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Tables of Contents for Coherent Atomic Matter Waves
Chapter/Section Title
Page #
Page Count
Lecturers
xi
Participants
xiii
Preface
xvii
Preface
xxi
Contents
xxv
Bose-Einstein Condensates in Atomic Gases: Simple Theoretical Results
1
136
Y. Castin
Introduction
5
3
1925: Einstein's prediction for the ideal Bose gas
5
1
Experimental proof?
6
1
Why interesting?
6
1
Simple systems for the theory
6
1
New features
7
1
The ideal Bose gas in a trap
8
12
Bose-Einstein condensation in a harmonic trap
8
1
In the basis of harmonic levels
8
3
Comparison with the exact calculation
11
1
In position space
11
3
Relation to Einstein's condition ρλ3dB = ζ
14
1
Bose-Einstein condensation in a more general trap
15
1
The Wigner distribution
15
1
Critical temperature in the semiclassical limit
16
2
Is the ideal Bose gas model sufficient: Experimental verdict
18
1
Condensed fraction as a function of temperature
18
1
Energy of the gas as a function of temperature and the number of particles
18
1
Density profile of the condensate
19
1
Response frequencies of the condensate
20
1
A model for the atomic interactions
20
12
Reminder of scattering theory
21
1
General results of scattering theory
22
2
Low energy limit for scattering by a finite range potential
24
1
Power law potentials
25
1
The model potential used in this lecture
25
1
Why not keep the exact interaction potential?
25
3
Scattering states of the pseudo-potential
28
1
Bound states of the pseudo-potential
29
1
Perturbative vs. non-perturbative regimes for the pseudo-potential
30
1
Regime of the Born approximation
30
1
Relevance of the pseudo-potential beyond the Born approximation
31
1
Interacting Bose gas in the Hartree-Fock approximation
32
9
BBGKY hierarchy
32
1
Few body-density matrices
32
1
Equations of the hierarchy
33
1
Hartree-Fock approximation for T > Tc
34
1
Mean field potential for the non-condensed particles
34
2
Effect of interactions on Tc
36
1
Hartree-Fock approximation in presence of a condensate
37
1
Improved Hartree-Fock Ansatz
37
1
Mean field seen by the condensate
38
1
At thermal equilibrium
38
1
Comparison of Hartree-Fock to exact results
39
1
Quantum Monte Carlo calculations
39
1
Experimental results for the energy of the gas
39
2
Properties of the condensate wavefunction
41
21
The Gross-Pitaevskii equation
42
1
From Hartree-Fock
42
1
Variational formulation
43
3
The fastest trick to recover the Gross-Pitaevskii equation
46
1
Gaussian Ansatz
46
1
Time-independent case
47
4
Time-dependent case
51
1
Strongly interacting regime: Thomas-Fermi approximation
52
1
Time-independent case
52
3
How to extend the Thomas-Fermi approximation to the time-dependent case?
55
1
Hydrodynamic equations
56
2
Classical hydrodynamic approximation
58
1
Recovering time-dependent experimental results
59
1
The scaling solution
59
2
Ballistic expansion of the condensate
61
1
Breathing frequencies of the condensate
61
1
What we learn from a linearization of the Gross-Pitaevskii equation
62
19
Linear response theory for the condensate wavefunction
63
1
Linearize the Gross-Pitaevskii solution around a steady-state solution
63
2
Extracting the ``relevant part'' from δ&phis;
65
1
Spectral properties of ℒ` and dynamical stability
66
1
Diagonalization of ℒ
67
2
General solution of the linearized problem
69
1
Link between eigenmodes of ℒGP and eigenmodes of ℒ
69
1
Examples of dynamical instabilities
70
1
Condensate in a box
70
4
Demixing instability
74
4
Linear response in the classical hydrodynamic approximation
78
1
Linearized classical hydrodynamic equations
78
1
Validity condition of the linearized classical hydrodynamic equations
79
1
Approximate spectrum in a harmonic trap
80
1
Bogoliubov approach and thermodynamical stability
81
19
Small parameter of the theory
82
1
Zeroth order in &epsis;: Gross-Pitaevskii equation
83
1
Next order in &epsis;: Linear dynamics of non-condensed particles
83
2
Bogoliubov Hamiltonian
85
2
Order &epsis;2: Corrections to the Gross-Pitaevskii equation
87
1
Thermal equilibrium of the gas of quasi-particles
88
1
Condensate depletion and the small parameter &(pa3)1/2
89
3
Fluctuations in the number of condensate particles
92
3
A simple reformulation of the thermodynamical stability condition
95
2
Thermodynamical stability implies dynamical stability
97
1
Examples of thermodynamical instability
97
1
Real condensate wavefunction with a node
97
1
Condensate with a vortex
98
2
Phase coherence properties of Bose-Einstein condensates
100
14
Interference between two BECs
101
1
A very simple model
102
1
A trap to avoid
103
2
A Monte Carlo simulation
105
1
Analytical solution
105
3
Moral of the story
108
1
What is the time evolution of an initial phase state?
108
1
Physical motivation
108
1
A quadratic approximation for the energy
109
1
State vector at time t
110
1
An indicator of phase coherence
111
3
Symmetry-breaking description of condensates
114
23
The ground state of spinor condensates
114
1
A model interaction potential
115
1
Ground state in the Hartree-Fock approximation
116
2
Exact ground state of the spinor part of the problem
118
3
Advantage of a symmetry-breaking description
121
2
Solitonic condensates
123
1
How to make a solitonic condensate?
123
4
Ground state of the one-dimensional attractive Bose gas
127
2
Physical advantage of the symmetry-breaking description
129
8
Spinor Condensates and Light Scattering from Bose-Einstein Condensates
137
82
D.M. Stamper-Kurn
W. Ketterle
Introduction
139
1
Optical properties of a Bose-Einstein condensate
140
27
Light scattering from a Bose-Einstein condensate
141
1
Elastic and inelastic light scattering
141
3
Light scattering from atomic beams and atoms at rest
144
1
Relation to the dynamic structure factor of a many-body system
145
1
The dynamic structure factor of a Bose-Einstein condensate
146
1
The homogeneous condensate
146
2
Bragg scattering as a probe of pair correlations in the condensate
148
2
Mean-field theory determination of S(q, ω)
150
2
The inhomogeneous condensate
152
2
Relevance of Doppler broadening
154
1
Experimental aspects of Bragg spectroscopy
155
2
Light scattering in the free-particle regime
157
1
Measurement of line shift and line broadening
157
4
A measurement of the coherence length of a Bose-Einstein condensate
161
2
Light scattering in the phonon regime
163
1
Experimental study
163
1
Suppression of light scattering from a Bose-Einstein condensate
164
3
Amplified scattering of light
167
15
Introduction
167
1
Superradiant Rayleigh scattering
167
1
Semiclassical derivation of the gain mechanism
167
2
Four-wave mixing of light and atoms
169
1
Bosonic stimulation by scattered atoms or scattered light?
170
3
Observation of directional emission of light and atoms
173
4
Relation to other non-linear phenomena
177
2
Phase-coherent amplification of matter waves
179
3
Spinor Bose-Einstein condensates
182
37
The implications of rotational symmetry
184
4
Tailoring the ground-state structure with magnetic fields
188
3
Spin-domain diagrams: A local density approximation to the spin structure of spinor condensates
191
2
Experimental methods for the study of spinor condensates
193
1
The formation of ground-state spin domains
194
3
Miscibility and immiscibility of spinor condensate components
197
1
Metastable states of spinor Bose-Einstein condensates
198
1
Metastable spin-domain structures
199
3
Metastable spin composition
202
1
Quantum tunneling
203
5
Magnetic field dependence of spin-domain boundaries
208
11
Field Theory for Trapped Atomic Gases
219
98
H.T.C. Stoof
Introduction
221
2
Equilibrium field theory
223
43
Second quantization
223
4
Grassmann variables and coherent states
227
4
Functional integrals
231
3
Ideal quantum gases
234
1
Semiclassical method
234
1
Matsubara expansion
235
2
Green's function method
237
3
Interactions and Feynmann diagrams
240
5
Hartree-Fock theory for an atomic Fermi gas
245
4
Landau theory of phase transitions
249
3
Superfluidity and superconductivity
252
1
Superfluidity
252
7
Some atomic physics
259
2
Superconductivity
261
5
Nonequilibrium field theory
266
45
Macroscopic quantum tunneling of a condensate
266
6
Phase diffusion
272
4
Quantum kinetic theory
276
1
Ideal Bose gas
276
6
Ideal Bose gas in contact with a reservoir
282
13
Condensate formation
295
1
Weak-coupling limit
296
6
Strong-coupling limit
302
5
Collective modes
307
4
Outlook
311
6
Atom Interferometry
317
54
S. Chu
Introduction
319
1
Basic principles
320
14
Ramsey interference
320
4
Interference due to different physical paths
324
1
Path integral description of interference
325
1
Atom optics
326
3
Interference with combined internal and external degrees of freedom
329
5
Beam splitters and interferometers
334
5
Interferometers based on microfabricated structures
334
3
Interferometers based on light-induced potentials
337
1
Diffraction from an optical standing wave
337
1
Interaction of atoms with light in the sudden approximation
338
1
An atom interferometry measurement of the acceleration due to gravity
339
13
Circumventing experimental obstacles
342
1
Stimulated Raman transitions
343
3
Frequency sweep and stability issues
346
1
Vibration isolation
347
1
Experimental results
348
4
Interferometry based on adiabatic transfer
352
11
Theory of adiabatic passage with time-delayed pulses
354
2
Atom interferometry using adiabatic transfer
356
3
A measurement of the photon recoil and h/M
359
4
Atom gyroscopes
363
8
A comparison of atom interferometers
364
1
Future prospects
365
6
Mesoscopic Light Scattering in Atomic Physics
371
44
B.A. van Tiggelen
Introduction
373
2
Mesoscopic wave physics
375
8
Mesoscopic quantum mechanics
375
3
Phenomenological radiative transfer
378
1
Mesoscopic physics with classical waves
379
1
Mesoscopic light scattering in atomic gases
380
3
Light scattering from simple atoms
383
13
Vector Green's function
384
1
An atom as a point scatterer
385
2
Polarization, cross-section and stored energy
387
2
Two atoms: Dipole-dipole coupling
389
4
Induced dipole force between two simple atoms
393
2
Van der Waals interaction
395
1
Applications in multiple scattering
396
19
Effective medium
397
1
Group and energy velocity
398
4
Dipole-dipole coupling in the medium
402
2
Coherent backscattering
404
4
Dependent scattering with quantum correlation
408
2
From weak towards strong localization
410
5
Quantum Chaos in Atomic Physics
415
66
D. Delande
What is quantum chaos?
417
13
Classical chaos
418
1
Quantum dynamics
419
2
Semiclassical dynamics
421
2
Physical situations of interest
423
2
A simple example: The hydrogen atom in a magnetic field
425
1
Hamiltonian
425
1
Classical scaling
426
1
Classical dynamics
427
1
Quantum scaling - Scaled spectroscopy
428
2
Time scales - Energy scales
430
5
Shortest periodic orbit
430
1
Ehrenfest time
430
2
Heisenberg time
432
2
Inelastic time
434
1
Statistical properties of energy levels - Random Matrix Theory
435
13
Level dynamics
435
2
Statistical analysis of the spectral fluctuations
437
1
Density of states
438
1
Unfolding the spectrum
438
1
Nearest-neighbor spacing distribution
439
1
Number variance
439
1
Regular regime
440
1
Chaotic regime - Random Matrix Theory
441
3
Usefulness of Random Matrix Theory
444
2
Other statistical ensembles
446
2
Semiclassical approximation
448
18
Regular systems - EBK/WKB quantization
448
4
Semiclassical propagator
452
2
Green's function
454
2
Trace formula
456
2
``Backward'' application of the trace formula
458
1
``Forward'' application of the trace formula
458
2
Scarring
460
1
Convergence properties of the trace formula
461
2
An example: The helium atom
463
1
Link with Random Matrix Theory
464
2
Transport properties - Localization
466
9
The classical kicked rotor
467
1
The quantum kicked rotor
468
1
Dynamical localization
469
2
Link with Anderson localization
471
1
Experimental observation of dynamical localization
472
2
The effect of noise and decoherence
474
1
Conclusion
475
6
Photonic Band Gap Materials: A New Frontier in Quantum and Nonlinear Optics
481
52
S. John
Introduction
483
3
The existence of photon localization
486
5
Independent scatterers and microscopic resonances
487
2
A new criterion for light localization
489
1
Photonic band gap formation
490
1
Quantum electrodynamics in a photonic band gap
491
9
Theory of the photon-atom bound state
491
7
Lifetime of the photon-atom bound state
498
2
Non-Markovian spontaneous emission dynamics near a photonic band edge
500
11
Single atom radiative dynamics
500
4
Collective time scale factors
504
4
Superradiance near a photonic band edge
508
3
Quantum and nonlinear optics in a three-dimensional PBG material
511
5
Low-threshold nonlinear optics
511
2
Collective switching and transistor effects
513
3
Resonant nonlinear dielectric response in a doped photonic band gap material
516
4
Collective switching and inversion without fluctuation in a colored vacuum
520
13
Environment-Induced Decoherence and the Transition from Quantum to Classical
533
82
J.P. Paz
W.H. Zurek
Introduction and overview
535
4
Quantum measurements
539
18
Bit-by-bit measurement and quantum entanglement
541
3
Interactions and the information transfer in quantum measurements
544
2
Monitoring by the environment and decoherence
546
2
One-bit environment for a bit-by-bit measurement
548
2
Decoherence of a single (qu) bit
550
4
Decoherence, einselection, and controlled shifts
554
3
Dynamics of quantum open systems: Master equations
557
17
Master equation: Perturbative evaluation
558
3
Perturbative master equation in quantum Brownian motion
561
3
Perturbative master equation for a two-level system coupled to a bosonic heat bath
564
2
Perturbative master equation for a particle interacting with a quantum field
566
2
Exact master equation for quantum Brownian motion
568
6
Einselection in quantum Brownian motion
574
7
Decoherence of a superposition of two coherent states
574
4
Predictability sieve and preferred states for QBM
578
2
Energy eigenstates can also be selected by the environment!
580
1
Deconstructing decoherence: Landscape beyond the standard models
581
8
Saturation of the decoherence rate at large distances
582
1
Decoherence at zero temperature
583
2
Preexisting correlations between the system and the environment
585
4
Decoherence and chaos
589
9
Quantum predictability horizon: How the correspondence is lost
589
2
Exponential instability vs. decoherence
591
2
The arrow of time: A price of classicality?
593
4
Decoherence, einselection, and the entropy production
597
1
How to fight against decoherence: Quantum error correcting codes
598
11
How to protect a classical bit
599
1
How to protect a quantum bit
599
7
Stabilizer quantum error-correcting codes
606
3
Discussion
609
6
Cavity QED Experiments, Entanglement and Quantum Measurement
615
44
M. Brune
Introduction
617
2
Microwave CQED experiments: The strong coupling regime
619
5
The experimental tools and orders of magnitude
620
1
Circular Rydberg atoms
620
1
The photon box
621
1
Resonant atom-field interaction: The vacuum Rabi oscillation
622
1
``Quantum logic'' operations based on the vacuum Rabi oscillation
622
2
Quantum non-demolition detection of a single photon
624
11
Quantum non-demolition strategies
624
1
The Ramsey interferometer for detecting a single photon
625
2
Experimental realization
627
1
Input-meter: Demonstrating the single photon phase shift
628
1
Meter-output correlation: Detecting the same photon twice
629
3
Input-output correlation: Quantifying the QND performance
632
3
Step-by-step synthesis of a three-particle entangled state
635
10
The SP-QND scheme as a quantum phase gate
635
3
Building step-by-step three-particle entanglement: Principle
638
2
Detection of the three-particle entanglement
640
5
Decoherence and quantum measurement
645
11
Quantum measurement theory
645
1
The postulates
645
1
Von Neumann's analysis of meters
646
2
Observing progressive decoherence during a measurement process
648
1
Measuring the atom state with the field phase
648
1
Characterizing the Schrodinger cat state
649
3
Theoretical analysis
652
3
Decoherence and interpretation of a quantum measurement
655
1
Conclusion and perspectives
656
3
Basic Concepts in Quantum Computation
659
40
A. Ekert
P.M. Hayden
H. Inamori
Qubits, gates and networks
663
5
Quantum arithmetic and function evaluations
668
4
Algorithms and their complexity
672
3
From interferometers to computers
675
4
The first quantum algorithms
679
3
Quantum search
682
2
Optimal phase estimation
684
2
Periodicity and quantum factoring
686
3
Cryptography
689
4
Conditional quantum dynamics
693
1
Decoherence and recoherence
694
5
Concluding remarks
699
1
Coherent Backscattering of Light from a Cold Atomic Cloud
699
16
G. Labeyrie
F. de Tomasi
J.-C. Bernard
C. Mueller
C. Miniatura
R. Kaiser
Introduction
705
1
Coherent backscattering
706
2
Principle of CBS
706
1
CBS with cold atoms
707
1
Description of the experiment
708
2
Preparation of the atomic sample
708
1
CBS detection setup
709
1
Results
710
3
Conclusion
713
2
Seminars by participants
715
2
Posters
717