Split Instability of a Vortex in an Attractive Bose-Einstein

An attractive Bose-Einstein condensate with a vortex splits into two pieces via the quadrupole dynamical instability, which arises at a weaker strength of interaction than the monopole and the dipole instabilities. The split pieces subsequently unite to re

SplitInstabilityofaVortexinanAttractiveBose-EinsteinCondensate

HirokiSaitoandMasahitoUeda

DepartmentofPhysics,TokyoInstituteofTechnology,Tokyo152-8551,Japan

(Dated:February1,2008)

AnattractiveBose-Einsteincondensatewithavortexsplitsintotwopiecesviathequadrupoledynamicalinstability,whicharisesataweakerstrengthofinteractionthanthemonopoleandthedipoleinstabilities.Thesplitpiecessubsequentlyunitetorestoretheoriginalvortexorcollapse.

PACSnumbers:03.75.Fi,05.30.Jp,32.80.Pj,67.40.Vs

QuantizedvorticesingaseousBose-Einsteinconden-2

sates(BECs)o eravisiblehallmarkofsuper uid-00ity[1,2,3],whererepulsiveinteratomicinteractionsplay2acrucialroleinthevortexstabilizationandlatticefor- lmation.AttractiveBECs,ontheotherhand,cannotholdvorticesinanythermodynamicallystablestate.AJu fundamentalissueofthedecayofamany-particlequan-2tumsystemmaybeaddressedifavortexiscreatedin2anattractiveBEC.Suchastatehasbecomepossibleow- 3ingtothedevelopmentoftheFeshbachtechnique[4]byvwhichthestrengthandthesignofinteractionscanbe3controlled[5,6,7].

6Supposethatasinglyquantizedvortexiscreatedin3aBECwithrepulsiveinteractionandthattheinterac-40tionisadiabaticallychangedfromrepulsivetoattrac-2tive.Accordingtopreviouswork[8],thevortexstate0remainsmetastableuntiladimensionlessstrengthofin-/tteractiongtobede nedlaterreachesacriticalvalueagcrmM(<0),andwhen|g|exceeds|gcramonopole(breathing-mode)instabilityM|,thesystemandcollapses.develops-dInthisLetter,weshowthatthisconclusionholdsonlynwhenthesystemhasexactaxisymmetry,andthatevenanin nitesimalsymmetry-breakingperturbationinducescothequadrupoledynamicalinstabilitythatappearsfor|g|

:vsmallerthan|gcr

iXbilities[9,10,11]M|.initiateWenotevortexthatsimilarnucleationdynamicalobservedinsta-bytheENSgroup[2].AdynamicalinstabilityisalsoshownratotransferavortexfromonetotheothercomponentofabinaryBECsystem[12].Here,weshowthatyetan-otherdynamicalinstabilitycausesavortextosplitintotwopiecesthatrevolvearoundthecenterofthetrap.Surprisingly,insomeparameterregimes,thepiecessub-sequentlyunitetorestoretheoriginalvortex,andthissplit-mergecyclerepeats.WereportbelowtheresultsofourstudiesonthecollapsingdynamicsofavortexinanattractiveBEC.

We rstinvestigatetheBogoliubovspectrumofasingle-vortexstate.Thesingle-vortexstateisdeter-minedsoastominimizetheGross-Pitaevskii(GP)en-ergyfunctionalwithintheaxisymmetricfunctionalspaceψ0=f(r,z)eiφwithr=(x2+y2)1/2,whereweignorethee ectofvortexbending[13,14,15].Inthefollowinganalysis,wenormalizethelength,time,tionsinunitsofd0≡(¯h/mω⊥)1/2,ω 1andwavefunc-1/2

istheradialtrapfrequency,⊥,and(N/d3andN,the0),whereω⊥num-

berofBECatoms.WeobtaintheBogoliubovspectrumbynumericallydiagonalizingtheBogoliubov-deGennesequations[16]

K+2g|ψ0K+2g|ψ|20|2 un+gψ2

vn=Enun,(1a)vn+gψ 0

2

un= Envn,(1b)whereK≡ 2/2+(r2+λ2z2)/2 µwithλ≡ωz/ω⊥,

andnistheindexoftheeigenmode.Here,g≡4πNa/d0characterizesthestrengthofinteraction,whereaisthes-wavescatteringlength.Foravortexstateψ0eachangularmomentumstateunvn∝ei(m 2)φ,andweshallrefer∝eimφiscoupled∝onlyeiφ,totomastheangularmomentumoftheexcitation.

We ndthatthereisatleastonenegativeeigenvalueinthem=0modeforanyg<0andλeveninthepresenceofarotatingdrive.Thevortexstatewithat-tractiveinteractionsisthereforethermodynamicallyun-stable,andeventuallydecaysintothenon-vortexgroundstatebydissipatingitsenergyandangularmomentum.Atsu cientlylowtemperaturesinahigh-vacuumcham-ber,however,thethermodynamicinstabilityisirrelevant,sincetheenergyandangularmomentumareconserved.Infact,recentexperiments[2]havedemonstratedthatthevortexstateinastationarytraphasalifetimeof～scales1s,ofwhichthedynamicsismuchlongerthatwethanshallthediscusscharacteristicbelow.

timeWhenthecomplexeigenvaluesemergeintheBogoli-ubovspectrum,theamplitudeofthecorrespondingmodegrowsexponentiallyintime.Asnoiseisinevitableinexperimentalsituations,suchdynamicalinstabilitiesaremoreimportantthanthethermodynamiconeatlowtem-perature.

Figure1showstherealandimaginarypartsofthelowesteigenvaluesofthem= 1and3excitationsinanisotropictrap.Theeigenvaluesbecomecomplexat

thecriticalstrengthofinteractiongcr

Q= 15.06,showingtheonsetofthedynamicalinstabilityinthequadrupolemode.Theimaginarypartofthecomplexeigenvalueproportionalto

is

An attractive Bose-Einstein condensate with a vortex splits into two pieces via the quadrupole dynamical instability, which arises at a weaker strength of interaction than the monopole and the dipole instabilities. The split pieces subsequently unite to re

0.5E

eR 1 1.5 2

15

10

g

50

FIG.1:Forg>gcr

Q= 15.06,thesolidcurvedenotestheeigenvalue(real)ofthem= 1mode,andthedashedonethatofthenegative-normbranchofthem=3modeisotropictrap.Thesetwobranchesmergeatg=gcr

inan

Q,belowwhichtheeigenvaluebecomescomplexandthedynamicalin-stabilitysetsin.Inset:Thesolidlineshowstheimaginary

partofthe complexeigenvaluesforg<gcr

Q,whichispropor-tionalto

2

λ/(2π)g3D

[20].In

the

2Dsystem,the

dynamicalinstabilityarisesinthequadrupole,dipole,

andmonopolemodesatg2DcrQ= 7.79,g2Dcr

D= 11.48,

andg2Dcr

M= 23.4.ThedependenciesofthecomplexeigenvaluesongaresimilartothatinFig.1.

Tounderstandthedynamicalinstabilitiesanalytically,letusconsidertheGPactionintegralin2D

S= dt drψ i

2

2+r

2|ψ|2 ψ.(2)Weassumethatthestateevolutionisdescribedψ=

bymcm(t)φm(r),where φm(r)isassumedtotaketheformofφm(r)=[r|m|/(

B,whereA≡[g/(8π)

1]/[1+g/(8π)]1/2

andB≡3+5g/(8π) …… 此处隐藏：11944字，全部文档内容请下载后查看。喜欢就下载吧 ……