Effect of atomic transfer on the decay of a Bose-Einstein condensate

EffectofatomictransferonthedecayofaBose-Einsteincondensate

arXiv:quant-ph/0303134v1 21 Mar 2003Pawe󰀳lZi´n*§,AndrzejDragan*§,SzymonCharzy´nski*‡,

NorbertHerschbach*,PaulTol*,WimHogervorst*andWimVassen*

*LaserCentreVrijeUniversiteit,DeBoelelaan1081,1081HVAmsterdam,TheNetherlands§Wydzia󰁊lFizyki,UniwersytetWarszawski,Ho˙za69,PL-00-681Warszawa,Poland‡CentrumFizykiTeoretycznej,Al.Lotnik´ow32/4602-668Warszawa,PolandAbstract.WepresentamodeldescribingthedecayofaBose-Einsteincondensate,whichassumesthesystemtoremaininthermalequilibriumduringthedecay.Weshowthatunderthisassumptiontransferofatomsoccursfromthecondensatetothethermalcloudenhancingthecondensatedecayrate.

PACSnumbers:03.75.Hh,05.30.Jp,34.50.Fa

Submittedto:J.Phys.B:At.Mol.Opt.Phys.

LettertotheEditor1.Introduction

2

RapidadvancesinexperimentaltechniquesoflasercoolingandtrappingmadeitpossibletoachieveBose-Einsteincondensation(BEC)inweaklyinteractingsystems.OneofthemanyinterestingaspectsofBECindilutegasesisthedynamicsofgrowthanddecayofthecondensate[1,2,3].Bose-Einsteincondensatescanhavelonglifetimesrangingfrom2s[4]uptomorethan10s[1,5].Thisfinitelifetimeismainlycausedbyinelasticcollisionsbetweencondensateatomsandbycollisionswithparticlesfromthebackgroundgas,resultinginatomlossandheatingofthesystem.Manyexperimentswithcondensatescanbeperformedonatimescaleshortcomparedtothecondensatelifetime,suchthatdecayeffectsarenotimportant.Thisis,however,notalwaysthecaseandsomeexperimentsspecificallyfocusoncondensatedecay[1].Hence,itisimportanttomodelthedecayofacondensateindetail,takingintoaccountallprocesseswhichsignificantlycontribute.Inliteraturecondensatedecayduetoinelasticcollisionsandcollisionswithbackgroundgasatomsaswellasheatingeffectsarediscussed.Toourknowledge,thetransferofatomsbetweenthecondensateandthethermalcloudoccuringdynamicallyasaconsequenceofthermalizationofthesystemhasnotyetbeentakenintoaccount.Wewillshowthatthisprocessplaysanimportantrolewhenthenumberofatomsinthecondensateisofthesameorderorsmallerthanthenumberinthethermalcloud.Itseffectcouldonlybeneglectedinexperimentswithlargecondensatefractionswhichareoftenobtainedbyremovalofmostofthethermalpartwithanrf-knife,orinsituationsinwhichourassumptionoffastthermalizationisnotfulfilled.

Thepresentworkconcentratesonthedecayofcondensatesinthepresenceofaconsiderablylargethermalfraction.WeassumethermalequilibriumduringthedecayofthecondensatewhichisjustifiedinmostexperimentswithBECindiluteatomicgases.Sinceelasticcollisionratesaretypicallylarge(󰀁103s−1[4,6]),oneexpectsthermalizationtooccurveryrapidly[7]comparedtotherateofchangeinthermodynamicvariablesduringthedecay.Usingthesimplecondensategrowthequation[2]weinvestigatednumericallytheeffectofdynamicaldisturbancescausedbyatomlossandheatingfrominelasticcollisionsonasystemoriginallyinthermalequilibrium.Wefoundthatthesystemstaysveryclosetothermalequilibriumunderconditionstypicallyencounteredintheexperimentswithdiluteatomicgases.

Althoughourmodelincorporatestwo-andthree-bodycollisionsitdoesnottakeintoaccountsubsequentsecondaryeffects.Forinstance,theeffectsofinducedlocalvariationsinthemean-fieldinterparticleinteraction[8]areneglected.Alsosecondarycollisionsofreactionproductswithatomsofthecondensateorthethermalcloudarenotaccountedfor.Avalanches,recentlydiscussedbySchusteretal[9],arethereforebeyondthescopeofthiswork.Theymaybeincorporatedlater.

Thisletterisorganizedasfollows.InSec.2wederiveasimpleanalyticalformulaforthedecayofacondensateofnon-interactingbosonsincludingonlylossesduetocollisionswithbackgroundgasparticles.Theexpressionsderivedinthissectionshowtheexistenceoftransferofatomsbetweencondensateandthermalcloudandthesignificant

LettertotheEditor3

reductionincondensatelifetimethatmayoccur.InSec.3wepresentamorecompletedynamicalmodelofcondensatedecayusingthemean-fieldtheoryforweakly-interactingbosonsandincludingalsolossesbyinelastictwo-andthree-bodycollisions.InSec.4wepresent,asanexample,numericalsimulationsofcondensatedecayinmetastablehelium.

2.Atomictransfer

Apartfromelasticcollisionsthatkeepthesysteminthermalequilibrium,inthissectionweassumetheatomstoundergoonlycollisionswithbackgroundgasparticles,inducingatomlosses.Forsimplicityweconsiderasystemofnon-interactingbosons.AtatemperatureTbelowthecriticaltemperatureTc=󰀁ω[N/g3(1)]1/3/kthenumberofatomsNTinthethermalcloudisgivenby[10]

󰀆kT

NT=g3(1)

30

󰀆

kT

τ

˙C+N˙T=−1N=N

τ

2

˙C+E˙T=−1E=E

󰀁(ωx+ωy+ωz)beingthegroundstateenergyofthetrap.UsingEq.(2)thetime

˙T=4derivativesE

NT

1−ε0NT/ET

4τNT,

(5)

τ

LettertotheEditor4

whereweusedε0NT/ET≪1,neglectingtheenergyofacondensateatomcomparedtotheaverageenergyperatominthethermalcloud.SubstitutinginEq.(3)yields

󰀈

1˙C=−NT.(6)N

4Thissimpleanalysisshowsthatforacondensatecoexistinginthermalequilibriumwith

˙C=−NC/τnorN˙T=−NT/τholdsforthedecayathermalcloudinatrapneitherN

inducedbycollisionswithbackgroundparticles.Conservationofenergyandnumberofatomscombinedwithrapidthermalizationinevitablyresultsintransferofatomsfromcondensatetothermalcloudtherebyenhancingthedecayrateofthecondensate.Especiallywithaconsiderablefractionofthermalatomsinthesystemthisaffectsthedecayofthecondensate.OnlyinthelimitofalargecondensatefractionEq.(6)becomes˙C=−NC/τ.N

3.Interactingmodel

Inordertoparametrizetheequilibriumstateofthegas,wewillusethetemperatureTandthenumberofatomsinthecondensateNCasindependentvariablesfullydescribingthestateofthesystem.Wewillstartwiththestationary“two-gas”modelproposedbyDoddetal[11],inwhichatomsofthethermalclouddonotaffectthecondensatedescribedbythestationaryGross-Pitaevskii(GP)equation.Thethermalcloudatomsdonotinteractwitheachother(exceptforthermalization)buttheyareinfluencedbythecondensatethroughthemean-fieldpotential2U0nC(r).Here,thecontactpotentialU0=4π󰀁2a/mexpressesthebinaryinteractionbetweenatomswithscatteringlengthaandmassm,andnC(r)denotesthespatialdensityofthecondensatewhichcanbecalculatedfromtheGPequation.Weadditionallysimplifythetwo-gasmodelbydescribingthethermalcloudsemiclassically,replacingdiscretestateswhenevaluatingstatisticalaveragesbyacontinuumofstates.ThiswayweobtainanalyticalformulasfortheatomicdensitynT(r)[1]andenergydensityeT(r)inthethermalcloud:

󰀂

d3p

󰀇󰀃2nT(r)=

expp

p2

(2π󰀁)3exp

−3

whereλ=

󰀁

󰀃󰀅−(Veff(r)−µ)/kT

kTλg5/2e+2󰀃󰀅

−(Veff(r)−µ)/kT−3

Veff(r)λg3/2e,

󰀇󰀃

p2

(8)

LettertotheEditor5

fromtheGPequation.TheydependonthenumberofatomsinthecondensateNC.ByintegratingthedensitiesnCandnToverspatialdegreesoffreedomweobtainthenumberofatomsinthethermalcloudNT(NC,T)andtheenergyofthethermalcloudET(NC,T).

WeareinterestedinthetimedependenceofNCandT.Asbefore,wewillfirstconsiderthedynamicsofthetotalnumberoftrappedatomsNandtotalenergyEasafunctionofNCandT.Theadvantageofthisapproachisthatwedonotneedtostatetransfertermsexplicitly;thetransferwillfollowfromouranalysisautomatically.

˙andE˙arerelatedtoN˙CandT˙viaThetotallossratesN

󰀈˙C˙=∂NT+1NN

∂NC

(9)

󰀈˙C,˙=∂ET+µNE

∂NCwithµ=∂EC/∂NC.ThereparametrizationisstraightforwardaswehavealreadyfoundexplicitexpressionsofalmostalltermsappearinginEq.(9).

Inordertodescribethedynamicsofthesystemwemustconsiderallrelevantprocessesaffectingitsstate.Weincludethreemaindynamicaleffectsthatmaycauselossesofatomsfromthesystem:two-bodyinelasticcollisions,three-bodyrecombinationsandcollisionswithbackgroundgas.Inthefollowingwewillneglect

˙isgivenby[12]:allsecondarycollisions.ThenthetotalatomiclossrateN󰀈

12˙=+n2−NC+2nCnT+nT

2!

󰀆󰀈󰀂

123

3ξd3rn2,(10)CnT+3nCnT+nT

2!whereχandξaretwo-andthree-bodycollisionrateconstantsandτisthelifetimeofthetrap.

EachterminEq.(10)correspondstoalossprocessthatmayoccurinthesystem.Forexamplethetermξ3

2!

󰀆

eT

n2n2µ+CT

LettertotheEditor

collisions.Finallyweendupwiththeexpressionforthetotalenergylossrate:

󰀌2󰀍1˙n+nCnT+−E=

nTT

󰀆1µ

6

nT

󰀆1

23!n3C+

LettertotheEditor

54NC󰀂1053210

T󰀃ΜK󰀁1.551.501.45

0

1t󰀃s󰀁

2

7

0.00.5

1.0t󰀃s󰀁

1.52.0

Figure1.DecayofacondensateofmetastableheliumatomswithNC(0)=5×105applyingourmodel(solidline)incomparisonwithasimplermodelneglectingthermalization(dashedline).TheinitialnumberofthermalatomsNT(0)=5×105[T(0)≈1.5µK].Thedottedlinerepresentsthedecayofapurecondensate:NT(0)=0(T=0K).Inset:dynamicsofthetemperatureinourmodel.

54NC󰀂1053210

T󰀃ΜK󰀁2.72.62.50.0t󰀃s󰀁

0.50.00.5

1.0t󰀃s󰀁

1.52.0

Figure2.Decayofacondensatewithalargethermalcloud:NT(0)=2×106[T(0)≈2.5µK].FurtherdetailsasinthecaptionofFig.1.

enhancethecondensatedecayrate.Thiseffectcouldbeseenbyanexperimentalexaminationofthedecayrateasafunctionofthefractionofthermalatoms.Acknowledgements

ThisworkwasdoneaspartofastudentprojectwithintheSocrates-Erasmusexchangeprogramme2000/2001attheVrijeUniversiteitinAmsterdam.P.Z.acknowledgesthesupportfromPolishKBNgrant2/PO3/BO7819.WewouldliketothankMarekTrippenbachandKazimierzRzazewskiforusefuldiscussions.

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