Tuning the dipolar interaction in quantum gases(2)

We have studied the tunability of the interaction between permanent dipoles in Bose-Einstein condensates. Based on time-dependent control of the anisotropy of the dipolar interaction, we show that even the very weak magnetic dipole coupling in alkali gases

andisthustime-dependent.

Ouranalysisisbasedonthemean- eldapproachintheThomas-Fermilimitalreadyusedin

the

context

of

dipolar

Bosegases[4–12].Such

a

descriptioncanbeac-complishedthroughthehydrodynamicequationsforsu-per uids[1]:

n

=

v2

M ,

(5)

Thomas-Fermiregimecon nedinanisotropicharmonictrapwithfrequencyω0(forexampleω0/2π=100Hz).Becauseεddissmallthedipole-dipolemean- eldpoten-tialΦddcanbecalculatedusingthesymmetricequilib-2

riumdensitydistributionn0=Mω0(R2 r2)/2g,whereRisitsradiusΦdd(r,t)=

2

εddMω0

t

5

R5

2

3(e(t)· r) 1

wherenisthedensity,vthevelocity eldandδµthe

di erencebetweenthemean- eldplusexternal eldsandthechemicalpotentialµgivenby

δµ=gn+Φdd+

22Mω0r

2(seeRef.[15]).Afterlinearizationof

theequationsofmotion(4)and(5)weobtain

12π¯ha

2

,(8) 2δn

M

whichisameasureofthestrengthofthedipole-dipole

interactionrelativetothes-wavescatteringenergy.Mod-i cationsinthegroundstatecondensatedensitycanonlybeexpectedifεddisappreciablydi erentfromzero.Forchromiumthes-wavescatteringlengthaCrisnotknown.IfweassumethataCrisequaltothesodiumscatteringlength(a=2.8nm)thedipolarstrengthpa-rameterbecomesεCrdd=0.29(m=6µB).Therefore,thedipolarforceisexpectedtoleadtovisiblemodi -cationinthedensityofachromiumBEC.Di erently,inbothrubidiumandsodiumthemagneticdipolaren-ergyisrathersmallcomparedtothemean- elds-wave

87

scatteringenergy.ThevaluesofεddareεRb=0.0064dd

Na

andεdd=0.0035,inthedoublyspin-polarizedground-state,wherem=µB.ForheteronuclearmoleculeswithapermanentelectricdipolemomentofoneDebyethecor-respondingdipolarcouplingwillbeincreasedbyapproxi-mativelytheinversesquareofthe nestructureconstant.Thisimpliesthatεdd[13]isontheorderof102.

Probingthequadrupolemodesinsodiumandrubid-ium—Thetimemodulationofthedipolarcoupling(3)canbeusedtoresonantlyexciteaquadrupolemodeevenforsodiumandrubidiumcondensates.Forsimplicitywerestrictourselvestothecaseofacondensateinthe

2

n

(δn)+

2+ω2ωXY

=

3

10g

r2sin2θsin(2φ 2 t) t,

(14)

whereφistheazimuthalangle(r<R).Itisnoteworthy

thattherotatingpart[14]of(9)couplesonlythe rstquadrupolemodeintheThomas-Fermilimit[15].Fromthestudyoftheevolutionofthecondensatedeformation

α=

x2 y2

5

εdd tsin(2 t)

(15)

wecanextractinformationabouttheratioofthedipolarcouplingversusthestandardmean- eldenergy.Ifthe