Bardou F., Bouchaud J., Aspect A. — Levy statistics and laser cooling :: Электронная библиотека попечительского совета мехмата МГУ

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

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Название: Levy statistics and laser cooling

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

Аннотация:

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.

Язык:

Рубрика: Физика/Физика твёрдого тела/Приложения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 199

Добавлена в каталог: 26.07.2005

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Предметный указатель

Absorbing wall      see “Wall”
Aging      59
Anomalous diffusion      1
Anomalous random walks      1
Arrhenius law      43
Broad distributions      42—59 (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 24—25 43
Central Limit Theorem, generalized      24 28 33 43—49
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      39—40
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      14—20 (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)      29—30 33
Devil’s 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      120—122
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      37—38 172—176
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      60—62 67 99
Ergodicity      see “Ensemble average” “Time “Non-ergodicity”
Experiments on subrecoil cooling, comparison with optimized cooling conditions      134
Experiments on subrecoil cooling, overview      102—103
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      116—120
Exponential distribution      43 52
Exponential model (for trapping times)      30—31 33
Feynman path integral      174—175
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 25—26 140 159
Friction, role in subrecoil cooling, tests of the statistical approach      120—122 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      69—70
Hierarchical structure      see “Levy sum”
Hypergeometric function      76
Interaction time      12 23
Isotropic random walk      see “Random walk (isotropic)”
Jump rate      3 9—12 20—21 147
Jump rate in p=0      see “Non-vanishing jump rate at p=0”
Jump rate, calculation for Raman cooling      164—168
Jump rate, inhomogeneous      9—11 22
Jump rate, models      25—28 (see also “Models”)
Jump rate, non-quadratic      32
Jump rate, quadratic      28—31
Jump rate, quantum calculations for Velocity Selective Coherent Population Trapping      146—154
Jump rate, simplified      20—21
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      1—2
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 42—59
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      49—55
Levy sum, repeatability      53
Logarithmic corrections when $\mu=1$       86 177—179
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      69—87 (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      72—73
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      71—75
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      14—16 (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 96—100 122
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      96—100
Non-stationarity, momentum distribution flatness as a signature of non-stationarity and non-ergodicity      96
Non-vanishing jump rate at p=0      93—96
Normal distribution      see “Gaussian distribution”
Optical molasses      8
Optimization of the peak of cooled atoms      124—136
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      133—135
Optimization of the peak of cooled atoms, intuitive explanation of the existence of an optimum      128—129
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      135—136
Optimization of the peak of cooled atoms, using Levy sums      131—132
Optimization of the peak of cooled atoms, using the expression of the height      130—131
Order statistics      see “Levy sum (hierarchical structure)”
Parameters of the statistical models for Raman cooling      168—170
Parameters of the statistical models for Velocity Selective Coherent Population Trapping      155—160
Parameters of the statistical models, correspondence with atomic and laser parameters      145—171
Parametrization of the cooling process      126—128
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      42—44 (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      14—15 (see also “Delay function” “Monte “Stochastic
Quasi-steady-state for the tails      see “Tails of the peak of cooled atoms”
Raman cooling      2 10—11 25 28 34
Raman cooling, correspondence between statistical parameters and atomic and laser parameters      164—171
Raman cooling, experiments      see “Experiments on subrecoil cooling”
Raman cooling, jump rate      164—168
Raman cooling, Monte Carlo simulations      105
Raman cooling, optimization parameter      127
Raman cooling, sequence of pulses      165—168
Raman cooling, single pulse excitation      164—165
Random generator      see “Power-law distribution”
Random recoil      9
Random walk      see “Diffusion”
Random walk in Hilbert space      16—19 142
Random walk in standard cooling      8
Random walk in subrecoil cooling      9—12 16
Random walk of the momentum      2 8 9 19—22
Random walk, anomalous      see “Anomalous random walks”
Random walk, inhomogeneous      9—12 16 22
Random walk, interpretation of optimum cooling      135—136
Random walk, isotropic      25
Rare events      see “Levy sum (hierarchy in)”
Rate equation for the momentum distribution      88—91
Recoil, limit      9 12
Recoil, random      8 9
Recoil, single photon      2
Recoil, temperature      see “Temperature”
recycling      22—28 34—41 “Confined “Unconfined “Doppler
Recycling, region      22—23 26
Recycling, time      23—25
Recycling, time distribution      34—41 162 171
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      113—120 Figure Figure Figure
Sisyphus cooling      2 8 159
Sojourn time      12
Spatial diffusion      74
Spontaneous emission      8 14
Sprinkling distribution      55—59
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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