Vortices in Bose—Einstein Condensates

211 Pages·2006·1.303 MB·English

by  Amandine Aftalion (auth.)

Progress in Nonlinear Differential Equations
and Their Applications
Volume67
Editor
HaimBrezis
Universite´ PierreetMarieCurie
Paris
and
RutgersUniversity
NewBrunswick,N.J.
EditorialBoard
AntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,Trieste
A.Bahri,RutgersUniversity,NewBrunswick
FelixBrowder,RutgersUniversity,NewBrunswick
LuisCaffarelli,TheUniversityofTexas,Austin
LawrenceC.Evans,UniversityofCalifornia,Berkeley
MarianoGiaquinta,UniversityofPisa
DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh
SergiuKlainerman,PrincetonUniversity
RobertKohn,NewYorkUniversity
P.L.Lions,UniversityofParisIX
JeanMawhin,Universite´ CatholiquedeLouvain
LouisNirenberg,NewYorkUniversity
LambertusPeletier,UniversityofLeiden
PaulRabinowitz,UniversityofWisconsin,Madison
JohnToland,UniversityofBath
Amandine Aftalion
Vortices in Bose–Einstein
Condensates
Birkha¨user
Boston • Basel • Berlin
AmandineAftalion
CNRS,LaboratoireJacques-LouisLions
Universite´PierreetMarieCurie(Paris6)
175rueduchevaleret
F-75013Paris
France
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Preface
Vortices are often associated with dramatic circumstances such as hurricanes; this
typeofvortexhasbeenstudiedextensivelyintheframeworkofclassicalfluids.Nev-
ertheless,itsquantumcounterparthasgainedmajorinterestinthepastfewyearsdue
totheexperimentalrealizationofBose–Einsteincondensates(BEC),anewstateof
matterpredictedbyEinsteinin1925.VorticesinBECarequantized,andtheirsize,
origin, and significance are quite different from those in normal fluids since they
exemplify“superfluid”properties.
SincethefirstexperimentalachievementofBose–Einsteincondensatesin1995
in alkali gases and the award of the Nobel Prize in Physics in 2001, the properties
ofthesegaseousquantumfluidshavebeenthefocusofinternationalinterestincon-
densedmatterphysics.Thisbookwasbothmotivatedbythisintenseactivity,espe-
ciallyinthegroupofJeanDalibardattheEcolenormalesupe´rieure,butalsobythe
constantdevelopmentofmathematicaltechniqueswhichcouldproveusefulintack-
ling these problems, in particular in the group of Haim Brezis. This monograph is
dedicatedtothemathematicalmodellingofsomespecificexperimentswhichdisplay
vorticesandtoarigorousanalysisoffeaturesemergingexperimentally.Itcanserve
as a reference for mathematical researchers and theoretical physicists interested in
superfluidity and quantum condensates, and can also complement a graduate semi-
narinellipticPDEsormodellingofphysicalexperiments.Therearetwointroductory
chapters:thefirstisrelatedtothephysicsbackground,whilethesecondisdevoted
tothepresentationofthemathematicalresultsdescribedinthebook.
Vortices have been observed experimentally by rotating the trap holding the
atoms in the condensate. In contrast to a classical fluid, for which the equilibrium
velocitycorrespondstosolidbodyrotation,aquantumfluidsuchasaBose–Einstein
condensatecanrotateonlythroughthenucleationofquantizedvorticesbeyondsome
criticalvelocity.Therearetwointerestingregimes:oneclosetothecriticalvelocity
wherethereisonlyonevortex,andanotherathighrotationvalues,forwhichadense
lattice is observed. Another experiment consists of a superfluid flow around an ob-
stacle: at low velocity, the flow is stationary; while at larger velocity, vortices are
nucleatedfromtheboundaryoftheobstacle.
vi Preface
One of the key issues is thus the existence of these quantized vortices. We ad-
dressthisissuemathematicallyandderiveinformationontheirshape,number,and
location.Inthedilutelimitoftheexperiments,thecondensateiswelldescribedby
ameanfieldtheoryandamacroscopicwavefunction,solvingtheso-calledGross–
Pitaevskiiequation.Themathematicaltoolsemployedareenergyestimates,Gamma
convergence,andhomogenizationtechniques.Weproveexistenceofsolutionswhich
have properties consistent with the experimental observations. Open problems re-
latedtorecentexperimentsarealsopresented.Theywillrequirethedevelopmentof
newtoolsrelatedforinstance,tomicrolocalanalysisortime–dependentproblems.
ThesuggestionforsettingdowntheseimportantideascamefromHaimBrezis,
andIwouldliketothankhimwarmlyforhisconstantenthusiasmandsupportwhile
Iwasworkingonit.Manytoolsusedherehavebeendevelopedbyeitherhimorhis
school.Iamgladtobeabletopresentanapplicationofthisbeautifulmathematics
totoday’sphysics.
I am also extremely grateful to Jean Dalibard, who has always been willing to
take time to share his experiments, his ideas, and his interests in how mathematics
cancontributetophysics.Workingtogetherwithhimandwritingajointpaperwasa
realpleasureandasourceofmathematicalproblemsformanyyearstocome.Iwould
alsoliketothankhimforhiscarefulreadingofthismanuscript.Beforeworkingwith
Jean,Ihadtheopportunityofmanyfruitfuldiscussionswithmembersofhisgroup,
in particular Vincent Bretin, Yvan Castin, and David Gue´ry-Odelin. I have always
appreciatedtheiropenmindsandinterestinmathematics.IwouldliketothankDavid
inparticularforhiscommentsleadingtoimprovementsintheintroductorypartsof
thisbook.
IowemypersonalinterestininterdisciplinarytopicstothejointeffortsofEtienne
Guyon,YvesPomeau,andHenriBerestycki,wholaunchedaprogramforstudents
attheEcolenormalesupe´rieuretosparkinterestinproblemsontheborderbetween
mathematicsandphysics.Thiseffortwasarealsuccess,aswerethevariousmaths-
physics meetings in Foljuif, a property of the Ecole normale supe´rieure. At one of
them,ImetYvanCastinandrealizedthatwehadmathematicaltoolsthatcouldhelp
inunderstandingproblemsemerginginrotatingBose–Einsteincondensates.Iwould
liketoagainmydeepgratitudetoEtienneGuyon,YvesPomeau,andmysupervisor
HenriBerestycki,forallthatIdiscoveredhasbeenthankstothem.
I would like to also thank, of course, all my collaborators on these topics, in
particular: Tristan Rivie`re, with whom this huge program started and the evidence
of vortex bending occurred; Bob Jerrard, whose involvement in understanding the
shape of vortices was quite influential and with whom it was a pleasure to work in
Vancouver,Milan,Istanbul,andMinneapolis(Ithankallthehostinginstitutions)and
whohasundertakenaverycarefulreadingofthebook;QiangDuandIonutDanaila,
who have performed, on different topics, beautiful numerical computations; Stan
Alama and Lia Bronsard, who came to Paris and became interested in these topics
when they discovered them; Xavier Blanc, with whom I am very happy to have
worked with on many projects; and very recently Francis Nier, who has allowed
metodiscovermicrolocalanalysisandBargmanntransforms,whichhaveprovedto
Preface vii
bequiteusefulintacklingtheseproblems.Itwasalsoveryrewardingtoworkwith
outstanding physicists, Jean Dalibard and Yves Pomeau. Many colleagues all over
theworldhavementionedthatIwasquiteluckytohavehadthisopportunityandto
havefoundacommonlanguagetospeak.Icertainlybelieveit.
Part of this monograph was taught as a Ph.D. course at Paris 6 in 2003–2004.
OneoftheresultspresentedherewasobtainedbytwoofthesePh.D.students,Radu
IgnatandVincentMillot,whomIjointlysupervisedwithHaimBrezis.Iampleased
todescribetheirworkinoneofthechapters.
The quality of the presentation of this book was greatly improved thanks to all
the lectures given in various universities or summer schools. I would like to thank
in particular: Luis Caffarelli and Irene Gamba, Peter Constantin, Peter Sternberg,
Miguel Escobedo, Gero Friesecke, Fang-Hua Lin, Stefan Muller, Tristan Rivie`re,
andJuan-LuisVazquez.IhavealsobenefitedfrominformaldiscussionswithFabrice
Bethuel,PetruMironescu,SylviaSerfaty,EtienneSandier,andDidierSmets.
I take the opportunity here to express my gratitude to all my colleagues in the
Laboratoire Jacques-Louis Lions, in particular: to Yvon Maday, who is a very en-
thusiasticheadofdepartment;tomyofficemateXavierBlanc;tomyofficeneigh-
bours, Edwige Godlewski and Francois Murat; to Olivier Glass, Fre´de´ric Hecht,
Simon Masnou, and the staff in the laboratory, Danielle Boulic, Michel Legendre,
JacquesPortes,andLilianeRuprecht.
The writing of this book was made possible by my position at CNRS, which
should be naturally associated with the outcome of such interdisciplinary effort.
My research during this period was supported by a CNRS grant for young re-
searchers and a French ministry of research grant, ACI “Nouvelles interfaces des
mathe´matiques.” Some of the open problems were derived during a maths–physics
meetingorganizedwithDavidGue´ry-Odelinatthe“FondationdesTreilles”inTour-
tour,andIwouldliketoacknowledgetheirwelcome.
Finally,Iwouldliketothankmyfamilyandfriendsfortheirconstantsupportin
thepreparationofthemanuscriptandallthepeopleatBirkha¨user,inparticularAnn
Kostant,fortheirhelp.
Paris,October31,2005, AmandineAftalion
Contents
Preface ......................................................... v
1 ThePhysicalExperimentsandTheirMathematicalModelling ...... 1
1.1 Ahintontheexperiments.................................... 1
1.1.1 Briefhistoricalsummary.............................. 1
1.1.2 HowtoachieveaBECexperimentally................... 2
1.1.3 Experimentalobservations ............................ 3
1.2 Themathematicalframework................................. 5
1.2.1 TheGross–Pitaevskiienergy........................... 5
1.2.2 TheThomas–Fermiregime............................ 7
1.2.3 Remarksontheoriginalproblem ....................... 12
1.2.4 Mean-fieldquantumHallregime ....................... 13
1.2.5 Flowaroundanobstacle .............................. 16
2 TheMathematicalSetting:ASurveyoftheMainTheorems ........ 19
2.1 Smallεproblem ........................................... 19
2.1.1 Thetwo-dimensionalsetting........................... 20
2.1.2 Thethree-dimensionalsetting ......................... 23
2.2 Vortexlattice .............................................. 24
2.3 Flowaroundanobstacle .................................... 27
3 Two-DimensionalModelforaRotatingCondensate ............... 29
3.1 Mainresults ............................................... 30
3.1.1 Single-vortexsolutionandlocationofvortices............ 31
3.1.2 Ideasoftheproof .................................... 32
3.2 Preliminaries .............................................. 35
3.2.1 Determiningthedensityprofile......................... 35
3.2.2 Existenceofaminimizerof Eε......................... 36
3.2.3 Splittingtheenergy .................................. 37
3.3 Boundednumberofvortices ................................. 38
3.3.1 Firstenergybound ................................... 39
x Contents
3.3.2 Vortexballs ......................................... 40
3.3.3 Therotationterm .................................... 41
3.3.4 Alowerboundexpansion ............................. 43
3.4 Refinedstructureofvortices ................................. 46
3.4.1 Somelocalestimates ................................. 47
3.4.2 Baddiscs........................................... 49
3.4.3 Nodegree-zerovortex ................................ 50
3.4.4 ProofofProposition3.12.............................. 53
3.5 Lowerbound .............................................. 55
3.6 Upperbound .............................................. 61
3.7 Finalexpansionandpropertiesofvortices ...................... 66
3.7.1 Vorticeshavedegreeone .............................. 66
3.7.2 Thesubcriticalcase .................................. 67
3.7.3 Thesupercriticalcase................................. 68
3.8 OpenQuestions ............................................ 75
3.8.1 Vorticesintheregionoflowdensity .................... 75
3.8.2 Othertrappingpotentials.............................. 75
3.8.3 Intermediate(cid:2) ...................................... 76
3.8.4 Time-dependentproblem.............................. 77
4 OtherTrappingPotentials..................................... 79
4.1 Nonradialharmonicpotential ................................ 80
4.2 Quarticpotential ........................................... 82
4.2.1 Giantvortex ........................................ 82
4.2.2 Circleofvortices .................................... 87
4.3 Openquestions ............................................ 98
4.3.1 Circleofvortices .................................... 98
4.3.2 Giantvortexorisolatedvortices........................ 98
5 High-VelocityandQuantumHallRegime ........................ 99
5.1 Introduction ............................................... 100
5.1.1 LowestLandaulevel ................................. 100
5.1.2 Constructionofanupperbound ........................ 101
5.1.3 Propertiesoftheminimizer............................ 106
5.1.4 Othertrappingpotentials.............................. 107
5.2 Regularlattice ............................................. 107
5.3 Distortedlattice ............................................ 111
5.4 Infinitenumberofzeros ..................................... 117
5.5 Othertrappingpotentials .................................... 119
5.6 Openquestions ............................................ 119
5.6.1 Lowerboundand(cid:3)convergence ....................... 119
5.6.2 RestrictiontotheLLL ................................ 120
5.6.3 Reductiontoatwo-dimensionalproblem ................ 121
5.6.4 Meanfieldmodel .................................... 121
Contents xi
6 Three-DimensionalRotatingCondensate ........................ 123
6.1 Numericalsimulations ...................................... 124
6.2 Formalderivationofthereducedenergy E[γ]................... 126
6.2.1 Thesolutionwithoutvortices .......................... 126
6.2.2 Decouplingtheenergy................................ 127
6.2.3 EstimateofGε(vε)................................... 128
6.2.4 Estimateof Iε(vε).................................... 131
6.2.5 Finalestimatefortheenergy........................... 131
6.3 (cid:3)convergenceresults ....................................... 132
6.3.1 Mainresults......................................... 132
6.3.2 Mainideasintheproof ............................... 134
6.4 SingleVortexline,studyof E[γ] ............................. 142
6.4.1 Settingofminimizationof E[γ]........................ 142
6.4.2 Thebentvortex...................................... 144
6.4.3 Propertiesofcriticalpoints ............................ 152
6.5 Afewopenquestions ....................................... 154
6.5.1 Smallvelocity....................................... 154
6.5.2 Criticalpointsof E(cid:2)[χ] .............................. 155
6.5.3 Finitenumberofvortices.............................. 155
6.5.4 Othertrappingpotentials.............................. 155
6.5.5 Wholespaceproblem................................. 155
6.5.6 Decayofvortices .................................... 156
7 SuperfluidFlowAroundanObstacle ............................ 157
7.1 Mathematicalsetting........................................ 158
7.1.1 Two-dimensionalflow ................................ 158
7.1.2 Three-dimensionalflowaroundacondensate ............. 162
7.2 ProofofTheorem7.1 ....................................... 167
7.2.1 Solutionsatc=0.................................... 169
7.2.2 Existenceofasolutionto I ........................... 171
R
7.2.3 Boundsonthesolutionsof I .......................... 172
R
7.2.4 Estimatingthemomentum............................. 174
7.2.5 ProofofTheorem7.4................................. 177
7.2.6 Limitatinfinity...................................... 177
7.3 ProofofTheorem7.2 ....................................... 180
7.3.1 ProofofTheorem7.14................................ 181
7.3.2 ProofofTheorem7.16................................ 185
8 FurtherOpenProblems....................................... 195
8.1 SettinginthewholespacefortheThomas–Fermiregime ......... 195
8.1.1 Three-dimensionalproblem ........................... 195
8.1.2 Two-dimensionalproblem............................. 195
8.1.3 Painleve´ boundarylayer............................... 196
8.1.4 Vorticesinthehole................................... 196
8.2 Otherscalings ............................................. 196