A Group-Theoretical Approach to Quantum

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.

A.B.Klimov is a professor at the Department of Physics, University of Guadalajara, Mexico. Professor Klimov obtained his academic degrees 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.

S.M.Chumakov is a physics professor at the University of Guadalajara, Mexico. He received his PhD degree in theoretical and mathematical physics from the Lebedev Physical Institute, Academy of Science 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
Atomic Kinematics	p. 1
Kinematics of an Atom with Two Energy Levels	p. 1
Dicke States	p. 5
Atomic Coherent States	p. 7
Squeezed Atomic States	p. 12
Atoms with n > 2 Energy Levels	p. 17
Systems with n Energy Levels	p. 17
Systems with Three Energy Levels	p. 20
Problems	p. 21
Atomic Dynamics	p. 23
Spin in a Constant Magnetic Field	p. 23
A Two-level Atom in a Linearly Polarized Field	p. 24
The Rotating Wave Approximation	p. 24
A Two-level Atom in a Circularly Polarized Field	p. 26
Evolution of the Bloch Vector	p. 28
Dynamics of the Two-level Atom without the RWA	p. 29
Collective Atomic Systems	p. 33
Atomic System in a Field of a Single Pulse	p. 39
Problems	p. 42
Quantized Electromagnetic Field	p. 45
Quantization of the Electromagnetic Field	p. 45
Coherent States	p. 47
Properties of the Coherent States	p. 48
Displacement Operator	p. 51
Squeezed States	p. 54
Thermal States	p. 58
Phase Operator	p. 58
Regularized Phase Operator	p. 63
Phase Distribution	p. 65
Problems	p. 69
Field Dynamics	p. 71
Evolution of a Field with Classical Pumping	p. 71
Linear Parametric Amplifier	p. 72
Evolution in the Kerr Medium	p. 75
Second Harmonic Generation in the Dispersive Limit	p. 77
Raman Dispersion	p. 79
Problems	p. 81
The Jaynes-Cummings Model	p. 83
The Interaction Hamiltonian	p. 83
The Spectrum and Wave Functions	p. 85
Evolution Operator	p. 87
The Classical Field Limit	p. 90
Collapses and Revivals	p. 92
The Dispersive Limit	p. 93
Exact Resonance	p. 95
The JCM with an Initial Thermal Field	p. 97
Trapping States	p. 99
Factorization of the Wave Function	p. 101
Evolution I n Field Phase Space	p. 104
The JCM without RWA	p. 105
Diagonalization of the Hamiltonian	p. 106
Atomic Inversion	p. 109
Classical Field Limit	p. 110
Problems	p. 111
Collective Interactions	p. 113
The Dicke Model (Exactly Solvable Examples)	p. 113
The Dicke Model (Symmetry Properties)	p. 118
The Dicke Model (Symmetric Case)	p. 121
The Zeroth-Order Approximation	p. 122
The Weak Field Case	p. 122
The Strong Field Case	p. 123
Perturbation Theory	p. 124
Revivals of the First and Second Orders	p. 128
Revivals of the Second Order	p. 130
Atom-Field Dynamics for Different Initial Conditions	p. 132
Initial Number States	p. 132
Coherent and Thermal Fields	p. 134
Three-Level Atoms Interacting with Two Quantum Field Modes	p. 136
Problems	p. 141
Atomic Systems in a Strong Quantum Field	p. 143
Dicke Model in a Strong Field	p. 143
Factorization of the Wave Function	p. 146
Evolution in Phase Space	p. 148
Dicke Model in the Presence of the Kerr Medium	p. 152
Generation of the Field Squeezed States	p. 154
Coherence Transfer Between Atoms and Field	p. 157
Resonant Fluorescence Spectrum	p. 159
Atomic Systems with n Energy Levels	p. 162
Cascade Configuration ¿	p. 167
A-Type Configuration	p. 168
V-Type Configuration	p. 169
Dicke Model in the Dispersive Limit	p. 169
Two-Photon Dicke Model	p. 172
Effective Transitions in Three-Level Atoms with A Configuration	p. 180
N-Level Atoms of Cascade Configuration	p. 183
Problems	p. 186
Quantum Systems Beyond the Rotating Wave Approximation	p. 189
Kinematic and Dynamic Resonances in Quantum Systems	p. 189
Kinematic Resonances: Generic-Atom Field Interactions	p. 192
Dynamic Resonances	p. 198
Atom-Quantized Field Interaction	p. 203
Atom-Classical Field Interaction	p. 204
Interaction of Atoms with the Quantum Field in the Presence of Classical Fields	p. 206
Dynamics of Slow and Fast Interacting Subsystems	p. 212
Effective Field Dynamics	p. 214
Effective Atomic Dynamics	p. 215
Problems	p. 216
Models with Dissipation	p. 217
Dissipation and Pumping of the Quantum Field	p. 217
Dicke Model with Dissipation and Pumping (Dispersive Limit)	p. 219
Dicke Model with Dissipation (Resonant Case)	p. 223
Initial Field Number State	p. 226
Initial Field Coherent State	p. 226
Factorized Dynamics	p. 229
Strong Dissipation	p. 231
Field-Field Interaction	p. 234
Atom-Field Interaction	p. 235
Problems	p. 235
Quasi-distributions in Quantum Optics	p. 237
Quantization and Quasi-distributions	p. 237
Weyl Quantization Method	p. 237
Moyal-Stratonovich-Weyl Quantization	p. 240
Ordering Problem in L(H)	p. 241
Star Product	p. 242
Phase-Space Representation and Quantum-Classical Correspondence	p. 243
Atomic Quasi-distributions	p. 245
P Function	p. 246
Q Function	p. 247
Stratonovich-Weyl Distribution	p. 250
s-Ordered Distributions	p. 251
Star Product	p. 252
Evolution Equations	p. 255
Large Representation Dimensions (Semiclassical Limit)	p. 256
Field Quasi-distributions	p. 262
P Function	p. 262
Q Function	p. 264
Wigner Function	p. 265
s-Ordered Distributions	p. 266
Miscellaneous Applications	p. 269
Kerr Hamiltonian	p. 269
The Dicke Hamiltonian	p. 271
Problems	p. 276
Appendices	p. 279
Lie Groups and Lie Algebras	p. 279
Groups: Basic Concepts	p. 279
Group Representations	p. 281
Lie Algebras	p. 282
Examples	p. 284
Coherent States	p. 294
Examples	p. 295
Linear Systems	p. 299
Diagonalization of the Time-independent Hamiltonian	p. 301
Evolution Operator	p. 302
Reference Formulas	p. 303
Lie Transformation Method	p. 304
Wigner d Function	p. 306
Irreducible Tensor Operators	p. 311
References	p. 315
Index	p. 321
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A Group-Theoretical Approach to Quantum Optics Models of Atom-Field Interactions

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