Controlled Manipulation with a Bose--Einstein Condensates N-

A model of the perturbed complex Toda chain (PCTC) to describe the dynamics of a Bose--Einstein condensate (BEC) N-soliton train trapped in an applied combined external potential consisting of both a weak harmonic and tilted periodic component is first dev

Vol.25,No.7(2008)2370

ControlledManipulationwithaBose–EinsteinCondensatesN-SolitonTrain

undertheIn uenceofHarmonicandTiltedPeriodicPotentials

ZONGFeng-De(宗丰德),ZHANGJie-Fang(张解放)

InstituteofNonlinearPhysics,ZhejiangNormalUniversity,Jinhua321004

(Received12January2008)

AmodeloftheperturbedcomplexTodachain(PCTC)todescribethedynamicsofaBose–Einsteincondensate(BEC)N-solitontraintrappedinanappliedcombinedexternalpotentialconsistingofbothaweakharmonicandtiltedperiodiccomponentis http://doc.guandang.netingthedevelopedtheory,theBECN-solitontraindynamicsisshowntobewellapproximatedby4Ncouplednonlineardi erentialequations,whichdescribethefundamentalinteractionsinthesystemarisingfromtheinterplayofamplitude,velocity,centre-of-massposition,andphase.Thesimpli edanalytictheoryallowsforane cientandconvenientmethodforcharacterizingtheBECN-solitontrainbehaviour.ItfurthergivesthecriticalvaluesofthestrengthofthepotentialforwhichoneormorelocalizedstatescanbeextractedfromasolitontrainanddemonstratesthattheBECN-solitontraincanmoveselectivelyfromonelatticesitetoanotherbysimplymanipulatingthestrengthofthepotential.

PACS:03.75.Lm,03.75.Kk,05.45.Yv

Recently,Bose–Einsteincondensate(BEC)soli-tonshaveattractedagreatdealofinterestbothfromthetheoreticalandtheexperimentalpointofview,[1,2]andhavebeenusedtostudysuchdi-versephenomenaasphasecoherence,[3 5]matter–wavedi raction,[6]quantumlogic,[7 9]matter–wavetransport,[10,11]matter–wavegratings,andpulsedmatter–wavelasers.[12]SomedynamicalissuesrelatedtoaBECN-solitontrain,suchasthedynamicsofatrainofmatter–wavesolitonscon nedtoaparabolictrap,opticallattice,andtiltedperiodicpotentials[13]havebeeninvestigated.However,sofarthecon-trolledmanipulationdynamicsofaBECN-solitontraintrappedinanappliedcombinedexternalpoten-tialhasnotyetbeenanalysed.InthisLetter,we rstgeneralizethemodeloftheperturbedcomplexTodachain(PCTC)[14 17]toaBECN-solitontrain,andfurtherconsiderthedynamicalevolutionofaBECN-solitontrainaswellashowtomanipulateandcon-trolitsubjecttothein uenceofbothharmonicandtiltedperiodictrappingpotentials.[3,4,18]Ouranalysischaracterizesthedynamictransitionfromtroughtotroughandfurtherpredictswhenmultiplelatticesitesarestable.Theanalysisalsosuggestshowtoselec-tivelymovetheBECN-solitontrainfromonelatticesitetoanotherbysimplymanipulatingthestrengthofthepotential.

Ithasbeenwellknownthatthemacroscopicde-scriptionoftheattractiveBECwavefunctionincigar-shapedBECtrapgeometriesisgovernedbythenon-linearSchr¨odingerequationwithexternalpotential

[19]

V(x)(orGross–Pitaevskiiequation[20,21])

u1 2u

+|u|2u=V(x)u,i+2 t2 x

Supported

whereu(x,t)isthenormalizedcomplex-valuedmean- eldvariable,V(x)isarealphysicalpotential.

Wewillconsideranappliedcombinedexternalpo-tentialconsistingofbothaharmonicandtiltedperi-odiccomponent,thusthepotentialV(x)inEq.(1)isgivenby

V(x)=V0x2+V1cos[ (x x¯)]+V2x,

(2)

whereV0,V1,andV2measurethestrengthsofthe

harmonic,periodic,andtiltedpotentials,respectively.Theparameterx¯measurestheo setofaminimumoftheperiodicpotentialwithrespecttotheminimumoftheharmonicpotential.

NotethatalthoughV0～10 5 10 3isasmallparameterintherangeofrealisticexperimen-talconditions,[22,23]foranynon-zerovaluesofV0andV2inEq.(2)theparabolicandtiledpotentialstendtoin nityas|x|→∞,andtheexternalpotential(2)isanunboundedoperator.However,foralatticeof nitesize L≤x≤LwithLbeingthe niteboundaryvalueofx,[24]theoperator(2)isalwaysbounded.Inthefollowing,weconsidertheaboveweakpotentialsintheaxialdirectionxandassumethemasperturbationsiR[u]=V(x)u(x,t).Withthee ectofallthesephysicalprocessesmentionedabove,theBECdynamicsisdescribedbytheperturbednonlin-earSchr¨odingerequation(NLS)

i

u1 2u

+|u|2u=iεR[u].+ t2 x

(3)

(1)

We rstgeneralizethePCTCmodeltoaBECwith

weakharmonicandtiltedperiodicexternalpotentialsintheadiabaticapproximation.WeconcentrateontheperturbedNLSEq.(3).By‘N-solitontrain’we

bytheNationalNaturalScienceFoundationofChinaunderGrantNo10672147,andtheNaturalScienceFoundation

ofZhejiangProvinceunderGrantNoY605312.

Towhomcorrespondenceshouldbeaddressed.Email:jfzhang@http://doc.guandang.net)c2008ChinesePhysicalSocietyandIOPPublishingLtd

A model of the perturbed complex Toda chain (PCTC) to describe the dynamics of a Bose--Einstein condensate (BEC) N-soliton train trapped in an applied combined external potential consisting of both a weak harmonic and tilted periodic component is first dev

No.7ZONGFeng-Deetal.2371

meanasolutionoftheperturbedNLSequation xedupbytheinitialcondition

u(x,t=0)=

Nu1ks

(x,t=0),

k=1

u1ks(x,t

=0)=2νksechζkeiφk,

ζk(x,t)=2νk[x ξk(t)],(4)

ξk(t)=2µkt+ξk,0,φµk

k(x,t)=

νζk+δk(t),k

δk(t)=wkt+δk,0,

(5)

whereamplitudeνk,velocityµk,c.m.positionξkandphaseδkfork=1,···,Narethe4Nsolitonparameters.BecauseintheprocessofformationoftheBECN-solitontrain,thenumberofsolitons,theamplitudeofaninpidualsoliton,andtheseparationbetweenneighbouringsolitonsproducedshouldvarywiththescatteringlengthandtheinitialsizeonthecondensate,[25 27]fortheadiabaticapproximationthesolitonparametersmustsatisfy[28]

|νk ν0| ν0,|µk µ0| µ0,|νk ν0||ξk+1,0 ξk,0| 1,

(6)

whereν1 0=Nνandµ=1 N

µkaretheaverageamplitudeNk=1k0

andvelocity,respectively.Nk=1

Theadiabaticapproximationusesasasmallparameterε0 1thesolitonoverlapwhichfallso exponentiallywiththedistancebetweenthesolitons.Thesmallparameterε0canberelatedtotheinitialdistancer …… 此处隐藏：13863字，全部文档内容请下载后查看。喜欢就下载吧 ……