Quantum Optics in Phase Space

by Schleich, Wolfgang P.

• ISBN13: 9783527294350
• ISBN10: 352729435X
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-03-30
• Publisher: Wiley-VCH

Summary

Quantum Optics in Phase Space provides a concise introduction to the rapidly moving field of quantum optics from the point of view of phase space. Modern in style and didactically skillful, Quantum Optics in Phase Space prepares students for their own research by presenting detailed derivations, many illustrations and a large set of workable problems at the end of each chapter. Often, the theoretical treatments are accompanied by the corresponding experiments. An exhaustive list of references provides a guide to the literature. Quantum Optics in Phase Space also serves advanced researchers as a comprehensive reference book.Starting with an extensive review of the experiments that define quantum optics and a brief summary of the foundations of quantum mechanics the author Wolfgang P. Schleich illustrates the properties of quantum states with the help of the Wigner phase space distribution function. His description of waves ala WKB connects semi-classical phase space with the Berry phase. These semi-classical techniques provide deeper insight into the timely topics of wave packet dynamics, fractional revivals and the Talbot effect.Whereas the first half of the book deals with mechanical oscillators such as ions in a trap or atoms in a standing wave the second half addresses problems where the quantization of the radiation field is of importance. Such topics extensively discussed include optical interferometry, the atom-field interaction, quantum state preparation and measurement, entanglement, decoherence, the one-atom maser and atom optics in quantized light fields.Quantum Optics in Phase Space presents the subject of quantum optics as transparently as possible. Giving wide-ranging references, it enables students to study and solve problems with modern scientific literature. The result is a remarkably concise yet comprehensive and accessible text- and reference book - an inspiring source of information and insight for students, teachers and researchers alike.

Author Biography

- Recipient of the Max Planck Research Award 2002

Table of Contents

What's Quantum Optics?	1	(34)
On the Road to Quantum Optics	1	(1)
Resonance Fluorescence	2	(5)
Elastic Peak: Light as a Wave	2	(1)
Mollow-Three-Peak Spectrum	3	(2)
Anti-Bunching	5	(2)
Squeezing the Fluctuations	7	(7)
What is a Squeezed State?	7	(2)
Squeezed States in the Optical Parametric Oscillator	9	(3)
Oscillatory Photon Statistics	12	(1)
Interference in Phase Space	13	(1)
Jaynes-Cummings-Paul Model	14	(2)
Single Two-Level Atom plus a Single Mode	15	(1)
Time Scales	15	(1)
Cavity QED	16	(6)
An Amazing Master	16	(3)
Cavity QED in the Optical Domain	19	(3)
de Broglie Optics	22	(4)
Electron and Neutron Optics	22	(1)
Atom Optics	23	(2)
Atom Optics in Quantized Light Fields	25	(1)
Quantum Motion in Paul Traps	26	(2)
Analogy to Cavity QED	26	(1)
Quantum Information Processing	26	(2)
Two-Photon Interferometry and More	28	(1)
Outline of the Book	29	(6)
Ante	35	(32)
Position and Momentum Eigenstates	36	(4)
Properties of Eigenstates	36	(2)
Derivative of Wave Function	38	(1)
Fourier Transform Connects x-and p-Space	39	(1)
Energy Eigenstate	40	(4)
Arbitrary Representation	41	(1)
Position Representation	42	(2)
Density Operator: A Brief Introduction	44	(9)
A State Vector is not Enough!	44	(4)
Definition and Properties	48	(1)
Trace of Operator	49	(2)
Examples of a Density Operator	51	(2)
Time Evolution of Quantum States	53	(14)
Motion of a Wave Packet	54	(1)
Time Evolution due to Interaction	55	(2)
Time Dependent Hamiltonian	57	(4)
Time Evolution of Density Operator	61	(6)
Wigner Function	67	(32)
Jump Start of the Wigner Function	68	(1)
Properties of the Wigner Function	69	(5)
Marginals	69	(2)
Overlap of Quantum States as Overlap in Phase Space	71	(1)
Shape of Wigner Function	72	(2)
Time Evolution of Wigner Function	74	(2)
von Neumann Equation in Phase Space	74	(1)
Quantum Liouville Equation	75	(1)
Wigner Function Determined by Phase Space	76	(2)
Definition of Moyal Function	76	(1)
Phase Space Equations for Moyal Functions	77	(1)
Phase Space Equations for Energy Eigenstates	78	(6)
Power Expansion in Planck's Constant	79	(2)
Model Differential Equation	81	(3)
Harmonic Oscillator	84	(3)
Wigner Function as Wave Function	85	(1)
Phase Space Enforces Energy Quantization	86	(1)
Evaluation of Quantum Mechanical Averages	87	(12)
Operator Ordering	88	(2)
Examples of Weyl-Wigner Ordering	90	(9)
Quantum States in Phase Space	99	(54)
Energy Eigenstate	100	(8)
Simple Phase Space Representation	100	(1)
Large-m Limit	101	(4)
Wigner Function	105	(3)
Coherent State	108	(11)
Definition of a Coherent State	109	(1)
Energy Distribution	110	(3)
Time Evolution	113	(6)
Squeezed State	119	(17)
Definition of a Squeezed State	121	(4)
Energy Distribution: Exact Treatment	125	(3)
Energy Distribution: Asymptotic Treatment	128	(4)
Limit Towards Squeezed Vacuum	132	(3)
Time Evolution	135	(1)
Rotated Quadrature States	136	(7)
Wigner Function of Position and Momentum States	137	(3)
Position Wave Function of Rotated Quadrature States	140	(2)
Wigner Function of Rotated Quadrature States	142	(1)
Quantum State Reconstruction	143	(10)
Tomographic Cuts through Wigner Function	143	(1)
Radon Transformation	144	(9)
Waves a la WKB	153	(18)
Probability for Classical Motion	153	(2)
Probability Amplitudes for Quantum Motion	155	(4)
An Educated Guess	156	(2)
Range of Validity of WKB Wave Function	158	(1)
Energy Quantization	159	(4)
Determining the Phase	159	(2)
Bohr-Sommerfeld-Kramers Quantization	161	(2)
Summary	163	(8)
Construction of Primitive WKB Wave Function	163	(1)
Uniform Asymptotic Expansion	164	(7)
WKB and Berry Phase	171	(18)
Berry Phase and Adiabatic Approximation	172	(4)
Adiabatic Theorem	172	(2)
Analysis of Geometrical Phase	174	(1)
Geometrical Phase as a Flux in Hilbert Space	175	(1)
WKB Wave Functions from Adiabaticity	176	(9)
Energy Eigenvalue Problem as Propagation Problem	177	(4)
Dynamical and Geometrical Phase	181	(2)
WKB Waves Rederived	183	(2)
Non-Adiabatic Berry Phase	185	(4)
Derivation of the Aharonov-Anandan Phase	186	(1)
Time Evolution in Harmonic Oscillator	187	(2)
Interference in Phase space	189	(16)
Outline of the Idea	189	(3)
Derivation of Area-of-Overlap Formalism	192	(8)
Jumps Viewed From Position Space	192	(5)
Jumps Viewed From Phase Space	197	(3)
Application to Franck-Condon Transitions	200	(1)
Generalization	201	(4)
Applications of Interference in Phase Space	205	(28)
Connection to Interference in Phase Space	205	(1)
Energy Eigenstates	206	(2)
Coherent State	208	(5)
Elementary Approach	209	(3)
Influence of Internal Structure	212	(1)
Squeezed State	213	(8)
Oscillations from Interference in Phase Space	213	(3)
Giant Oscillations	216	(2)
Summary	218	(3)
The Question of Phase States	221	(12)
Amplitude and Phase in a Classical Oscillator	221	(2)
Definition of a Phase State	223	(4)
Phase Distribution of a Quantum State	227	(6)
Wave Packet Dynamics	233	(22)
What are Wave Packets?	233	(1)
Fractional and Full Revivals	234	(3)
Natural Time Scales	237	(4)
Hierarchy of Time Scales	237	(2)
Generic Signal	239	(2)
New Representations of the Signal	241	(5)
The Early Stage of the Evolution	241	(3)
Intermediate Times	244	(2)
Fractional Revivals Made Simple	246	(9)
Gauss Sums	246	(1)
Shape Function	246	(9)
Field Quantization	255	(36)
Wave Equations for the Potentials	256	(6)
Derivation of the Wave Equations	256	(1)
Gauge Invariance of Electrodynamics	257	(3)
Solution of the Wave Equation	260	(2)
Mode Structure in a Box	262	(4)
Solutions of Helmholtz Equation	262	(1)
Polarization Vectors from Gauge Condition	263	(1)
Discreteness of Modes from Boundaries	264	(1)
Boundary Conditions on the Magnetic Field	264	(1)
Orthonormality of Mode Functions	265	(1)
The Field as a Set of Harmonic Oscillators	266	(6)
Energy in the Resonator	267	(2)
Quantization of the Radiation Field	269	(3)
The Casimir Effect	272	(6)
Zero-Point Energy of a Rectangular Resonator	272	(2)
Zero-Point Energy of Free Space	274	(1)
Difference of Two Infinite Energies	275	(1)
Casimir Force: Theory and Experiment	276	(2)
Operators of the Vector Potential and Fields	278	(3)
Vector Potential	278	(2)
Electric Field Operator	280	(1)
Magnetic Field Operator	281	(1)
Number States of the Radiation Field	281	(10)
Photons and Anti-Photons	282	(1)
Multi-Mode Case	282	(1)
Superposition and Entangled States	283	(8)
Field States	291	(30)
Properties of the Quantized Electric Field	291	(4)
Photon Number States	292	(1)
Electromagnetic Field Eigenstates	293	(2)
Coherent States Revisited	295	(11)
Eigenvalue Equation	295	(2)
Coherent State as a Displaced Vacuum	297	(1)
Photon Statistics of a Coherent State	298	(1)
Electric Field Distribution of a Coherent State	299	(2)
Over-completeness of Coherent States	301	(2)
Expansion into Coherent States	303	(2)
Electric Field Expectation Values	305	(1)
Schrodinger Cat State	306	(15)
The Original Cat Paradox	306	(1)
Definition of the Field Cat State	307	(1)
Wigner Phase Space Representation	307	(3)
Photon Statistics	310	(11)
Phase Space Functions	321	(28)
There is more than Wigner Phase Space	321	(3)
Who Needs Phase Space Functions?	321	(1)
Another Description of Phase Space	322	(2)
The Husimi-Kano Q-Function	324	(6)
Definition of Q-Function	324	(1)
Q-Functions of Specific Quantum States	324	(6)
Averages Using Phase Space Functions	330	(7)
Heuristic Argument	330	(3)
Rigorous Treatment	333	(4)
The Glauber-Sudarshan P-Distribution	337	(12)
Definition of P-Distribution	337	(1)
Connection between Q- and P-Function	338	(1)
P-Function from Q-Function	339	(2)
Examples of P-Distributions	341	(8)
Optical Interferometry	349	(32)
Beam Splitter	350	(7)
Classical Treatment	350	(2)
Symmetric Beam Splitter	352	(1)
Transition to Quantum Mechanics	353	(1)
Transformation of Quantum States	353	(3)
Count Statistics at the Exit Ports	356	(1)
Homodyne Detector	357	(4)
Classical Considerations	357	(1)
Quantum Treatment	358	(3)
Eight-Port Interferometer	361	(9)
Quantum State of the Output Modes	361	(2)
Photon Count Statistics	363	(2)
Simultaneous Measurement and EPR	365	(2)
Q-Function Measurement	367	(3)
Measured Phase Operators	370	(11)
Measurement of Classical Trigonometry	370	(2)
Measurement of Quantum Trigonometry	372	(2)
Two-Mode Phase Operators	374	(7)
Atom-Field Interaction	381	(32)
How to Construct the Interaction?	382	(1)
Vector Potential-Momentum Coupling	382	(7)
Gauge Principle Determines Minimal Coupling	383	(3)
Interaction of an Atom with a Field	386	(3)
Dipole Approximation	389	(4)
Expansion of Vector Potential	389	(1)
A · p-Interaction	390	(1)
Various Forms of the A · p Interaction	390	(2)
Higher Order Corrections	392	(1)
Electric Field-Dipole Interaction	393	(2)
Dipole Approximation	393	(1)
Rontgen Hamiltonians and Others	393	(2)
Subsystems, Interaction and Entanglement	395	(1)
Equivalence of A · p and r · E	396	(4)
Classical Transformation of Lagrangian	397	(2)
Quantum Mechanical Treatment	399	(1)
Matrix elements of A · p and r · E	399	(1)
Equivalence of Hamiltonians H(1) and H(1)	400	(2)
Simple Model for Atom-Field Interaction	402	(11)
Derivation of the Hamiltonian	402	(4)
Rotating-Wave Approximation	406	(7)
Jaynes-Cummings-Paul Model: Dynamics	413	(22)
Resonant Jaynes-Cummings-Paul Model	413	(7)
Time Evolution Operator Using Operator Algebra	414	(2)
Interpretation of Time Evolution Operator	416	(2)
State Vector of Combined System	418	(1)
Dynamics Represented in State Space	418	(2)
Role of Detuning	420	(3)
Atomic and Field States	420	(2)
Rabi Equations	422	(1)
Solution of Rabi Equations	423	(3)
Laplace Transformation	424	(1)
Inverse Laplace Transformation	425	(1)
Discussion of Solution	426	(9)
General Considerations	427	(1)
Resonant Case	427	(2)
Far Off-Resonant Case	429	(6)
State Preparation and Entanglement	435	(38)
Measurements on Entangled Systems	435	(9)
How to Get Probabilities	436	(3)
State of the Subsystem after a Measurement	439	(1)
Experimental Setup	440	(4)
Collapse, Revivals and Fractional Revivals	444	(7)
Inversion as Tool for Measuring Internal Dynamics	444	(3)
Experiments on Collapse and Revivals	447	(4)
Quantum State Preparation	451	(3)
State Preparation with a Dispersive Interaction	451	(3)
Generation of Schrodinger Cats	454	(1)
Quantum State Engineering	454	(19)
Outline of the Method	454	(4)
Inverse Problem	458	(3)
Example: Preparation of a Phase State	461	(12)
Paul Trap	473	(34)
Basics of Trapping Ions	474	(5)
No Static Trapping in Three Dimensions	474	(1)
Dynamical Trapping	475	(4)
Laser Cooling	479	(1)
Motion of an Ion in a Paul Trap	480	(14)
Reduction to Classical Problem	481	(2)
Motion as a Sequence of Squeezing and Rotations	483	(3)
Dynamics in Wigner Phase Space	486	(4)
Floquet Solution	490	(4)
Model Hamiltonian	494	(6)
Transformation to Interaction Picture	495	(1)
Lamb-Dicke Regime	496	(2)
Multi-Phonon Jaynes-Cummings-Paul Model	498	(2)
Effective Potential Approximation	500	(7)
Damping and Amplification	507	(42)
Damping and Amplification of a Cavity Field	508	(1)
Density Operator of a Subsystem	509	(2)
Coarse-Grained Equation of Motion	509	(2)
Time Independent Hamiltonian	511	(1)
Reservoir of Two-Level Atoms	511	(11)
Approximate Treatment	512	(2)
Density Operator in Number Representation	514	(5)
Exact Master Equation	519	(3)
Summary	522	(1)
One-Atom Maser	522	(10)
Density Operator Equation	523	(1)
Equation of Motion for the Photon Statistics	524	(5)
Phase Diffusion	529	(3)
Atom-Reservoir Interaction	532	(17)
Model and Equation of Motion	532	(1)
First Order Contribution	533	(2)
Bloch Equations	535	(2)
Second Order Contribution	537	(2)
Lamb Shift	539	(1)
Weisskopf-Wigner Decay	540	(9)
Atom Optics in Quantized Light Fields	549	(30)
Formulation of Problem	549	(5)
Dynamics	549	(3)
Time Evolution of Probability Amplitudes	552	(2)
Reduction to One-Dimensional Scattering	554	(3)
Slowly Varying Approximation	554	(1)
From Two Dimensions to One	555	(1)
State Vector	556	(1)
Raman-Nath Approximation	557	(2)
Heuristic Arguments	557	(1)
Probability Amplitudes	558	(1)
Deflection of Atoms	559	(12)
Measurement Schemes and Scattering Conditions	559	(3)
Kapitza-Dirac Regime	562	(6)
Kapitza-Dirac Scattering with a Mask	568	(3)
Interference in Phase Space	571	(8)
How to Represent the Quantum State?	572	(1)
Area of Overlap	572	(1)
Expression for Probability Amplitude	573	(6)
Wigner Functions in Atom Optics	579	(18)
Model	579	(2)
Equation of Motion for Wigner Functions	581	(1)
Motion in Phase Space	582	(5)
Harmonic Approximation	583	(1)
Motion of the Atom in the Cavity	583	(2)
Motion of the Atom outside the Cavity	585	(1)
Snap Shots of the Wigner Function	586	(1)
Quantum Lens	587	(3)
Distributions of Atoms in Space	587	(2)
Focal Length and Deflection Angle	589	(1)
Photon and Momentum Statistics	590	(2)
Heuristic Approach	592	(5)
Focal Length	592	(2)
Focal Size	594	(3)
A Energy Wave Functions of Harmonic Oscillator	597	(8)
Polynomial Ansatz	597	(2)
Asymptotic Behavior	599	(6)
Energy Wave Function as a Contour Integral	600	(1)
Evaluation of the Integral Im	600	(3)
Asymptotic Limit of fm	603	(1)
Bohr's Correspondence Principle	603	(2)
B Time Dependent Operators	605	(6)
Caution when Differentiating Operators	605	(1)
Time Ordering	606	(5)
Product of Two Terms	607	(1)
Product of n Terms	608	(3)
C Sßumann Measure	611	(4)
Why Other Measures Fail	611	(1)
One Way out of the Problem	612	(1)
Generalization to Higher Dimensions	613	(2)
D Phase Space Equations	615	(6)
Formulation of the Problem	615	(1)
Fourier Transform of Matrix Elements	616	(1)
Kinetic Energy Terms	617	(2)
Potential Energy Terms	619	(1)
Summary	620	(1)
E Airy Function	621	(8)
Definition and Differential Equation	621	(1)
Asymptotic Expansion	622	(7)
Oscillatory Regime	623	(1)
Decaying Regime	624	(1)
Stokes Phenomenon	625	(4)
F Radial Equation	629	(4)
G Asymptotics of a Poissonian	633	(2)
H Toolbox for Integrals	635	(8)
Method of Stationary Phase	635	(4)
One-Dimensional Integrals	635	(2)
Multi-Dimensional Integrals	637	(2)
Cornu Spiral	639	(4)
I Area of Overlap	643	(6)
Diamond Transformed into a Rectangle	643	(1)
Area of Diamond	644	(2)
Area of Overlap as Probability	646	(3)
J P-Distributions	649	(6)
Thermal State	649	(1)
Photon Number State	650	(1)
Squeezed State	651	(4)
K Homodyne Kernel	655	(4)
Explicit Evaluation of Kernel	655	(1)
Strong Local Oscillator Limit	656	(3)
L Beyond the Dipole Approximation	659	(10)
First Order Taylor Expansion	659	(2)
Expansion of the Hamiltonian	659	(2)
Extension to Operators	661	(1)
Classical Gauge Transformation	661	(3)
Lagrangian with Center-of-Mass Motion	662	(1)
Complete Time Derivative	663	(1)
Hamiltonian Including Center-of-Mass Motion	663	(1)
Quantum Mechanical Gauge Transformation	664	(5)
Gauge Potential	664	(3)
Schrodinger equation for Φ	667	(2)
M Effective Hamiltonian	669	(2)
N Oscillator Reservoir	671	(8)
Second Order Contribution	671	(2)
Evaluation of Double Commutator	671	(2)
Trace over Reservoir	673	(1)
Symmetry Relations in Trace	673	(2)
Complex Conjugates	674	(1)
Commutator Between Field Operators	674	(1)
Master Equation	675	(1)
Explicit Expressions for &Gamma, β and G	676	(1)
Integration over Time	677	(2)
O Bessel Functions	679	(4)
Definition	679	(1)
Asymptotic Expansion	680	(3)
P Square Root of δ	683	(2)
Q Further Reading	685	(3)
Index	688