相空間中的量子光學

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