The Theory of Open Quantum Systems Book

I Probability in Classical and Quantum Physics

1 Classical probability theory and stochastic processes 3

1.1 The probability space 3

1.1.1 The [sigma]-algebra of events 3

1.1.2 Probability measures and Kolmogorov axioms 4

1.1.3 Conditional probabilities and independence 5

1.2 Random variables 5

1.2.1 Definition of random variables 5

1.2.2 Transformation of random variables 8

1.2.3 Expectation values and characteristic function 9

1.3 Stochastic processes 11

1.3.1 Formal definition of a stochastic process 11

1.3.2 The hierarchy of joint probability distributions 12

1.4 Markov processes 13

1.4.1 The Chapman-Kolmogorov equation 14

1.4.2 Differential Chapman-Kolmogorov equation 17

1.4.3 Deterministic processes and Liouville equation 19

1.4.4 Jump processes and the master equation 20

1.4.5 Diffusion processes and Fokker-Planck equation 27

1.5 Piecewise deterministic processes 31

1.5.1 The Liouville master equation 32

1.5.2 Waiting time distribution and sample paths 33

1.5.3 Path integral representation of PDPs 36

1.5.4 Stochastic calculus for PDPs 38

1.6 Levy processes 44

1.6.1 Translation invariant processes 44

1.6.2 The Levy-Khintchine formula 46

1.6.3 Stable Levy processes 50

References 55

2 Quantum probability 57

2.1 The statistical interpretation of quantum mechanics 57

2.1.1 Self-adjoint operators and the spectral theorem 57

2.1.2 Observables and random variables 61

2.1.3 Pure states and statistical mixtures 63

2.1.4 Joint probabilities in quantum mechanics 68

2.2 Composite quantum systems 72

2.2.1 Tensor product 72

2.2.2 Schmidt decomposition and entanglement 75

2.3 Quantum entropies 76

2.3.1 Von Neumannentropy 76

2.3.2 Relative entropy 78

2.3.3 Linear entropy 80

2.4 The theory of quantum measurement 80

2.4.1 Ideal quantum measurements 81

2.4.2 Operations and effects 83

2.4.3 Representation theorem for quantum operations 85

2.4.4 Quantum measurement and entropy 89

2.4.5 Approximate measurements 90

2.4.6 Indirect quantum measurements 93

2.4.7 Quantum non-demolition measurements 99

References 101

II Density Matrix Theory

3 Quantum master equations 105

3.1 Closed and open quantum systems 106

3.1.1 The Liouville-von Neumann equation 106

3.1.2 Heisenberg and interaction picture 108

3.1.3 Dynamics of open systems 110

3.2 Quantum Markov processes 113

3.2.1 Quantum dynamical semigroups 113

3.2.2 The Markovian quantum master equation 115

3.2.3 The adjoint quantum master equation 120

3.2.4 Multi-time correlation functions 121

3.2.5 Irreversibility and entropy production 123

3.3 Microscopic derivations 125

3.3.1 Weak-coupling limit 126

3.3.2 Relaxation to equilibrium 132

3.3.3 Singular-coupling limit 133

3.3.4 Low-density limit 134

3.4 The quantum optical master equation 136

3.4.1 Matter in quantized radiation fields 136

3.4.2 Decay of a two-level system 141

3.4.3 Decay into a squeezed field vacuum 144

3.4.4 More general reservoirs 147

3.4.5 Resonance fluorescence 148

3.4.6 The damped harmonic oscillator 155

3.5 Non-selective, continuous measurements 160

3.5.1 The quantum Zeno effect 161

3.5.2 Density matrix equation 162

3.6 Quantum Brownian motion 166

3.6.1 The Caldeira-Leggett model 167

3.6.2 High-temperature master equation 168

3.6.3 The exact Heisenberg equations of motion 176

3.6.4 The influence functional 186

3.7 Non-linear quantum master equations 195

3.7.1 Quantum Boltzmann equation 195

3.7.2 Mean field master equations 197

3.7.3 Mean field laser equations 199

3.7.4 Non-linear Schrodinger equation 202

3.7.5 Super-radiance 204

References 209

4 Decoherence 212

4.1 The decoherence function 213

4.2 An exactly solvable model 218

4.2.1 Time evolution of the total system 218

4.2.2 Decay of coherences and the decoherence factor 220

4.2.3 Coherent subspaces and system-size dependence 223

4.3 Markovian mechanisms of decoherence 225

4.3.1 The decoherence rate 225

4.3.2 Quantum Brownian motion 226

4.3.3 Internal degrees of freedom 227

4.3.4 Scattering of particles 230

4.4 The damped harmonic oscillator 234

4.4.1 Vacuum decoherence 234

4.4.2 Thermal noise 238

4.5 Electromagnetic field states 242

4.5.1 Atoms interacting with a cavity field mode 243

4.5.2 Schrodinger cat states 248

4.6 Caldeira-Leggett model 254

4.6.1 General decoherence formula 254

4.6.2 Ohmic environments 256

4.7 Decoherence and quantum measurement 261

4.7.1 Dynamical selection of a pointer basis 261

4.7.2 Dynamical model for a quantum measurement 267

References 270

III Stochastic Processes in Hilbert Space

5 Probability distributions on Hilbert space 275

5.1 The state vector as a random variable in Hilbert space 275

5.1.1 A new type of quantum mechanical ensemble 275

5.1.2 Stern-Gerlach experiment 280

5.2 Probability density functionals on Hilbert space 283

5.2.1 Probability measures on Hilbert space 283

5.2.2 Distributions on projective Hilbert space 286

5.2.3 Expectation values 289

5.3 Ensembles of mixtures 290

5.3.1 Probability density functionals on state space 291

5.3.2 Description of selective quantum measurements 292

References 293

6 Stochastic dynamics in Hilbert space 295

6.1 Dynamical semigroups and PDPs in Hilbert space 296

6.1.1 Reduced system dynamics as a PDP 296

6.1.2 The Hilbert space path integral 303

6.1.3 Diffusion approximation 305

6.1.4 Multi-time correlation functions 307

6.2 Stochastic representation of continuous measurements 312

6.2.1 Stochastic time evolution of [epsilon subscript P]-ensembles 313

6.2.2 Short-time behaviour of the propagator 313

6.3 Direct photodetection 316

6.3.1 Derivation of the PDP 316

6.3.2 Path integral solution 322

6.4 Homodyne photodetection 326

6.4.1 Derivation of the PDP for homodyne detection 327

6.4.2 Stochastic Schrodinger equation 331

6.5 Heterodyne photodetection 333

6.5.1 Stochastic Schrodinger equation 333

6.5.2 Stochastic collapse models 336

6.6 Stochastic density matrix equations 339

6.7 Photodetection on a field mode 341

6.7.1 The photocounting formula 341

6.7.2 QND measurement of a field mode 345

References 349

7 The stochastic simulation method 352

7.1 Numerical simulation algorithms for PDPs 353

7.1.1 Estimation of expectation values 353

7.1.2 Generation of realizations of the process 354

7.1.3 Determination of the waiting time 355

7.1.4 Selection of the jumps 357

7.2 Algorithms for stochastic Schrodinger equations 358

7.2.1 General remarks on convergence 359

7.2.2 The Euler scheme 360

7.2.3 The Heun scheme 361

7.2.4 The fourth-order Runge-Kutta scheme 361

7.2.5 A second-order weak scheme 362

7.3 Examples 363

7.3.1 The damped harmonic oscillator 363

7.3.2 The driven two-level system 366

7.4 A case study on numerical performance 371

7.4.1 Numerical efficiency and scaling laws 371

7.4.2 The damped driven Morse oscillator 373

References 379

8 Applications to quantum optical systems 381

8.1 Continuous measurements in QED 382

8.1.1 Constructing the microscopic Hamiltonian 382

8.1.2 Determination of the QED operation 384

8.1.3 Stochastic dynamics of multipole radiation 387

8.1.4 Representation of incomplete measurements 389

8.2 Dark state resonances 391

8.2.1 Waiting time distribution and trapping state 392

8.2.2 Measurement schemes and stochastic evolution 394

8.3 Laser cooling and Levy processes 399

8.3.1 Dynamics of the atomic wave function 401

8.3.2 Coherent population trapping 406

8.3.3 Waiting times and momentum distributions 411

8.4 Strong field interaction and the Floquet picture 418

8.4.1 Floquet theory 419

8.4.2 Stochastic dynamics in the Floquet picture 421

8.4.3 Spectral detection and the dressed atom 424

References 427

IV Non-Markovian Quantum Processes

9 Projection operator techniques 431

9.1 The Nakajima-Zwanzig projection operator technique 432

9.1.1 Projection operators 432

9.1.2 The Nakajima-Zwanzig equation 433

9.2 The time-convolutionless projection operator method 435

9.2.1 The time-local master equation 436

9.2.2 Perturbation expansion of the TCL generator 437

9.2.3 The cumulant expansion 441

9.2.4 Perturbation expansion of the inhomogeneity 442

9.2.5 Error analysis 445

9.3 Stochastic unravelling in the doubled Hilbert space 446

References 448

10 Non-Markovian dynamics in physical systems 450

10.1 Spontaneous decay of a two-level system 451

10.1.1 Exact master equation and TCL generator 451

10.1.2 Jaynes-Cummings model on resonance 456

10.1.3 Jaynes-Cummings model with detuning 461

10.1.4 Spontaneous decay into a photonic band gap 464

10.2 The damped harmonic oscillator 465

10.2.1 The model and frequency renormalization 465

10.2.2 Factorizing initial conditions 466

10.2.3 The stationary state 471

10.2.4 Non-factorizing initial conditions 475

10.2.5 Disregarding the inhomogeneity 479

10.3 The spin-boson system 480

10.3.1 Microscopic model 480

10.3.2 Relaxation of an initially factorizing state 481

10.3.3 Equilibrium correlation functions 485

10.3.4 Transition from coherent to incoherent motion 486

References 487

V Relativistic Quantum Processes

11 Measurements in relativistic quantum mechanics 491

11.1 The Schwinger-Tomonaga equation 492

11.1.1 States as functionals of spacelike hypersurfaces 492

11.1.2 Foliations of space-time 496

11.2 The measurement of local observables 497

11.2.1 The operation for a local measurement 498

11.2.2 Relativistic state reduction 500

11.2.3 Multivalued space-time amplitudes 504

11.2.4 The consistent hierarchy of joint probabilities 507

11.2.5 EPR correlations 511

11.2.6 Continuous measurements 512

11.3 Non-local measurements and causality 516

11.3.1 Entangled quantum probes 517

11.3.2 Non-local measurement by EPR probes 520

11.3.3 Quantum state verification 525

11.3.4 Non-local operations and the causality principle 528

11.3.5 Restrictions on the measurability of operators 534

11.3.6 QND verification of non-local states 539

11.3.7 Preparation of non-local states 543

11.3.8 Exchange measurements 544

11.4 Quantum teleportation 546

11.4.1 Coherent transfer of quantum states 546

11.4.2 Teleportation and Bell-state measurement 549

11.4.3 Experimental realization 551

References 555

12 Open quantum electrodynamics 557

12.1 Density matrix theory for QED 558

12.1.1 Field equations and correlation functions 558

12.1.2 The reduced density matrix 565

12.2 The influence functional of QED 566

12.2.1 Elimination of the radiation degrees of freedom 566

12.2.2 Vacuum-to-vacuum amplitude 572

12.2.3 Second-order equation of motion 574

12.3 Decoherence by emission of bremsstrahlung 577

12.3.1 Introducing the decoherence functional 577

12.3.2 Physical interpretation 582

12.3.3 Evaluation of the decoherence functional 585

12.3.4 Path integral approach 595

12.4 Decoherence of many-particle states 602

References 605

Index 607