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1、8.514:Many-bodyphenomenaincondensedmatterandatomicphysicsLastmodi ed:erSeptemb29,20031Lecture6.Vortices,supyer
uidit.Trappedgases.BECat nitetemperature.TotreatydrohdynamicsandBECinspatiallyaryingvbackground,needamoregeneralhapproacthatdoesnotassumecondensationintoaparticularplaneevawst
2、ate.OnecanulateformtheoryofBECsothattheefunctionvawofthecondensateisanarbitraryfunctioninspace-timethatisdeterminedtlyself-consistenfromaclassicalnonlinear eldequation,knownastheGross-Pitaevskiiequation.1.1Gross-PitaevskiiequationLetusstartagainfromtheHamiltonianofeaklywteractinginBose
3、gaswithshort-rangeteraction,in!!Z2h+2++H='^(x);r+U(x)'x)^(+'^(x)'^(x)'x)^('x)^(dx(1)x2m2Toulateformthecondensatedynamics,startwiththebHeisenergevolution,ih@'^=t2h2+2+['^H]=;r+U(x)'^+'^'^,andreplace'^,'^ybclassicalariablesv',',2mhwhicgivesaclassical elddynamicsproblem,!2h22ih
4、@'=;r+U(x)'+''(2)t2m!2h22;ih@'=;r+U(x)'+''(3)t2mcalledtheGross-Pitaevskiiequations.iItiseinstructivtowriteeqs.separatelyforthemodulusandphase'=j'je.hh22@n+rj=0n=j'jj=('r';'r')=j'jr(4)t2mimand!2h2h@=;U(x)+n+jr'j(5)t2mComparingtheeabvoexpressionsforthedensityandcurren
5、t,obtainthesupwer
oveloycithv=r(6)mThe
owisirrotational,rv=0(thisistrueyawafromsingularitiesin),and2ph22pm@v=;r~+mv=2~=U(x)+n+rn(7)t2mn(comparetotheEulerequation).11.2Supyer
uidit.Vortices.Letusconsiderthecirculationofvelocityinasuper
ow.Itfollowsfromtherelationbetweentheelo
6、yvcitandthephase,Eq.(6),thatthecirculationaroundyancontourCobeysIh;=vdr=2l(8)mCwithsometegerinl.WeseethatThecirculationistizedquaninultiplesmofh=mThews
oinulti-connectedmgeometries,suchasaring-shapetube,arediscretetumQuanleapsarerequiredtohangecaow.Thefactthatasupw,er
oduetothe
7、dicretenessofcirculation,cannotbedissipatedgradually,butonlyindicretesteps,istheoriginofsupyer
uidit.Theonlyyawtoeliminateasupwer
oistoproduceexcitationswithdiscreteyorticitvandthenvremoethem(alongwiththey)orticitvfromthesystem.0AlsotionmentheLandaucriterionforsupy:er
uiditThequasipa
