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Bardou F., Bouchaud J., Aspect A. Levy statistics and laser cooling
Bardou F., Bouchaud J., Aspect A.  Levy statistics and laser cooling







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: Levy statistics and laser cooling

: Bardou F., Bouchaud J., Aspect A.

:

This is a book about laser cooling, a new research field with many potential applications. The authors present an original approach, using the tools and concepts of statistical physics. A new understanding of laser cooling, both intuitive and quantitative, is obtained. The volume also comprises a case study allowing non-Gaussian (Lévy) statistics, a technique being used more frequently in many different fields.


: en

: / //

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ed2k: ed2k stats

: 2002

: 199

: 26.07.2005

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Absorbing wall      see Wall
Aging      59
Anomalous diffusion      1
Anomalous random walks      1
Arrhenius law      43
Broad distributions      4259 (see also Power-law distribution Sprinkling
Broad distributions, connection with non-ergodicity      97
Broad distributions, generalized Central Limit Theorem      44
Broadband Doppler cooling on a narrow transition      141
Central limit theorem      3 2425 43
Central Limit Theorem, generalized      24 28 33 4349
Central Limit Theorem, generalized, proof      45
Change of variable as origin of broad distribution      43
Characteristic momentum      69
CLT      see Central Limit Theorem
Coefficient of variation      52
Coherence length      13
Confined model (of recycling) for Raman cooling      168 169
Confined model (of recycling) for Velocity Selective Coherent Population Trapping      159 163
Confined model (of recycling), definition      26
Confined model (of recycling), distribution of recycling times      3940
Confined model (of recycling), tests of the statistical approach      111 113 116
Cooled atoms      see Height (width) of the peak of cooled atoms Momentum Trapped
Cooled atoms fraction      see Momentum distribution (important features) Trapped
Cooled atoms fraction, definition      74
Cooled atoms fraction, optimization      135
Cooled atoms fraction, relation to non-ergodicity      97 98
Cooling      see Laser cooling
Damping (of momentum)      see Friction
Dark state      2 3 10 11 165 166
Delay function      1420 (see also Monte Carlo simulations Quantum Stochastic
Delay function in Velocity Selective Coherent Population Trapping      146
Delay function, method      14 104
Deterministic model (for trapping times)      2930 33
Devils staircase      54
Diffusion      2 8 9 Random Spatial
Dimensionality, influence on recycling time distribution      39
Dimensionality, influence on trapping time distribution      32
Dimensionality, role in subrecoil cooling, tests of the statistical approach      120122
Discrete Laplace transform      see Laplace transform
Dissipation      8
Distributions      see Broad distributions Exponential Levy Narrow Power-law Sprinkling
Domination of rare events      see Levy sum (hierarchy in)
Doppler cooling      2 8 25
Doppler cooling, broadband      see Broadband Doppler cooling
Doppler cooling, narrow transition      see Broadband Doppler cooling
Doppler effect      8 152
Doppler model (of recycling) for Velocity Selective Coherent Population Trapping      157 158 162
Doppler model (of recycling), definition      26
Doppler model (of recycling), distribution of recycling times      3738 172176
Doppler model (of recycling), tests of the statistical approach      106
Doppler shift      11 148 150
Elementary step of the momentum random walk      22 28 160 169
Ensemble average      61
Ensemble average, versus time average      6062 67 99
Ergodicity      see Ensemble average Time Non-ergodicity
Experiments on subrecoil cooling, comparison with optimized cooling conditions      134
Experiments on subrecoil cooling, overview      102103
Experiments on subrecoil cooling, role of friction in higher dimensions      121
Experiments on subrecoil cooling, role of friction in one dimension      120
Experiments on subrecoil cooling, width and shape of the peak of cooled atoms      116120
Exponential distribution      43 52
Exponential model (for trapping times)      3031 33
Feynman path integral      174175
First return time      see Recycling time
Fluctuations      8 (see also Levy sum)
Fluorescence      see Spontaneous emission
Fluorescence, rate      2 147
Friction      25 (see also Confined model)
Friction in standard laser cooling      2 7
Friction in subrecoil cooling      10 11 2526 140 159
Friction, role in subrecoil cooling, tests of the statistical approach      120122 Fig.
Gaussian distribution      4 44
Generalized CLT      see Central Limit Theorem
Generalized momentum      11
Generalized Optical Bloch Equations      13 (see also Velocity Selective Coherent Population Trapping)
Generalized Optical Bloch Equations for the Doppler model      107 109
Generalized Optical Bloch Equations for the unconfined model      110
Generalized Optical Bloch Equations, tests of the role of friction in two dimensions      121
Generalized Optical Bloch Equations, trapped atoms proportion      107 109 110
Generalized Optical Bloch Equations, width and height of the cooled peak      116
Glasses      59
GOBE      see Generalized Optical Bloch Equations
Half-width      see Width of the peak of cooled atoms
Hamiltonian, effective      14 147
Hamiltonian, effective, exact diagonalization      149
harmonic oscillator      174
Height of the peak of cooled atoms      see Momentum distribution (important features)
Height of the peak of cooled atoms for finite $\langle\tau\rangle$ and $\langle\hat{\tau}\rangle$      81
Height of the peak of cooled atoms for infinite $\langle\tau\rangle$ and $\langle\hat{\tau}\rangle$      85
Height of the peak of cooled atoms for infinite $\langle\tau\rangle$ and finite $\langle\hat{\tau}\rangle$      76
Height of the peak of cooled atoms in optimized conditions      134
Height of the peak of cooled atoms, definition      73
Height of the peak of cooled atoms, optimization      see Optimization of the peak of cooled atoms
Height of the peak of cooled atoms, physical interpretation and connection with the sprinkling distribution      74 92
Height of the peak of cooled atoms, relation to non-ergodicity      99
Heuristic arguments on subrecoil cooling      6970
Hierarchical structure      see Levy sum
Hypergeometric function      76
Interaction time      12 23
Isotropic random walk      see Random walk (isotropic)
Jump rate      3 912 2021 147
Jump rate in p=0      see Non-vanishing jump rate at p=0
Jump rate, calculation for Raman cooling      164168
Jump rate, inhomogeneous      911 22
Jump rate, models      2528 (see also Models)
Jump rate, non-quadratic      32
Jump rate, quadratic      2831
Jump rate, quantum calculations for Velocity Selective Coherent Population Trapping      146154
Jump rate, simplified      2021
Jump, momentum      see Momentum (jumps)
Jump, quantum      see Quantum jump
Kundt tube      10
Laplace transform      36
Laplace transform of a Levy distribution      45
Laplace transform of a power-law distribution      47
Laplace transform, discrete      36
Laplace transform, notation      45
Largest term      see Levy sum
Laser cooling      see Doppler cooling Non-ergodic Raman Sisyphus Standard Subrecoil VSCPT
Laser cooling, experiments      see Experiments on subrecoil cooling
Laser cooling, introduction to      12
Laser wavelength      13
Law of Large Numbers      4 (see also Levy sum N-dependence)
Levy distribution      4 45
Levy distribution, Laplace transform      45
Levy distribution, properties      48
Levy statistics      4 4259
Levy sum      44
Levy sum, definition      44
Levy sum, fluctuations      52
Levy sum, hierarchy in      50 54 55 70
Levy sum, largest term      51
Levy sum, N-dependence      49
Levy sum, numerical illustration      53
Levy sum, predictability      53
Levy sum, properties      4955
Levy sum, repeatability      53
Logarithmic corrections when $\mu=1$      86 177179
Many-atom effects      142
MCWF      see Monte Carlo (Wave Function)
Median momentum      73 (see also Momentum distribution (important features))
Models of recycling      see Recycling (region) Unconfined Confined Doppler
Models of the inhomogeneous random walk      Section 3.2
Models of trapping      see Trapping region Trapping Trapping
Molasses      8
Momentum distribution      6987 (see also Heuristic arguments on subrecoil cooling)
Momentum distribution for finite $\langle\tau\rangle$ and finite $\langle\hat{\tau}\rangle$      79
Momentum distribution for infinite $\langle\hat{\tau}\rangle$      83
Momentum distribution for infinite $\langle\tau\rangle$ and finite $\langle\hat{\tau}\rangle$      75
Momentum distribution, along a given axis      7273
Momentum distribution, characteristic momentum      see Characteristic momentum
Momentum distribution, cooled atoms fraction      see Cooled atoms fraction
Momentum distribution, explicit forms      75 80 84
Momentum distribution, expressions of      7175
Momentum distribution, flatness around p=0      76 81 96 119 Fig.
Momentum distribution, height      see Height of the peak of cooled atoms
Momentum distribution, important features      77 82 85 86
Momentum distribution, median      see Median momentum
Momentum distribution, modulus distribution      71
Momentum distribution, overview of main results      86
Momentum distribution, rate equation      see Rate equation
Momentum distribution, self-similarity      77 82 85 118 Fig.
Momentum distribution, shape      see Shape of the peak of cooled atoms
Momentum distribution, tails      see Tails of the peak of cooled atoms
Momentum distribution, tests of the statistical approach      see Tests (of the statistical approach)
Momentum distribution, trapped atoms      see Trapped atoms
Momentum distribution, width      see Width of the peak of cooled atoms
Momentum of a single photon      2
Momentum, confinement      see Friction
Momentum, damping      see Friction
Momentum, diffusion      see Diffusion
Momentum, generalized      see Generalized momentum
Momentum, jumps      28
Momentum, median      see Median
Momentum, random walk      see Random walk
Momentum, trapping      see Dark state
Monte Carlo, simulations      1416 (see also Delay function Quantum Raman Stochastic
Monte Carlo, Wave Function      16
Narrow distribution      see Sprinkling distribution
Narrow distribution, Central Limit Theorem      44
Non-ergodic cooling      3 100
Non-ergodicity      24 67 96100 122
Non-ergodicity with non-vanishing jump rate at p=0      96
Non-ergodicity, connection with broad distributions      97
Non-ergodicity, cross-over between non-ergodic and steady-state behaviour with non-vanishing jump rate at p=0      96
Non-ergodicity, fraction-limited      97 98 100
Non-ergodicity, global      97 98 100
Non-ergodicity, non-ergodic versus ergodic histories      98 99
Non-ergodicity, various degrees      97
Non-stationarity      96100
Non-stationarity, momentum distribution flatness as a signature of non-stationarity and non-ergodicity      96
Non-vanishing jump rate at p=0      9396
Normal distribution      see Gaussian distribution
Optical molasses      8
Optimization of the peak of cooled atoms      124136
Optimization of the peak of cooled atoms, comparison with experiments      134
Optimization of the peak of cooled atoms, cooled atoms fraction      135
Optimization of the peak of cooled atoms, features of the optimized cooling      133135
Optimization of the peak of cooled atoms, intuitive explanation of the existence of an optimum      128129
Optimization of the peak of cooled atoms, optimization parameter      126 127
Optimization of the peak of cooled atoms, parametrization      see Parametrization of the cooling process
Optimization of the peak of cooled atoms, possible improvements      140
Optimization of the peak of cooled atoms, random walk interpretation of the optimized solution      135136
Optimization of the peak of cooled atoms, using Levy sums      131132
Optimization of the peak of cooled atoms, using the expression of the height      130131
Order statistics      see Levy sum (hierarchical structure)
Parameters of the statistical models for Raman cooling      168170
Parameters of the statistical models for Velocity Selective Coherent Population Trapping      155160
Parameters of the statistical models, correspondence with atomic and laser parameters      145171
Parametrization of the cooling process      126128
Peak of the momentum distribution, contribution to the sprinkling distribution      see Sprinkling distribution
Peak of the momentum distribution, height      see Height of the peak of cooled atoms
Peak of the momentum distribution, tails      see Momentum distribution Sprinkling
Peak of the momentum distribution, width      see Width of the peak of cooled atoms
Phase space density      74
Poisson process      59
Power-law distribution      4244 (see also Broad distribution)
Power-law distribution, Laplace transform      47
Power-law distribution, random generator      53
Power-law tails      43
Predictability      see Levy sum
Proportion of trapped atoms      see Trapped atoms
Quadratic jump rate      see Jump rate
Quantum Jump      14
Quantum jump simulations      see Delay function Velocity
Quantum jump simulations for the confined model      112
Quantum jump simulations for the Doppler model      107
Quantum jump simulations for the unconfined model      109
Quantum jump simulations, tests of the role of friction in one dimension      120
Quantum jump simulations, trapped atoms proportion      107 109 112
Quantum jump simulations, width and shape of the peak of cooled atoms      113
Quantum jump, description      1415 (see also Delay function Monte Stochastic
Quasi-steady-state for the tails      see Tails of the peak of cooled atoms
Raman cooling      2 1011 25 28 34
Raman cooling, correspondence between statistical parameters and atomic and laser parameters      164171
Raman cooling, experiments      see Experiments on subrecoil cooling
Raman cooling, jump rate      164168
Raman cooling, Monte Carlo simulations      105
Raman cooling, optimization parameter      127
Raman cooling, sequence of pulses      165168
Raman cooling, single pulse excitation      164165
Random generator      see Power-law distribution
Random recoil      9
Random walk      see Diffusion
Random walk in Hilbert space      1619 142
Random walk in standard cooling      8
Random walk in subrecoil cooling      912 16
Random walk of the momentum      2 8 9 1922
Random walk, anomalous      see Anomalous random walks
Random walk, inhomogeneous      912 16 22
Random walk, interpretation of optimum cooling      135136
Random walk, isotropic      25
Rare events      see Levy sum (hierarchy in)
Rate equation for the momentum distribution      8891
Recoil, limit      9 12
Recoil, random      8 9
Recoil, single photon      2
Recoil, temperature      see Temperature
recycling      2228 3441 Confined Unconfined Doppler
Recycling, region      2223 26
Recycling, time      2325
Recycling, time distribution      3441 162 171
Recycling, versus trapping      see Trapping
Renewal density      58 (see also Sprinkling distribution)
Renewal process      58 138
Renewal process in quantum optics      58
Repeatability      see Levy sum
Return time      see Recycling (time)
Scaling      4 (see also Law of large numbers Levy N-dependence)
Scattering rate      2 (see also Jump rate)
Second-order correlation function      58 59
Self-similarity      see Momentum distribution
Self-similarity in Levy sums      55
Shape of the peak of cooled atoms      see Momentum distribution (important features) Quantum
Shape of the peak of cooled atoms for finite $\langle\tau\rangle$ and $\langle\hat{\tau}\rangle$      81
Shape of the peak of cooled atoms for infinite $\langle\tau\rangle$ and $\langle\hat{\tau}\rangle$      85
Shape of the peak of cooled atoms for infinite $\langle\tau\rangle$ and finite $\langle\hat{\tau}\rangle$      76
Shape of the peak of cooled atoms, results of the statistical approach for VSCPT (confined)      113
Shape of the peak of cooled atoms, tests of the statistical approach      113120 Figure Figure Figure
Sisyphus cooling      2 8 159
Sojourn time      12
Spatial diffusion      74
Spontaneous emission      8 14
Sprinkling distribution      5559
Sprinkling distribution as a source term for the momentum distribution rate equation      89
Sprinkling distribution for infinite $\langle\tau\rangle$ and $\langle\hat{\tau}\rangle$      66
Sprinkling distribution for infinite $\langle\tau\rangle$ and finite $\langle\hat{\tau}\rangle$      65
Sprinkling distribution of a broad distribution      59
Sprinkling distribution of a narrow distribution      58
Sprinkling distribution of exit events      62
Sprinkling distribution of return events      61 62
Sprinkling distribution, contributions of the peak and of the tails      89
Sprinkling distribution, definition      55
Sprinkling distribution, examples      57
Sprinkling distribution, interpretation of the time dependence      90
Sprinkling distribution, Laplace transform      55
Sprinkling distribution, logarithmic corrections when $\mu=1$      178
Sprinkling distribution, role in the height of the peak of cooled atoms      see Height of the peak of cooled atoms (physical interpretation)
Sprinkling distribution, role in the momentum distribution expressions      75
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