Coherent atomic matter waves

Coherent atomic matter waves......Page 1
Preface......Page 2
CONTENTS......Page 5
1......Page 17
1.1 1925: Einstein’s prediction for the ideal Bose gas......Page 20
1.3.1 Simple systems for the theory......Page 21
1.3.2 New features......Page 22
2.1.1 In the basis of harmonic levels......Page 23
2.1.3 In position space......Page 26
2.1.4 Relation to Einstein’s condition $ρλ3_{dB} = ζ(3/2)$......Page 29
2.2.1 The Wigner distribution......Page 30
2.2.2 Critical temperature in the semiclassical limit......Page 31
2.3.2 Energy of the gas as a function of temperature and the number of particles......Page 33
2.3.3 Density profile of the condensate......Page 34
3 A model for the atomic interactions......Page 35
3.1 Reminder of scattering theory......Page 36
3.1.1 General results of scattering theory......Page 37
3.1.2 Low energy limit for scattering by a finite range potential......Page 39
3.2.1 Why not keep the exact interaction potential?......Page 40
3.2.2 Scattering states of the pseudo-potential......Page 43
3.2.3 Bound states of the pseudo-potential......Page 44
3.3.1 Regime of the Born approximation......Page 45
3.3.2 Relevance of the pseudo-potential beyond the Born approximation......Page 46
4.1.1 Few body-density matrices......Page 47
4.1.2 Equations of the hierarchy......Page 48
4.2.1 Mean field potential for the non-condensed particles......Page 49
4.2.2 Effect of interactions on $T_c$......Page 51
4.3.1 Improved Hartree–Fock Ansatz......Page 52
4.3.3 At thermal equilibrium......Page 53
4.4.2 Experimental results for the energy of the gas......Page 54
5 Properties of the condensate wavefunction......Page 56
5.1.1 From Hartree–Fock......Page 57
5.1.2 Variational formulation......Page 58
5.2 Gaussian Ansatz......Page 61
5.2.1 Time–independent case......Page 62
5.2.2 Time–dependent case......Page 66
5.3.1 Time–independent case......Page 67
5.3.2 How to extend the Thomas–Fermi approximation to the time–dependent case?......Page 70
5.3.3 Hydrodynamic equations......Page 71
5.3.4 Classical hydrodynamic approximation......Page 73
5.4.1 The scaling solution......Page 74
5.4.3 Breathing frequencies of the condensate......Page 76
6 What we learn from a linearization of the Gross–Pitaevskii equation......Page 77
6.1.1 Linearize the Gross–Pitaevskii solution around a steady–state solution......Page 78
6.1.2 Extracting the “relevant part” from $δφ$......Page 80
6.1.3 Spectral properties of $\mathcal{L}$ and dynamical stability......Page 81
6.1.4 Diagonalization of $\mathcal{L}$ with eigenvalue $\epsilon_k$......Page 82
6.1.6 Link between eigenmodes of LGP and eigenmodes of L......Page 84
6.2.1 Condensate in a box......Page 85
6.2.2 Demixing instability......Page 89
6.3.1 Linearized classical hydrodynamic equations......Page 93
6.3.2 Validity condition of the linearized classical hydrodynamic equations......Page 94
6.3.3 Approximate spectrum in a harmonic trap......Page 95
7 Bogoliubov approach and thermodynamical stability......Page 96
7.1 Small parameter of the theory......Page 97
7.3 Next order in $ε$: Linear dynamics of non-condensed particles......Page 98
7.4 Bogoliubov Hamiltonian......Page 100
7.5 Order $ε^2$: Corrections to the Gross–Pitaevskii equation......Page 102
7.6 Thermal equilibrium of the gas of quasi-particles......Page 103
7.7 Condensate depletion and the small parameter $(ρa^3)^1/2$......Page 104
7.8 Fluctuations in the number of condensate particles......Page 107
7.9 A simple reformulation of the thermodynamical stability condition......Page 110
7.11.1 Real condensate wavefunction with a node......Page 112
7.11.2 Condensate with a vortex......Page 113
8 Phase coherence properties of Bose–Einstein condensates......Page 115
8.1 Interference between two BECs......Page 116
8.1.1 A very simple model......Page 117
8.1.2 A trap to avoid......Page 118
8.1.4 Analytical solution......Page 120
8.2.1 Physical motivation......Page 123
8.2.2 A quadratic approximation for the energy......Page 124
8.2.3 State vector at time t......Page 125
8.2.4 An indicator of phase coherence......Page 126
9.1 The ground state of spinor condensates......Page 129
9.1.1 A model interaction potential......Page 130
9.1.2 Ground state in the Hartree–Fock approximation......Page 131
9.1.3 Exact ground state of the spinor part of the problem......Page 133
9.1.4 Advantage of a symmetry–breaking description......Page 136
9.2.1 How to make a solitonic condensate?......Page 138
9.2.2 Ground state of the one-dimensional attractive Bose gas......Page 142
9.2.3 Physical advantage of the symmetry–breaking description......Page 144
References......Page 148
2......Page 152
1 Introduction......Page 154
2 Optical properties of a Bose–Einstein condensate......Page 155
2.1.1 Elastic and inelastic light scattering......Page 156
2.1.2 Light scattering from atomic beams and atoms at rest......Page 159
2.1.3 Relation to the dynamic structure factor of a many-body system......Page 160
2.2.1 The homogeneous condensate......Page 161
2.2.2 Bragg scattering as a probe of pair correlations in the condensate......Page 163
2.2.3 Mean-field theory determination of $S(\vec q, \omega)$......Page 165
2.2.4 The inhomogeneous condensate......Page 167
2.2.5 Relevance of Doppler broadening......Page 169
2.3 Experimental aspects of Bragg spectroscopy......Page 170
2.4.1 Measurement of line shift and line broadening......Page 172
2.4.2 A measurement of the coherence length of a Bose–Einstein condensate......Page 176
2.5.1 Experimental study......Page 178
2.5.2 Suppression of light scattering from a Bose–Einstein condensate......Page 179
3.2.1 Semiclassical derivation of the gain mechanism......Page 182
3.2.2 Four-wave mixing of light and atoms......Page 184
3.2.3 Bosonic stimulation by scattered atoms or scattered light?......Page 185
3.2.4 Observation of directional emission of light and atoms......Page 188
3.2.5 Relation to other non-linear phenomena......Page 192
3.3 Phase-coherent amplification of matter waves......Page 194
4 Spinor Bose–Einstein condensates......Page 197
4.1 The implications of rotational symmetry......Page 199
4.2 Tailoring the ground-state structure with magnetic fields......Page 203
4.3 Spin-domain diagrams: A local density approximation to the spin structure of spinor condensates......Page 206
4.4 Experimental methods for the study of spinor condensates......Page 208
4.5 The formation of ground-state spin domains......Page 209
4.6 Miscibility and immiscibility of spinor condensate components......Page 212
4.7 Metastable states of spinor Bose–Einstein condensates......Page 213
4.7.1 Metastable spin-domain structures......Page 214
4.7.2 Metastable spin composition......Page 217
4.8 Quantum tunneling......Page 218
4.9 Magnetic field dependence of spin-domain boundaries......Page 223
Conclusions......Page 226
References......Page 229
3......Page 233
1 Introduction......Page 235
2.1 Second quantization......Page 237
2.2 Grassmann variables and coherent states......Page 241
2.3 Functional integrals......Page 245
2.4.1 Semiclassical method......Page 248
2.4.2 Matsubara expansion......Page 249
2.4.3 Green’s function method......Page 251
2.5 Interactions and Feynmann diagrams......Page 254
2.6 Hartree–Fock theory for an atomic Fermi gas......Page 259
2.7 Landau theory of phase transitions......Page 263
2.8.1 Superfluidity......Page 266
2.8.2 Some atomic physics......Page 273
2.8.3 Superconductivity......Page 275
3.1 Macroscopic quantum tunneling of a condensate......Page 280
3.2 Phase diffusion......Page 286
3.3.1 Ideal Bose gas......Page 290
3.3.2 Ideal Bose gas in contact with a reservoir......Page 296
3.4 Condensate formation......Page 309
3.4.2 Strong-coupling limit......Page 316
3.5 Collective modes......Page 321
References......Page 325
4......Page 330
1 Introduction......Page 332
2.1 Ramsey interference......Page 333
2.2 Interference due to different physical paths......Page 337
2.3 Path integral description of interference......Page 338
2.4 Atom optics......Page 339
2.5 Interference with combined internal and external degrees of freedom......Page 342
3.1 Interferometers based on microfabricated structures......Page 347
3.2.1 Diffraction from an optical standing wave......Page 350
3.2.2 Interaction of atoms with light in the sudden approximation......Page 351
4 An atom interferometry measurement of the acceleration due to gravity......Page 352
4.1 Circumventing experimental obstacles......Page 355
4.2 Stimulated Raman transitions......Page 356
4.3 Frequency sweep and stability issues......Page 359
4.4 Vibration isolation......Page 360
4.5 Experimental results......Page 361
5 Interferometry based on adiabatic transfer......Page 365
5.1 Theory of adiabatic passage with time-delayed pulses......Page 367
5.2 Atom interferometry using adiabatic transfer......Page 369
5.3 A measurement of the photon recoil and $hbar/M$......Page 372
6 Atom gyroscopes......Page 376
6.1 A comparison of atom interferometers......Page 377
6.2 Future prospects......Page 378
References......Page 380
5......Page 384
1 Introduction......Page 386
2.1 Mesoscopic quantum mechanics......Page 388
2.2 Phenomenological radiative transfer......Page 391
2.3 Mesoscopic physics with classical waves......Page 392
2.4 Mesoscopic light scattering in atomic gases......Page 393
3 Light scattering from simple atoms......Page 396
3.1 Vector Green’s function......Page 397
3.2 An atom as a point scatterer......Page 398
3.3 Polarization, cross-section and stored energy......Page 400
3.4 Two atoms: Dipole–dipole coupling......Page 402
3.5 Induced dipole force between two simple atoms......Page 406
3.6 Van der Waals interaction......Page 408
4 Applications in multiple scattering......Page 409
4.1 Effective medium......Page 410
4.2 Group and energy velocity......Page 411
4.3 Dipole–dipole coupling in the medium......Page 415
4.4 Coherent backscattering......Page 417
4.5 Dependent scattering with quantum correlation......Page 421
4.6 From weak towards strong localization......Page 423
References......Page 425
6......Page 428
1 What is quantum chaos?......Page 430
1.1 Classical chaos......Page 431
1.2 Quantum dynamics......Page 432
1.3 Semiclassical dynamics......Page 434
1.4 Physical situations of interest......Page 436
1.5.1 Hamiltonian......Page 438
1.5.2 Classical scaling......Page 439
1.5.4 Quantum scaling – Scaled spectroscopy......Page 441
2.2 Ehrenfest time......Page 443
2.3 Heisenberg time......Page 445
2.4 Inelastic time......Page 447
3.1 Level dynamics......Page 448
3.2 Statistical analysis of the spectral .uctuations......Page 450
3.2.2 Unfolding the spectrum......Page 451
3.2.4 Number variance......Page 452
3.3 Regular regime......Page 453
3.4 Chaotic regime – Random Matrix Theory......Page 454
3.5 Usefulness of Random Matrix Theory......Page 457
3.6 Other statistical ensembles......Page 459
4.1 Regular systems – EBK/WKB quantization......Page 461
4.2 Semiclassical propagator......Page 465
4.3 Green’s function......Page 467
4.4 Trace formula......Page 469
4.5 “Backward” application of the trace formula......Page 471
4.7 Scarring......Page 473
4.8 Convergence properties of the trace formula......Page 474
4.9 An example: The helium atom......Page 476
4.10 Link with Random Matrix Theory......Page 477
5 Transport properties – Localization......Page 479
5.1 The classical kicked rotor......Page 480
5.2 The quantum kicked rotor......Page 481
5.3 Dynamical localization......Page 482
5.4 Link with Anderson localization......Page 484
5.5 Experimental observation of dynamical localization......Page 485
5.6 The effect of noise and decoherence......Page 487
6 Conclusion......Page 488
References......Page 490
7......Page 493
1 Introduction......Page 495
2 The existence of photon localization......Page 498
2.1 Independent scatterers and microscopic resonances......Page 499
2.2 A new criterion for light localization......Page 501
2.3 Photonic band gap formation......Page 502
3.1 Theory of the photon–atom bound state......Page 503
3.2 Lifetime of the photon–atom bound state......Page 510
4.1 Single atom radiative dynamics......Page 512
4.2 Collective time scale factors......Page 516
4.3 Superradiance near a photonic band edge......Page 520
5.1 Low-threshold nonlinear optics......Page 523
5.2 Collective switching and transistor e.ects......Page 525
6 Resonant nonlinear dielectric response in a doped photonic band gap material......Page 528
7 Collective switching and inversion without fluctuation in a colored vacuum......Page 532
References......Page 540
8......Page 544
1 Introduction and overview......Page 546
2 Quantum measurements......Page 550
2.1 Bit-by-bit measurement and quantum entanglement......Page 552
2.2 Interactions and the information transfer in quantum measurements......Page 555
2.3 Monitoring by the environment and decoherence......Page 557
2.4 One-bit environment for a bit-by-bit measurement......Page 559
2.5 Decoherence of a single (qu)bit......Page 561
2.6 Decoherence, einselection, and controlled shifts......Page 565
3 Dynamics of quantum open systems: Master equations......Page 568
3.1 Master equation: Perturbative evaluation......Page 569
3.2 Example 1: Perturbative master equation in quantum Brownian motion......Page 572
3.3 Example 2: Perturbative master equation for a two-level system coupled to a bosonic heat bath......Page 575
3.4 Example 3: Perturbative master equation for a particle interacting with a quantum field......Page 577
3.5 Exact master equation for quantum Brownian motion......Page 579
4.1 Decoherence of a superposition of two coherent states......Page 585
4.2 Predictability sieve and preferred states for QBM......Page 589
4.3 Energy eigenstates can also be selected by the environment!......Page 591
5 Deconstructing decoherence: Landscape beyond the standard models......Page 592
5.1 Saturation of the decoherence rate at large distances......Page 593
5.2 Decoherence at zero temperature......Page 594
5.3 Preexisting correlations between the system and the environment......Page 596
6.1 Quantum predictability horizon: How the correspondence is lost......Page 600
6.2 Exponential instability vs. decoherence......Page 602
6.3 The arrow of time: A price of classicality?......Page 604
6.4 Decoherence, einselection, and the entropy production.......Page 608
7 How to fight against decoherence: Quantum error correcting codes......Page 609
7.2 How to protect a quantum bit......Page 610
7.3 Stabilizer quantum error-correcting codes......Page 617
8 Discussion......Page 620
References......Page 622
9......Page 626
1 Introduction......Page 628
2 Microwave CQED experiments: The strong coupling regime......Page 630
2.1.1 Circular Rydberg atoms......Page 631
2.1.2 The photon box......Page 632
2.3 “Quantum logic” operations based on the vacuum Rabi oscillation......Page 633
3.1 Quantum non-demolition strategies......Page 635
3.2 The Ramsey interferometer for detecting a single photon......Page 636
3.3 Experimental realization......Page 638
3.3.1 Input–meter: Demonstrating the single photon phase shift......Page 639
3.3.2 Meter–output correlation: Detecting the same photon twice......Page 640
3.3.3 Input–output correlation: Quantifying the QND performance......Page 643
4.1 The SP-QND scheme as a quantum phase gate......Page 646
4.2 Building step-by-step three-particle entanglement: Principle......Page 649
4.3 Detection of the three-particle entanglement......Page 651
5.1.1 The postulates......Page 656
5.1.2 Von Neumann’s analysis of meters......Page 657
5.2.1 Measuring the atom state with the .eld phase......Page 659
5.2.2 Characterizing the Schrödinger cat state......Page 660
5.3 Theoretical analysis......Page 663
5.4 Decoherence and interpretation of a quantum measurement......Page 666
6 Conclusion and perspectives......Page 667
References......Page 668
10......Page 672
1 Qubits, gates and networks......Page 674
2 Quantum arithmetic and function evaluations......Page 679
3 Algorithms and their complexity......Page 683
4 From interferometers to computers......Page 686
5 The first quantum algorithms......Page 690
6 Quantum search......Page 693
7 Optimal phase estimation......Page 695
8 Periodicity and quantum factoring......Page 697
9 Cryptography......Page 700
10 Conditional quantum dynamics......Page 704
11 Decoherence and recoherence......Page 705
12 Concluding remarks......Page 710
References......Page 711
11......Page 713
1 Introduction......Page 715
2.1 Principle of CBS......Page 716
2.2 CBS with cold atoms......Page 717
3.1 Preparation of the atomic sample......Page 718
3.2 CBS detection setup......Page 719
4 Results......Page 720
5 Conclusion......Page 723
References......Page 724