ISBN: 3540410473
TITLE: Coherent Atomic Matter Waves
AUTHOR: Kaiser, Robin; Westbrook, Christoph; David, Francois (Eds.)
TOC:

Lecturers xi
Participants xiii
Preface xvii
Preface xxi
Contents xxv
Course 1. Bose-Einstein Condensates in Atomic Gases: Simple Theoretical Results
by Y. Castin 1
1 Introduction 5
1.1 1925: Einstein's prediction for the ideal Bose gas 5
1.2 Experimental proof? 6
1.3 Why interesting? 6
1.3.1 Simple systems for the theory 6
1.3.2 New features 7
2 The ideal Bose gas in a trap 8
2.1 Bose-Einstein condensation in a harmonic trap 8
2.1.1 In the basis of harmonic levels 8
2.1.2 Comparison with the exact calculation 11
2.1.3 In position space 11
2.1.4 Relation to Einstein's condition roh lamda^3_db=zeta(3/2) 3
2.2 Bose-Einstein condensation in a more general trap 15
2.2.1 The Wigner distribution 15
2.2.2 Critical temperature in the semi classical limit 16
2.3 Is the ideal Bose gas model sufficient: Experimental verdict 18
2.3.1 Condensed fraction as a function of temperature 18
2.3.2 Energy of the gas as a function of temperature and the number of particles 18
2.3.3 Density profile of the condensate 19
2.3.4 Response frequencies of the condensate 20
3 A model for the atomic interactions 20
3.1 Reminder of scattering theory 21
3.1.1 General results of scattering theory 22
3.1.2 Low energy limit for scattering by a finite range potential 24
3.1.3 Power law potentials 25
3.2 The model potential used in this lecture 25
3.2.1 Why not keep the exact interaction potential? 25
3.2.2 Scattering states of the pseudo-potential 28
3.2.3 Bound states of the pseudo-potential 29
3.3 Perturbative vs. non-perturbative regimes for the pseudo-potential 30
3.3.1 Regime of the Born approximation 30
3.3.2 Relevance of the pseudo-potential beyond the Born approximation 31
4 Interacting Bose gas in the Hartree-Fock approximation 32
4.1 BBGKY hierarchy 32
4.1.1 Few body-density matrices 32
4.1.2 Equations of the hierarchy 33
4.2 Hartree-Fock approximation for T>T_c 34
4.2.1 Mean field potential for the non-condensed particles 34
4.2.2 Effect of interactions on T_c 36
4.3 Hartree-Fock approximation in presence of a condensate 37
4.3.1 Improved Hartree-Fock Ansatz 37
4.3.2 Mean field seen by the condensate 38
4.3.3 At thermal equilibrium 38
4.4 Comparison of Hartree-Fock to exact results 39
4.4.1 Quantum Monte Carlo calculations 39
4.4.2 Experimental results for the energy of the gas 39
5 Properties of the condensate wave function 41
5.1 The Gross-Pitaevskii equation 42
5.1.1 From Hartree-Fock 42
5.1.2 Variational formulation 43
5.1.3 The fastest trick to recover the Gross-Pitaevskii equation 46
5.2 Gaussian Ansatz 46
5.2.1 Time-independent case 47
5.2.2 Time-dependent case 51
5.3 Strongly interacting regime: Thomas-Fermi approximation 52
5.3.1 Time-independent case 52
5.3.2 How to extend the Thomas Fermi approximation to the time dependent case? 55
5.3.3 Hydrodynamic equations 56
5.3.4 Classical hydrodynamic approximation 58
5.4 Recovering time-dependent experimental results 59
5.4.1 The scaling solution 59
5.4.2 Ballistic expansion of the condensate 61
5.4.3 Breathing frequencies of the condensate 61

6 What we learn from a linearization of the Gross-Pitaevskii equation 62
6.1 Linear response theory for the condensate wave function 63
6.1.1 Linearize the Gross-Pitaevskii solution around a steady state solution 63
6.1.2 Extracting the "relevant part" from delta phi 65
6.1.3 Spectral properties of L and dynamical stability 66
6.1.4 Diagonalization of L 67
6.1.5 General solution of the linearized problem 69
6.1.6 Link between eigenmodes of L_GP and eigenmodes of L 69
6.2 Examples of dynamical instabilities 70

6.2.1 Condensate in a box 70
6.2.2 Demixing instability 74
6.3 Linear response in the classical hydrodynamic approximation 78
6.3.1 Linearized classical hydrodynamic equations 78
6.3.2 Validity condition of the linearized classical hydrodynamic equations 79
6.3.3 Approximate spectrum in a harmonic trap 80
7 Bogoliubov approach and thermodynamical stability 81
7.1 Small parameter of the theory 82
7.2 Zeroth order in epsilon: Gross-Pitaevskii equation 83
7.3 Next order in epsilon: Linear dynamics of non-condensed particles 83
7.4 Bogoliubov Hamiltonian 85
7.5 Order epsilon^2: Corrections to the Gross-Pitaevskii equation 87
7.6 Thermal equilibrium of the gas of quasi-particles 88
7.7 Condensate depletion and the small parameter (rho a^3)^1/2 89
7.8 Fluctuations in the number of condensate particles 92
7.9 A simple reformulation of the thermodynamical stability condition 95
7.10 Thermodynamical stability implies dynamical stability 97
7.11 Examples of thermodynamical instability 97
7.11.1 Real condensate wave function with a node 97
7.11.2 Condensate with a vortex 98
8 Phase coherence properties of Bose-Einstein condensates 100
8.1 Interference between two BECs 101
8.1.1 A very simple model 102
8.1.2 A trap to avoid 103
8.1.3 A Monte Carlo simulation 105
8.1.4 Analytical solution 105
8.1.5 Moral of the story 108
8.2 What is the time evolution of an initial phase state? 108
8.2.1 Physical motivation 108
8.2.2 Aquadratic approximation for the energy 109
8.2.3 State vector at time t 110
8.2.4 An indicator of phase coherence 111
9 Symmetry-breaking description of condensates 114

9.1 The ground state of spinor condensates 114
9.1.1 A model interaction potential 115
9.1.2 Ground state in the Hartree-Fock approximation 116
9.1.3 Exact ground state of the spinor part of the problem 118
9.1.4 Advantage of a symmetry breaking description 121
9.2 Solitonic condensates 123
9.2.1 How to make a solitonic condensate? 123
9.2.2 Ground state of the one-dimensional attractive Bose gas 127
9.2.3 Physical advantage of the symmetry breaking description 129
Course 2. Spinor Condensates and Light Scattering from Bose-Einstein Condensates
by D.M. Stamper-Kurn and W. Ketterle 137
1 Introduction 139
2 Optical properties of a Bose-Einstein condensate 140
2.1 Light scattering from a Bose-Einstein condensate 141
2.1.1 Elastic and inelastic light scattering 141
2.1.2 Light scattering from atomic beams and atoms at rest 144
2.1.3 Relation to the dynamic structure factor of a many-body system 145
2.2 The dynamic structure factor of a Bose-Einstein condensate 146
2.2.1 The homogeneous condensate 146
2.2.2 Bragg scattering as a probe of pair correlations in the condensate 148
2.2.3 Mean-field theory determination of S(q,omega) 150
2.2.4 The inhomogeneous condensate 152
2.2.5 Relevance of Doppler broadening 154
2.3 Experimental aspects of Bragg spectroscopy 155
2.4 Light scattering in the free-particle regime 157
2.4.1 Measurement of line shift and line broadening 157
2.4.2 A measurement of the coherence length of a Bose-Einstein condensate 161
2.5 Light scattering in the phonon regime 163
2.5.1 Experimental study 163
2.5.2 Suppression of light scattering from a Bose-Einstein condensate 164
3 Amplified scattering of light 167
3.1 Introduction 167
3.2 Superradiant Rayleigh scattering 167
3.2.1 Semiclassical derivation of the gain mechanism 167
3.2.2 Four-wave mixing of light and atoms 169
3.2.3 Bosonic stimulation by scattered atoms or scattered light? 170
3.2.4 Observation of directional emission of light and atoms 173
3.2.5 Relation to other non-linear phenomena 177
3.3 Phase-coherent amplification of matter waves 179
4 Spinor Bose-Einstein condensates 182
4.1 The implications of rotational symmetry 184
4.2 Tailoring the ground-state structure with magnetic fields 188
4.3 Spin-domain diagrams: A local density approximation to the spin structure of spinor condensates 191
4.4 Experimental methods for the study of spinor condensates 193
4.5 The formation of ground-state spin domains 194
4.6 Miscibility and immiscibility of spinor condensate components 197
4.7 Metastable states of spinor Bose-Einstein condensates 198
4.7.1 Metastable spin-domain structures 199
4.7.2 Metastable spin composition 202
4.8 Quantum tunneling 203
4.9 Magnetic field dependence of spin-domain boundaries 208
Course 3. Field Theory for Trapped Atomic Gases
by H.T.C. Stoof 219
1 Introduction 221
2 Equilibrium field theory 223
2.1 Second quantization 223
2.2 Grassmann variables and coherent states 227
2.3 Functional integrals 231
2.4 Ideal quantum gases 234
2.4.1 Semiclassical method 234
2.4.2 Matsubara expansion 235
2.4.3 Green's function method 237
2.5 Interactions and Feynmann diagrams 240
2.6 Hartree-Fock theory for an atomic Fermi gas 245
2.7 Landau theory of phase transitions 249
2.8 Superfluidity and superconductivity 252
2.8.1 Superfluidity 252
2.8.2 Some atomic physics 259
2.8.3 Superconductivity 261
3 Nonequilibrium field theory 266
3.1 Macroscopic quantum tunneling of a condensate 266
3.2 Phase diffusion 272
3.3 Quantum kinetic theory 276
3.3.1 Ideal Bose gas 276
3.3.2 Ideal Bose gas in contact with a reservoir 282
3.4 Condensate formation 295
3.4.1 Weak-coupling limit 296
3.4.2 Strong-coupling limit 302
3.5 Collective modes 307
4 Outlook 311
Course 4. Atom Interferometry
by S. Chu 317
1 Introduction 319
2 Basic principles 320
2.1 Ramsey interference 320
2.2 Interference due to different physical paths 324
2.3 Path integral description of interference 325
2.4 Atom optics 326
2.5 Interference with combined internal and external degrees of freedom 329
3 Beam splitters and interferometers 334
3.1 Interferometers based on microfabricated structures 334
3.2 Interferometers based on light-induced potentials 337
3.2.1 Diffraction from an optical standing wave 337
3.2.2 Interaction of atoms with light in the sudden approximation 338
4 An atom interferometry measurement of the acceleration due to gravity 339
4.1 Circumventing experimental obstacles 342
4.2 Stimulated Raman transitions 343
4.3 Frequency sweep and stability issues 346
4.4 Vibration isolation 347
4.5 Experimental results 348
5 Interferometry based on adiabatic transfer 352
5.1 Theory of adiabatic passage with time-delayed pulses 354
5.2 Atom interferometry using adiabatic transfer 356
5.3 A measurement of the photon recoil and h/M 359
6 Atom gyroscopes 363
6.1 A comparison of atom interferometers 364
6.2 Future prospects 365
Course 5. Mesoscopic Light Scattering in Atomic Physics
by B.A. van Tiggelen 371
1 Introduction 373
2 Mesoscopic wave physics 375
2.1 Mesoscopic quantum mechanics 375
2.2 Phenomenological radiative transfer 378
2.3 Mesoscopic physics with classical waves 379
2.4 Mesoscopic light scattering in atomic gases 380
3 Light scattering from simple atoms 383
3.1 Vector Green's function 384
3.2 An atoma a point scatterer 385
3.3 Polarization, cross-section and stored energy 387
3.4 Two atoms: Dipole-dipole coupling 389
3.5 Induced dipole force between two simple atoms 393
3.6 Van der Waals interaction 395
4 Applications in multiple scattering 396
4.1 Effective medium 397
4.2 Group and energy velocity 398
4.3 Dipole-dipole coupling in the medium 402
4.4 Coherent back-scattering 404
4.5 Dependent scattering with quantum correlation 408
4.6 From weak towards strong localization 410
Course 6. Quantum Chaos in Atomic Physics
by D. Delande 415
1 What is quantum chaos? 417
1.1 Classical chaos 418
1.2 Quantum dynamics 419
1.3 Semi-classical dynamics 421
1.4 Physical situations of interest 423
1.5 A simple example: The hydrogen atom in a magnetic field 425
1.5.1 Hamiltonian 425
1.5.2 Classical scaling 426
1.5.3 Classical dynamics 427
1.5.4 Quantum scaling-Scaled spectroscopy 428
2 Time scales-Energy scales 430
2.1 Shortest periodic orbit 430
2.2 Ehrenfest time 430
2.3 Heisenberg time 432
2.4 Inelastic time 434
3 Statistical properties of energy levels - Random Matrix Theory 435
3.1 Level dynamics 435
3.2 Statistical analysis of the spectral fluctuations 437
3.2.1 Density of states 438
3.2.2 Unfolding the spectrum 438
3.2.3 Nearest-neighbour spacing distribution 439
3.2.4 Number variance 439
3.3 Regular regime 440
3.4 Chaotic regime-Random Matrix Theory 441
3.5 Usefulness of Random Matrix Theory 444
3.6 Other statistical ensembles 446
4 Semiclassical approximation 448
4.1 Regular systems-EBK/WKB quantization 448
4.2 Semiclassical propagator 452
4.3 Green's function 454
4.4 Trace formula 456
4.5 "Backward" application of the trace formula 458
4.6 "Forward" application of the trace formula 458
4.7 Scarring 460
4.8 Convergence properties of the trace formula 461
4.9 An example: The helium atom 463
4.10 Link with Random Matrix Theory 464
5 Transport properties-Localization 466
5.1 The classical kicked rotor 467

5.2 The quantum kicked rotor 468
5.3 Dynamical localization 469
5.4 Link with Anderson localization 471
5.5 Experimental observation of dynamical localization 472
5.6 The effect of noise and decoherence 474
6 Conclusion 475
Course 7. Photonic Band Gap Materials:
A New Frontier in Quantum and Nonlinear Optics
by S. John 481
1 Introduction 483
2 The existence of photon localization 486
2.1 Independent scatterers and microscopic resonances 487
2.2 A new criterion for light localization 489
2.3 Photonic band gap formation 490
3 Quantum electrodynamics in a photonic band gap 491
3.1 Theory of the photon atom boundstate 491
3.2 Life time of the photon atom boundstate 498
4 Non-Markovian spontaneous emission dynamics near a photonic band edge 500

4.1 Single atom radiative dynamics 500
4.2 Collective time scale factors 504
4.3 Superradiance near a photonic band edge 508
5 Quantum and nonlinear optics in a three-dimensional PBG material 511
5.1 Low-threshold nonlinear optics 511
5.2 Collective switching and transistor effects 513
6 Resonant nonlinear dielectric response in a doped photonic band gap material 516
7 Collective switching and inversion without fluctuation in a colored vacuum 520
Course 8. Environment-Induced Decoherence and the Transition from Quantum to Classical
by J.P. Paz and W.H. Zurek 533
1 Introduction and overview 535
2 Quantum measurements 539
2.1 Bit-by-bit measurement and quantum entanglement 541
2.2 Interactions and the information transfer in quantum measurements 544
2.3 Monitoring by the environment and decoherence 546
2.4 One-bit environment for a bit-by-bit measurement 548
2.5 Decoherence of a single(qu)bit 550
2.6 Decoherence, einselection, and controlled shifts 554
3 Dynamics of quantum open systems: Master equations 557
3.1 Master equation: Perturbative evaluation 558
3.2 Example 1: Perturbative master equation in quantum Brownian motion 561
3.3 Example 2: Perturbative master equation for a two-level system coupled to a bosonic heat bath 564
3.4 Example 3: Perturbative master equation for a particle interacting with a quantum field 566
3.5 Exact master equation for quantum Brownian motion 568
4 Einselection in quantum Brownian motion 574
4.1 Decoherence of a superposition of two coherent states 574
4.2 Predictability sieve and preferred states for QBM 578
4.3 Energy eigenstates can also be selected by the environment! 580
5 Deconstructing decoherence: Landscape beyond the standard models 581
5.1 Saturation of the decoherence rate at large distances 582
5.2 Decoherence at zero temperature 583
5.3 Preexisting correlations between the system and the environment 585
6 Decoherence and chaos 589
6.1 Quantum predictability horizon: How the correspondence is lost 589
6.2 Exponential instability vs. decoherence 591
6.3 The arrow of time: A price of classicality? 593
6.4 Decoherence, einselection, and the entropy production 597
7 How to fight against decoherence: Quantum error correcting codes 598
7.1 How to protect a classical bit 599
7.2 How to protect a quantum bit 599
7.3 Stabilizer quantum error-correcting codes 606
8 Discussion 609 
Course 9. Cavity QED Experiments, Entanglement and Quantum Measurement
by M. Brune 615
1 Introduction 617
2 Microwave CQED experiments: The strong coupling regime 619
2.1 The experimental tools and orders of magnitude 620
2.1.1 Circular Rydberg atoms 620
2.1.2 The photon box 621
2.2 Resonant atom-field interaction: The vacuum Rabi oscillation 622
2.3 "Quantum logic" operations based on the vacuum Rabi oscillation 622
3 Quantum non-demolition detection of a single photon 624
3.1 Quantum non-demolition strategies 624
3.2 The Ramsey interferometer for detecting a single photon 625
3.3 Experimental realization 627
3.3.1 Input-meter: Demonstrating the single photon phase shift 628
3.3.2 Meter-output correlation: Detecting the same photon twice 629
3.3.3 Input-output correlation: Quantifying the QND
performance 632
4 Step-by-step synthesis of a three-particle entangled state 635
4.1 The SP-QND scheme as a quantum phase gate 635
4.2 Building step-by-step three-particle entanglement: Principle 638
4.3 Detection of the three-particle entanglement 640
5 Decoherence and quantum measurement 645
5.1 Quantum measurement theory 645
5.1.1 The postulates 645
5.1.2 Von Neumann's analysis of meters 646
5.2 Observing progressive decoherence during a measurement process 648
5.2.1 Measuring the atom state with the field phase 648
5.2.2 Characterizing the Schrdinger cat state 649
5.3 Theoretical analysis 652
5.4 Decoherence and interpretation of a quantum measurement 655
6 Conclusion and perspectives 656
Course 10. Basic Concepts in Quantum Computation
by A. Ekert, P.M. Hayden and H. Inamori 659
1 Qubits, gates and networks 663
2 Quantum arithmetic and function evaluations 668
3 Algorithms and their complexity 672
4 From interferometers to computers 675
5 The first quantum algorithms 679
6 Quantum search 682
7 Optimal phase estimation 684

8 Periodicity and quantum factoring 686
9 Cryptography 689
10 Conditional quantum dynamics 693
11 Decoherence and recoherence 694
12 Concluding remarks 699
Course 11. Coherent Backscattering of Light from a Cold Atomic Cloud
by G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. Mueller, C. Miniatura and R. Kaiser 699
1 Introduction 705
2 Coherent backscattering 706
2.1 Principle of CBS 706
2.2 CBS with cold atoms 707
3 Description of the experiment 708
3.1 Preparation of the atomic sample 708
3.2 CBS detection setup 709
4 Results 710
5 Conclusion 713
Seminars by participants 715
Posters 717
END
