1 WhatS Quantum Optics? 1.1 On the Road to Quantum Optics 1.2 Resonance Fluorescence 1.2.1 Elastic Peak：Light as a Wave 1.2.2 Mollow.Three-Peak Spectrum 1.2.3 Anti-Bunching 1.3 Squeezing the Fluctuations 1.3.1 What iS a Squeezed State7 1.3.2 Squeezed States in the Optical Parametric Oscillator 1.3.3 Oscillatory Photon Statistics 1.3.4 Interference in Phase Space 1.4 Jaynes.Cummings-Paul Model 1.4.1 Single Two-Level Atom plus a Single Mode 1.4.2 Time Scales 1.5 Cavity QED 1.5.1 An Amazing Muser 1.5.2 Cavity QED in the Optical Domain 1.6 de Broglie Optics 1.6.1 Electron and Neutron Optics 1.6.2 Atom Optics 1.6.3 Atom Optics in Quantized Light Fields 1.7 Quantum Motion in Paul Traps 1.7.1 Analogy to Cavity QED 1.7.2 Quantum Information Processing 1.8 Two-Photon Interferometry and More 1.9 Outline of the Book 2 Ante 2.1 Position and Momentum Eigenstates 2.1.1 Properties of Eigenstates 2.1.2 Derivative of Wlave Function 2.1.3 Fourier Transform Connects x-and p-Space 2.2 Energy Eigenstate 2.2.1 Arbitrary ReDresentation 2.2.2 Position Representation 2.3 Density Operator：A Brief Introduction 2.3.1 A State Vector is not Enough！ 2.3.2 Definition and Properties 2.3.3 Trace of Operator 2.3.4 Examples of a Density Operator 2.4 Time Evolution of Quantum States 2.4.1 Motion of a Wave Packet 2.4.2 Time Evolution due to Interaction 2.4.3 Time Dependent Hamiltonian 2.4.4 Time Evolution of Density Operator 3 Wigner Function 3.1 Jump Start of the Wigner Function 3.2 Properties of the Wigner Function 3.2.1 Marginals 3.2.2 Overlap of Quantum States as Overlap in Phase SDace 3.2.3 Shape of Wigner Function 3.3 Time Evolution of Wigner Function 3.3.1 von Neumann Equation in Phase Space 3.3.2 Quantum Liouville Equation 3.4 Wigner Function Determined by Phase Space 3.4.1 Definition of Moyal Function 3.4.2 Phase Space Equations for Moyal Functions 3.5 Phase Space Equations for Energy Eigenstates 3.5.1 Power Expansion in PlanckS Constant 3.5.2 Model Differential Equation 3.6 Harmonic Oscillator 3.6.1 Wigner Function as Wave Function 3.6.2 Phase Space Enforces Energy Quantization 3.7 Evaluation of Quantum Mechanical Averages 3.7.1 Operator Ordering 3.7.2 Examples of Weyl-Wigner Ordering 4 Quantum States in Phase Space 4.1 Energy Eigenstate 4.1.1 Simple Phase Space Representation 4.1.2 Large-m Limit 4.1.3 Wigner Function 4.2 Coherent State 4.2.1 Definition of a Coherent State 4.2.2 Energy Distribution 4.2.3 Time Evolution 4.3 Squeezed State 4.3.1 Definition of a Squeezed State 4.3.2 Energy Distribution：Exact Treatment 4.3.3 Energy Distribution：Asymptotic Treatment 4.3.4 Limit Towards Squeezed Vacuum 4.3.5 Time Evolution 4.4 R0tated Quadrature States 4.4.1 Wigner Function of Position and Momentum States 4.4.2 Position wave Function of Rotated Quadrature States 4.4.3 Wigner Function of Rotated Quadrature States 4.5 Quantum State Reconstruction 4.5.1 Tomographic Cuts through Wigner Function 4.5.2 Radon Transformation 5 Waves A la WKB 5.1 Probability for Classical Motion 5.2 Probability Amplitudes for Quantum Motion 5.2.1 An Educated Guess 5.2.2 Range of Validity of WKB Wave Function 5.3 Energy Quantization 5.3.1 Determining the Phase 5.3.2 Bohr.Sommerfeld.Kramers Quantization 5.4 Summary 5.4.1 Construction of Primitive WKB Wave Function 5.4.2 Uniform Asymptotic Expansion 6 WKB and Berry Phase 6.1 Berry Phase and Adiabatic Approximation 6.1.1 Adiabatic Theorem 6.1.2 Analysis of Geometrical Phase 6.1.3 Geometrical Phase as a Flux in Hilbert Space 6.2 WKB Wave FUnctions from Adiabaticity 6.2.1 Energy Eigenvalue Problem as Propagation Problem 6.2.2 Dynamical and Geometrical Phase 6.2.3 WKB Waves Rederived 6.3 Non-Adiabatic Berry Phase 6.3.1 Derivation of the Aharonov-Anandan Phase 6.3.2 Time Evolution in Harmonic Oscillator 7 Interference in Phase space 7.1 0utline of the Idea 7.2 Derivation of Area.of-Overlap Formalism 7.2.1 Jumps Viewed From Position Space 7.2.2 Jumps Viewed From Phase Space 7.3 Application to Franck-Condon Transitions 7.4 Generalization 8 Applications of Interference in Phase Space 8.1 Connection to Interference in Phase Space 8.2 Energy Eigenstates 8.3 Coherent State 8.3.1 Elementary Approach 8.3.2 Influence of Internal Structure 8.4 Squeezed State. 8.4.1 Oscillations from Interference in Phase Space 8.4.2 Giant Oscillations 8.4.3 Summary 8.5 The Question of Phase States 8.5.1 Amplitude and Phase in a Classical Oscillator 8.5.2 Definition of a Phase State 8.5.3 Phase Distribution of a Quantum State 9 Wave Packet Dynamics 9.1 What are Wave Packets7 9.2 Fractional and Full Revivals 9.3 Natural Time Scales 9.3.1 Hierarchy of Time Scales 9.3.2 Generic Signal 9.4 New Representations of the Signal 9.4.1 The Early Stage of the Evolution 9.4.2 Intermediate Times 9.5 Fractional Revivals Made Simple 9.5.1 Gauss Sums 9.5.2 Shape Function 10 Field Quantization 10.1 Wave Equations for the Potentials 10.1.1 Derivation of the Wavee Equations 10.1.2 Gauge Invariance of Electrodynamics 10.1.3 Solution of the Wlave Equation 10.2 Mode Structure in a Box 10.2.1 Solutions of Helmholtz Equation 10.2.2 Polarization Vectors from Gauge Condition 10.2.3 Discreteness of Modes from Boundaries 10.2.4 Boundary Conditions on the Magnetic Field 10.2.5 Orthonormality of Mode Functions 10.3 The Field as a Set of Harmonic Oscillators 10.3.1 Energy in the Resonator 10.3.2 Quantization of the Radiation Field 10.4 The Casimir Efiect 10.4.1 Zero-Point Energy of a Rectangular Resonator 10.4.2 Zero.Point Energy of Free Space 10.4.3 Difierence of Two Infinite Energies 10.4.4 Casimir Force：Theory and Experiment 10.5 Operators of the Vector Potential and Fields 10.5.1 Vector Potential 10.5.2 Electric Field Operator 10.5.3 Magnetic Field Operator 10.6 Number States of the Radiation Field 11 Field States 11.1 Properties of the Quantized Electric Field 11.2 Coherent States Revisited 11.3 SchrSdinger Cat State 12 Phase Space Functions 12.1 There is more 12.2 The Husimi-Kano Q-Function 12.3 Averages Using Phase Space Functions 12.4 The Glauber-Sudarshan P-Distribution 13 Optical Interferometry 13.1 Beam Splitter 13.2 Homodyne Detector 13.3 Eight-Port Interferometer 13.4 Measured Phase Operators 14 Atom-Field Interaction 14.1 How to Construct the Interaction? 14.2 Vector Potential-Momentum Coupling 14.3 Dipole Approximation 14.4 Electric Field-Dipole Interaction 14.5 Subsystems， Interaction and Entanglement 14.6 Equivalence of 14.7 Equivalence of Hamiltonians 14.8 Simple Model for Atom-Field Interaction 15 Jaynes-Cummings-Paul Model： Dynamics 15.1 Resonant Jaynes-Cummings-Paul Model 15.2 Role of Detuning 15.3 Solution of Rabi Equations 15.4 Discussion of Solution 16 State Preparation and Entanglement 16.1 Measurements on Entangled Systems 16.2 Collapse， Revivals and Fractional Revivals 16.3 Quantum State Preparation 16.4 Quantum State Engineering 17 Paul Trap 17.1 Basics of Trapping Ions 17.2 Laser Cooling 17.3 Motion of an Ion in a Paul Trap 17.4 Model Hamiltonian 17.5 Effective Potential Approximation 18 Damping and Amplification 18.1 Damping and Amplification of a Cavity Field 18.2 Density Operator of a Subsystem 18.3 Reservoir of Two-Level Atoms 18.4 One-Atom Maser 18.5 Atom-Reservoir Interaction 19 Atom Optics in Quantized Light Fields 19.1 Formulation of Problem 19.2 Reduction to One-Dimensional Scattering 19.3 Raman-Nath Approximation 19.4 Deflection of Atoms 19.5 Interference in Phase Space 20 Wigner Functions in Atom Optics 20.1 Model 20.2 Equation of Motion for Wigner Functions 20.3 Motion in Phase Space 20.4 Quantum Lens 20.5 Photon and Momentum Statistics 20.6 Heuristic Approach A Energy Wave Functions of Harmonic Oscillator A.1 Polynomial Ansatz A.2 Asymptotic Behavior B Time Dependent Operators B.1 Caution when Differentiating Operators B.2 Time Ordering C SiiBmann Measure C.1 Why Other Measures Fail C.2 One Way out of the Problem C.3 Generalization to Higher Dimensions D Phase Space Equations D.1 Formulation of the Problem D.2 Fourier Transform of Matrix Elements D.3 Kinetic Energy Terms D.4 Potential Energy Terms D.5 Summary E Airy Function E. 1 Definition and Differential Equation E.2 Asymptotic Expansion F Radial Equation O Asymptotics of a Poissonian H Toolbox for Integrals H.1 Method of Stationary Phase H.2 Cornu Spiral Area of Overlap 1.1 Diamond Transformed into a Rectangle 1.2 Area of Diamond 1.3 Area of Overlap as Probability J P-Distributions J.1 Thermal State J.2 Photon Number State J.3 Squeezed State K Homodyne Kernel K.1 Explicit Evaluation of Kernel K.2 Strong Local Oscillator Limit L Beyond the Dipole Approximation L.1 First Order Taylor Expansion L.2 Classical Gauge Transformation L.3 Quantum Mechanical Gauge Transformation M Effective Hamiltonian N Oscillator Reservoir N.1 Second Order Contribution N.2 Symmetry Relations in Trace N.3 Master Equation N.4 Explicit Expressions for Γ，β and G N.5 Integration over Time O Bessel Functions O.1 Definition O.2 Asymptotic Expansion P Square Root of Q Further Reading Index
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