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

213 Pages·2002·1.468 MB·English

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

Le´vyStatisticsandLaserCooling
HowRareEventsBringAtomstoRest
Laser cooling of atoms provides an ideal case study for the application of Le´vy
statistics in a privileged situation where statistical models can be derived from
first principles. This book establishes profitable connections between these two
research fields, and demonstrates how the most efficient laser cooling techniques
can be simply and quantitatively understood in terms of non-ergodic random pro-
cessesdominatedbyafewrareevents.
Le´vy statistics is now recognized as the proper framework for analysing many
different problems (in physics, biology, earth sciences, finance, etc.) for which
standard Gaussian statistics is inadequate. Le´vy statistics involves random vari-
ableswithsuchbroaddistributionsthattheusualCentralLimitTheoremnolonger
holds. Laser cooling allows atoms to be cooled to very low temperatures and
brought to rest, and is a new research field with many applications. It provides
a fruitful example of how approaches based on Le´vy statistics can yield analytic
predictionsthatcanthenbecomparedwithbothmicroscopicquantumopticstreat-
mentsandexperimentalresults.
The authors of this book are world leaders in the fields of laser cooling, light–
atom interactions and statistical physics, and are also renowned for their clarity
of exposition. Since the subject of this book embraces several different research
areas,theauthorshavemadeeveryefforttoensurethatitremainscomprehensible
tothenon-specialist. Theyexplaintheimportantconceptsoflasercoolingandgive
an introduction to the concept of random walks and Le´vy statistics, such that no
detailedpriorknowledgeisrequired. Thisbookwillthereforebeofgreatinterestto
researchers in the fields of atomic physics, quantum optics and statistical physics,
as well as to engineers and mathematicians interested in stochastic processes. It
willalsobemostusefulforillustratinggraduatecoursesonthesetopics.
FRANC¸OIS BARDOUisaresearcherattheCentreNationaldelaRechercheSci-
entifique (CNRS) and currently works on problems in quantum stochastics at the
Institut de Physique et de Chimie des Mate´riaux de Strasbourg. He received the
1995Aime´ Cottonprize(AtomicPhysicsprizeoftheFrenchPhysicalSociety)for
his experimental and theoretical studies of laser cooling performed at the E´cole
Normale Supe´rieure de Paris. These studies, in collaboration with his co-authors,
providedthebasisforthisbook.
JEAN-PHILIPPE BOUCHAUD is a Senior Expert at the Service de Physique de
l’E´tat Condense´ and at CEA-Saclay. In 1994 he founded his own company, ‘Sci-
ence and Finance’, and continues to have diverse research interests which include
statistical physics, granular matter and theoretical finance. In 1996 he won the
CNRS Silver Medal. He is in charge of a number of statistical physics and fi-
nance courses in various Grandes E´coles, Paris, and is the co-author of Theory of
FinancialRisk(CambridgeUniversityPress,2000).
ALAIN ASPECTisaDirectorofResearchatCNRSandaProfessorattheE´cole
Polytechnique, Palaiseau. After completing, in the early 1980s, a series of ex-
periments on the foundations of quantum mechanics, he joined Claude Cohen-
TannoudjiattheE´coleNormaleSupe´rieuretoworkonlasercoolingofatoms. He
is now head of the Atom Optics group of Institut d’Optique at Orsay and is the
co-author of Introduction to Lasers and Quantum Optics (Cambridge University
Press,inpreparation).
CLAUDE COHEN-TANNOUDJIisProfessorofAtomicandMolecularPhysicsat
theColle`gedeFranceinParisandwashonouredwiththeNobelPrizeforPhysics
in 1997 for his work on the development of methods to cool and trap atoms with
laser light. He is also the co-author of three other books: Quantum Mechanics
(1992), Photons and Atoms: Introduction to Quantum Electrodynamics (1989),
andAtom–PhotonInteractions: BasicProcessesandApplications(1998).
Le´vy Statistics and Laser Cooling
HowRareEventsBringAtomstoRest
FRANC¸OISBARDOU, JEAN-PHILIPPEBOUCHAUD,
ALAINASPECT and CLAUDECOHEN-TANNOUDJI
PUBLISHEDBYTHEPRESSSYNDICATEOFTHEUNIVERSITYOFCAMBRIDGE
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http://www.cambridge.org
(cid:176)c Franc¸oisBardou,Jean-PhilippeBouchaud,AlainAspect,ClaudeCohen-Tannoudji2002
Thisbookisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2002
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
TypefaceTimes11/14pt. SystemLATEX2"[DBD]
AcataloguerecordofthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Le´vystatisticsandlasercooling:howrareeventsbringatomstorest/Franc¸oisBardou... [etal.].
p. cm.
Includesbibliographicalreferencesandindex.
ISBN0521808219–ISBN0521004225(pb.)
1. Lasermanipulation(Nuclearphysics). 2. Lasercooling.
3. Atoms–Cooling. 4. Le´vyprocesses. I. Bardou,Franc¸ois,1967–
QC689.5.L35L482001
539.7–dc21 2001025939
ISBN 0521808219 hardback
ISBN 0521004225 paperback
Contents
Foreword pagexi
Acknowledgements xiii
1 Introduction 1
1.1 Lasercooling 1
1.2 Subrecoillasercooling 2
1.3 SubrecoilcoolingandLe´vystatistics 3
1.4 Contentofthebook 5
2 Subrecoillasercoolingandanomalousrandomwalks 7
2.1 Standardlasercooling: frictionforcesandtherecoillimit 7
2.1.1 Frictionforcesandcooling 7
2.1.2 Therecoillimit 9
2.2 Lasercoolingbasedoninhomogeneousrandomwalksinmomentum
space 9
2.2.1 Physicalmechanism 9
2.2.2 Howtocreateaninhomogeneousrandomwalk 10
2.2.3 Expectedcoolingproperties 11
2.3 Quantumdescriptionofsubrecoillasercooling 12
2.3.1 Wavenatureofatomicmotion 12
2.3.2 Difficultiesofthestandardquantumtreatment 13
2.3.3 Quantumjumpdescription. Thedelayfunction 14
2.3.4 Simulationoftheatomicmomentumstochasticevolution 15
2.3.5 Generalization. Stochastic wave functions and random walks
inHilbertspace 16
2.4 Fromquantumopticstoclassicalrandomwalks 19
2.4.1 Fictitiousclassicalparticleassociatedwiththequantumrandom
walk 19
2.4.2 Simplifiedjumprate 20
2.4.3 Discussion 21
v
vi Contents
3 Trappingandrecycling. Statisticalproperties 22
3.1 Trappingandrecyclingregions 22
3.2 Modelsofinhomogeneousrandomwalks 25
3.2.1 Friction 25
3.2.2 Trappingregion 25
3.2.3 Recyclingregion 26
3.2.4 Momentumjumps 28
3.2.5 Discussion 28
3.3 Probabilitydistributionofthetrappingtimes 28
3.3.1 One-dimensionalquadraticjumprate 28
3.3.2 Generalizationtohigherdimensions 32
3.3.3 Generalizationtoanon-quadraticjumprate 32
3.3.4 Discussion 33
3.4 Probabilitydistributionoftherecyclingtimes 34
3.4.1 Presentationoftheproblem: firstreturntimeinBrownianmotion 34
3.4.2 Theunconfinedmodelinonedimension 35
3.4.3 TheDopplermodelinonedimension 37
3.4.4 Theconfinedmodel: randomwalkwithwalls 39
3.4.5 Discussion 40
4 BroaddistributionsandLe´vystatistics: abriefoverview 42
4.1 Power-lawdistributions. Whendotheyoccur? 42
4.2 GeneralizedCentralLimitTheorem 44
4.2.1 Le´vysums. AsymptoticbehaviourandLe´vydistributions 44
4.2.2 SketchoftheproofofthegeneralizedCLT 45
4.2.3 Afewmathematicalresults 47
4.3 QualitativediscussionofsomepropertiesofLe´vysums 49
4.3.1 DependenceofaLe´vysumonthenumberoftermsforµ < 1 49
4.3.2 HierarchicalstructureinaLe´vysum 50
4.3.3 Largefluctuations 52
4.3.4 Illustrationwithnumericalsimulations 53
4.4 Sprinklingdistribution 55
4.4.1 Definition. Laplacetransform 55
4.4.2 Examplestakenfromotherfields 57
4.4.3 Asymptoticbehaviour. Broadversusnarrowdistributions 58
5 Theproportionofatomstrappedinquasi-darkstates 60
5.1 Ensembleaveragesversustimeaverages 60
5.1.1 Timeaverage: fractionoftimespentinthetrap 60
5.1.2 Ensembleaverage: trappedproportion 61
Contents vii
5.2 Calculationoftheproportionoftrappedatoms 62
5.2.1 Laplace transforms of the sprinkling distributions associated
withthereturnandexittimes 62
5.2.2 Laplacetransformoftheproportionoftrappedatoms 63
5.2.3 Results for a finite average trapping time and a finite average
recyclingtime 64
5.2.4 Resultsforaninfiniteaveragetrappingtimeandafiniteaverage
recyclingtime 64
5.2.5 Results for an infinite average trapping time and an infinite
averagerecyclingtime 66
5.3 Discussion: non-ergodicbehaviourofthetrappedpopulation 67
6 Themomentumdistribution 69
6.1 Briefsurveyofpreviousheuristicarguments 69
6.2 Expressionsofthemomentumdistributionandofrelatedquantities 71
6.2.1 Distributionofthemomentummodulus 71
6.2.2 Momentumdistributionalongagivenaxis 72
6.2.3 Characterizationofthecooledatoms’momentumdistribution 73
6.3 Caseofaninfiniteaveragetrappingtimeandafiniteaveragerecycling
time 75
6.3.1 Explicitformofthemomentumdistribution 75
6.3.2 Importantfeaturesofthemomentumdistribution 77
6.4 Case of a finite average trapping time and a finite average recycling
time 79
6.4.1 Explicitformofthemomentumdistribution 80
6.4.2 Importantfeaturesofthemomentumdistribution 82
6.5 Caseswithaninfiniteaveragerecyclingtime 83
6.6 Overviewofmainresults 86
7 Physicaldiscussion 88
7.1 Equivalencewitharateequationdescription 88
7.1.1 Rateequationforthemomentumdistribution 88
7.1.2 Re-interpretation of the sprinkling distribution of return times
asasourceterm 89
7.1.3 Whichatomscontributetothesprinklingdistributionofreturn
times? 89
7.1.4 Interpretation of the time dependence of the sprinkling
distributionofreturntimes 90
7.2 Tailsofthemomentumdistribution 91
7.2.1 Steady-stateversusquasi-steady-state 91
7.2.2 Dependenceonthevariousparameters 92
viii Contents
7.3 Heightofthepeakofthemomentumdistribution 92
7.4 Effectofanon-vanishingjumprateatzeromomentum 93
7.4.1 Existenceofasteady-stateforlongtimes 94
7.4.2 Intermediatetimes 95
7.5 Non-stationarityandnon-ergodicity 96
7.5.1 Flatnessofthemomentumdistributionaroundzeromomentum 96
7.5.2 Variousdegreesofnon-ergodicity 97
7.5.3 Connectionwithbroaddistributions 97
8 Testsofthestatisticalapproach 101
8.1 Motivation 101
8.2 Overviewofotherapproaches 102
8.2.1 Experiments 102
8.2.2 QuantumopticscalculationsforVSCPT 103
8.2.3 MonteCarlosimulationsofRamancooling 105
8.3 Proportionoftrappedatomsinone-dimensional(cid:190)+/(cid:190)¡ VSCPT 105
8.3.1 Dopplermodel 106
8.3.2 Unconfinedmodel 109
8.3.3 Confinedmodel 111
8.4 Widthandshapeofthepeakofcooledatoms 113
8.4.1 Statisticalpredictions 113
8.4.2 Comparisontoquantumcalculations 113
8.4.3 Experimentaltests 116
8.5 Roleoffrictionandofdimensionality 120
8.5.1 One-dimensionalcase 120
8.5.2 Higherdimensionalcase 120
8.6 Conclusion 122
9 Exampleofapplication: optimizationofthepeakofcooledatoms 124
9.1 Introduction 124
9.2 Parametrization 126
9.3 Whyisthereanoptimumparameter? 128
9.4 Optimizationusingtheexpressionoftheheight 130
9.5 OptimizationusingLe´vysums 131
9.6 Featuresoftheoptimizedcooling 133
9.7 Randomwalkinterpretationoftheoptimizedsolution 135
10 Conclusion 137
10.1 Whathasbeendoneinthisbook 137
10.2 Significanceandimportanceoftheresults 138
10.2.1 FromthepointofviewofLe´vystatistics 138
Contents ix
10.2.2 Fromthepointofviewoflasercooling 139
10.3 Possibleextensions 140
10.3.1 Improvingtheoptimization 140
10.3.2 Moreprecisemodeloffriction-assistedVSCPT 140
10.3.3 Extensiontoothercoolingschemes 140
10.3.4 Extensiontotrappedatoms 141
10.3.5 Inclusionofmany-atomeffects 142
AppendixA Correspondencebetweenparametersofthestatistical
modelsandatomicandlaserparameters 145
A.1 VelocitySelectiveCoherentPopulationTrapping 145
A.1.1 Quantumcalculationofthejumprate 146
A.1.1.1 EffectiveHamiltonian 147
A.1.1.2 Exactdiagonalization 149
A.1.1.3 Expansionaround p = 0 151
A.1.1.4 Behaviouroutofthetrappingdip 152
A.1.1.5 CaseofanegligibleDopplereffect 153
A.1.2 Parametersoftherandomwalkmodels 155
A.1.2.1 Trappingregionandplateau: p and¿ 155
0 0
A.1.2.2 Dependenceonlaserintensity 156
A.1.2.3 Dopplertail: p 157
D
A.1.2.4 Discussion: comparison between quantum calculations and
statisticalmodels 158
A.1.2.5 Confiningwalls: p 159
max
A.1.2.6 Elementarystepoftherandomwalk: 1p 160
A.1.3 Trappingtimedistribution: ¿ 161
b
A.1.4 Recyclingtimedistribution 162
A.1.4.1 Dopplermodel: ¿ˆ 162
b
A.1.4.2 Unconfinedmodel: ¿ˆ 163
b
A.1.4.3 Confinedmodel: h¿ˆi 164
A.2 Ramancooling 164
A.2.1 Jumprate 164
A.2.2 Parametersoftherandomwalkmodels 168
A.2.2.1 Trappingregionandplateau: p and¿ 169
0 0
A.2.2.2 Confiningwalls: p 169
max
A.2.2.3 Elementarystepoftherandomwalk: 1p 169
A.2.3 Trappingtimedistribution: ¿ 170
b
A.2.4 Recyclingtimedistribution: h¿ˆi 171
AppendixB TheDopplercase 172
B.1 Motivations 172
x Contents
B.2 Settingthestage 172
B.3 Feynmanpathintegralandmappingtotheharmonicoscillator 174
B.4 Backtothereturntimeprobability 175
AppendixC Thespecialcaseµ = 1 177
References 181
Indexofmainnotation 189
Index 195