Effect of quantum fluctuations on the dipolar motion of Bose

We revisit dipolar motion of condensate atoms in one-dimensional optical lattices and harmonic magnetic traps including quantum fluctuations within the truncated Wigner approximation. In the strong tunneling limit we reproduce the meanfield results with a

E ectofquantum uctuationsonthedipolarmotionofBose-Einsteincondensatesin

opticallattices

AnatoliPolkovnikovandDaw-WeiWang

PhysicsDepartment,HarvardUniversity,Cambridge,MA02138

(Dated:February2,2008)

arXiv:cond-mat/0311492v3 [cond-mat.soft] 30 Jul 2004

Werevisitdipolarmotionofcondensateatomsinone-dimensionalopticallatticesandharmonicmagnetictrapsincludingquantum uctuationswithinthetruncatedWignerapproximation.Inthestrongtunnelinglimitwereproducethemean eldresultswithasharpdynamicaltransitionatthecriticaldisplacement.Whenthetunnelingisreduced,onthecontrary,strongquantum uctuationsleadto nitedampingofcondensateoscillationsevenatin nitesimaldisplacement.Wearguethatthereisasmoothcrossoverbetweenthechaoticclassicaltransitionat nitedisplacementandthesuper uid-to-insulatorphasetransitionatzerodisplacement.Wefurtheranalyzethetimedependenceofthedensity uctuationsandofthecoherenceofthecondensateand ndseveralnontrivialdynamicale ects,whichcanbeobservedinthepresentexperimentalconditions.

ThestudyofBose-Einsteincondensatesofultracoldatomshasbeengrowingrapidlyinrecentyears1.Load-ingbosenicatomsintoanopticallatticeandenhancingthelaserintensity,itispossibletostronglysuppressthekineticenergyoftheatomsresultinginasuper uid-to-Mott-insulator(SF-MI)quantumphasetransition2,3:inthestrongtunnelinglimitthecondensateisinthesuper- uidphasewith nitephasesti ness,whileitisdriventotheMott-insulatorphaseandlosesthephasecoherencebecauseofquantum uctuationsifthetunnelingbecomessmallenough.Typicalinterferencepatternsobtainedintime-of- ightexperimentscannotprovidesu cientinfor-mationtoseeasharptransitionboundary3,4.

Inone-dimensionalsystemsthecondensatedynam-icshasbeenobservedinadipolarmotionexperiment5,wherethecenter-of-mass(CM)ofthecondensateoscil-latesafterasuddendisplacementofthemagnetictrapwithrespecttotheopticallattice.Thisdynamicscanbedescribedbythemean eldGross-Pitaveskiiequations(GPE)inthestrongtunnelingregime5,6.Whenthedis-placementisbeyondacertaincriticalvalue,thecon-densateoscillationsareoverdamped7andthemotionbe-comescompletelydecoherent,indicatingaclassicallocal-izationtransition8,9.Suchatransitionwasrecentlyob-servedexperimentally10.Itisplausiblethatasthequan-tum uctuationsincrease,thereisasmoothcrossoverbe-tweenthequantum(SF-MI)andtheclassicaltransitions.However,weemphasizethatthe rstoneisasecondor-dertransitioncharacterizedbyreversiblephasecoherenceifthesystemisdriventotheinsulatingphaseandthenbacktotheSF3.Ontheotherhandtheclassicaltransi-tionisirreversible9duetothechaoticexcitationsinsidethecondensatecloud.

Motivatedbythisinterestingandimportantquestion,inthisletterweinvestigatethequantum uctuationef-fectsontheCMmotionofacondensateinaparabolicpotentialinaone-dimensionalopticallatticeusingthetruncatedWignerapproximation(TWA).Inthestrongtunneling(orweaklyinteracting)regime,wereproducethesharpdynamicaltransitionofamean eldcalcula-tion9withacriticaldisplacementproportionaltothesquarerootofthetunnelingamplitude.Whenthetun-

nelingisweakthequantum uctuationsstronglymodifythedynamicsofthecondensateinthefollowingways:(i)theCMmotionisdampedevenforanin nitesimalinitialdisplacement;(ii)theoscillationsbecomeoverdampedatacriticaldisplacement,Dc,whichisbelowtheclas-sicalvalueatthesametunneling;(iii)thecondensatedipolarmotionisfrictionlessfromoneendtoanotherwiththeperiodindependentofdamping,whilethesu-per uidfractionandthephasecoherencedropwhenthecondensatepassesthroughthecenteroftheharmonicwell.(iv)Wefurthershowthatatagivendamping,thelossofcoherenceislessforsmallerdisplacements(orstrongerquantum uctuations).Soasweincreasequantum uctuationsthelocalizationtransitionbecomesmorereversible.Ourresultshencesuggestthatthereisasmoothcrossoverbetweentheclassicallocalizationtran-sitionandthequantumsuper uid-to-insulator(SF-IN)transition11asthedisplacementgoestozero.

ThetruncatedWignerapproximation12hasbeenwellknowninquantumopticsforawhile.Recently,itwasappliedtothesystemsofinteractingbosons13,14.InRef.[14]itwasarguedthatTWAisequivalenttothesemiclassicalapproximationandnaturallyappearsinthequantumexpansionofthetimeevolutionofthesystem.TheideabehindTWAisthattheexpectationvalueofanobservable canbefoundaccordingto:

(t) ≈dψ0dψ0P(ψ0,ψ0) cl(ψ(t),ψ (t),t),(1)whereψandψ denotebosonic eldsobeyingclassicaldiscreteGross-Pitaevskiiequationsofmotion15,16:i ψj

2j2ψj+

U

We revisit dipolar motion of condensate atoms in one-dimensional optical lattices and harmonic magnetic traps including quantum fluctuations within the truncated Wigner approximation. In the strong tunneling limit we reproduce the meanfield results with a

2

theoperator evaluatedontheclassical eldsψ(t)andψ (t).ItisimportanttorealizethatTWAconsiderablyimprovesBogoluibov’stheory,especiallyiftheclassicaldynamicsbecomesunstable14.ThewayweimplementTWAinthisletterisoutlinedinRef.[14].

)

Displacement (D

Inverse Tunneling (1/J)

FIG.1:DynamicalphasediagramforUN=50,andK=0.02(theoccupancyofthecentralsite,N0,isapproximately5%ofthetotalnumberofbosons,N).Thesolidandthedashedlinescorrespondtodampingγ=0.11andγ=0.36respectively.Thelargeseparationbetweenthetwocurvesforstrongerquantum uctuations(smallerN)impliesbroaden-ingofthetransition.TheinsetshowsthetypicaltemporalCMoscillationsforγ≈0.36(dashedline)andγ≈2.1(solidline).

Inadipolarmotionexperiment,thecondensateisini-tially(t<0)preparedinthesuper uidgroundstate.Att=0thetrappositionissuddenlydisplacedbythedistanceD0fromtheoriginandthecondensatestartsto

move.To ndtheWignertransformP(ψ0,ψ0)ofthein-teractinggroundstate,westartfromthenoninteracting

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