1 What's Quantum Optics? 1
1.1 On the Road to Quantum Optics 1
1.2 Resonance Fluorescence 2
1.2.1 Elastic Peak:Light as a Wave 2
1.2.2 Mollow-Three-Peak Spectrum 3
1.2.3 Anti-Bunching 5
1.3 Squeezing the Fluctuations 7
1.3.1 What is a Squeezed State? 7
1.3.2 Squeezed States in the Optical Parametric Oscillator 9
1.3.3 Oscillatory Photon Statistics 12
1.3.4 Interference in Phase Space 13
1.4 Jaynes-Cummings-Paul Model 14
1.4.1 Single Two-Level Atom plus a Single Mode 15
1.4.2 Time Scales 15
1.5 Cavity QED 16
1.5.1 An Amazing Maser 16
1.5.2 Cavity QED in the Optical Domain 19
1.6 de Broglie Optics 22
1.6.1 Electron and Neutron Optics 22
1.6.2 Atom Optics 23
1.6.3 Atom Optics in Quantized Light Fields 25
1.7 Quantum Motion in Paul Traps 26
1.7.1 Analogy to Cavity QED 26
1.7.2 Quantum Information Processing 26
1.8 Two-Photon Interferometry and More 28
1.9 Outline of the Book 29
2 Ante 35
2.1 Position and Momentum Eigenstates 36
2.1.1 Properties of Eigenstates 36
2.1.2 Derivative of Wave Function 38
2.1.3 Fourier Transform Connects x-and p-Space 39
2.2 Energy Eigenstate 40
2.2.1 Arbitrary Representation 41
2.2.2 Position Representation 42
2.3 Density Operator:A Brief Introduction 44
2.3.1 A State Vector is not Enough! 44
2.3.2 Definition and Properties 48
2.3.3 Trace of Operator 49
2.3.4 Examples of a Density Operator 51
2.4 Time Evolution of Quantum States 53
2.4.1 Motion of a Wave Packet 54
2.4.2 Time Evolution due to Interaction 55
2.4.3 Time Dependent Hamiltonian 57
2.4.4 Time Evolution of Density Operator 61
3 Wigner Function 67
3.1 Jump Start of the Wigner Function 68
3.2 Properties of the Wigner Function 69
3.2.1 Marginals 69
3.2.2 Overlap of Quantum States as Overlap in Phase Space 71
3.2.3 Shape of Wigner Function 72
3.3 Time Evolution of Wigner Function 74
3.3.1 von Neumann Equation in Phase Space 74
3.3.2 Quantum Liouville Equation 75
3.4 Wigner Function Determined by Phase Space 76
3.4.1 Definition of Moyal Function 76
3.4.2 Phase Space Equations for Moyal Functions 77
3.5 Phase Space Equations for Energy Eigenstates 78
3.5.1 Power Expansion in Planck's Constant 79
3.5.2 Model Differential Equation 81
3.6 Harmonic Oscillator 84
3.6.1 Wigner Function as Wave Function 85
3.6.2 Phase Space Enforces Energy Quantization 86
3.7 Evaluation of Quantum Mechanical Averages 87
3.7.1 Operator Ordering 88
3.7.2 Examples of Weyl-Wigner Ordering 90
4 Quantum States in Phase Space 99
4.1 Energy Eigenstate 100
4.1.1 Simple Phase Space Representation 100
4.1.2 Large-m Limit 101
4.1.3 Wigner Function 105
4.2 Coherent State 108
4.2.1 Definition of a Coherent State 109
4.2.2 Energy Distribution 110
4.2.3 Time Evolution 113
4.3 Squeezed State 119
4.3.1 Definition of a Squeezed State 121
4.3.2 Energy Distribution:Exact Treatment 125
4.3.3 Energy Distribution:Asymptotic Treatment 128
4.3.4 Limit Towards Squeezed Vacuum 132
4.3.5 Time Evolution 135
4.4 Rotated Quadrature States 136
4.4.1 Wigner Function of Position and Momentum States 137
4.4.2 Position Wave Function of Rotated Quadrature States 140
4.4.3 Wigner Function of Rotated Quadrature States 142
4.5 Quantum State Reconstruction 143
4.5.1 Tomographic Cuts through Wigner Function 143
4.5.2 Radon Transformation 144
5 Waves à la WKB 153
5.1 Probability for Classical Motion 153
5.2 Probability Amplitudes for Quantum Motion 155
5.2.1 An Educated Guess 156
5.2.2 Range of Validity of WKB Wave Function 158
5.3 Energy Quantization 159
5.3.1 Determining the Phase 159
5.3.2 Bohr-Sommerfeld-Kramers Quantization 161
5.4 Summary 163
5.4.1 Construction of Primitive WKB Wave Function 163
5.4.2 Uniform Asymptotic Expansion 164
6 WKB and Berry Phase 171
6.1 Berry Phase and Adiabatic Approximation 172
6.1.1 Adiabatic Theorem 172
6.1.2 Analysis of Geometrical Phase 174
6.1.3 Geometrical Phase as a Flux in Hilbert Space 175
6.2 WKB Wave Functions from Adiabaticity 176
6.2.1 Energy Eigenvalue Problem as Propagation Problem 177
6.2.2 Dynamical and Geometrical Phase 181
6.2.3 WKB Waves Rederived 183
6.3 Non-Adiabatic Berry Phase 185
6.3.1 Derivation of the Aharonov-Anandan Phase 186
6.3.2 Time Evolution in Harmonic Oscillator 187
7 Interference in Phase space 189
7.1 Outline of the Idea 189
7.2 Derivation of Area-of-Overlap Formalism 192
7.2.1 Jumps Viewed From Position Space 192
7.2.2 Jumps Viewed From Phase Space 197
7.3 Application to Franck-Condon Transitions 200
7.4 Generalization 201
8 Applications of Interference in Phase Space 205
8.1 Connection to Interference in Phase Space 205
8.2 Energy Eigenstates 206
8.3 Coherent State 208
8.3.1 Elementary Approach 209
8.3.2 Influence of Internal Structure 212
8.4 Squeezed State 213
8.4.1 Oscillations from Interference in Phase Space 213
8.4.2 Giant Oscillations 216
8.4.3 Summary 218
8.5 The Question of Phase States 221
8.5.1 Amplitude and Phase in a Classical Oscillator 221
8.5.2 Definition of a Phase State 223
8.5.3 Phase Distribution of a Quantum State 227
9 Wave Packet Dynamics 233
9.1 What are Wave Packets? 233
9.2 Fractional and Full Revivals 234
9.3 Natural Time Scales 237
9.3.1 Hierarchy of Time Scales 237
9.3.2 Generic Signal 239
9.4 New Representations of the Signal 241
9.4.1 The Early Stage of the Evolution 241
9.4.2 Intermediate Times 244
9.5 Fractional Revivals Made Simple 246
9.5.1 Gauss Sums 246
9.5.2 Shape Function 246
10 Field Quantization 255
10.1 Wave Equations for the Potentials 256
10.1.1 Derivation of the Wave Equations 256
10.1.2 Gauge Invariance of Electrodynamics 257
10.1.3 Solution of the Wave Equation 260
10.2 Mode Structure in a Box 262
10.2.1 Solutions of Helmholtz Equation 262
10.2.2 Polarization Vectors from Gauge Condition 263
10.2.3 Discreteness of Modes from Boundaries 264
10.2.4 Boundary Conditions on the Magnetic Field 264
10.2.5 Orthonormality of Mode Functions 265
10.3 The Field as a Set of Harmonic Oscillators 266
10.3.1 Energy in the Resonator 267
10.3.2 Quantization of the Radiation Field 269
10.4 The Casimir Effect 272
10.4.1 Zero-Point Energy of a Rectangular Resonator 272
10.4.2 Zero-Point Energy of Free Space 274
10.4.3 Difference of Two Infinite Energies 275
10.4.4 Casimir Force:Theory and Experiment 276
10.5 Operators of the Vector Potential and Fields 278
10.5.1 Vector Potential 278
10.5.2 Electric Field Operator 280
10.5.3 Magnetic Field Operator 281
10.6 Number States of the Radiation Field 281
10.6.1 Photons and Anti-Photons 282
10.6.2 Multi-Mode Case 282
10.6.3 Superposition and Entangled States 283
11 Field States 291
11.1 Properties of the Quantized Electric Field 291
11.1.1 Photon Number States 292
11.1.2 Electromagnetic Field Eigenstates 293
11.2 Coherent States Revisited 295
11.2.1 Eigenvalue Equation 295
11.2.2 Coherent State as a Displaced Vacuum 297
11.2.3 Photon Statistics of a Coherent State 298
11.2.4 Electric Field Distribution of a Coherent State 299
11.2.5 Over-completeness of Coherent States 301
11.2.6 Expansion into Coherent States 303
11.2.7 Electric Field Expectation Values 305
11.3 Schr？dinger Cat State 306
11.3.1 The Original Cat Paradox 306
11.3.2 Definition of the Field Cat State 307
11.3.3 Wigner Phase Space Representation 307
11.3.4 Photon Statistics 310
12 Phase Space Functions 321
12.1 There is more than Wigner Phase Space 321
12.1.1 Who Needs Phase Space Functions? 321
12.1.2 Another Description of Phase Space 322
12.2 The Husimi-Kano Q-Function 324
12.2.1 Definition of Q-Function 324
12.2.2 Q-Functions of Specific Quantum States 324
12.3 Averages Using Phase Space Functions 330
12.3.1 Heuristic Argument 330
12.3.2 Rigorous Treatment 333
12.4 The Glauber-Sudarshan P-Distribution 337
12.4.1 Definition of P-Distribution 337
12.4.2 Connection between Q-and P-Function 338
12.4.3 P-Function from Q-Function 339
12.4.4 Examples of P-Distributions 341
13 Optical Interferometry 349
13.1 Beam Splitter 350
13.1.1 Classical Treatment 350
13.1.2 Symmetric Beam Splitter 352
13.1.3 Transition to Quantum Mechanics 353
13.1.4 Transformation of Quantum States 353
13.1.5 Count Statistics at the Exit Ports 356
13.2 Homodyne Detector 357
13.2.1 Classical Considerations 357
13.2.2 Quantum Treatment 358
13.3 Eight-Port Interferometer 361
13.3.1 Quantum State of the Output Modes 361
13.3.2 Photon Count Statistics 363
13.3.3 Simultaneous Measurement and EPR 365
13.3.4 Q-Function Measurement 367
13.4 Measured Phase Operators 370
13.4.1 Measurement of Classical Trigonometry 370
13.4.2 Measurement of Quantum Trigonometry 372
13.4.3 Two-Mode Phase Operators 374
14 Atom-Field Interaction 381
14.1 How to Construct the Interaction? 382
14.2 Vector Potential-Momentum Coupling 382
14.2.1 Gauge Principle Determines Minimal Coupling 383
14.2.2 Interaction of an Atom with a Field 386
14.3 Dipole Approximation 389
14.3.1 Expansion of Vector Potential 389
14.3.2 ？·？-Interaction 390
14.3.3 Various Forms of the ？·？ Interaction 390
14.3.4 Higher Order Corrections 392
14.4 Electric Field-Dipole Interaction 393
14.4.1 Dipole Approximation 393
14.4.2 R？ntgen Hamiltonians and Others 393
14.5 Subsystems,Interaction and Entanglement 395
14.6 Equivalence of ？·？ and ？·？ 396
14.6.1 Classical Transformation of Lagrangian 397
14.6.2 Quantum Mechanical Treatment 399
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 Interaction 402
14.8.1 Derivation of the Hamiltonian 402
14.8.2 Rotating-Wave Approximation 406
15 Jaynes-Cummings-Paul Model:Dynamics 413
15.1 Resonant Jaynes-Cummings-Paul Model 413
15.1.1 Time Evolution Operator Using Operator Algebra 414
15.1.2 Interpretation of Time Evolution Operator 416
15.1.3 State Vector of Combined System 418
15.1.4 Dynamics Represented in State Space 418
15.2 Role of Detuning 420
15.2.1 Atomic and Field States 420
15.2.2 Rabi Equations 422
15.3 Solution of Rabi Equations 423
15.3.1 Laplace Transformation 424
15.3.2 Inverse Laplace Transformation 425
15.4 Discussion of Solution 426
15.4.1 General Considerations 427
15.4.2 Resonant Case 427
15.4.3 Far Off-Resonant Case 429
16 State Preparation and Entanglement 435
16.1 Measurements on Entangled Systems 435
16.1.1 How to Get Probabilities 436
16.1.2 State of the Subsystem after a Measurement 439
16.1.3 Experimental Setup 440
16.2 Collapse,Revivals and Fractional Revivals 444
16.2.1 Inversion as Tool for Measuring Internal Dynamics 444
16.2.2 Experiments on Collapse and Revivals 447
16.3 Quantum State Preparation 451
16.3.1 State Preparation with a Dispersive Interaction 451
16.3.2 Generation of Schr？dinger Cats 454
16.4 Quantum State Engineering 454
16.4.1 Outline of the Method 454
16.4.2 Inverse Problem 458
16.4.3 Example:Preparation of a Phase State 461
17 Paul Trap 473
17.1 Basics of Trapping Ions 474
17.1.1 No Static Trapping in Three Dimensions 474
17.1.2 Dynamical Trapping 475
17.2 Laser Cooling 479
17.3 Motion of an Ion in a Paul Trap 480
17.3.1 Reduction to Classical Problem 481
17.3.2 Motion as a Sequence of Squeezing and Rotations 483
17.3.3 Dynamics in Wigner Phase Space 486
17.3.4 Floquet Solution 490
17.4 Model Hamiltonian 494
17.4.1 Transformation to Interaction Picture 495
17.4.2 Lamb-Dicke Regime 496
17.4.3 Multi-Phonon Jaynes-Cummings-Paul Model 498
17.5 Effective Potential Approximation 500
18 Damping and Amplification 507
18.1 Damping and Amplification of a Cavity Field 508
18.2 Density Operator of a Subsystem 509
18.2.1 Coarse-Grained Equation of Motion 509
18.2.2 Time Independent Hamiltonian 511
18.3 Reservoir of Two-Level Atoms 511
18.3.1 Approximate Treatment 512
18.3.2 Density Operator in Number Representation 514
18.3.3 Exact Master Equation 519
18.3.4 Summary 522
18.4 One-Atom Maser 522
18.4.1 Density Operator Equation 523
18.4.2 Equation of Motion for the Photon Statistics 524
18.4.3 Phase Diffusion 529
18.5 Atom-Reservoir Interaction 532
18.5.1 Model and Equation of Motion 532
18.5.2 First Order Contribution 533
18.5.3 Bloch Equations 535
18.5.4 Second Order Contribution 537
18.5.5 Lamb Shift 539
18.5.6 Weisskopf-Wigner Decay 540
19 Atom Optics in Quantized Light Fields 549
19.1 Formulation of Problem 549
19.1.1 Dynamics 549
19.1.2 Time Evolution of Probability Amplitudes 552
19.2 Reduction to One-Dimensional Scattering 554
19.2.1 Slowly Varying Approximation 554
19.2.2 From Two Dimensions to One 555
19.2.3 State Vector 556
19.3 Raman-Nath Approximation 557
19.3.1 Heuristic Arguments 557
19.3.2 Probability Amplitudes 558
19.4 Deflection of Atoms 559
19.4.1 Measurement Schemes and Scattering Conditions 559
19.4.2 Kapitza-Dirac Regime 562
19.4.3 Kapitza-Dirac Scattering with a Mask 568
19.5 Interference in Phase Space 571
19.5.1 How to Represent the Quantum State? 572
19.5.2 Area of Overlap 572
19.5.3 Expression for Probability Amplitude 573
20 Wigner Functions in Atom Optics 579
20.1 Model 579
20.2 Equation of Motion for Wigner Functions 581
20.3 Motion in Phase Space 582
20.3.1 Harmonic Approximation 583
20.3.2 Motion of the Atom in the Cavity 583
20.3.3 Motion of the Atom outside the Cavity 585
20.3.4 Snap Shots of the Wigner Function 586
20.4 Quantum Lens 587
20.4.1 Distributions of Atoms in Space 587
20.4.2 Focal Length and Deflection Angle 589
20.5 Photon and Momentum Statistics 590
20.6 Heuristic Approach 592
20.6.1 Focal Length 592
20.6.2 Focal Size 594
A Energy Wave Functions of Harmonic Oscillator 597
A.1 Polynomial Ansatz 597
A.2 Asymptotic Behavior 599
A.2.1 Energy Wave Function as a Contour Integral 600
A.2.2 Evaluation of the Integral Im 600
A.2.3 Asymptotic Limit of fm 603
A.2.4 Bohr's Correspondence Principle 603
B Time Dependent Operators 605
B.1 Caution when Differentiating Operators 605
B.2 Time Ordering 606
B.2.1 Product of Two Terms 607
B.2.2 Product of n Terms 608
C Süβmann Measure 611
C.1 Why Other Measures Fail 611
C.2 One Way out of the Problem 612
C.3 Generalization to Higher Dimensions 613
D Phase Space Equations 615
D.1 Formulation of the Problem 615
D.2 Fourier Transform of Matrix Elements 616
D.3 Kinetic Energy Terms 617
D.4 Potential Energy Terms 619
D.5 Summary 620
E Airy Function 621
E.1 Definition and Differential Equation 621
E.2 Asymptotic Expansion 622
E.2.1 Oscillatory Regime 623
E.2.2 Decaying Regime 624
E.2.3 Stokes Phenomenon 625
F Radial Equation 629
G Asymptotics of a Poissonian 633
H Toolbox for Integrals 635
H.1 Method of Stationary Phase 635
H.1.1 One-Dimensional Integrals 635
H.1.2 Multi-Dimensional Integrals 637
H.2 Cornu Spiral 639
I Area of Overlap 643
I.1 Diamond Transformed into a Rectangle 643
I.2 Area of Diamond 644
I.3 Area of Overlap as Probability 646
J P-Distributions 649
J.1 Thermal State 649
J.2 Photon Number State 650
J.3 Squeezed State 651
K Homodyne Kernel 655
K.1 Explicit Evaluation of Kernel 655
K.2 Strong Local Oscillator Limit 656
L Beyond the Dipole Approximation 659
L.1 First Order Taylor Expansion 659
L.1.1 Expansion of the Hamiltonian 659
L.1.2 Extension to Operators 661
L.2 Classical Gauge Transformation 661
L.2.1 Lagrangian with Center-of-Mass Motion 662
L.2.2 Complete Time Derivative 663
L.2.3 Hamiltonian Including Center-of-Mass Motion 663
L.3 Quantum Mechanical Gauge Transformation 664
L.3.1 Gauge Potential 664
L.3.2 Schr？dinger equation for ？ 667
M Effctive Hamiltonian 669
N Oscillator Reservoir 671
N.1 Second Order Contribution 671
N.1.1 Evaluation of Double Commutator 671
N.1.2 Trace over Reservoir 673
N.2 Symmetry Relations in Trace 673
N.2.1 Complex Conjugates 674
N.2.2 Commutator Between Field Operators 674
N.3 Master Equation 675
N.4 Explicit Expressions for Г，β and ？ 676
N.5 Integration over Time 677
O Bessel Functions 679
O.1 Definition 679
O.2 Asymptotic Expansion 680
P Square Root of δ 683
Q Further Reading 685
Index 688