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.

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