相空间中的量子光学 英文版PDF|Epub|txt|kindle电子书版本网盘下载

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1 What's Quantum Optics?1

1.1 On the Road to Quantum Optics1

1.2 Resonance Fluorescence2

1.2.1 Elastic Peak:Light as a Wave2

1.2.2 Mollow-Three-Peak Spectrum3

1.2.3 Anti-Bunching5

1.3 Squeezing the Fluctuations7

1.3.1 What is a Squeezed State?7

1.3.2 Squeezed States in the Optical Parametric Oscillator9

1.3.3 Oscillatory Photon Statistics12

1.3.4 Interference in Phase Space13

1.4 Jaynes-Cummings-Paul Model14

1.4.1 Single Two-Level Atom plus a Single Mode15

1.4.2 Time Scales15

1.5 Cavity QED16

1.5.1 An Amazing Maser16

1.5.2 Cavity QED in the Optical Domain19

1.6 de Broglie Optics22

1.6.1 Electron and Neutron Optics22

1.6.2 Atom Optics23

1.6.3 Atom Optics in Quantized Light Fields25

1.7 Quantum Motion in Paul Traps26

1.7.1 Analogy to Cavity QED26

1.7.2 Quantum Information Processing26

1.8 Two-Photon Interferometry and More28

1.9 Outline of the Book29

2 Ante35

2.1 Position and Momentum Eigenstates36

2.1.1 Properties of Eigenstates36

2.1.2 Derivative of Wave Function38

2.1.3 Fourier Transform Connects x-and p-Space39

2.2 Energy Eigenstate40

2.2.1 Arbitrary Representation41

2.2.2 Position Representation42

2.3 Density Operator:A Brief Introduction44

2.3.1 A State Vector is not Enough!44

2.3.2 Definition and Properties48

2.3.3 Trace of Operator49

2.3.4 Examples of a Density Operator51

2.4 Time Evolution of Quantum States53

2.4.1 Motion of a Wave Packet54

2.4.2 Time Evolution due to Interaction55

2.4.3 Time Dependent Hamiltonian57

2.4.4 Time Evolution of Density Operator61

3 Wigner Function67

3.1 Jump Start of the Wigner Function68

3.2 Properties of the Wigner Function69

3.2.1 Marginals69

3.2.2 Overlap of Quantum States as Overlap in Phase Space71

3.2.3 Shape of Wigner Function72

3.3 Time Evolution of Wigner Function74

3.3.1 von Neumann Equation in Phase Space74

3.3.2 Quantum Liouville Equation75

3.4 Wigner Function Determined by Phase Space76

3.4.1 Definition of Moyal Function76

3.4.2 Phase Space Equations for Moyal Functions77

3.5 Phase Space Equations for Energy Eigenstates78

3.5.1 Power Expansion in Planck's Constant79

3.5.2 Model Differential Equation81

3.6 Harmonic Oscillator84

3.6.1 Wigner Function as Wave Function85

3.6.2 Phase Space Enforces Energy Quantization86

3.7 Evaluation of Quantum Mechanical Averages87

3.7.1 Operator Ordering88

3.7.2 Examples of Weyl-Wigner Ordering90

4 Quantum States in Phase Space99

4.1 Energy Eigenstate100

4.1.1 Simple Phase Space Representation100

4.1.2 Large-m Limit101

4.1.3 Wigner Function105

4.2 Coherent State108

4.2.1 Definition of a Coherent State109

4.2.2 Energy Distribution110

4.2.3 Time Evolution113

4.3 Squeezed State119

4.3.1 Definition of a Squeezed State121

4.3.2 Energy Distribution:Exact Treatment125

4.3.3 Energy Distribution:Asymptotic Treatment128

4.3.4 Limit Towards Squeezed Vacuum132

4.3.5 Time Evolution135

4.4 Rotated Quadrature States136

4.4.1 Wigner Function of Position and Momentum States137

4.4.2 Position Wave Function of Rotated Quadrature States140

4.4.3 Wigner Function of Rotated Quadrature States142

4.5 Quantum State Reconstruction143

4.5.1 Tomographic Cuts through Wigner Function143

4.5.2 Radon Transformation144

5 Waves à la WKB153

5.1 Probability for Classical Motion153

5.2 Probability Amplitudes for Quantum Motion155

5.2.1 An Educated Guess156

5.2.2 Range of Validity of WKB Wave Function158

5.3 Energy Quantization159

5.3.1 Determining the Phase159

5.3.2 Bohr-Sommerfeld-Kramers Quantization161

5.4 Summary163

5.4.1 Construction of Primitive WKB Wave Function163

5.4.2 Uniform Asymptotic Expansion164

6 WKB and Berry Phase171

6.1 Berry Phase and Adiabatic Approximation172

6.1.1 Adiabatic Theorem172

6.1.2 Analysis of Geometrical Phase174

6.1.3 Geometrical Phase as a Flux in Hilbert Space175

6.2 WKB Wave Functions from Adiabaticity176

6.2.1 Energy Eigenvalue Problem as Propagation Problem177

6.2.2 Dynamical and Geometrical Phase181

6.2.3 WKB Waves Rederived183

6.3 Non-Adiabatic Berry Phase185

6.3.1 Derivation of the Aharonov-Anandan Phase186

6.3.2 Time Evolution in Harmonic Oscillator187

7 Interference in Phase space189

7.1 Outline of the Idea189

7.2 Derivation of Area-of-Overlap Formalism192

7.2.1 Jumps Viewed From Position Space192

7.2.2 Jumps Viewed From Phase Space197

7.3 Application to Franck-Condon Transitions200

7.4 Generalization201

8 Applications of Interference in Phase Space205

8.1 Connection to Interference in Phase Space205

8.2 Energy Eigenstates206

8.3 Coherent State208

8.3.1 Elementary Approach209

8.3.2 Influence of Internal Structure212

8.4 Squeezed State213

8.4.1 Oscillations from Interference in Phase Space213

8.4.2 Giant Oscillations216

8.4.3 Summary218

8.5 The Question of Phase States221

8.5.1 Amplitude and Phase in a Classical Oscillator221

8.5.2 Definition of a Phase State223

8.5.3 Phase Distribution of a Quantum State227

9 Wave Packet Dynamics233

9.1 What are Wave Packets?233

9.2 Fractional and Full Revivals234

9.3 Natural Time Scales237

9.3.1 Hierarchy of Time Scales237

9.3.2 Generic Signal239

9.4 New Representations of the Signal241

9.4.1 The Early Stage of the Evolution241

9.4.2 Intermediate Times244

9.5 Fractional Revivals Made Simple246

9.5.1 Gauss Sums246

9.5.2 Shape Function246

10 Field Quantization255

10.1 Wave Equations for the Potentials256

10.1.1 Derivation of the Wave Equations256

10.1.2 Gauge Invariance of Electrodynamics257

10.1.3 Solution of the Wave Equation260

10.2 Mode Structure in a Box262

10.2.1 Solutions of Helmholtz Equation262

10.2.2 Polarization Vectors from Gauge Condition263

10.2.3 Discreteness of Modes from Boundaries264

10.2.4 Boundary Conditions on the Magnetic Field264

10.2.5 Orthonormality of Mode Functions265

10.3 The Field as a Set of Harmonic Oscillators266

10.3.1 Energy in the Resonator267

10.3.2 Quantization of the Radiation Field269

10.4 The Casimir Effect272

10.4.1 Zero-Point Energy of a Rectangular Resonator272

10.4.2 Zero-Point Energy of Free Space274

10.4.3 Difference of Two Infinite Energies275

10.4.4 Casimir Force:Theory and Experiment276

10.5 Operators of the Vector Potential and Fields278

10.5.1 Vector Potential278

10.5.2 Electric Field Operator280

10.5.3 Magnetic Field Operator281

10.6 Number States of the Radiation Field281

10.6.1 Photons and Anti-Photons282

10.6.2 Multi-Mode Case282

10.6.3 Superposition and Entangled States283

11 Field States291

11.1 Properties of the Quantized Electric Field291

11.1.1 Photon Number States292

11.1.2 Electromagnetic Field Eigenstates293

11.2 Coherent States Revisited295

11.2.1 Eigenvalue Equation295

11.2.2 Coherent State as a Displaced Vacuum297

11.2.3 Photon Statistics of a Coherent State298

11.2.4 Electric Field Distribution of a Coherent State299

11.2.5 Over-completeness of Coherent States301

11.2.6 Expansion into Coherent States303

11.2.7 Electric Field Expectation Values305

11.3 Schr？dinger Cat State306

11.3.1 The Original Cat Paradox306

11.3.2 Definition of the Field Cat State307

11.3.3 Wigner Phase Space Representation307

11.3.4 Photon Statistics310

12 Phase Space Functions321

12.1 There is more than Wigner Phase Space321

12.1.1 Who Needs Phase Space Functions?321

12.1.2 Another Description of Phase Space322

12.2 The Husimi-Kano Q-Function324

12.2.1 Definition of Q-Function324

12.2.2 Q-Functions of Specific Quantum States324

12.3 Averages Using Phase Space Functions330

12.3.1 Heuristic Argument330

12.3.2 Rigorous Treatment333

12.4 The Glauber-Sudarshan P-Distribution337

12.4.1 Definition of P-Distribution337

12.4.2 Connection between Q-and P-Function338

12.4.3 P-Function from Q-Function339

12.4.4 Examples of P-Distributions341

13 Optical Interferometry349

13.1 Beam Splitter350

13.1.1 Classical Treatment350

13.1.2 Symmetric Beam Splitter352

13.1.3 Transition to Quantum Mechanics353

13.1.4 Transformation of Quantum States353

13.1.5 Count Statistics at the Exit Ports356

13.2 Homodyne Detector357

13.2.1 Classical Considerations357

13.2.2 Quantum Treatment358

13.3 Eight-Port Interferometer361

13.3.1 Quantum State of the Output Modes361

13.3.2 Photon Count Statistics363

13.3.3 Simultaneous Measurement and EPR365

13.3.4 Q-Function Measurement367

13.4 Measured Phase Operators370

13.4.1 Measurement of Classical Trigonometry370

13.4.2 Measurement of Quantum Trigonometry372

13.4.3 Two-Mode Phase Operators374

14 Atom-Field Interaction381

14.1 How to Construct the Interaction?382

14.2 Vector Potential-Momentum Coupling382

14.2.1 Gauge Principle Determines Minimal Coupling383

14.2.2 Interaction of an Atom with a Field386

14.3 Dipole Approximation389

14.3.1 Expansion of Vector Potential389

14.3.2 ？·？-Interaction390

14.3.3 Various Forms of the ？·？ Interaction390

14.3.4 Higher Order Corrections392

14.4 Electric Field-Dipole Interaction393

14.4.1 Dipole Approximation393

14.4.2 R？ntgen Hamiltonians and Others393

14.5 Subsystems,Interaction and Entanglement395

14.6 Equivalence of ？·？ and ？·？396

14.6.1 Classical Transformation of Lagrangian397

14.6.2 Quantum Mechanical Treatment399

14.6.3 Matrix elements of ？·？ and ？·？399

14.7 Equivalence of Hamiltonians H(1) and ？(1)400

14.8 Simple Model for Atom-Field Interaction402

14.8.1 Derivation of the Hamiltonian402

14.8.2 Rotating-Wave Approximation406

15 Jaynes-Cummings-Paul Model:Dynamics413

15.1 Resonant Jaynes-Cummings-Paul Model413

15.1.1 Time Evolution Operator Using Operator Algebra414

15.1.2 Interpretation of Time Evolution Operator416

15.1.3 State Vector of Combined System418

15.1.4 Dynamics Represented in State Space418

15.2 Role of Detuning420

15.2.1 Atomic and Field States420

15.2.2 Rabi Equations422

15.3 Solution of Rabi Equations423

15.3.1 Laplace Transformation424

15.3.2 Inverse Laplace Transformation425

15.4 Discussion of Solution426

15.4.1 General Considerations427

15.4.2 Resonant Case427

15.4.3 Far Off-Resonant Case429

16 State Preparation and Entanglement435

16.1 Measurements on Entangled Systems435

16.1.1 How to Get Probabilities436

16.1.2 State of the Subsystem after a Measurement439

16.1.3 Experimental Setup440

16.2 Collapse,Revivals and Fractional Revivals444

16.2.1 Inversion as Tool for Measuring Internal Dynamics444

16.2.2 Experiments on Collapse and Revivals447

16.3 Quantum State Preparation451

16.3.1 State Preparation with a Dispersive Interaction451

16.3.2 Generation of Schr？dinger Cats454

16.4 Quantum State Engineering454

16.4.1 Outline of the Method454

16.4.2 Inverse Problem458

16.4.3 Example:Preparation of a Phase State461

17 Paul Trap473

17.1 Basics of Trapping Ions474

17.1.1 No Static Trapping in Three Dimensions474

17.1.2 Dynamical Trapping475

17.2 Laser Cooling479

17.3 Motion of an Ion in a Paul Trap480

17.3.1 Reduction to Classical Problem481

17.3.2 Motion as a Sequence of Squeezing and Rotations483

17.3.3 Dynamics in Wigner Phase Space486

17.3.4 Floquet Solution490

17.4 Model Hamiltonian494

17.4.1 Transformation to Interaction Picture495

17.4.2 Lamb-Dicke Regime496

17.4.3 Multi-Phonon Jaynes-Cummings-Paul Model498

17.5 Effective Potential Approximation500

18 Damping and Amplification507

18.1 Damping and Amplification of a Cavity Field508

18.2 Density Operator of a Subsystem509

18.2.1 Coarse-Grained Equation of Motion509

18.2.2 Time Independent Hamiltonian511

18.3 Reservoir of Two-Level Atoms511

18.3.1 Approximate Treatment512

18.3.2 Density Operator in Number Representation514

18.3.3 Exact Master Equation519

18.3.4 Summary522

18.4 One-Atom Maser522

18.4.1 Density Operator Equation523

18.4.2 Equation of Motion for the Photon Statistics524

18.4.3 Phase Diffusion529

18.5 Atom-Reservoir Interaction532

18.5.1 Model and Equation of Motion532

18.5.2 First Order Contribution533

18.5.3 Bloch Equations535

18.5.4 Second Order Contribution537

18.5.5 Lamb Shift539

18.5.6 Weisskopf-Wigner Decay540

19 Atom Optics in Quantized Light Fields549

19.1 Formulation of Problem549

19.1.1 Dynamics549

19.1.2 Time Evolution of Probability Amplitudes552

19.2 Reduction to One-Dimensional Scattering554

19.2.1 Slowly Varying Approximation554

19.2.2 From Two Dimensions to One555

19.2.3 State Vector556

19.3 Raman-Nath Approximation557

19.3.1 Heuristic Arguments557

19.3.2 Probability Amplitudes558

19.4 Deflection of Atoms559

19.4.1 Measurement Schemes and Scattering Conditions559

19.4.2 Kapitza-Dirac Regime562

19.4.3 Kapitza-Dirac Scattering with a Mask568

19.5 Interference in Phase Space571

19.5.1 How to Represent the Quantum State?572

19.5.2 Area of Overlap572

19.5.3 Expression for Probability Amplitude573

20 Wigner Functions in Atom Optics579

20.1 Model579

20.2 Equation of Motion for Wigner Functions581

20.3 Motion in Phase Space582

20.3.1 Harmonic Approximation583

20.3.2 Motion of the Atom in the Cavity583

20.3.3 Motion of the Atom outside the Cavity585

20.3.4 Snap Shots of the Wigner Function586

20.4 Quantum Lens587

20.4.1 Distributions of Atoms in Space587

20.4.2 Focal Length and Deflection Angle589

20.5 Photon and Momentum Statistics590

20.6 Heuristic Approach592

20.6.1 Focal Length592

20.6.2 Focal Size594

A Energy Wave Functions of Harmonic Oscillator597

A.1 Polynomial Ansatz597

A.2 Asymptotic Behavior599

A.2.1 Energy Wave Function as a Contour Integral600

A.2.2 Evaluation of the Integral Im600

A.2.3 Asymptotic Limit of fm603

A.2.4 Bohr's Correspondence Principle603

B Time Dependent Operators605

B.1 Caution when Differentiating Operators605

B.2 Time Ordering606

B.2.1 Product of Two Terms607

B.2.2 Product of n Terms608

C Süβmann Measure611

C.1 Why Other Measures Fail611

C.2 One Way out of the Problem612

C.3 Generalization to Higher Dimensions613

D Phase Space Equations615

D.1 Formulation of the Problem615

D.2 Fourier Transform of Matrix Elements616

D.3 Kinetic Energy Terms617

D.4 Potential Energy Terms619

D.5 Summary620

E Airy Function621

E.1 Definition and Differential Equation621

E.2 Asymptotic Expansion622

E.2.1 Oscillatory Regime623

E.2.2 Decaying Regime624

E.2.3 Stokes Phenomenon625

F Radial Equation629

G Asymptotics of a Poissonian633

H Toolbox for Integrals635

H.1 Method of Stationary Phase635

H.1.1 One-Dimensional Integrals635

H.1.2 Multi-Dimensional Integrals637

H.2 Cornu Spiral639

I Area of Overlap643

I.1 Diamond Transformed into a Rectangle643

I.2 Area of Diamond644

I.3 Area of Overlap as Probability646

J P-Distributions649

J.1 Thermal State649

J.2 Photon Number State650

J.3 Squeezed State651

K Homodyne Kernel655

K.1 Explicit Evaluation of Kernel655

K.2 Strong Local Oscillator Limit656

L Beyond the Dipole Approximation659

L.1 First Order Taylor Expansion659

L.1.1 Expansion of the Hamiltonian659

L.1.2 Extension to Operators661

L.2 Classical Gauge Transformation661

L.2.1 Lagrangian with Center-of-Mass Motion662

L.2.2 Complete Time Derivative663

L.2.3 Hamiltonian Including Center-of-Mass Motion663

L.3 Quantum Mechanical Gauge Transformation664

L.3.1 Gauge Potential664

L.3.2 Schr？dinger equation for ？667

M Effctive Hamiltonian669

N Oscillator Reservoir671

N.1 Second Order Contribution671

N.1.1 Evaluation of Double Commutator671

N.1.2 Trace over Reservoir673

N.2 Symmetry Relations in Trace673

N.2.1 Complex Conjugates674

N.2.2 Commutator Between Field Operators674

N.3 Master Equation675

N.4 Explicit Expressions for Г，β and ？676

N.5 Integration over Time677

O Bessel Functions679

O.1 Definition679

O.2 Asymptotic Expansion680

P Square Root of δ683

Q Further Reading685

Index688