Three-dimensional solitary and vortex structures in Bose-Einstein condensates are studied in the framework of Gross-Pitaevskii model including the simultaneous action of local cubic-quintic nonlinearity and nonlocal dipole-dipole interactions. Nonlocal int

Three-dimensionalsolitonsandvorticesindipolarBose-Einsteincondensates

InstituteforNuclearResearch,Kiev03680,Ukraine

DepartmentofPhysics,TarasShevchenkoNationalUniversity,Kiev03022,Ukraine

1

Yu.A.Zaliznyak1,A.I.Yakimenko1,2

2

Three-dimensionalsolitaryandvortexstructuresinBose-EinsteincondensatesarestudiedintheframeworkofGross-Pitaevskiimodelincludingthesimultaneousactionoflocalcubic-quinticnonlinearityandnonlocaldipole-dipoleinteractions.Nonlocalinteractionsareshowntochangesigni cantlytheformationthresholdandthenumbersofatomscon nedintothecoherentstructures.Anappearanceofrobusthigh-order(m=2)three-dimensionalvorticesisrevealed.

PACSnumbers:03.75.Lm,05.45.Yv

7

00I.INTRODUCTION

2 guRecentprogressinexperimentalandtheoreticalstud-AiesofBose-Einsteincondensates(BECs)hasopenedup anewopportunitytoinvestigatenonlinearinteractions7ofatomicmatterwaves[1,2].Byapplyinganexternal1 magnetic eldandusingtheFeshbachresonance,thes- ]wavescatteringlengthcanbetuned,thusitispossibleStoexploreextremeregimesofinteractionfromstronglyPrepulsivetostronglyattractive.Theresultingnonlinear.nevolutionofmatterwavesgivesthepossibilitytoobserveithenonlineare ectssuchasatomicself-focusingandfor-lnmationofsolitons.

[ Darkandbrightsolitons5,6]havealreadybeen 1createdinBECs.Intheabsenceofanexternaltrap,vsuch3Dstructuresarealwaysunstable:theyeithercol-5lapse,ifnumberofatomsexceedssomecriticalvalue,or5spreadoutintheoppositecase.Duringanimplosion3ofBEC,theatomdensitybecomeshigh,andrepulsive2.three-particleinteractionscomeintoplayandshouldbe8takenintoaccount,whichgivesrisetotheadditionallocal0quinticnonlineartermincorrespondingGross-Pitaevskii7equation(GPE)10].InRef.[11]3Dnon-spinningsoli-:0tonsandvortexsolitonshavebeenstudiedintheframe-vworkofnonlinearcubic-quintic(CQ)Scr¨odingerequa-iXtionwiththeapplicationtothebulknonlinearopticalrmedia.Thecompetitionbetweenthecubicattractiveaandquinticrepulsivetermsisabletoarrestthecollapseandtostabilize3Dsolitons.Incontrasttonon-spinningsolitons,vortexsolitonscanbeunstablewithrespecttoazimuthalperturbationswhichbrake3Dvortexintosev-eral laments.InconservativeCQmedia,single-charged3Dvorticescanbestabilized,whilehigher-ordervorticesremainunstable[11].Veryrecently,stabledouble-chargevortexsolitonswerefoundintheframeworkofthecom-plexGinzburg-Landauequation[12].Asthematteroffact,dissipativesolitarystructuresmaybemorecom-pactandmorerobust,thusitmaybeeasierto ndstablelocalizedvorticesindissipativesystemsthanintheirconservativecounterparts.Inthepresentpaperwepresentthe rstexampleofstable3Dspinninghigher-order(m>1)solitonsinaconservativenonlinearsys-tem.Thestabilizationisachievedduetotheactionofthenonlocalnonlinearityassociatedwithlong-rangein-

terparticleinteractionsindipolarBEC.

Nonlocalnonlinearmediaresponsenaturallyappearsinawidevarietyofphysicalsystemssuchasplasmas[15,21,Bose-EinsteinBEC[26],op-ticalmedialiquidcrystals[23,24],andsoftmat-terInthe2Dcase,nonlocalnonlinearinteractionswereshowntosuppressanazimuthalinstabilityandtostabilizevortexsolitonsItwasfoundre-cently[28]thatspinning3Dsolitonsareunstableagainstsplittingintoasetofstablefundamentalsolitonsinthemediumwithnonlocalthermalnonlinearity.

Indegeneratedipolargases,thenonlocalnonlinear-ityarisesfromlong-range,partiallyattractive,andanisotropicdipole-dipole(DD)interactions.Thetheoret-icalinvestigationshaveshownthatthestabilityofdipo-largasescruciallydependsonthetrapgeometryHowever,itisnotalwaysnecessarytoincludeanexternaltrapsinceself-actionofBECcanbeenoughtoprovidetheconditionsforformationofstablelocalizedcoherentstructures.Inthepresentpaperweaddressthefollowingquestion:howthenonlocalDDinteractionsa ecttheparametersandstabilitypropertiesoftheself-consistedsolitonsandvortexsolitonsformedinthetrap-freeBEC.

II.

MODELEQUATIONS

WeconsiderBECofatomswithelectricdipoled(themodelisequallyvalidformagneticdipoles)orientedinthezdirectionbyasu cientlystrongexternal eld,andthathenceinteractviaadipole-dipolepotential

Vd(r)=gd(1 3cos2 )/r3,

(1)

wheregd=αd2/4πε0,ε0isthevacuumpermittivity, istheangleformedbythevectorjoiningtheinteractingparticlesandthedipoledirection.Theparameterαcanbetuned(seee.g.intherange 1/2≤α≤1.

Inthemean eldapproximation,adipolarBECatsu cientlylowtemperaturesisdescribedbyaGPEwithnonlocalnonlinearity:

i¯h Ψ

Ψ gΨ|Ψ|2

+gKΨ|Ψ|

4

+gdΨ 2M

Vd(r r′)|Ψ(r′,t)|2d3r′=0,

(2)