Instability of a Bose-Einstein Condensate with Att

arXiv:cond-mat/9908229v116Aug1999InstabilityofaBose-EinsteinCondensatewithAttractiveInteractionAntoniosEleftheriouandKersonHuangDepartmentofPhysicsandCenterforTheoreticalPhysicsMassachusettsInstituteofTechnologyCambridge,MA02139,USA(February1,2008)03.75.Fi,42.65.Jx,32.80.PjMIT-CTP#2886WestudythestabilityofaBose-Einsteincondensateofharmonicallytrappedatomswithnegativescatteringlength,specifically7Li.Ourmethodistosolvethetime-dependentnonlinearSchr¨odingerequationnumerically.Foranisolatedcondensate,withnogainorloss,wefindthatthesystemisstable(apartfromquantumtunneling)iftheparticlenumberNislessthanacriticalnumberNc.ForNNc,thesystemcollapsestohigh-densityclumpsinaregionnearthecenterofthetrap.Thetimefortheonsetofcollapseisontheorderof1trapperiod.Withinnumericaluncertainty,theresultsareconsistentwiththeformationofa“blackhole”ofinfinitedensityfluctuations,aspredictedbyUedaandHuang[16].WeobtainnumericallyNc≈1251.Wethenincludegain-lossmechanisms,i.e.,thegainofatomsfromasurrounding“thermalcloud”,andthelossduetotwo-andthree-bodycollisions.ThenumberNnowoscillatesinasteadystate,withaperiodofabout145trapperiods.WeobtainNc≈1260asthemaximumvalueintheoscillations.I.INTRODUCTIONANDSUMMARYBose-Einsteincondensationhasbeenobservedinmagneticallytrappeddilutevaporsofthealkalielements87Rb[1],23Na[2],7Li[4],and1H[3].Atthenanodegreetemperaturesoftheseexperiments,thesystemswouldhavefrozensolidlongagoweretheyinfreespace.Intheconfiningtrap,however,zero-pointmotionkeepstheatomsapart,andthesystemsremaingaseous.Thecaseof7Liisspecial,however,inthattheinteratomicinteractionispredominantlyattractive,asindicatedbyanegativescatteringlength.Thus,thecondensatein7Lishouldbelessstablethantheothercases.Thepurposeofthispaperistostudythenatureoftheinstability,itsonset,andmanifestations.Theobjectofstudyisthecondensatewavefunctionψ(r,t),whichgivestheproba-bilityamplitudeforannihilatingoneparticleinthecondensateatrattimet.Weusethetime-dependentGross-Pitaevskii(GP)equation,ornonlinearSchr¨odingerequation(NLSE),whichcorrespondstoamean-fieldapproximation:i~∂ψ∂t=−~22m∇2+V(r)−U0|ψ|2ψU0=4π~2|a|m(1)1whereaisanegativescatteringlength,andtheexternalpotentialistakentobeharmonic:V(r)=12mω2r2(2)Thisdefinesacharacteristiclengthd0,thewidthoftheunperturbedground-statewavefunction:d0=r~mω(3)ThenumberofcondensateparticlesentersthroughthenormalizationN=Zd3r|ψ|2(4)whichisaconstantofthemotion.AnotherconstantofthemotionistheHamiltonianH=Zd3r−~22mψ∗∇2ψ+V(r)ψ∗ψ−U02(ψ∗ψ)2(5)Theparametersusedin7Liexperimentscorrespondto[4]a=−1.45nmd0≈3.16μmω≈908s−1(6)whereωistakentobeapproximatelyequaltothegeometricmeanofthethreecircularfrequenciesoftheexperimentaltrap.Equation(1)doesnottakeintoaccountthecouplingbetweenthecondensateandthe“thermalcloud”ofuncondensedatoms,nordoesitdescribethelossofatomsfromthetrapduetocollisions.Theseeffectswillbeconsideredlater.Infreespace,theNLSEwithattractiveinteractionshasaninstabilityknowninplasmaphysicsas“self-focusing”[5],wherebytheinitialwavefunctiondevelopsasingularityinfinitetime,correspondingtoalocalcollapsetoastateofinfinitedensity.Inanexternaltrap,however,thegroundstateisapparentlystable,aslongasNisnottoolarge,asindicatedbyvariationalcalculationsusingaGaussiantrialwavefunction[6].ThewidthoftheGaussiannarrowswithincreasingN,andcollapsestozeroatsomecriticalvalueNc.Thisconclusionisborneoutbyotherstudies[7–11].Inparticular,KimandZubarevobtainanexactupperboundforNc,whichforaGaussianwavefunctiongivesNc≦0.671d0|a|(7)UedaandLeggett[9]obtaintheupperboundinavariationalcalculation.Forthe7Lipa-rameters(6),thisformulagivesNc≦1463.Inthenumericalcalculationsdescribedlater,weobtainNc=0.574d0|a|(8)2orNc=1251forthe7Liparameters.Thisnumberbecomesslightlylargerwhengainandlosseffectsaretakenintoaccount.Actually,evenignoringcollisionalloss,thecondensateisonlymetastableforNNc,foritcandecayviaquantumtunneling.ThiseffectisnotdescribedbytheNLSE,al-thoughanapproximatedecayamplitudecanbeobtainedbycontinuationoftheNLSEtoimaginarytime.Kaganetal.[12]estimatedtheamplitudebycalculatinganoverlapinte-gralbetweeninitialandfinalstatesrepresentedbyGaussianwavefunctions,andobtained(df/di)3N/2,wheredianddfarerespectivelythewidthsoftheinitialandfinalwavefunctions.Throughnumericalcalculations,Shuryak[13]foundthatthedecayrateisproportionaltoexp[−const.(Nc−N)].Stoof[14]wrotedownaWKBformulaforthedecayrate,butdidnotexplicitlyevaluateit.Usingvariationalwavefunctions,UedaandLeggett[9]obtainedarateproportionaltoexp[−const.(Nc−N)5/4].TheseestimatesindicatethatthetunnelingprobabilityisnegligibleunlessN≈Nc.However,theWKBapproximationbreaksdowninthisneighborhood,andreliablecalculationsbecomedifficult.Inexperimentalterms,itisalsodifficulttoobservetunnelinginthepresentsystem,becausetheeffectismaskedbycollisionalloss,especiallynearN=Nc[15].Forthesereasons,weshallnotcalculatethetunnelingamplitudeinthispaper;butwewillgiveaqualitativediscussionofthephenomenon,inasfarasitpertainstotheinstabilityofthesystem.UedaandHuang[16]formulatedanapproachtothestabilityproblem,includingtun-neling,intermsoftheFeynmanpathintegralforatra