ISBN: 3540678026
TITLE: Mathematical Methods of Quantum Optics
AUTHOR: Puri, Ravinder R.
TOC:

1. Basic Quantum Mechanics 1
1.1 Postulates of Quantum Mechanics 1
1.1.1 Postulate 1 1
1.1.2 Postulate 2 11
1.1.3 Postulate 3 11
1.1.4 Postulate 4 11
1.1.5 Postulate 5 13
1.2 Geometric Phase 16
1.2.1 Geometric Phase of a Harmonic Oscillator 18
1.2.2 Geometric Phase of a Two-Level System 18
1.2.3 Geometric Phase in Adiabatic Evolution 18
1.3 Time-Dependent Approximation Method 19
1.4 Quantum Mechanics of a Composite System 20
1.5 Quantum Mechanics of a Subsystem and Density Operator 21
1.6 Systems of One and Two Spin-1/2s 23
1.7 WaveParticle Duality 26
1.8 Measurement Postulate and Paradoxes of Quantum Theory 29
1.8.1 The Measurement Problem 30
1.8.2 Schrdinger's Cat Paradox 31
1.8.3 EPR Paradox 32
1.9 Local Hidden Variables Theory 34
2. Algebra of the Exponential Operator 37
2.1 Parametric Differentiation of the Exponential 37
2.2 Exponential of a Finite-Dimensional Operator 38
2.3 Lie Algebraic Similarity Transformations 39
2.3.1 Harmonic Oscillator Algebra 41
2.3.2 The SU(2) Algebra 42
2.3.3 The SU(1,1) Algebra 43
2.3.4 The SU(m) Algebra 45
2.3.5 The SU(m, n) Algebra 45
2.4 Disentangling an Exponential 48
2.4.1 The Harmonic Oscillator Algebra 49
2.4.2 The SU(2) Algebra 50
2.4.3 SU(1,1) Algebra 51
2.5 Time-Ordered Exponential Integral 52
2.5.1 Harmonic Oscillator Algebra 52
2.5.2 SU(2) Algebra 53
2.5.3 The SU(1,1) Algebra 53
3. Representations of Some Lie Algebras 55
3.1 Representation by Eigenvectors and Group Parameters 55
3.1.1 Bases Constituted by Eigenvectors 55
3.1.2 Bases Labeled by Group Parameters 56
3.2 Representations of Harmonic Oscillator Algebra 60
3.2.1 Orthonormal Bases 60
3.2.2 Minimum Uncertainty Coherent States 61
3.3 Representations of SU(2) 68
3.3.1 Orthonormal Representation 68
3.3.2 Minimum Uncertainty Coherent States 70
3.4 Representations of SU(1, 1) 76
3.4.1 Orthonormal Bases 76
3.4.2 Minimum Uncertainty Coherent States 77
4. Quasiprobabilities and Non-classical States 81
4.1 Phase Space Distribution Functions 81
4.2 Phase Space Representation of Spins 88
4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components 93
4.4 Classical and Non-classical States 95
4.4.1 Non-classical States of Electromagnetic Field 95
4.4.2 Non-classical States of Spin-1/2s 97
5. Theory of Stochastic Processes 99
5.1 Probability Distributions 99
5.2 Markov Processes 102
5.3 Detailed Balance 105
5.4 Liouville and FokkerPlanck Equations 106
5.4.1 Liouville Equation 107
5.4.2 The FokkerPlanck Equation 107
5.5 Stochastic Differential Equations 109
5.6 Linear Equations with Additive Noise 110
5.7 Linear Equations with Multiplicative Noise 112
5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations 113
5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations 114
5.8 The Poisson Process 115
5.9 Stochastic Differential Equation Driven by Random Telegraph Noise 116
6. The Electromagnetic Field 119
6.1 Free Classical Field 119
6.2 Field Quantization 121
6.3 Statistical Properties of Classical Field 123
6.3.1 First-Order Correlation Function 125
6.3.2 Second-Order Correlation Function 126
6.3.3 Higher-Order Correlations 126
6.3.4 Stable and Chaotic Fields 127
6.4 Statistical Properties of Quantized Field 130
6.4.1 First-Order Correlation 131
6.4.2 Second-Order Correlation 132
6.4.3 Quantized Coherent and Thermal Fields 132
6.5 Homodyned Detection 134
6.6 Spectrum 135
7. AtomField Interaction Hamiltonians 137
7.1 Dipole Interaction 137
7.2 Rotating Wave and Resonance Approximations 140
7.3 Two-Level Atom 144
7.4 Three-Level Atom 145
7.5 Effective Two-Level Atom 146
7.6 Multi-channel Models 149
7.7 Parametric Processes 150
7.8 Cavity QED 151
7.9 Moving Atom 153
8. Quantum Theory of Damping 155
8.1 The Master Equation 155
8.2 Solving a Master Equation 160
8.3 Multi-Time Average of System Operators 162
8.4 Bath of Harmonic Oscillators 163
8.4.1 Thermal Reservoir 164
8.4.2 Squeezed Reservoir 166
8.4.3 Reservoir of the Electromagnetic Field 167
8.5 Master Equation for a Harmonic Oscillator 168
8.6 Master Equation for Two-Level Atoms 170
8.6.1 Two-Level Atom in a Monochromatic Field 171
8.6.2 Collisional Damping 172
8.7 Master Equation for a Three-Level Atom 173
8.8 Master Equation for Field Interacting with a Reservoir of Atoms 174
9. Linear and Nonlinear Response of a System in an External Field 177
9.1 Steady State of a System in an External Field 177
9.2 Optical Susceptibility 179
9.3 Rate of Absorption of Energy 181
9.4 Response in a Fluctuating Field 183
10. Solution of Linear Equations: Method of Eigenvector Expansion 185
10.1 Eigenvalues and Eigenvectors 186
10.2 Generalized Eigenvalues and Eigenvectors 189
10.3 Solution of Two-Term Difference-Differential Equation 191
10.4 Exactly Solvable Two- and Three-Term Recursion Relations 192
10.4.1 Two-Term Recursion Relations 192
10.4.2 Three-Term Recursion Relations 193
11. Two-Level and Three-Level Hamiltonian Systems 199
11.1 Exactly Solvable Two-Level Systems 199
11.1.1 Time-Independent Detuning and Coupling 202
11.1.2 On-Resonant Real Time-Dependent Coupling 208
11.1.3 Fluctuating Coupling 208
11.2 N Two-Level Atoms in a Quantized Field 210
11.3 Exactly Solvable Three-Level Systems 210
11.4 Effective Two-Level Approximation 212
12. Dissipative Atomic Systems 215
12.1 Two-Level Atom in a Quasimonochromatic Field 215
12.1.1 Time-Dependent Evolution Operator Reducible to SU(2) 217
12.1.2 Time-Independent Evolution Operator 219
12.1.3 Nonlinear Response in a Bichromatic Field 223
12.2 N Two-Level Atoms in a Monochromatic Field 224
12.3 Two-Level Atoms in a Fluctuating Field 236
12.4 Driven Three-Level Atom 237
13. Dissipative Field Dynamics 239
13.1 Down-Conversion in a Damped Cavity 239
13.1.1 Averages and Variances of the Cavity Field Operators 240
13.1.2 Density Matrix 242
13.2 Field Interacting with a Two-Photon Reservoir 245
13.2.1 Two-Photon Absorption 245
13.2.2 Two-Photon Generation and Absorption 247
13.3 Reservoir in the L ambda Configuration 248
14. Dissipative Cavity QED 251
14.1 Two-Level Atoms in a Single-Mode Cavity 251
14.2 Strong AtomField Coupling 252
14.2.1 Single Two-Level Atom 252
14.3 Response to an External Field 255
14.3.1 Linear Response to a Monochromatic Field 256
14.3.2 Nonlinear Response to a Bichromatic Field 257
14.4 The Micromaser 259
14.4.1 Density Operator of the Field 259
14.4.2 Two-Level Atomic Micromaser 263
14.4.3 Atomic Statistics 266
Appendices 267
A. Some Mathematical Formulae 267
B. Hypergeometric Equation270
C. Solution of Two- and Three-Dimensional Linear Equations 272
D. Roots of a Polynomial 273
References 277
Index 283
END
