Generating vortex rings in Bose-Einstein condensat

arXiv:cond-mat/0202312v119Feb2002GeneratingvortexringsinBose-Einsteincondensatesintheline-sourceapproximationM.Guilleumas1,D.M.Jezek2,R.Mayol1,M.Pi1,M.Barranco11Departamentd’EstructuraiConstituentsdelaMat`eria,FacultatdeF´ısica,UniversitatdeBarcelona,E-08028Barcelona,Spain2DepartamentodeF´ısica,FacultaddeCienciasExactasyNaturales,UniversidaddeBuenosAires,RA-1428BuenosAires,andConsejoNacionaldeInvestigacionesCient´ıficasyT´ecnicas,Argentina(February1,2008)AbstractWepresentanumericalmethodforgeneratingvortexringsinBose-Einsteincondensatesconfinedinaxiallysymmetrictraps.Thevortexringisgeneratedusingtheline-sourceapproximationforthevorticity,i.e.,therotationalofthesuperfluidvelocityfieldisdifferentfromzeroonlyonacir-cumferenceofgivenradiuslocatedonaplaneperpendiculartothesymmetryaxisandcoaxialwithit.Theparticledensityisobtainedbysolvingamodi-fiedGross-Pitaevskiiequationthatincorporatestheeffectofthevelocityfield.Wediscusstheappearanceofdensityprofiles,thevortexcorestructureandthevortexnucleationenergy,i.e.,theenergydifferencebetweenvorticalandground-stateconfigurations.Thisisusedtopresentaqualitativedescriptionofthevortexdynamics.03.75.Fi,05.30.Jp,32.80.Pj,67.40.Vs,67.40.DbTypesetusingREVTEX1I.INTRODUCTIONSince1999,whenvortexlinesinatrappedBose-Einsteincondensate(BEC)werefirstexperimentallyobtained[1],theirstudyhasreceivedgreatexperimentalandtheoreticalinterestasitconstitutesaclearsignatureofsuperfluidityeffectsintheseconfinedsystems.Someremarkableexperimentalachievementsare,amongothers,thestudyofthedynamicsofsinglevortexlines[2]andtheformationofsmall[3]andlarge[4]vortexarrays.AreviewoftheresearchdoneinthisfieldispresentedinRef.[5].ItisworthtonotethattheexperimentalsituationinBECregardingtheformationanddetectionofvorticesisatvariancewiththatinheliumIIdroplets.Forthesedrops,aparadigmofsuperfluidfinitesystems,theproblemofnucleatingvorticesandtheirstabilityhasonlybeenaddressedveryrecentlyfromthetheoreticalpointofview(seeRefs.[6–8]),andtheirexperimentaldetectionisstillanopenquestion.Vortexringsarevorticeswhosecoreisaclosedloopwithquantizedcirculationaroundit[9].Theyarecomplextopologicalstructuresthathaveattractedandwillcontinuetoattractsomeexperimentalandtheoreticalinterest.Basedonnumericalsimulations,differentmeth-odshavebeenproposedtogeneratevortexringsinBEC.Asinbulkliquidhelium,vortexringsmaybeproducedintroducinganimpurityinthecondensatewithadefinitevelocity,whosedisplacementcausesavortexring[10].Anothermechanism[11]consistsinusingdy-namicalinstabilitiesinthecondensatetocausedarksolitonstodecayintovortexrings.Awellcontrolledmethodtoproducevortexringsbyelectromagneticallyinducedatomictran-sitionsintwo-componentcondensateshasalsobeenputforward[12].ThemethodproposedinRef.[11]hasbeensuccessfullyappliedinRef.[13]togeneratevortexringsexperimentally.Ratherthanproposingamethodthatcouldbeimplementedtocreateavortexringexperimentally,ouraimhereistosetupanumericalmethodsimpleyetaccurateenoughtogeneratequantizedvortexringswithadefiniteradiusRinone-componentcondensates.Werestrictourstudytovortexstatessuchthatthedivergenceoftheirvelocityisvanishinglysmall.Thisassumptionyieldsanalyticalexpressionsforthevelocityfield,andprovidesafairapproximationtothesuperfluidflowaroundthevortexcoreforringsinthebulkoflargecondensates(Thomas-Fermilimit).Wehaveconsideredlargecondensatesatzerotemperatureandtherefore,dissipationhasnotbeentakenintoaccount.Theyareaxiallysymmetricaboutthezaxis,andhavethez=0planeassymmetryplane,andmayhostavortexringofradiusR,coaxialwiththesymmetryaxisofthetrap,andplacedonaplaneatadistanceZ≥0fromthesymmetryplane.Althoughgeneralizationtosituationswithmorethanonesuchvortexringsisstraightforward,wehavenotconsideredthispossibility.Thispaperisorganizedasfollows.InSect.IIwepresentthemethodusedtogeneratevortices.ThewaytoobtainthevelocityfieldofavortexringisdescribedinSect.III,andtheexplicitexpressionsofthevelocitycomponentsaregiveninanAppendix.Sect.IVisdevotedtotheanalysisoftheparticledensityprofilesandvortexnucleationenergies,whichallowsaqualitativedescriptionoftheirdynamics.Wealsopresentresultsobtainedfromacompletelydifferent,moreinvolvedmethodwehavesetuptogenerateringvorticesthatpermitstotesttheapproximationofzerodivergenceofthevelocity.Finally,abriefsummaryoftheresultsispresentedinSect.V.2II.ENERGYFUNCTIONALWeconsideraweaklyinteractingBose-condensedgasconfinedinaharmonictrapVext(r)atzerotemperature.IntheGross-Pitaevskii(GP)theory,thegroundstate(g.s.)energyofthecondensateisgivenbythefunctional[14]E[Ψ]=Zdr¯h22m|∇Ψ|2+Vext(r)|Ψ|2+g2|Ψ|4#,(1)whereΨ(r)isthecondensatewavefunction.ThefirstterminEq.(1)isthekineticenergyofthecondensate,thesecondtermistheharmonicoscillatorenergyarisingfromthetrappingpotential,andthethirdtermisthemean-fieldinteractionenergy.Thecouplingconstantisg=4π¯h2a/m,whereaisthes-wavescatteringlength,andmistheatomicmass.ThenumberofatomsinthecondensateisRdr|Ψ|2=N.Theg.s.wavefunctionisdeterminedbysolvingtheGPequationobtainedminimizingtheenergyfunctional.ThewavefunctionΨcanbewrittenintermsoftheparticledensityρ(r)=|Ψ(r)|2andphaseS(r)asΨ(r)=qρ(r)e