Lévy Statistics and Laser Cooling : How Rare Events Bring Atoms to Rest Book

	Foreword
	Acknowledgements
1	Introduction	1
1.1	Laser cooling	1
1.2	Subrecoil laser cooling	2
1.3	Subrecoil cooling and Levy statistics	3
1.4	Content of the book	5
2	Subrecoil laser cooling and anomalous random walks	7
2.1	Standard laser cooling: friction and the recoil limit	7
2.2	Laser cooling based on inhomogeneous random walks in momentum space	9
2.3	Quantum description of subrecoil laser cooling	12
2.4	From quantum optics to classical random walks	19
3	Trapping and recycling. Statistical properties	22
3.1	Trapping and recycling regions	22
3.2	Models of inhomogeneous random walks	25
3.3	Probability distribution of the trapping times	28
3.4	Probability distribution of the recycling times	34
4	Broad distributions and Levy statistics: a brief overview	42
4.1	Power-law distributions. When do they occur?	42
4.2	Generalized Central Limit Theorem	44
4.3	Qualitative discussion of some properties of Levy sums	49
4.4	Sprinkling distribution	55
5	The proportion of atoms trapped in quasi-dark states	60
5.1	Ensemble averages versus time averages	60
5.2	Calculation of the proportion of trapped atoms	62
5.3	Discussion: non-ergodic behaviour of the trapped population	67
6	The momentum distribution	69
6.1	Brief survey of previous heuristic arguments	69
6.2	Expressions of the momentum distribution and of related quantities	71
6.3	Case of an infinite average trapping time and a finite average recycling time	75
6.4	Case of a finite average trapping time and a finite average recycling time	79
6.5	Cases with an infinite average recycling time	83
6.6	Overview of main results	86
7	Physical discussion	88
7.1	Equivalence with a rate equation description	88
7.2	Tails of the momentum distribution	91
7.3	Height of the peak of the momentum distribution	92
7.4	Effect of a non-vanishing jump rate at zero momentum	93
7.5	Non-stationarity and non-ergodicity	96
8	Tests of the statistical approach	101
8.1	Motivation	101
8.2	Overview of other approaches	102
8.3	Proportion of trapped atoms in one-dimensional [sigma][subscript +]/[sigma][subscript -] VSCPT	105
8.4	Width and shape of the peak of cooled atoms	113
8.5	Role of friction and of dimensionality	120
8.6	Conclusion
9	Example of application: optimization of the peak of cooled atoms	124
9.1	Introduction	124
9.2	Parametrization	126
9.3	Why is there an optimum parameter?	128
9.4	Optimization using the expression of the height	130
9.5	Optimization using Levy sums	131
9.6	Features of the optimized cooling	133
9.7	Random walk interpretation of the optimized solution	135
10	Conclusion	137
10.1	What has been done in this book	137
10.2	Significance and importance of the results	138
10.3	Possible extensions	140
App. A	Correspondence between parameters of the statistical models and atomic and laser parameters	145
App. B	The Doppler case	172
App. C	The special case [mu] = 1	177
	References	181
	Index of main notation	189
	Index	195