The quantum world of ultra-cold atoms and light. Book 1, Foundations of quantum optics

This book is the first in a trilogy designed to make available to those working in the field, and to advanced students, a full suite of theoretical techniques needed for quantum technologies. These have been developed within the paradigm of quantum optics, a field based on the use of atoms and light. As participants in the theoretical aspects of quantum optics, ultra-cold atoms, and quantum information, we have worked both together and independently on most of the significant themes which have arisen in the last thirty years, and during this period have both taught classes and supervised research projects in the field. Gardiner, C. W. ,Zoller P. The quantum world of ultra-cold atoms and light, book 1, Foundations of quantum optics (ICP,2014 ) ......Page 4 Copyright ......Page 5 Preface ......Page 7 Acknowledgments ......Page 9 Contents ......Page 10 I The Physical Background ......Page 18 1. Controlling the Quantum World 3 ......Page 20 1.1.1 The Development of the Laser 5 ......Page 22 1.1.2 The Development of Quantum Optics 6 ......Page 23 1.1.3 The Elementary Objects of Ideal Quantum Optics 8 ......Page 25 1.2.1 The Einstein-Podolsky-Rosen Paradox 9 ......Page 26 1.2.3 The Quantum Computer 10 ......Page 27 1.2.5 Quantum Computing with Trapped Ions 11 ......Page 28 1.2.6 Engineered Systems for Quantum Optics 13 ......Page 30 2. Describing the Quantum World 14 ......Page 31 2.1.1 Probabilities, Paths and Correlations 15 ......Page 32 2.1.4 Correlation Functions and Spectra 16 ......Page 33 2.2.1 Quantum Fields and “Stripped-Down” Quantum Electrodynamics 17 ......Page 34 2.2.4 The Input-Output Formalism 18 ......Page 35 2.3.1 Quantum Markov Description 19 ......Page 36 2.4 Ultra-Cold Atoms 20 ......Page 37 2.4.1 Field Operators and Hamiltonians 21 ......Page 38 10.5.2 The Harmonic Oscillator 122 ......Page 39 II Classical Stochastic Methods ......Page 42 3.1 Brownian Motion and the Thermal Origin of Noise 27 ......Page 44 3.1.2 Markov Processes 28 ......Page 45 3.2.1 The Langevin Equation 29 ......Page 46 3.2.2 Solution of the Langevin Equation 30 ......Page 47 3.2.3 Fluctuation-Dissipation Theorem 31 ......Page 48 3.2.4 Diffusion as a Result of Brownian Motion 32 ......Page 49 3.3.1 Quantities Most Commonly Considered 34 ......Page 51 3.3.2 The Regression Theorem 35 ......Page 52 3.3.4 Spectrum and Autocorrelation Function 36 ......Page 53 3.3.5 Fourier Analysis of Fluctuating Functions 38 ......Page 55 4.1 Ito Stochastic Differential Equation 39 ......Page 56 4.1.1 Calculus of Stochastic Differential Equations 40 ......Page 57 4.1.2 Change of Variables: Ito’s Formula 41 ......Page 58 4.3 The Stratonovich Stochastic Differential Equation 42 ......Page 59 4.3.1 Change of Variables for the Stratonovich Stochastic Differential Equation 43 ......Page 60 4.3.3 Comparison of Ito and Stratonovich Formalisms 44 ......Page 61 4.4.2 Complex Variable Systems 45 ......Page 62 4.5 Numerical Simulation of Stochastic Differential Equations 46 ......Page 63 5.1 Fokker-Planck Equation in One Dimension 47 ......Page 64 5.1.1 Boundary Conditions 48 ......Page 65 5.1.2 Deterministic Motion 50 ......Page 67 5.2.1 Construction of Eigenfunctions 51 ......Page 68 5.2.3 Eigenfunctions for the Ornstein-Uhlenbeck Process 53 ......Page 70 5.3.1 Boundary Conditions 54 ......Page 71 5.3.2 Stationary Solutions and Potential Conditions 55 ......Page 72 6.1 The Master Equation 57 ......Page 74 6.1.1 The Continuous Time Random Walk 59 ......Page 76 6.1.2 The Poisson Process 61 ......Page 78 6.1.3 The Random Telegraph Process 62 ......Page 79 6.1.4 Example—Simulating Jumps in a Two-Level Atom 63 ......Page 80 7.1.1 Ornstein-Uhlenbeck Process in One Dimension 66 ......Page 83 7.1.2 Many-Variable Ornstein-Uhlenbeck Process 68 ......Page 85 7.1.3 Stationary Correlation Functions and Spectrum 69 ......Page 86 7.2 Johnson Noise 70 ......Page 87 7.3 Complex Variable Oscillator Processes 71 ......Page 88 7.3.1 Line Broadening in a Random Frequency Oscillator 72 ......Page 89 7.3.2 The Thermalized Oscillator 73 ......Page 90 7.3.3 Equations for Phase and Amplitude 74 ......Page 91 7.3.4 The van der Pol Laser Equation 75 ......Page 92 8.1 The White Noise Limit 78 ......Page 95 8.1.2 The Projector Formalism 79 ......Page 96 8.1.4 Evaluation of the Diffusion Coefficient in Terms of Correlation Functions 82 ......Page 99 8.2 Interpretation and Generalizations of the White Noise Limit 84 ......Page 101 8.2.3 Time Dependence of Coefficients 85 ......Page 102 8.3 Linear Non-Markovian Stochastic Differential Equations 86 ......Page 103 8.3.1 The Complex Oscillator with a Noisy Frequency 87 ......Page 104 8.3.2 Approximation Methods for Multivariable Systems 89 ......Page 106 8.3.3 Example—The Two-Dimensional Oscillator Driven by Random Telegraph Noise 91 ......Page 108 8.3.4 Driving with Other Kinds of Noise 94 ......Page 111 9.1 Slow and Fast Variables 97 ......Page 114 9.1.1 A Simplified Laser Model 98 ......Page 115 9.1.2 Stochastic Elimination of the Fast Variable 100 ......Page 117 9.2.1 A Stochastic Model of Trapped Atoms 104 ......Page 121 9.2.2 Motional Narrowing 106 ......Page 123 III Fields, Quanta and Atoms ......Page 126 10.1.1 Bosons 111 ......Page 128 10.1.3 The Hamiltonian and Total Number Operators 112 ......Page 129 10.2.1 Canonical Ensemble 113 ......Page 130 10.2.2 The Grand Canonical Ensemble 114 ......Page 131 10.3.1 Moments of the Number Operator 115 ......Page 132 10.3.2 Many Modes 116 ......Page 133 10.4.1 Hartree-Fock Factorization 117 ......Page 134 10.4.2 Generalized Hartree-Fock Factorization for Quantum Gaussian Density Operators 118 ......Page 135 10.5.1 Properties of the Coherent States 119 ......Page 136 10.6.2 Fermi-Gaussian Systems 123 ......Page 140 10.7 Two-Level Systems and Pauli Matrices 124 ......Page 141 10.7.1 Pauli Matrix Properties 125 ......Page 142 11.1.1 Matter Wave Fields 127 ......Page 144 11.1.2 Soundwaves 129 ......Page 146 11.1.3 The Electromagnetic Field 130 ......Page 147 11.1.4 Monochromatic Electromagnetic Waves 133 ......Page 150 11.1.5 States of Quantized Fields 134 ......Page 151 11.2 Coherence and Correlation Functions 135 ......Page 152 11.2.1 Interference of Classical Waves 136 ......Page 153 11.2.2 Quantum Interference 138 ......Page 155 11.2.3 Summary—Phase and Interference 141 ......Page 158 12. Atoms, Light and their Interaction 143 ......Page 160 12.1.1 Hamiltonian and Schrodinger Equation 144 ......Page 161 12.1.3 The Rotating Wave Approximation 147 ......Page 164 12.2.1 Wavefunction and Initial Condition 148 ......Page 165 12.2.2 Solutions for Atomic Decay and Radiated Field 152 ......Page 169 12.3.1 Interaction Hamiltonian 154 ......Page 171 12.3.2 Solution of the Schrodinger Equation 155 ......Page 172 12.3.3 Optical Pulses 156 ......Page 173 12.3.4 Effective Potential on a Ground State Atom 158 ......Page 175 12.4 Interaction of a Two-Level Atom with a Single Mode 159 ......Page 176 12.4.1 Quantum Collapses and Revivals 160 ......Page 177 IV Quantum Stochastic Processes ......Page 178 13.1.1 The Quantum-Mechanical Master Equation 163 ......Page 180 13.1.2 System and Heat Bath 164 ......Page 181 13.1.3 The Master Equation 165 ......Page 182 13.2.1 Description of Projection Method 168 ......Page 185 13.2.2 Explicit Formulation as a Quantum Master Equation 172 ......Page 189 13.3 More General Heat Baths 174 ......Page 191 13.3.1 Generalized Bath Operators 175 ......Page 192 13.4.2 Multitime Averages 177 ......Page 194 13.4.3 Quantum Regression Theorem 178 ......Page 195 13.4.4 Spectrum and Quantum Correlation Functions 179 ......Page 196 14.1 A Two-Level Atom Interacting with a Thermal Heat Bath 181 ......Page 198 14.1.1 Frequency Shifts 182 ......Page 199 14.1.2 Equations of Motion 183 ......Page 200 14.1.3 Master Equation for the Occupation Numbers 184 ......Page 201 14.1.4 Comparison with Classical Damping 185 ......Page 202 14.2 The Two-Level Atom Driven by a Coherent Light Field 186 ......Page 203 14.2.1 The Resonant Optical Bloch Equations 187 ......Page 204 14.2.2 Correlation Functions and Spectrum 190 ......Page 207 14.3.1 Damping and Gain with a Harmonic Oscillator 191 ......Page 208 14.3.2 Formulation in Terms of Density Operator Matrix Elements 194 ......Page 211 14.3.3 The Phase-Damped Oscillator 195 ......Page 212 14.4 A Simple Model of Laser Cooling 197 ......Page 214 14.4.1 Formulation of the Model 198 ......Page 215 14.4.2 Doppler Cooling 200 ......Page 217 14.4.4 Fluctuations 201 ......Page 218 V Phase Space Methods ......Page 222 15. Phase Space Representations for Bosons 207 ......Page 224 15.1.3 Symmetrically Ordered Quantum Characteristic Function 208 ......Page 225 15.1.5 Properties of the Quantum Characteristic Functions 209 ......Page 226 15.2.1 Wigner Function 210 ......Page 227 15.2.2 The P-Function 212 ......Page 229 15.2.4 Multitime Correlation Functions 213 ......Page 230 16.1 Operator Correspondences and Equations of Motion 215 ......Page 232 16.1.1 The Driven Harmonic Oscillator 216 ......Page 233 16.1.2 The Anharmonic Oscillator 218 ......Page 235 16.1.3 The Bogoliubov Hamiltonian 219 ......Page 236 16.2.1 The Harmonic Oscillator Including Loss or Gain 220 ......Page 237 16.2.2 The Phase-Damped Harmonic Oscillator 224 ......Page 241 16.3 The Wigner Distribution Function f(x, p) 225 ......Page 242 16.3.1 The Wigner Distribution Function as a Quasiprobability 226 ......Page 243 16.3.2 Operator Mappings for the Wigner Distribution Function 227 ......Page 244 16.3.3 Motion in a Potential 229 ......Page 246 16.3.4 A Free Particle in One Dimension 230 ......Page 247 16.3.5 Quantum Brownian Motion 231 ......Page 248 16.4 Quantum Fluctuations in Equations of Motion 232 ......Page 249 17.1 Operator Correspondences and Equations of Motion 234 ......Page 251 17.1.1 The Driven Harmonic Oscillator 235 ......Page 252 17.2.1 The Harmonic Oscillator Including Loss or Gain 236 ......Page 253 17.3.1 A Simple Laser Model 238 ......Page 255 17.3.2 Implementation of the Laser Equation 241 ......Page 258 17.3.3 Solutions of the Laser Equations 242 ......Page 259 VI Quantum Measurement Theory ......Page 264 18.1 Formulations of Quantum Mechanics 249 ......Page 266 18.1.2 Status of the Postulates 250 ......Page 267 18.2.1 Excitation of an Atom 251 ......Page 268 18.2.3 Entanglement 254 ......Page 271 18.2.4 Interpretation as Collapse of the Wavefunction 255 ......Page 272 18.3 Formal Quantum Measurement Theory 257 ......Page 274 18.3.1 Measurement Operators 258 ......Page 275 18.3.3 Example—The Two-Level Atom 259 ......Page 276 18.4.1 Sequences of Measurements 260 ......Page 277 18.4.3 General Correlation Functions 262 ......Page 279 19.1 Photon Counting 263 ......Page 280 19.1.2 Measurement as a Physical Process 264 ......Page 281 19.2.1 Wavefunction Evolution 265 ......Page 282 19.2.2 The Stochastic Schrodinger Equation 266 ......Page 283 19.2.3 Multiple Detection 267 ......Page 284 19.3.1 Interference of Independent Bose-Einstein Condensates 269 ......Page 286 19.4 Damping of Quantum Coherence 271 ......Page 288 19.4.1 Stochastic Schrodinger Equation Treatment 272 ......Page 289 19.4.2 Interference and the Quantum Characteristic Function 273 ......Page 290 19.4.3 The Linear Loss-Gain Model 274 ......Page 291 19.4.4 The Stability and Robustness of Coherent States 277 ......Page 294 19.5.1 Modelling Quantum Measurement 278 ......Page 295 19.5.3 Relationship to van Kamperis Formulation of Measurement Theory 280 ......Page 297 20.1 Theoretical Basis for the Quantum Zeno Effect 281 ......Page 298 20.1.1 Connection with Continuous Measurement Theory 282 ......Page 299 20.2 A Quantum Model of Trapped Atoms 284 ......Page 301 20.2.1 Stationary State and Projectors 285 ......Page 302 20.2.2 The Master Equation in the Strong Dissipation Limit 286 ......Page 303 20.3.1 P-Representation Solution 288 ......Page 305 20.3.2 Implementation of Fast Loss Mechanism 290 ......Page 307 References 293 ......Page 310 Author Index 297 ......Page 314 Subject Index 299 ......Page 316 cover......Page 1 This century has seen the development of technologies for manipulating and controlling matter and light at the level of individual photons and atoms, a realm in which physics is fully quantum mechanical. The dominant experimental technology is the laser, and the theoretical paradigm is quantum optics.The Quantum World of Ultra-Cold Atoms and Light is a trilogy, which presents the quantum optics way of thinking and its applications to quantum devices. This book -- Foundations of Quantum Optics -- provides an introductory text on the theoretical techniques of quantum optics, containing the elements of what one needs to teach, learn, and "think" about quantum optics. There is a particular emphasis on the classical and quantum stochastic methods which have come to dominate the field.Book II will cover applications to quantum devices, such as quantum computers and simulators, and will include the more advanced techniques necessary to describe non-classical light fields. Book III will cover the field of ultra-cold atoms, for which the quantum-optical paradigm has proved to be highly applicable for quantitative work. This century has seen the development of technologies for manipulating and controlling matter and light at the level of individual photons and atoms, a realm in which physics is fully quantum mechanical. The dominant experimental technology is the laser, and the theoretical paradigm is quantum optics. The Quantum World of Ultra-Cold Atoms and Light is a trilogy, which presents the quantum optics way of thinking and its applications to quantum devices. This book - Foundations of Quantum Optics - provides an introductory text on the theoretical techniques of quantum optics, containing the elements of what one needs to teach, learn, and "think" about quantum optics. There is a particular emphasis on the classical and quantum stochastic methods which have come to dominate the field. Book II will cover applications to quantum devices, such as quantum computers and simulators, and will include the more advanced techniques necessary to describe non-classical light fields. Book III will cover the field of ultra-cold atoms, for which the quantum-optical paradigm has proved to be highly applicable for quantitative work