L©vy Statistics and Laser Cooling

How Rare Events Bring Atoms to Rest

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Hardcover | 20 May 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.

Industry Reviews

'... 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

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Non-Fiction Engineering & Technology Other Technologies & Applied Sciences Applied Optics Laser Technology & Holography Science Physics Nuclear Physics Atomic & Molecular Physics Optical Physics Electricity Applied Physics

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Published: 7th February 2002

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Foreword	p. xi
Acknowledgements	p. xiii
Introduction	p. 1
Laser cooling	p. 1
Subrecoil laser cooling	p. 2
Subrecoil cooling and Levy statistics	p. 3
Content of the book	p. 5
Subrecoil laser cooling and anomalous random walks	p. 7
Standard laser cooling: friction forces and the recoil limit	p. 7
Friction forces and cooling	p. 7
The recoil limit	p. 9
Laser cooling based on inhomogeneous random walks in momentum space	p. 9
Physical mechanism	p. 9
How to create an inhomogeneous random walk	p. 10
Expected cooling properties	p. 11
Quantum description of subrecoil laser cooling	p. 12
Wave nature of atomic motion	p. 12
Difficulties of the standard quantum treatment	p. 13
Quantum jump description. The delay function	p. 14
Simulation of the atomic momentum stochastic evolution	p. 15
Generalization. Stochastic wave functions and random walks in Hilbert space	p. 16
From quantum optics to classical random walks	p. 19
Fictitious classical particle associated with the quantum random walk	p. 19
Simplified jump rate	p. 20
Discussion	p. 21
Trapping and recycling. Statistical properties	p. 22
Trapping and recycling regions	p. 22
Models of inhomogeneous random walks	p. 25
Friction	p. 25
Trapping region	p. 25
Recycling region	p. 26
Momentum jumps	p. 28
Discussion	p. 28
Probability distribution of the trapping times	p. 28
One-dimensional quadratic jump rate	p. 28
Generalization to higher dimensions	p. 32
Generalization to a non-quadratic jump rate	p. 32
Discussion	p. 33
Probability distribution of the recycling times	p. 34
Presentation of the problem: first return time in Brownian motion	p. 34
The unconfined model in one dimension	p. 35
The Doppler model in one dimension	p. 37
The confined model: random walk with walls	p. 39
Discussion	p. 40
Broad distributions and Levy statistics: a brief overview	p. 42
Power-law distributions. When do they occur?	p. 42
Generalized Central Limit Theorem	p. 44
Levy sums. Asymptotic behaviour and Levy distributions	p. 44
Sketch of the proof of the generalized CLT	p. 45
A few mathematical results	p. 47
Qualitative discussion of some properties of Levy sums	p. 49
Dependence of a Levy sum on the number of terms for [mu] [ 1	p. 49
Hierarchical structure in a Levy sum	p. 50
Large fluctuations	p. 52
Illustration with numerical simulations	p. 53
Sprinkling distribution	p. 55
Definition. Laplace transform	p. 55
Examples taken from other fields	p. 57
Asymptotic behaviour. Broad versus narrow distributions	p. 58
The proportion of atoms trapped in quasi-dark states	p. 60
Ensemble averages versus time averages	p. 60
Time average: fraction of time spent in the trap	p. 60
Ensemble average: trapped proportion	p. 61
Calculation of the proportion of trapped atoms	p. 62
Laplace transforms of the sprinkling distributions associated with the return and exit times	p. 62
Laplace transform of the proportion of trapped atoms	p. 63
Results for a finite average trapping time and a finite average recycling time	p. 64
Results for an infinite average trapping time and a finite average recycling time	p. 64
Results for an infinite average trapping time and an infinite average recycling time	p. 66
Discussion: non-ergodic behaviour of the trapped population	p. 67
The momentum distribution	p. 69
Brief survey of previous heuristic arguments	p. 69
Expressions of the momentum distribution and of related quantities	p. 71
Distribution of the momentum modulus	p. 71
Momentum distribution along a given axis	p. 72
Characterization of the cooled atoms' momentum distribution	p. 73
Case of an infinite average trapping time and a finite average recycling time	p. 75
Explicit form of the momentum distribution	p. 75
Important features of the momentum distribution	p. 77
Case of a finite average trapping time and a finite average recycling time	p. 79
Explicit form of the momentum distribution	p. 80
Important features of the momentum distribution	p. 82
Cases with an infinite average recycling time	p. 83
Overview of main results	p. 86
Physical discussion	p. 88
Equivalence with a rate equation description	p. 88
Rate equation for the momentum distribution	p. 88
Re-interpretation of the sprinkling distribution of return times as a source term	p. 89
Which atoms contribute to the sprinkling distribution of return times?	p. 89
Interpretation of the time dependence of the sprinkling distribution of return times	p. 90
Tails of the momentum distribution	p. 91
Steady-state versus quasi-steady-state	p. 91
Dependence on the various parameters	p. 92
Height of the peak of the momentum distribution	p. 92
Effect of a non-vanishing jump rate at zero momentum	p. 93
Existence of a steady-state for long times	p. 94
Intermediate times	p. 95
Non-stationarity and non-ergodicity	p. 96
Flatness of the momentum distribution around zero momentum	p. 96
Various degrees of non-ergodicity	p. 97
Connection with broad distributions	p. 97
Tests of the statistical approach	p. 101
Motivation	p. 101
Overview of other approaches	p. 102
Experiments	p. 102
Quantum optics calculations for VSCPT	p. 103
Monte Carlo simulations of Raman cooling	p. 105
Proportion of trapped atoms in one-dimensional [sigma subscript +]/[sigma subscript -] VSCPT	p. 105
Doppler model	p. 106
Unconfined model	p. 109
Confined model	p. 111
Width and shape of the peak of cooled atoms	p. 113
Statistical predictions	p. 113
Comparison to quantum calculations	p. 113
Experimental tests	p. 116
Role of friction and of dimensionality	p. 120
One-dimensional case	p. 120
Higher dimensional case	p. 120
Conclusion	p. 122
Example of application: optimization of the peak of cooled atoms	p. 124
Introduction	p. 124
Parametrization	p. 126
Why is there an optimum parameter?	p. 128
Optimization using the expression of the height	p. 130
Optimization using Levy sums	p. 131
Features of the optimized cooling	p. 133
Random walk interpretation of the optimized solution	p. 135
Conclusion	p. 137
What has been done in this book	p. 137
Significance and importance of the results	p. 138
From the point of view of Levy statistics	p. 138
From the point of view of laser cooling	p. 139
Possible extensions	p. 140
Improving the optimization	p. 140
More precise model of friction-assisted VSCPT	p. 140
Extension to other cooling schemes	p. 140
Extension to trapped atoms	p. 141
Inclusion of many-atom effects	p. 142
Correspondence between parameters of the statistical models and atomic and laser parameters	p. 145
Velocity Selective Coherent Population Trapping	p. 145
Quantum calculation of the jump rate	p. 146
Effective Hamiltonian	p. 147
Exact diagonalization	p. 149
Expansion around p = 0	p. 151
Behaviour out of the trapping dip	p. 152
Case of a negligible Doppler effect	p. 153
Parameters of the random walk models	p. 155
Trapping region and plateau: p[subscript 0] and [tau subscript 0]	p. 155
Dependence on laser intensity	p. 156
Doppler tail: p[subscript D]	p. 157
Discussion: comparison between quantum calculations and statistical models	p. 158
Confining walls: p[subscript max]	p. 159
Elementary step of the random walk: [Delta]p	p. 160
Trapping time distribution: [tau subscript b]	p. 161
Recycling time distribution	p. 162
Doppler model: [tau subscript b]	p. 162
Unconfined model: [tau subscript b]	p. 163
Confined model: ([tau])	p. 164
Raman cooling	p. 164
Jump rate	p. 164
Parameters of the random walk models	p. 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]p	p. 169
Trapping time distribution: [tau subscript b]	p. 170
Recycling time distribution: ([tau])	p. 171
The Doppler case	p. 172
Motivations	p. 172
Setting the stage	p. 172
Feynman path integral and mapping to the harmonic oscillator	p. 174
Back to the return time probability	p. 175
The special case [mu] = 1	p. 177
References	p. 181
Index of main notation	p. 189
Index	p. 195
Table of Contents provided by Syndetics. All Rights Reserved.

L©vy Statistics and Laser Cooling

How Rare Events Bring Atoms to Rest

At a Glance

Hardcover

Industry Reviews

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