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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