Lévy statistics and laser cooling : how rare events bring atoms to rest / François Bardou ... [and others].

Format:

Book

Contributor:

Bardou, François, 1967-

Language:

English

Subjects (All):

Laser manipulation (Nuclear physics).

Laser cooling.

Atoms--Cooling.

Atoms.

Lévy processes.

Physical Description:

xiii, 199 pages : illustrations ; 26 cm

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2002.

Summary:

A graduate-level book demonstrating the application of Levy statistics to understand laser cooling of atoms.

Contents:

1.1 Laser cooling 1

1.2 Subrecoil laser cooling 2

1.3 Subrecoil cooling and Levy statistics 3

2 Subrecoil laser cooling and anomalous random walks 7

2.1 Standard laser cooling: friction forces and the recoil limit 7

2.1.1 Friction forces and cooling 7

2.1.2 The recoil limit 9

2.2 Laser cooling based on inhomogeneous random walks in momentum space 9

2.2.1 Physical mechanism 9

2.2.2 How to create an inhomogeneous random walk 10

2.2.3 Expected cooling properties 11

2.3 Quantum description of subrecoil laser cooling 12

2.3.1 Wave nature of atomic motion 12

2.3.2 Difficulties of the standard quantum treatment 13

2.3.3 Quantum jump description. The delay function 14

2.3.4 Simulation of the atomic momentum stochastic evolution 15

2.3.5 Generalization. Stochastic wave functions and random walks in Hilbert space 16

2.4 From quantum optics to classical random walks 19

2.4.1 Fictitious classical particle associated with the quantum random walk 19

2.4.2 Simplified jump rate 20

3 Trapping and recycling. Statistical properties 22

3.1 Trapping and recycling regions 22

3.2 Models of inhomogeneous random walks 25

3.2.1 Friction 25

3.2.2 Trapping region 25

3.2.3 Recycling region 26

3.2.4 Momentum jumps 28

3.3 Probability distribution of the trapping times 28

3.3.1 One-dimensional quadratic jump rate 28

3.3.2 Generalization to higher dimensions 32

3.3.3 Generalization to a non-quadratic jump rate 32

3.4 Probability distribution of the recycling times 34

3.4.1 Presentation of the problem: first return time in Brownian motion 34

3.4.2 The unconfined model in one dimension 35

3.4.3 The Doppler model in one dimension 37

3.4.4 The confined model: random walk with walls 39

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

4.2.1 Levy sums. Asymptotic behaviour and Levy distributions 44

4.2.2 Sketch of the proof of the generalized CLT 45

4.2.3 A few mathematical results 47

4.3 Qualitative discussion of some properties of Levy sums 49

4.3.1 Dependence of a Levy sum on the number of terms for [mu] < 1 49

4.3.2 Hierarchical structure in a Levy sum 50

4.3.3 Large fluctuations 52

4.3.4 Illustration with numerical simulations 53

4.4 Sprinkling distribution 55

4.4.1 Definition. Laplace transform 55

4.4.2 Examples taken from other fields 57

4.4.3 Asymptotic behaviour. Broad versus narrow distributions 58

5 The proportion of atoms trapped in quasi-dark states 60

5.1 Ensemble averages versus time averages 60

5.1.1 Time average: fraction of time spent in the trap 60

5.1.2 Ensemble average: trapped proportion 61

5.2 Calculation of the proportion of trapped atoms 62

5.2.1 Laplace transforms of the sprinkling distributions associated with the return and exit times 62

5.2.2 Laplace transform of the proportion of trapped atoms 63

5.2.3 Results for a finite average trapping time and a finite average recycling time 64

5.2.4 Results for an infinite average trapping time and a finite average recycling time 64

5.2.5 Results for an infinite average trapping time and an infinite average recycling time 66

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

6.2.1 Distribution of the momentum modulus 71

6.2.2 Momentum distribution along a given axis 72

6.2.3 Characterization of the cooled atoms' momentum distribution 73

6.3 Case of an infinite average trapping time and a finite average recycling time 75

6.3.1 Explicit form of the momentum distribution 75

6.3.2 Important features of the momentum distribution 77

6.4 Case of a finite average trapping time and a finite average recycling time 79

6.4.1 Explicit form of the momentum distribution 80

6.4.2 Important features of the momentum distribution 82

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

7.1.1 Rate equation for the momentum distribution 88

7.1.2 Re-interpretation of the sprinkling distribution of return times as a source term 89

7.1.3 Which atoms contribute to the sprinkling distribution of return times? 89

7.1.4 Interpretation of the time dependence of the sprinkling distribution of return times 90

7.2 Tails of the momentum distribution 91

7.2.1 Steady-state versus quasi-steady-state 91

7.2.2 Dependence on the various parameters 92

7.3 Height of the peak of the momentum distribution 92

7.4 Effect of a non-vanishing jump rate at zero momentum 93

7.4.1 Existence of a steady-state for long times 94

7.4.2 Intermediate times 95

7.5 Non-stationarity and non-ergodicity 96

7.5.1 Flatness of the momentum distribution around zero momentum 96

7.5.2 Various degrees of non-ergodicity 97

7.5.3 Connection with broad distributions 97

8 Tests of the statistical approach 101

8.1 Motivation 101

8.2 Overview of other approaches 102

8.2.1 Experiments 102

8.2.2 Quantum optics calculations for VSCPT 103

8.2.3 Monte Carlo simulations of Raman cooling 105

8.3 Proportion of trapped atoms in one-dimensional [sigma subscript +]/[sigma subscript -] VSCPT 105

8.3.1 Doppler model 106

8.3.2 Unconfined model 109

8.3.3 Confined model 111

8.4 Width and shape of the peak of cooled atoms 113

8.4.1 Statistical predictions 113

8.4.2 Comparison to quantum calculations 113

8.4.3 Experimental tests 116

8.5 Role of friction and of dimensionality 120

8.5.1 One-dimensional case 120

8.5.2 Higher dimensional case 120

9 Example of application: optimization of the peak of cooled atoms 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

10.1 What has been done in this book 137

10.2 Significance and importance of the results 138

10.2.1 From the point of view of Levy statistics 138

10.2.2 From the point of view of laser cooling 139

10.3 Possible extensions 140

10.3.1 Improving the optimization 140

10.3.2 More precise model of friction-assisted VSCPT 140

10.3.3 Extension to other cooling schemes 140

10.3.4 Extension to trapped atoms 141

10.3.5 Inclusion of many-atom effects 142

Appendix A Correspondence between parameters of the statistical models and atomic and laser parameters 145

A.1 Velocity Selective Coherent Population Trapping 145

A.1.1 Quantum calculation of the jump rate 146

A.1.1.1 Effective Hamiltonian 147

A.1.1.2 Exact diagonalization 149

A.1.1.3 Expansion around p = 0 151

A.1.1.4 Behaviour out of the trapping dip 152

A.1.1.5 Case of a negligible Doppler effect 153

A.1.2 Parameters of the random walk models 155

A.1.2.1 Trapping region and plateau: p[subscript 0] and [tau subscript 0] 155

A.1.2.2 Dependence on laser intensity 156

A.1.2.3 Doppler tail: p[subscript D] 157

A.1.2.4 Discussion: comparison between quantum calculations and statistical models 158

A.1.2.5 Confining walls: p[subscript max] 159

A.1.2.6 Elementary step of the random walk: [Delta]p 160

A.1.3 Trapping time distribution: [tau subscript b] 161

A.1.4 Recycling time distribution 162

A.1.4.1 Doppler model: [tau subscript b] 162

A.1.4.2 Unconfined model: [tau subscript b] 163

A.1.4.3 Confined model: ([tau]) 164

A.2 Raman cooling 164

A.2.1 Jump rate 164

A.2.2 Parameters of the random walk models 168

A.2.2.1 Trapping region and plateau: p[subscript 0] and [tau subscript 0] 169

A.2.2.2 Confining walls: p[subscript

max] 169

A.2.2.3 Elementary step of the random walk: [Delta]p 169

A.2.3 Trapping time distribution: [tau subscript b] 170

A.2.4 Recycling time distribution: ([tau]) 171

Appendix B The Doppler case 172

B.1 Motivations 172

B.3 Feynman path integral and mapping to the harmonic oscillator 174

B.4 Back to the return time probability 175

Appendix C The special case [mu] = 1 177.

Notes:

Includes bibliographical references (pages 181-187) and indexes.

ISBN:

0521808219

0521004225

OCLC:

46640834