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Levy Statistics and Laser Cooling : How Rare Events Bring Atoms to Rest - Francois Bardou

Levy Statistics and Laser Cooling

How Rare Events Bring Atoms to Rest


Published: 7th February 2002
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Laser cooling of atoms provides an ideal case study for the application of Levy statistics in a privileged situation where the statistical model can be derived from first principles. This book demonstrates how the most efficient laser cooling techniques can be simply and quantitatively understood in terms of non-ergodic random processes dominated by a few rare events. Levy statistics are now recognised as the proper tool for analysing many different problems for which standard Gaussian statistics are inadequate. Laser cooling provides a simple example of how Levy statistics can yield analytic predictions that can be compared to other theoretical approaches and experimental results. The authors of this book are world leaders in the fields of laser cooling and light-atom interactions, and are renowned for their clear presentation. This book will therefore hold much interest for graduate students and researchers in the fields of atomic physics, quantum optics, and statistical physics.

'... a beautifully concise yet complete introduction to the logic of this incredible technique ... students of physics and other scientists interested in laser cooling will find this book hard to beat for insight and conceptual clarity.' Mark Buchanan, New Scientist '... an excellent and readable account that will be of considerable use not only to people interested in laser cooling, but also to those wishing to see this important set of techniques make an impact in studies of ultracold matter ... a significant addition to the literature in both laser cooling and statistical physics. It is rare to have such a lucid and convincing account of a technique that will be new to most scientists. It will be greatly welcomed both by workers in the field of ultracold atom physics and by those who want to see an important theoretical apparatus used in practice.' Keith Burnett, Nature '... hard to beat for insight and conceptual clarity.' New Scientist

Forewordp. xi
Acknowledgementsp. xiii
Introductionp. 1
Laser coolingp. 1
Subrecoil laser coolingp. 2
Subrecoil cooling and Levy statisticsp. 3
Content of the bookp. 5
Subrecoil laser cooling and anomalous random walksp. 7
Standard laser cooling: friction forces and the recoil limitp. 7
Friction forces and coolingp. 7
The recoil limitp. 9
Laser cooling based on inhomogeneous random walks in momentum spacep. 9
Physical mechanismp. 9
How to create an inhomogeneous random walkp. 10
Expected cooling propertiesp. 11
Quantum description of subrecoil laser coolingp. 12
Wave nature of atomic motionp. 12
Difficulties of the standard quantum treatmentp. 13
Quantum jump description. The delay functionp. 14
Simulation of the atomic momentum stochastic evolutionp. 15
Generalization. Stochastic wave functions and random walks in Hilbert spacep. 16
From quantum optics to classical random walksp. 19
Fictitious classical particle associated with the quantum random walkp. 19
Simplified jump ratep. 20
Discussionp. 21
Trapping and recycling. Statistical propertiesp. 22
Trapping and recycling regionsp. 22
Models of inhomogeneous random walksp. 25
Frictionp. 25
Trapping regionp. 25
Recycling regionp. 26
Momentum jumpsp. 28
Discussionp. 28
Probability distribution of the trapping timesp. 28
One-dimensional quadratic jump ratep. 28
Generalization to higher dimensionsp. 32
Generalization to a non-quadratic jump ratep. 32
Discussionp. 33
Probability distribution of the recycling timesp. 34
Presentation of the problem: first return time in Brownian motionp. 34
The unconfined model in one dimensionp. 35
The Doppler model in one dimensionp. 37
The confined model: random walk with wallsp. 39
Discussionp. 40
Broad distributions and Levy statistics: a brief overviewp. 42
Power-law distributions. When do they occur?p. 42
Generalized Central Limit Theoremp. 44
Levy sums. Asymptotic behaviour and Levy distributionsp. 44
Sketch of the proof of the generalized CLTp. 45
A few mathematical resultsp. 47
Qualitative discussion of some properties of Levy sumsp. 49
Dependence of a Levy sum on the number of terms for [mu] [ 1p. 49
Hierarchical structure in a Levy sump. 50
Large fluctuationsp. 52
Illustration with numerical simulationsp. 53
Sprinkling distributionp. 55
Definition. Laplace transformp. 55
Examples taken from other fieldsp. 57
Asymptotic behaviour. Broad versus narrow distributionsp. 58
The proportion of atoms trapped in quasi-dark statesp. 60
Ensemble averages versus time averagesp. 60
Time average: fraction of time spent in the trapp. 60
Ensemble average: trapped proportionp. 61
Calculation of the proportion of trapped atomsp. 62
Laplace transforms of the sprinkling distributions associated with the return and exit timesp. 62
Laplace transform of the proportion of trapped atomsp. 63
Results for a finite average trapping time and a finite average recycling timep. 64
Results for an infinite average trapping time and a finite average recycling timep. 64
Results for an infinite average trapping time and an infinite average recycling timep. 66
Discussion: non-ergodic behaviour of the trapped populationp. 67
The momentum distributionp. 69
Brief survey of previous heuristic argumentsp. 69
Expressions of the momentum distribution and of related quantitiesp. 71
Distribution of the momentum modulusp. 71
Momentum distribution along a given axisp. 72
Characterization of the cooled atoms' momentum distributionp. 73
Case of an infinite average trapping time and a finite average recycling timep. 75
Explicit form of the momentum distributionp. 75
Important features of the momentum distributionp. 77
Case of a finite average trapping time and a finite average recycling timep. 79
Explicit form of the momentum distributionp. 80
Important features of the momentum distributionp. 82
Cases with an infinite average recycling timep. 83
Overview of main resultsp. 86
Physical discussionp. 88
Equivalence with a rate equation descriptionp. 88
Rate equation for the momentum distributionp. 88
Re-interpretation of the sprinkling distribution of return times as a source termp. 89
Which atoms contribute to the sprinkling distribution of return times?p. 89
Interpretation of the time dependence of the sprinkling distribution of return timesp. 90
Tails of the momentum distributionp. 91
Steady-state versus quasi-steady-statep. 91
Dependence on the various parametersp. 92
Height of the peak of the momentum distributionp. 92
Effect of a non-vanishing jump rate at zero momentump. 93
Existence of a steady-state for long timesp. 94
Intermediate timesp. 95
Non-stationarity and non-ergodicityp. 96
Flatness of the momentum distribution around zero momentump. 96
Various degrees of non-ergodicityp. 97
Connection with broad distributionsp. 97
Tests of the statistical approachp. 101
Motivationp. 101
Overview of other approachesp. 102
Experimentsp. 102
Quantum optics calculations for VSCPTp. 103
Monte Carlo simulations of Raman coolingp. 105
Proportion of trapped atoms in one-dimensional [sigma subscript +]/[sigma subscript -] VSCPTp. 105
Doppler modelp. 106
Unconfined modelp. 109
Confined modelp. 111
Width and shape of the peak of cooled atomsp. 113
Statistical predictionsp. 113
Comparison to quantum calculationsp. 113
Experimental testsp. 116
Role of friction and of dimensionalityp. 120
One-dimensional casep. 120
Higher dimensional casep. 120
Conclusionp. 122
Example of application: optimization of the peak of cooled atomsp. 124
Introductionp. 124
Parametrizationp. 126
Why is there an optimum parameter?p. 128
Optimization using the expression of the heightp. 130
Optimization using Levy sumsp. 131
Features of the optimized coolingp. 133
Random walk interpretation of the optimized solutionp. 135
Conclusionp. 137
What has been done in this bookp. 137
Significance and importance of the resultsp. 138
From the point of view of Levy statisticsp. 138
From the point of view of laser coolingp. 139
Possible extensionsp. 140
Improving the optimizationp. 140
More precise model of friction-assisted VSCPTp. 140
Extension to other cooling schemesp. 140
Extension to trapped atomsp. 141
Inclusion of many-atom effectsp. 142
Correspondence between parameters of the statistical models and atomic and laser parametersp. 145
Velocity Selective Coherent Population Trappingp. 145
Quantum calculation of the jump ratep. 146
Effective Hamiltonianp. 147
Exact diagonalizationp. 149
Expansion around p = 0p. 151
Behaviour out of the trapping dipp. 152
Case of a negligible Doppler effectp. 153
Parameters of the random walk modelsp. 155
Trapping region and plateau: p[subscript 0] and [tau subscript 0]p. 155
Dependence on laser intensityp. 156
Doppler tail: p[subscript D]p. 157
Discussion: comparison between quantum calculations and statistical modelsp. 158
Confining walls: p[subscript max]p. 159
Elementary step of the random walk: [Delta]pp. 160
Trapping time distribution: [tau subscript b]p. 161
Recycling time distributionp. 162
Doppler model: [tau subscript b]p. 162
Unconfined model: [tau subscript b]p. 163
Confined model: ([tau])p. 164
Raman coolingp. 164
Jump ratep. 164
Parameters of the random walk modelsp. 168
Trapping region and plateau: p[subscript 0] and [tau subscript 0]p. 169
Confining walls: p[subscript max]p. 169
Elementary step of the random walk: [Delta]pp. 169
Trapping time distribution: [tau subscript b]p. 170
Recycling time distribution: ([tau])p. 171
The Doppler casep. 172
Motivationsp. 172
Setting the stagep. 172
Feynman path integral and mapping to the harmonic oscillatorp. 174
Back to the return time probabilityp. 175
The special case [mu] = 1p. 177
Referencesp. 181
Index of main notationp. 189
Indexp. 195
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521004220
ISBN-10: 0521004225
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 214
Published: 7th February 2002
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 24.7 x 17.4  x 1.5
Weight (kg): 0.46