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

by François Bardou , Jean-Philippe Bouchaud , Alain Aspect , Claude Cohen-Tannoudji

• ISBN13: 9780521808217
• ISBN10: 0521808219
• Format: Hardcover
• Copyright: 2002-01-21
• Publisher: Cambridge University Press

Summary

Laser cooling of atoms provides an ideal case study for the application of Lèvy 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. Lèvy 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 Lèvy 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.

Table of Contents

Foreword	xi
Acknowledgements	xiii
Introduction	1	(6)
Laser cooling	1	(1)
Subrecoil laser cooling	2	(1)
Subrecoil cooling and Levy statistics	3	(2)
Content of the book	5	(2)
Subrecoil laser cooling and anomalous random walks	7	(15)
Standard laser cooling: friction forces and the recoil limit	7	(2)
Friction forces and cooling	7	(2)
The recoil limit	9	(1)
Laser cooling based on inhomogeneous random walks in momentum space	9	(3)
Physical mechanism	9	(1)
How to create an inhomogeneous random walk	10	(1)
Expected cooling properties	11	(1)
Quantum description of subrecoil laser cooling	12	(7)
Wave nature of atomic motion	12	(1)
Difficulties of the standard quantum treatment	13	(1)
Quantum jump description. The delay function	14	(1)
Simulation of the atomic momentum stochastic evolution	15	(1)
Generalization. Stochastic wave functions and random walks in Hilbert space	16	(3)
From quantum optics to classical random walks	19	(3)
Fictitious classical particle associated with the quantum random walk	19	(1)
Simplified jump rate	20	(1)
Discussion	21	(1)
Trapping and recycling. Statistical properties	22	(20)
Trapping and recycling regions	22	(3)
Models of inhomogeneous random walks	25	(3)
Friction	25	(1)
Trapping region	25	(1)
Recycling region	26	(2)
Momentum jumps	28	(1)
Discussion	28	(1)
Probability distribution of the trapping times	28	(6)
One-dimensional quadratic jump rate	28	(4)
Generalization to higher dimensions	32	(1)
Generalization to a non-quadratic jump rate	32	(1)
Discussion	33	(1)
Probability distribution of the recycling times	34	(8)
Presentation of the problem: first return time in Brownian motion	34	(1)
The unconfined model in one dimension	35	(2)
The Doppler model in one dimension	37	(2)
The confined model: random walk with walls	39	(1)
Discussion	40	(2)
Broad distributions and Levy statistics: a brief overview	42	(18)
Power-law distributions. When do they occur?	42	(2)
Generalized Central Limit Theorem	44	(5)
Levy sums. Asymptotic behaviour and Levy distributions	44	(1)
Sketch of the proof of the generalized CLT	45	(2)
A few mathematical results	47	(2)
Qualitative discussion of some properties of Levy sums	49	(6)
Dependence of a Levy sum on the number of terms for μ < 1	49	(1)
Hierarchical structure in a Levy sum	50	(2)
Large fluctuations	52	(1)
Illustration with numerical simulations	53	(2)
Sprinkling distribution	55	(5)
Definition. Laplace transform	55	(2)
Examples taken from other fields	57	(1)
Asymptotic behaviour. Broad versus narrow distributions	58	(2)
The proportion of atoms trapped in quasi-dark states	60	(9)
Ensemble averages versus time averages	60	(2)
Time average: fraction of time spent in the trap	60	(1)
Ensemble average: trapped proportion	61	(1)
Calculation of the proportion of trapped atoms	62	(5)
Laplace transforms of the sprinkling distributions associated with the return and exit times	62	(1)
Laplace transform of the proportion of trapped atoms	63	(1)
Results for a finite average trapping time and a finite average recycling time	64	(1)
Results for an infinite average trapping time and a finite average recycling time	64	(2)
Results for an infinite average trapping time and an infinite average recycling time	66	(1)
Discussion: non-ergodic behaviour of the trapped population	67	(2)
The momentum distribution	69	(19)
Brief survey of previous heuristic arguments	69	(2)
Expressions of the momentum distribution and of related quantities	71	(4)
Distribution of the momentum modulus	71	(1)
Momentum distribution along a given axis	72	(1)
Characterization of the cooled atoms' momentum distribution	73	(2)
Case of an infinite average trapping time and a finite average recycling time	75	(4)
Explicit form of the momentum distribution	75	(2)
Important features of the momentum distribution	77	(2)
Case of a finite average trapping time and a finite average recycling time	79	(4)
Explicit form of the momentum distribution	80	(2)
Important features of the momentum distribution	82	(1)
Cases with an infinite average recycling time	83	(3)
Overview of main results	86	(2)
Physical discussion	88	(13)
Equivalence with a rate equation description	88	(3)
Rate equation for the momentum distribution	88	(1)
Re-interpretation of the sprinkling distribution of return times as a source term	89	(1)
Which atoms contribute to the sprinkling distribution of return times?	89	(1)
Interpretation of the time dependence of the sprinkling distribution of return times	90	(1)
Tails of the momentum distribution	91	(1)
Steady-state versus quasi-steady-state	91	(1)
Dependence on the various parameters	92	(1)
Height of the peak of the momentum distribution	92	(1)
Effect of a non-vanishing jump rate at zero momentum	93	(3)
Existence of a steady-state for long times	94	(1)
Intermediate times	95	(1)
Non-stationarity and non-ergodicity	96	(5)
Flatness of the momentum distribution around zero momentum	96	(1)
Various degrees of non-ergodicity	97	(1)
Connection with broad distributions	97	(4)
Tests of the statistical approach	101	(23)
Motivation	101	(1)
Overview of other approaches	102	(3)
Experiments	102	(1)
Quantum optics calculations for VSCPT	103	(2)
Monte Carlo simulations of Raman cooling	105	(1)
Proportion of trapped atoms in one-dimensional σ+/σ- VSCPT	105	(8)
Doppler model	106	(3)
Unconfined model	109	(2)
Confined model	111	(2)
Width and shape of the peak of cooled atoms	113	(7)
Statistical predictions	113	(1)
Comparison to quantum calculations	113	(3)
Experimental tests	116	(4)
Role of friction and of dimensionality	120	(2)
One-dimensional case	120	(1)
Higher dimensional case	120	(2)
Conclusion	122	(2)
Example of application: optimization of the peak of cooled atoms	124	(13)
Introduction	124	(2)
Parametrization	126	(2)
Why is there an optimum parameter?	128	(2)
Optimization using the expression of the height	130	(1)
Optimization using Levy sums	131	(2)
Features of the optimized cooling	133	(2)
Random walk interpretation of the optimized solution	135	(2)
Conclusion	137	(8)
What has been done in this book	137	(1)
Significance and importance of the results	138	(2)
From the point of view of Levy statistics	138	(1)
From the point of view of laser cooling	139	(1)
Possible extensions	140	(5)
Improving the optimization	140	(1)
More precise model of friction-assisted VSCPT	140	(1)
Extension to other cooling schemes	140	(1)
Extension to trapped atoms	141	(1)
Inclusion of many-atom effects	142	(3)
Appendix A Correspondence between parameters of the statistical models and atomic and laser parameters	145	(27)
A.1 Velocity Selective Coherent Population Trapping	145	(19)
A.1.1 Quantum calculation of the jump rate	146	(1)
A.1.1.1 Effective Hamiltonian	147	(2)
A.1.1.2 Exact diagonalization	149	(2)
A.1.1.3 Expansion around p = 0	151	(1)
A.1.1.4 Behaviour out of the trapping dip	152	(1)
A.1.1.5 Case of a negligible Doppler effect	153	(2)
A.1.2 Parameters of the random walk models	155	(1)
A.1.2.1 Trapping region and plateau: p0 and τ0	155	(1)
A.1.2.2 Dependence on laser intensity	156	(1)
A.1.2.3 Doppler tail: pD	157	(1)
A.1.2.4 Discussion: comparison between quantum calculations and statistical models	158	(1)
A.1.2.5 Confining walls: Pmax	159	(1)
A.1.2.6 Elementary step of the random walk: Δp	160	(1)
A.1.3 Trapping time distribution: τb	161	(1)
A.1.4 Recycling time distribution	162	(1)
A.1.4.1 Doppler model: τ	162	(1)
A.1.4.2 Unconfined model: τb	163	(1)
A.1.4.3 Confined model: τ	164	(1)
A.2 Raman cooling	164	(8)
A.2.1 Jump rate	164	(4)
A.2.2 Parameters of the random walk models	168	(1)
A.2.2.1 Trapping region and plateau: p0 and τ0	169	(1)
A.2.2.2 Confining walls: Pmax	169	(1)
A.2.2.3 Elementary step of the random walk: τp	169	(1)
A.2.3 Trapping time distribution: τb	170	(1)
A.2.4 Recycling time distribution: τ	171	(1)
Appendix B The Doppler case	172	(5)
B.1 Motivations	172	(1)
B.2 Setting the stage	172	(2)
B.3 Feynman path integral and mapping to the harmonic oscillator	174	(1)
B.4 Back to the return time probability	175	(2)
Appendix C The special case &mu: = 1	177	(4)
References	181	(8)
Index of main notation	189	(6)
Index	195