Delocalizing transition in one-dimensional condensates in op

It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in delocalizing transition in one dimension, what sharply contrasts to the known behavior of discrete and periodic systems with homo

Delocalizingtransitioninone-dimensionalcondensatesinopticallatticesdueto

inhomogeneousinteractions

CentrodeF´ sicaTe´oricaeComputacional,UniversidadedeLisboa,

ComplexoInterdisciplinar,AvenidaProfessorGamaPinto2,Lisboa1649-003,Portugal

2

DepartamentodeF´ sica,UniversidadedeLisboa,

CampoGrande,Ed.C8,Piso6,Lisboa1749-016,Portugal3

DepartamentodeMatem´aticas,E.T.S.deIngenierosIndustriales,UniversidaddeCastilla-LaMancha13071CiudadReal,Spain

1

Yu.V.Bludov1, V.A.Brazhnyi1, andV.V.Konotop1,2,3

arXiv:0706.0079v1 [cond-mat.other] 1 Jun 2007

ItisshownthatinhomogeneousnonlinearinteractionsinaBose-Einsteincondensateloadedinanopticallatticecanresultindelocalizingtransitioninonedimension,whatsharplycontraststotheknownbehaviorofdiscreteandperiodicsystemswithhomogeneousnonlinearity.Thetransitioncanbeoriginatedeitherbydecreasingtheamplitudeofthelinearperiodicpotentialorbythechangeofthemeanvalueoftheperiodicnonlinearity.Thedynamicsofthedelocalizingtransitionisstudied.

PACSnumbers:03.75.Lm,03.75Kk,03.75Ss

I.INTRODUCTION

Spatiallocalizationoftheenergyisafundamentalphysicalphenomenonwhichattractsagreatdealofat-tentioninverydi erentphysicalapplicationsandmathe-maticalstatements.Whenlocalizedstatesareoriginatedbythebalancebetweennonlinearityandnaturaldisper-sionofthemedium,theyarereferredtoassolitons.Iflocalizationiscausedbytheinterplaybetweennonlin-earityandperiodicity,inducedeitherbyvariationsofparametersofthemediumorbydiscreteness,therespec-tivesolutionsareusuallytermedlocalizedmodes(orgapsolitons).Localizedmodesareknowntoexistinnumer-oussystemspossessingthetwomainingredients–non-linearityandperiodicity.Discretesystems[1],photoniccrystals[2],Bose-Einsteincondensates(BECs)loadedinopticallattices(seee.g.[3,4])areonlyfewexampleswherethelocalizedmodeshavebeendiscovered.

InthiscontextBECsinopticallatticeshaveattractedaspecialattention,justaftertheyhavebeencreatedex-perimentally[5].Thisisdueto exibilityofopticallat-ticesallowingonetochangetheirparametersintimeandinspacebysimplemanagementofgeometryandintensityoflaserbeams,aswellasduetopossibilitiesofexperi-mentalrealizationofone–,two–,andthree–dimensional(below1D,2D,and3D,respectively)periodicsystems.Naturally,aquestionaboutsurvivalsoflocalizedstatessubjectedtovariationofthelatticeparameters,orinotherwordsaboutexistenceofdelocalizingtransitionhasraised.ThisissuehasbeenaddressedwithintheframeworkofthediscretenonlinearSchr¨odingerequa-tion[6],whichmodelstheGross-Pitaevskii(GP)equa-tionintermsoftheWannierfunctionexpansion[7],anddirectlywithintheframeworkoftheGPequationwitha

It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in delocalizing transition in one dimension, what sharply contrasts to the known behavior of discrete and periodic systems with homo

Theorganizationofthepaperisasfollows.InSec.IIweintroducethemodelanddescribethephysicalargu-mentssupportingexistenceofthedelocalizatingtransi-tion.InSec.IIIwediscusshowtheparametersofthesystemshouldbechosenanddescribedetailsofthedy-namicsofthedelocalization.Sec.IVisdevotedtoanewmechanismofthedelocalizingtransition,whichisbasedonthechangeofthenonlinearitycontrollingparameter(whichisreferredsimplyasanumberofparticles).TheoutcomesaresummarizedinConclusion.

II.

THEMODEL

Toexplainthephysicsofthephenomenonwestartwiththedimensionless1DGPequation

iψt= ψxx+U(x)ψ+G(x)|ψ|2ψ,

(1)

whereU(x)isaπ-periodicpotential,U(x)=U(x+π),andG(x)isthecoordinate-dependentnonlinearity,whichisalsoπ-periodic:G(x)=G(x+π).Insuchascalingtheenergy2ismeasuredintheunitsoftherecoilenergyEr= π2/(2md2),wheredisthelatticeconstant,andthespatialandtemporalvariablesintheunitsd/πand /Er,respectively.ParticularphysicalunitsofG(x)dependontheapplicationofthemodel(seee.g.[13,14]).

Eq.(1)possessesstationarylocalizedmodesolu-tions[14,16,17],ψ=φ(x)e iµt.Hereµisachemi-calpotential.Itmust belongeitherEn,En toa niten-thgap

(n=1,2...):i.e.µ∈( )(+),whereEn(σ)

iseitherlower(σstandsfor” ”)orupper(σstandsfortothesemi-in nitegapµ∈ofthespectrumoftheoperatorL= d2/dx ”+”)edgeofthegap,or ∞,E(+)

2+U(x),

i.e.L (nσ)=En(σ) (nσ),where (nσ)

Blochstate: (x)isanormalized

π (nσ)

0|(x)|2dx=1.Allbranchesofsuchsolutionsareparameterizedbythechemicalpotential.Theyeithercollapseatsomevaluesofµinsideagaporendupatagapedge.Thelowestbranchmaybifur-catefromtheextendedBlochstate.Ifthishappens,themoderepresentsasmallamplitudegapsolitongovernedbythenonlinearSchr¨odingerequation(seee.g.[3,iAτ=

2Mn(σ) 19])

Aξξ+χ(nσ)

|A|2Aforaslowenve-lopeamplitudeA(τ,ξ),whichisintroducedthroughthe

representationψ≈ A(τ,ξ) (nσ)

(x)anddependsonslowvariablesξ= xandτ= 2t.Thesmallparameter isproportionaltotheenergydetuningtowardsthegaptheedgeEn(σ); ～|µ En(σ)|,M(σ)thee ectivemassandχ(nσ)

= n=π

d2En(σ)

/dk

2(σ) from

1is0G(x)| n(x)|4dxisthee ectivenonlinearity.

Inageneralsituation,whenthelatticeamplitudede-creases,thewidthofagapdecreasesaswellandallgapscollapseatU(x)≡0.InitsturnasmallgapmayallowfortheexistenceofonlysmallamplitudesolitonswhichtendtoextendedBlochwavesinthelimitwhenthechemical

2

potentialapproachesthegapedge.Existenceofthesoli-tonspreventsdelocalizingtransition,becauseadiabaticdecreaseofthelatticeamplitudeleadstoincreaseofthesolitonwidthandsubsequentincreaseofthepotentialresultsinrecoveringalocalizedstate[6,8].

Thus,inordertoobtaindelocalizingtransitiononehastoc …… 此处隐藏：17572字，全部文档内容请下载后查看。喜欢就下载吧 ……