A group-theoretical approach to quantum optics : models of atom-field interactions

Written by major contributors to the field who are well known within the community, this is the first comprehensive summary of the many results generated by this approach to quantum optics to date. As such, the book analyses selected topics of quantum optics, focusing on atom-field interactions from a group-theoretical perspective, while discussing the principal quantum optics models using algebraic language. The overall result is a clear demonstration of the advantages of applying algebraic methods to quantum optics problems, illustrated by a number of end-of-chapter problems.
An invaluable source for atomic physicists, graduates and students in physics.

Author Notes

Andrei B. Klimov is a professor of physics at the Department of Physics, University of Guadalajara, Mexico. He received his PhD degree in theoretical physics from the Moscow Institute of Physics and Technology in 1991, and worked at the Lebedev Physical Institute, Moscow, and at the National University of Mexico (UNAM) before accepting his present appointment at the University of Guadalajara. Professor Klimov is the author of over 100 scientific publications, including two book chapters.
Sergei M. Chumakov is a professor of physics at the University of Guadalajara, Mexico. He received his PhD degree in theoretical and mathematical physics from the Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, in 1986. Professor Chumakov worked at the Central Bureau for Design of Unique Devices, Moscow, and at the National University of Mexico (UNAM) before starting his present appointment at the University of Guadalajara in 1998. He has authored over 80 scientific publications, including two book chapters.

Preface	p. IX
1 Atomic Kinematics	p. 1
1.1 Kinematics of an Atom with Two Energy Levels	p. 1
1.2 Dicke States	p. 5
1.3 Atomic Coherent States	p. 7
1.4 Squeezed Atomic States	p. 12
1.5 Atoms with n > 2 Energy Levels	p. 17
1.5.1 Systems with n Energy Levels	p. 17
1.5.2 Systems with Three Energy Levels	p. 20
1.6 Problems	p. 21
2 Atomic Dynamics	p. 23
2.1 Spin in a Constant Magnetic Field	p. 23
2.2 A Two-level Atom in a Linearly Polarized Field	p. 24
2.2.1 The Rotating Wave Approximation	p. 24
2.3 A Two-level Atom in a Circularly Polarized Field	p. 26
2.4 Evolution of the Bloch Vector	p. 28
2.5 Dynamics of the Two-level Atom without the RWA	p. 29
2.6 Collective Atomic Systems	p. 33
2.7 Atomic System in a Field of a Single Pulse	p. 39
2.8 Problems	p. 42
3 Quantized Electromagnetic Field	p. 45
3.1 Quantization of the Electromagnetic Field	p. 45
3.2 Coherent States	p. 47
3.3 Properties of the Coherent States	p. 48
3.4 Displacement Operator	p. 51
3.5 Squeezed States	p. 54
3.6 Thermal States	p. 58
3.7 Phase Operator	p. 58
3.8 Regularized Phase Operator	p. 63
3.9 Phase Distribution	p. 65
3.10 Problems	p. 69
4 Field Dynamics	p. 71
4.1 Evolution of a Field with Classical Pumping	p. 71
4.2 Linear Parametric Amplifier	p. 72
4.3 Evolution in the Kerr Medium	p. 75
4.4 Second Harmonic Generation in the Dispersive Limit	p. 77
4.5 Raman Dispersion	p. 79
4.6 Problems	p. 81
5 The Jaynes-Cummings Model	p. 83
5.1 The Interaction Hamiltonian	p. 83
5.2 The Spectrum and Wave Functions	p. 85
5.3 Evolution Operator	p. 87
5.4 The Classical Field Limit	p. 90
5.5 Collapses and Revivals	p. 92
5.5.1 The Dispersive Limit	p. 93
5.5.2 Exact Resonance	p. 95
5.6 The JCM with an Initial Thermal Field	p. 97
5.7 Trapping States	p. 99
5.8 Factorization of the Wave Function	p. 101
5.9 Evolution I n Field Phase Space	p. 104
5.10 The JCM without RWA	p. 105
5.10.1 Diagonalization of the Hamiltonian	p. 106
5.10.2 Atomic Inversion	p. 109
5.10.3 Classical Field Limit	p. 110
5.11 Problems	p. 111
6 Collective Interactions	p. 113
6.1 The Dicke Model (Exactly Solvable Examples)	p. 113
6.2 The Dicke Model (Symmetry Properties)	p. 118
6.3 The Dicke Model (Symmetric Case)	p. 121
6.4 The Zeroth-Order Approximation	p. 122
6.4.1 The Weak Field Case	p. 122
6.4.2 The Strong Field Case	p. 123
6.5 Perturbation Theory	p. 124
6.6 Revivals of the First and Second Orders	p. 128
6.6.1 Revivals of the Second Order	p. 130
6.7 Atom-Field Dynamics for Different Initial Conditions	p. 132
6.7.1 Initial Number States	p. 132
6.7.2 Coherent and Thermal Fields	p. 134
6.8 Three-Level Atoms Interacting with Two Quantum Field Modes	p. 136
6.9 Problems	p. 141
7 Atomic Systems in a Strong Quantum Field	p. 143
7.1 Dicke Model in a Strong Field	p. 143
7.2 Factorization of the Wave Function	p. 146
7.3 Evolution in Phase Space	p. 148
7.4 Dicke Model in the Presence of the Kerr Medium	p. 152
7.5 Generation of the Field Squeezed States	p. 154
7.6 Coherence Transfer Between Atoms and Field	p. 157
7.7 Resonant Fluorescence Spectrum	p. 159
7.8 Atomic Systems with n Energy Levels	p. 162
7.8.1 Cascade Configuration ¿	p. 167
7.8.2 A-Type Configuration	p. 168
7.8.3 V-Type Configuration	p. 169
7.9 Dicke Model in the Dispersive Limit	p. 169
7.10 Two-Photon Dicke Model	p. 172
7.11 Effective Transitions in Three-Level Atoms with A Configuration	p. 180
7.12 N-Level Atoms of Cascade Configuration	p. 183
7.13 Problems	p. 186
8 Quantum Systems Beyond the Rotating Wave Approximation	p. 189
8.1 Kinematic and Dynamic Resonances in Quantum Systems	p. 189
8.2 Kinematic Resonances: Generic-Atom Field Interactions	p. 192
8.3 Dynamic Resonances	p. 198
8.3.1 Atom-Quantized Field Interaction	p. 203
8.3.2 Atom-Classical Field Interaction	p. 204
8.3.3 Interaction of Atoms with the Quantum Field in the Presence of Classical Fields	p. 206
8.4 Dynamics of Slow and Fast Interacting Subsystems	p. 212
8.4.1 Effective Field Dynamics	p. 214
8.4.2 Effective Atomic Dynamics	p. 215
8.5 Problems	p. 216
9 Models with Dissipation	p. 217
9.1 Dissipation and Pumping of the Quantum Field	p. 217
9.2 Dicke Model with Dissipation and Pumping (Dispersive Limit)	p. 219
9.3 Dicke Model with Dissipation (Resonant Case)	p. 223
9.3.1 Initial Field Number State	p. 226
9.3.2 Initial Field Coherent State	p. 226
9.3.3 Factorized Dynamics	p. 229
9.4 Strong Dissipation	p. 231
9.4.1 Field-Field Interaction	p. 234
9.4.2 Atom-Field Interaction	p. 235
9.5 Problems	p. 235
10 Quasi-distributions in Quantum Optics	p. 237
10.1 Quantization and Quasi-distributions	p. 237
10.1.1 Weyl Quantization Method	p. 237
10.1.2 Moyal-Stratonovich-Weyl Quantization	p. 240
10.1.3 Ordering Problem in L(H)	p. 241
10.1.4 Star Product	p. 242
10.1.5 Phase-Space Representation and Quantum-Classical Correspondence	p. 243
10.2 Atomic Quasi-distributions	p. 245
10.2.1 P Function	p. 246
10.2.2 Q Function	p. 247
10.2.3 Stratonovich-Weyl Distribution	p. 250
10.2.4 s-Ordered Distributions	p. 251
10.2.5 Star Product	p. 252
10.2.6 Evolution Equations	p. 255
10.2.7 Large Representation Dimensions (Semiclassical Limit)	p. 256
10.3 Field Quasi-distributions	p. 262
10.3.1 P Function	p. 262
10.3.2 Q Function	p. 264
10.3.3 Wigner Function	p. 265
10.3.4 s-Ordered Distributions	p. 266
10.4 Miscellaneous Applications	p. 269
10.4.1 Kerr Hamiltonian	p. 269
10.4.2 The Dicke Hamiltonian	p. 271
10.5 Problems	p. 276
11 Appendices	p. 279
11.1 Lie Groups and Lie Algebras	p. 279
11.1.1 Groups: Basic Concepts	p. 279
11.1.2 Group Representations	p. 281
11.1.3 Lie Algebras	p. 282
11.1.4 Examples	p. 284
11.2 Coherent States	p. 294
11.2.1 Examples	p. 295
11.3 Linear Systems	p. 299
11.3.1 Diagonalization of the Time-independent Hamiltonian	p. 301
11.3.2 Evolution Operator	p. 302
11.3.3 Reference Formulas	p. 303
11.4 Lie Transformation Method	p. 304
11.5 Wigner d Function	p. 306
11.6 Irreducible Tensor Operators	p. 311
References	p. 315
Index	p. 321

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