Mathematical methods of quantum optics

R.R.Puri MATHEMATICAL METHODS OF QUANTUM OPTICS Berlin: Springer, 2001, pp. XIII+285

This book provides an accessible introduction to the mathematical methods of quantum optics. Starting from first principles, it reveals how a given system of atoms and a field is mathematically modelled. The method of eigenfunction expansion and the Lie algebraic method for solving equations are outlined. Analytically exactly solvable classes of equations are identified. The text also discusses consequences of Lie algebraic properties of Hamiltonians, such as the classification of their states as coherent, classical or non-classical based on the generalized uncertainty relation and the concept of quasiprobability distributions. A unified approach is developed for determining the dynamics of two-level and a three- level atom in combinations of quantized fields under certain conditions. Simple methods for solving a variety of linear and nonlinear dissipative master equations are given. Contents 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-l/2s 23 1.7 Wave-Particle Duality 26 1.8 Measurement Postulate and Paradoxes of Quantum Theory 29 1.8.1 The Measurement Problem 30 1.8.2 Schrödinger'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 2.4 Disentangling an Exponential 2.4.1 The Harmonic Oscillator Algebra 2.4.2 The SU(2) Algebra 2.4.3 SU(1,1) Algebra 2.5 Time-Ordered Exponential Integral 2.5.1 Harmonic Oscillator Algebra 2.5.2 SU(2) Algebra 2.5.3 The SU(1, 1) Algebra 3. Representations of Some Lie Algebras 3.1 Representation by Eigenvectors and Group Parameters 3.1.1 Bases Constituted by Eigenvectors 3.1.2 Bases Labeled by Group Parameters 3.2 Representations of Harmonic Oscillator Algebra 3.2.1 Orthonormal Bases 3.2.2 Minimum Uncertainty Coherent States 3.3 Representations of SU(2) 3.3.1 Orthonormal Representation 3.3.2 Minimum Uncertainty Coherent States 3.4 Representations of SU(1, 1) 3.4.1 Orthonormal Bases 3.4.2 Minimum Uncertainty Coherent States 4. Quasiprobabilities and Non-classical States 4.1 Phase Space Distribution Functions 4.2 Phase Space Representation of Spins 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components 4.4 Classical and Non-classical States 4.4.1 Non-classical States of Electromagnetic Field 4.4.2 Non-classical States of Spin-l/2s 5. Theory of Stochastic Processes 5.1 Probability Distributions 5.2 Markov Processes 5.3 Detailed Balance 5.4 Liouville and Fokker-Planck Equations 5.4.1 Liouville Equation 5.4.2 The Fokker-Planck Equation 5.5 Stochastic Differential Equations 5.6 Linear Equations with Additive Noise 5.7 Linear Equations with Multiplicative Noise 5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations 5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations 5.8 The Poisson Process 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise 6. The Electromagnetic Field

45 48 49 50 51 52 52 53 53 55 55 55 56 60 60 61 68 68 70 76 76 77 81 81 88 93 95 95 97 99 99 102 105 106 107 107 109 110 112 113 114 115 116 119

6.1 Free Classical Field 6.2 Field Quantization 6.3 Statistical Properties of Classical Field 6.3.1 First-Order Correlation Function 6.3.2 Second-Order Correlation Function 6.3.3 Higher-Order Correlations 6.3.4 Stable and Chaotic Fields 6.4 Statistical Properties of Quantized Field 6.4.1 First-Order Correlation 6.4.2 Second-Order Correlation 6.4.3 Quantized Coherent and Thermal Fields 6.5 Homodyned Detection 6.6 Spectrum 7. Atom— Field Interaction Hamiltonians 7.1 Dipole Interaction 7.2 Rotating Wave and Resonance Approximations 7.3 Two-Level Atom 7.4 Three-Level Atom 7.5 Effective Two-Level Atom 7.6 Multi-channel Models 7.7 Parametric Processes 7.8 Cavity QED 7.9 Moving Atom 8. Quantum Theory of Damping 8.1 The Master Equation 8.2 Solving a Master Equation 8.3 Multi-Time Average of System Operators 8.4 Bath of Harmonic Oscillators 8.4.1 Thermal Reservoir 8.4.2 Squeezed Reservoir 8.4.3 Reservoir of the Electromagnetic Field 8.5 Master Equation for a Harmonic Oscillator 8.6 Master Equation for Two-Level Atoms 8.6.1 Two-Level Atom in a Monochromatic Field 8.6.2 Collisional Damping 8.7 Master Equation for a Three-Level Atom 8.8 Master Equation for Field Interacting with a Reservoir of Atoms 9. Linear and Nonlinear Response of a System in an External Field 9.1 Steady State of a System in an External Field 9.2 Optical Susceptibility 9.3 Rate of Absorption of Energy 9.4 Response in a Fluctuating Field 10. Solution of Linear Equations: Method of Eigenvector Expansion 10.1 Eigenvalues and Eigenvectors

119 121 123 125 126 126 127 130 131 132 132 134 135 137 137 140 144 145 146 149 150 151 153 155 155 160 162 163 164 166 167 168 170 171 172 173 174 177 177 179 181 183 185 186

10.2 Generalized Eigenvalues and Eigenvectors 10.3 Solution of Two- Term Difference-Differential Equation 10.4 Exactly Solvable Two- and Three-Term Recursion Relations 10.4.1 Two- Term Recursion Relations 10.4.2 Three- Term Recursion Relations 11. Two-Level and Three-Level Hamiltonian Systems 11.1 Exactly Solvable Two-Level Systems 11.1.1 Time-Independent Detuning and Coupling 11.1.2 On- Resonant Real Time-Dependent Coupling 11.1.3 Fluctuating Coupling 11.2 N Two-Level Atoms in a Quantized Field 11.3 Exactly Solvable Three- Level Systems 11.4 Effective Two-Level Approximation 12. Dissipative Atomic Systems 12.1 Two-Level Atom in a Quasimonochromatic Field 12.1.1 Time-Dependent Evolution Operator Reducible to SU(2) 12.1.2 Time-Independent Evolution Operator 12.1.3 Nonlinear Response in a Dichromatic Field 12.2 N Two-Level Atoms in a Monochromatic Field 12.3 Two-Level Atoms in a Fluctuating Field 12.4 Driven Three-Level Atom 13. Dissipative Field Dynamics 13.1 Down-Conversion in a Damped Cavity 13.1.1 Averages and Variances of the Cavity Field Operators 13.1.2 Density Matrix 13.2 Field Interacting with a Two-Photon Reservoir 13.2.1 Two-Photon Absorption 13.2.2 Two-Photon Generation and Absorption. 13.3 Reservoir in the Lambda Configuration 14. Dissipative Cavity QED 14.1 Two-Level Atoms in a Single-Mode Cavity 14.2 Strong Atom-Field Coupling 14.2.1 Single Two-Level Atom. 14.3 Response to an External Field 14.3.1 Linear Response to a Monochromatic Field 14.3.2 Nonlinear Response to a Bichromatic Field 14.4 The Micromaser 14.4.1 Density Operator of the Field 14.4.2 Two-Level Atomic Micromaser 14.4.3 Atomic Statistics Appendices A. Some Mathematical Formulae B. Hypergeometric Equation C. Solution of Two-and Three-Dimensional Linear Equations

189 191 192 192 193 199 199 202 208 208 210 210 212 215 215 217 219 223 224 236 237 239 239 240 242 245 245 247 248 251 251 252 252 255 256 257 259 259 263 266 267 267 270 272

D. Roots of a Polynomial References Index

273 277 283 Index

absorption spectrum 182 ac Stark splitting 223 algebra - harmonic oscillator 41 - SU(1,1) 43 - SU(2) 42 - SU(m) 44 - SU(m,n) 45 antibunching 132 antinormal ordering 49, 83 Bargmann representation 64 Bell's inequality 35 Bloch-Siegert shift 142 Bloembergen resonances 224 Born approximation 157 bunching 134 cat paradox 31 cavity QED 151 Chapman-Kolmogorov equation 103 characteristic function 100 coherent multiphoton process 148 coherent states 58 - generalized 57 - Glauber 58,62 - of e.m. field 58, 133 - of harmonic oscillator 62 - of spins 70 - of SU(2) 70 - pair 78, 248 - Perelomov 57 coherent states, completeness relation 57 - for harmonic oscillator 62 - for SU(1,1) 78 - for SU(2) 72 coherent states, minimum uncertainty 59 - of harmonic oscillator 64 - of spins 70, 72 - of SU(1,1) 77

coherent states, uncorrelated equal variance minimum uncertainty 59 - of harmonic oscillator 62 - of spins 70 - of SU(1,1) 77 collapses and revivals 206 collisional damping 172 complementarity 12,27 cumulants 101 density operator 21 Descarte's rule 277 detailed balance 106, 108, 160, 225 differentiation, parametric 8 - of exponential operator 37 - of operator product 8 disentangling an exponential 48 - harmonic oscillator algebra 49 - SU(1,1) algebra 51 - SU(2) algebra 50 down conversion 151,239 dressed states 204 e.m. field - chaotic, classical 127 - chaotic, quantum 133 - coherence time 130 - coherent 127,133 - correlation functions, classical 123 - correlation functions, quantum 130 - quantization 121 effective two-level approximation 212 effective two-level atom 147 eigenvalue 7,186 - generalized 189 eigenvector 7,186 - generalized 189 entangled state 20, 25 EPR Paradox 32 equal variance minimum uncertainty state 13 Fokker-Planck equation 107

four-wave mixing 223, 257 - collision induced resonances 224, 257 - quantum resonances 258 Gaussian process 102 geometric phase 16 - in adiabatic evolution 18 - of a harmonic oscillator 18 - of a two-level system 18 Hamilton-Cayley theorem 190 Heisenberg equation 14 hidden variables theory 34 - local 34 Hilbert space 1 homodyned detection 134 Hurwitz criterion 277 Husimi function 87 incompatibility 12 interaction picture 15 interference 26,27 Jaynes-Cummings model 143, 144, 204 Jordan canonical form 189 Lie algebra 40 Lie group 56 Markov approximation 158 Markov process 103 master equation 104, 105, 158 measurement problem 30 micromaser 259 - trapping condition 264 minimum uncertainty states 12, 59 - of harmonic oscillator 61, 67 - of spins 70 - of 5E7(M) 77 mixed state 22 moments 100 multi time joint probability 15 multi-channel models 149 noise - additive 110 - coloured 109 - delta correlated 109 - Gaussian white 109 - multiplicative 110, 112 - white 109

non-classical states 95 - of e.m. field 95 - ofspin-l/2s 97 normal ordering 49, 83 Ornstein-Uhlenbeck process 110-112 P-function 86 - for spins 91 parametric processes 150 phase - dynamic 16 - geometric (see geometric phase) 16 photon 122 Poisson process 115 probability amplitude 11 probability density 11,99 - conditional 103 - joint 99 pure state 22 Q-function 87 - for spins 92 quantum eraser 29 quasiprobability distribution 83 - for spins 89 Rabi frequency 142, 221 random telegraph noise 116 regression theorem 105,163 representations - by eigenvectors 55 - equivalent 56 - labeled by group parameters 56 - of harmonic oscillator algebra 60 - of SU(1,1) algebra 76 - of SU(2) algebra 68 resonance approximation 144 resonance fluorescence 171, 219 - collective 225 rotating wave approximation 143, 181 Rydberg atom 152 s-ordering 83 Schmidt decomposition 25 Schrödinger equation 13 Schwarz inequality 2 - generalized 3

secular approximation 162, 227, 253, 256 semiclassical approximation 139 similarity transformation 39 - harmonic oscillator 41 - SU(1,1) 43 - SU(2) 42 - SU(m) 44 - SU(m,n) 45 Sneddon's formula 37 spectroscopic squeezing 75 spectrum 136 - absorption 223,256 - emission 222 spin operators - collective 69 - lowering 23 - raising 23 squeezed reservoir 166 squeezed states - of harmonic oscillator 67 - of spins 73, 74 squeezed vacuum 166 squeezing operator 65 Stark shift 214 stationary process 100 stochastic differential equation 109 sub-Poissonian distribution 132 superoperator 10 - adjoint 10 superposition, principle of 26 susceptibility 179 - optical 180 symmetric ordering 83

- for spins 94 thermal reservoir 164 three level atom 145 time-ordered exponential integration - harmonic oscillator algebra 52 - SU(1,1) algebra 53 - SU(2) algebra 53 trace 6 transition probability 103 two-channel Raman-coupled model 150, 207 two-level atom 144 two-photon process 146 two-photon reservoir - in ladder configuration 245 - in Lambda configuration 248 uncertainty relation 12 uncorrelated equal variance minimum uncertainty state 13, 62 vacuum field Rabi oscillations 205 vacuum field Rabi splitting 257 vacuum fluctuations 122 wave mixing 149, 181 wave-particle duality 26 welcher weg 28 which path 28 Wiener process 110, 111 Wiener-Khintchine theorem 136 Wigner function 87 - for spins 92 Zeno effect 16