 60
3.4 Displacement Operator......Page 63
3.5 Squeezed States......Page 66
3.7 Phase Operator......Page 70
3.8 Regularized Phase Operator......Page 75
3.9 Phase Distribution......Page 77
3.10 Problems......Page 81
4.1 Evolution of a Field with Classical Pumping......Page 83
4.2 Linear Parametric Amplifier......Page 84
4.3 Evolution in the Kerr Medium......Page 87
4.4 Second Harmonic Generation in the Dispersive Limit......Page 89
4.5 Raman Dispersion......Page 91
4.6 Problems......Page 93
5.1 The Interaction Hamiltonian......Page 95
5.2 The Spectrum and Wave Functions......Page 97
5.3 Evolution Operator......Page 99
5.4 The Classical Field Limit......Page 102
5.5 Collapses and Revivals......Page 104
5.5.1 The Dispersive Limit......Page 105
5.5.2 Exact Resonance......Page 107
5.6 The JCM with an Initial Thermal Field......Page 109
5.7 Trapping States......Page 111
5.8 Factorization of the Wave Function......Page 113
5.9 Evolution in Field Phase Space......Page 116
5.10 The JCM without RWA......Page 117
5.10.1 Diagonalization of the Hamiltonian......Page 118
5.10.2 Atomic Inversion......Page 121
5.10.3 Classical Field Limit......Page 122
5.11 Problems......Page 123
6.1 The Dicke Model (Exactly Solvable Examples)......Page 125
6.2 The Dicke Model (Symmetry Properties)......Page 130
6.3 The Dicke Model (Symmetric Case)......Page 133
6.4.1 The Weak Field Case......Page 134
6.4.2 The Strong Field Case......Page 135
6.5 Perturbation Theory......Page 136
6.6 Revivals of the First and Second Orders......Page 140
6.6.1 Revivals of the Second Order......Page 142
6.7.1 Initial Number States......Page 144
6.7.2 Coherent and Thermal Fields......Page 146
6.8 Three-Level Atoms Interacting with Two Quantum Field Modes......Page 148
6.9 Problems......Page 153
7.1 Dicke Model in a Strong Field......Page 155
7.2 Factorization of the Wave Function......Page 158
7.3 Evolution in Phase Space......Page 160
7.4 Dicke Model in the Presence of the Kerr Medium......Page 164
7.5 Generation of the Field Squeezed States......Page 166
7.6 Coherence Transfer Between Atoms and Field......Page 169
7.7 Resonant Fluorescence Spectrum......Page 171
7.8 Atomic Systems with n Energy Levels......Page 174
7.8.1 Cascade Configuration......Page 179
7.8.2 Λ-Type Configuration......Page 180
7.9 Dicke Model in the Dispersive Limit......Page 181
7.10 Two-Photon Dicke Model......Page 184
7.11 Effective Transitions in Three-Level Atoms with   Configuration......Page 192
7.12 N-Level Atoms of Cascade Configuration......Page 195
7.13 Problems......Page 199
8.1 Kinematic and Dynamic Resonances in Quantum Systems......Page 201
8.2 Kinematic Resonances: Generic Atom–Field Interactions......Page 204
8.3 Dynamic Resonances......Page 210
8.3.1 Atom–Quantized Field Interaction......Page 215
8.3.2 Atom–Classical Field Interaction......Page 216
8.3.3 Interaction of Atoms with the Quantum Field in the Presence of Classical Fields......Page 218
8.4 Dynamics of Slow and Fast Interacting Subsystems......Page 224
8.4.1 Effective Field Dynamics......Page 226
8.4.2 Effective Atomic Dynamics......Page 227
8.5 Problems......Page 228
9.1 Dissipation and Pumping of the Quantum Field......Page 229
9.2 Dicke Model with Dissipation and Pumping (Dispersive Limit)......Page 231
9.3 Dicke Model with Dissipation (Resonant Case)......Page 235
9.3.2 Initial Field Coherent State......Page 238
9.3.3 Factorized Dynamics......Page 241
9.4 Strong Dissipation......Page 243
9.4.1 Field–Field Interaction......Page 246
9.5 Problems......Page 247
10.1.1 Weyl Quantization Method......Page 249
10.1.2 Moyal–Stratonovich–Weyl Quantization......Page 252
10.1.3 Ordering Problem in L(H)......Page 253
10.1.4 Star Product......Page 254
10.1.5 Phase–Space Representation and Quantum–Classical Correspondence......Page 255
10.2 Atomic Quasi-distributions......Page 257
10.2.1 P Function......Page 258
10.2.2 Q Function......Page 259
10.2.3 Stratonovich–Weyl Distribution......Page 262
10.2.4 s-Ordered Distributions......Page 263
10.2.5 Star Product......Page 264
10.2.6 Evolution Equations......Page 267
10.2.7 Large Representation Dimensions (Semiclassical Limit)......Page 268
10.3.1 P Function......Page 274
10.3.2 Q Function......Page 276
10.3.3 Wigner Function......Page 277
10.3.4 s-Ordered Distributions......Page 278
10.4.1 Kerr Hamiltonian......Page 281
10.4.2 The Dicke Hamiltonian......Page 283
10.5 Problems......Page 288
11.1.1 Groups: Basic Concepts......Page 291
11.1.2 Group Representations......Page 293
11.1.3 Lie Algebras......Page 294
11.1.4 Examples......Page 296
11.2 Coherent States......Page 306
11.2.1 Examples......Page 307
11.3 Linear Systems......Page 311
11.3.1 Diagonalization of the Time-independent Hamiltonian......Page 313
11.3.2 Evolution Operator......Page 314
11.3.3 Reference Formulas......Page 315
11.4 Lie Transformation Method......Page 316
11.5 Wigner d Function......Page 318
11.6 Irreducible Tensor Operators......Page 323
References......Page 327
Index......Page 333