Superfluid-Mott Insulator Transition of Spin-2 Cold Bosons i

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Super uid-MottInsulatorTransitionofSpin-2ColdBosonsinan

OpticalLatticeinaMagneticField

arXiv:cond-mat/0312456v1 [cond-mat.soft] 18 Dec 2003ShuoJin1 ,Jing-MinHou1 ,Bing-HaoXie2,Li-JunTian1,andMo-LinGe11TheoreticalPhysicsDivision,NankaiInstituteofMathematics,NankaiUniversity,Tianjin,300071,China,andLiuhuiCenterforAppliedMathematics,Tianjin,300071,China2LaboratoryforComputationalPhysics,InstituteofAppliedPhysicsandComputationalMathematics,Beijing,100088,ChinaAbstractThesuper uid-Mottinsulatortransitionofspin-2bosonatomswithrepulsiveinteractioninanopticallatticeinamagnetic eldispresented.Byusingthemean eldtheory,Mottgroundstatesandphasediagramsofsuper uid-Mottinsulatortransitionatzerotemperaturearerevealed.Appliedmagnetic eldleadstosomephaseboundariessplitting.ForalltheinitialMottgroundstatescon-tainingmultiplespincomponents,di erentspincomponentstakeondi erentphaseboundaries.Itisfoundthatinthissystemthephaseboundarieswithdi erentmagnetizationcanbemovedindi erentwaysbyonlychangingtheintensityoftheappliedmagnetic eld.PACSnumber(s):03.75.Kk,03.75.Lm,03.75.Mn,32.80.Pj

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

I.INTRODUCTION

Recentremarkableexperiments[1,2,3,4]onthesuper uid(SF)toMottinsulator(MI)transitioninasystemofultracoldatomsinanopticallatticeopenintriguingprospectsforstudyingmany-bodyphenomena,associatedwithstronglycorrelatedsystemsinahighlycon-trollableenvironment.Theopticallattices[5,6]—arraysofmicroscopicpotentialsinducedbytheacstarke ectofinterferinglaserbeams—provideidealconditionsforthestudyofthelasercoolingandthequantumphasetransitionsofthecon nedcoldatoms.Thedynamicsofthecon nedatomsinopticallatticesisadequatelydescribedbytheBose-Hubbardmodel

[7,8],whichpredictsSF-MItransitionatlowtemperaturewithincreasingtheratiooftheon-siteinteractiontothehoppingmatrixelement.Besidesmanyexperimentale ortsmadetorealizeSF-MItransition,alargenumberoftheoreticalstudieshaveappeared[7,8,9,10].Inreference[9]anappropriatemean- eldapproximationwasdevelopedfortheHamiltonianofspinlessorpolarizedbosonsinanopticallattice,andinordertodescribethezero-temperaturephasetransitionfromthesuper uidtotheMott-insulatingphase,thephasediagramswerecalculated.

Sinceopticaltraps[11,12,13]liberatethespindegreesoffreedomandmakepossiblecondensationofspinorbosons,extensiveinterestshavebeenstimulatedinthestudyofmulti-componentspinorBEC.ThequantumphasetransitioninspinorBEC,aswellasavarietyofothernovelphenomena[14,15,16,17,18]werewellstudied.Subsequently,inspiredbytheseworks,thetheoreticalresearchesabouttheSF-MItransitionofthespinorbosonstrappedinanopticallatticearise.Mostrecently,DemlerandZhou[19]havestudiedspin-1Boseatomsinanopticallatticeandobtainedseveraluniqueproperties.Tsuchiyaetal.[20],HouandGe[21]haveinvestigatedthespin-1andspin-2bosonsinanopticallatticewiththemean- eldapproximationmethodandobtainedthephasediagramsshowingatransitionfromMottinsulatortosuper uid,respectively.

Ontheotherhand,theresponsetoexternalmagnetic eldofBECisalsoatopicwithinterests[17,18,22,23,24,25].Theexperimentalistshaveconcentratedoninvestigatingthesystemsinanappliedmagnetic eldbecausethephasetransitioncanbetunedbyadjustingthemagnetic eldratherthanchangingthesamplesmeasured[26,27,28,29].Thusthetheoreticalstudyinthisaspectisnecessary.Sofar,UedaandKoashi[17,18]havediscussedthemagneticresponseofspin-1andspin-2BECinamesoscopicregime;astothesysteminanopticallattice,SvicainskyandChui[30]havestudiedthespin-1bosonsinamagnetic eldandshown

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

somee ectsinducedbythemagnetic eld.However,thespin-2caseinamagnetic eldhasnotbeendiscussedyet.Whataboutthegroundstates,quantumphasetransitionandthein uenceofthemagnetic eldonthissystem?Thisisourmissioninthepresentletter.First,ignoringthehoppingtermoftheHamiltonian,wegetthesite-independentHamiltonian,itsenergyeigenvaluesandtheMottgroundstatesfordi erentcases.Then,applyingmean- eldapproximationandregardingthehoppingtermasaperturbation,weperformthecalculationsinsecond-orderanddrawthephasediagrams.Theresponseofthephasediagramstotheappliedmagnetic eldisqualitativelyanalyzedsubsequently.Finally,wegivesomeremarksandtheconclusion.

II.THEMODEL

Weconsideradilutegasofbosonatomswithhyper nespinF=2,suchas23Na,87Rbor85Rbsubjecttoanexternalmagnetic eldinanopticallattice.BasedonRef.[21],theHamiltonianofspin-2bosonswithrepulsiveinteractioninanopticallatticeincludingamagnetic eldtermcanbewritteninthesecond-quantizedform:

H=HA+HB,

(1)

Ψ αΨβΨβΨαHA=

+dr[h¯2c¯1

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Forsimplicity,weassumeauniformmagnetic eldappliedalongthez-direction,anditisweakenoughtoignorethequadraticZeemane ect.Expandingthe eldoperatorsintheWannierbasisandkeepingonlythelowestvibrationalstates,Ψα=

reducestothegeneralizedBose-HubbardHamiltonian

H= t ibiαw(r ri),Eq.(1)b ibj

<i,j> µ in i+c02 i 2 6 (Fni)+i2c2

2n ( n 1)+c1

5 +S pF z.S(5)

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