Three-dimensional solitons and vortices in dipolar Bose-Eins(4)

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

elongationoftheplateau.Inthepreviousworks

[13,14],stabilizationofthe2Dvorticeswasassociatedwiththeappearanceofsuchplateau.ItisremarkablethatintheBECsdescribedbytheGPmodelwithadditionalnon-localnonlinearity,stabilizationofvorticesoccursbeforetheplateauisformed.AtgivennonlocalityparameterC,thegrowthratesvanishatsomethresholdvalueofλduetotheactionofnonlocaldipole-dipoleinteractions.Forinstance,atC=0.3,thesingle-chargedvorticesgetthestabilitywindowatλ>0.26anddouble-chargedvortices(m=2)–atλ>0.42,thesethresholdvaluesaremarkedbythearrowsinFig.2.Asforvorticeswithm>2,nostabilizationwasobserved.

Theevolutionandthedynamicalstabilityofsolitaryandvortexstructureswassimulatednumericallyusingthewell-knownsplit-stepFouriermethod[20],wherethenonlocalDDintegraltermwascalculatedinthespectralspace.Weusedperturbedsolutionsfoundbythestation-arysolverasaninitialconditionforthenonstationaryproblem.Forthestructureswithplateau,Petviashvili’sapproachmayfailandisunabletoreproducetheN(λ)diagramcompletelyinthehigh-energyregion.Inthiscase,weusedtheappropriateinitialconditionwiththeparametersextractedfromthevariationalanalysisasaninputforthenon-stationarysolvertocheckanexistenceandstabilityofsolitonsandvorticesforvaluesofNwhereourrelaxativesolverisine cient.

FIG.5:Decayofunstablevortex(m=2,C=0.3,λ=0.25):isosurfacesofconstantparticledensity|ψ|2aregivenfordi erenttimes.

t=0.0t=22.5t=37.5

FIG.6:Decayofunstablevortex(m=2,C= 0.3,λ=0.08):isosurfacesofconstantparticledensity|ψ|2aregivenfordi erenttimes.

Directnumericalsimulationscon rmedthepredictionsobtainedbythelinearstabilityanalysis.Solitonswerefoundtobestableeverywhereintheregion N/ λ>

4

0,stabilitythresholdsforvorticescoincidewiththosepredictedbythelinearstabilityanalysis.Intheunstableregion,destructionofavortexmayoccurinthedi erentways,dependingonthenumberofatomscon nedintothestructureandontheappliedperturbation.TypicalunstableevolutionsnapshotsareplottedinFig.5andFig.6fordi erentsignsofnonlocalityparameterC.

V.

CONCLUSIONS

Inthepresentpaperwestudythein uenceofnonlocaldipole-dipoleinterparticleinteractionon3DsolitonsandvorticesinBECdescribedbyGPEwithlocalcontactat-tractivetwo-particleandrepulsivethree-particleinterac-tion.Nonlocalanisotropicdipole-dipoleinteractionleadstoquantitative,aswellastoqualitativechangesinscales,energiesandstabilitypropertiesofsupportedsolitarystructures.Theformationthreshold(minimumnumberofatoms,whichisneededtoformastructure)goesdowncomparedtothecaseofabsenceofDDinteractions,andpossiblerangeofchemicalpotentialssigni cantlyelon-gates.Whilesolitonstructureshavingzerotopologicalchargearestableif N/ λ>0,vorticesmayexhibitaz-imuthalinstability.Weprovedthestabilizationof3Dvorticeswithm=1andm=2andrevealedtheirsta-bilizationthresholds.Allhigherorder(m>2)vorticeswerefoundtobeunstable.Uptoourknowledge,thisisa rstexampleof3Dstablevortexwiththetopologicalchargelargerthanoneinaconservativesystem.

Acknowledgments

WearegratefultoYuriKivsharandVolodymyrLashkinforfruitfuldiscussionsandcommentsaboutthispaper.