从宏观量子电动力学分析色散力毕业论文外文翻译.docx
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从宏观量子电动力学分析色散力毕业论文外文翻译
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英文原文
DispersionForceswithintheFrameworkofMacroscopicQED
ChristianRaabeandDirk-GunnarWelsch
Abstract.Dispersionforces,whichmaterialobjectsinthegroundstatearesubjectto,originatefromtheLorentzforcewithwhichthefluctuating,object-assistedelectromagneticvacuumactsonthefluctuatingchargeandcurrentdensitiesassociatedwiththeobjects.Wecalculatethemwithintheframe-workofmacroscopicQED,consideringmagnetodielectricobjectsdescribedintermsofspatiallyvaryingpermittivitiesandpermeabilitieswhicharecomplexfunctionsoffrequency.Theresultenablesustogiveaunifiedapproachtodispersionforcesonbothmacroscopicandmicroscopiclevels.
Keywords:
dispersionforces,Lorentz-forceapproach,QEDinlinearcausalmedia
1.Introduction
Asknown,electromagneticfieldscanexertforcesonelectricallyneutral,unpolar-izedandunmagnetizedmaterialobjects,providedthatthesearepolarizableand/ormagnetizable.Classically,itisthelackofpreciseknowledgeofthestateofthesourcesofafieldwhatletsoneresorttoaprobabilisticdescriptionofthefield,sothat,asamatterofprinciple,aclassicalfieldcanbenon-fluctuating.Inpractice,thiswouldbethecasewhenthesources,andthusthefield,wereunderstrictde-terministiccontrol.Inquantummechanics,thesituationisquitedifferent,asfieldfluctuationsarepresentevenifcompleteknowledgeofthequantumstatewouldbeachieved;astrictlynon-probabilisticregimesimplydoesnotexist.Similarly,polarizationandmagnetizationofanymaterialobjectarefluctuatingquantitiesinquantummechanics.Asaresult,theinteractionofthefluctuatingelectromagneticvacuumwiththefluctuatingpolarizationandmagnetizationofmaterialobjectsinthegroundstatecangiverisetonon-vanishingLorentzforces;thesearecommonlyreferredtoasdispersionforces.
Inthefollowingwewillrefertodispersionforcesactingbetweenatoms,betweenatomsandbodies,andbetweenbodiesasvanderWaals(vdW)forces,Casimir-Polder(CP)forcesandCasimirforces,respectively.Thisterminologyalsoreflectsthefactthat,althoughthethreetypesofforceshavethesamephysicalorigin,differentmethodstocalculatethemhavebeendeveloped.TheCPforcethatactsonanatom(HamiltonianRA)inanenergyeigenstatela)(RAla)==nwala))atpositionrAinthepresenceof(linearlyresponding)macroscopicbodiesiscornmonlyregardedasbeingthenegativegradientoftheposition-dependentpartoftheshiftoftheenergyoftheoverallsystem,~Ea,withtheatombeinginthestatela)andthebody-assistedelectromagneticfieldbeinginthegroundstate.Theinteractionoftheatornwiththefield,whichisresponsiblefortheenergyshift,istypicallytreatedintheelectric-dipoleapproximation,Le.Hint==-d.E(rA)inthemultipolarcouplingscheme,andtheenergyshiftiscalculatedinleading-orderperturbationtheory.Inthisway,onefinds[1,2]
(1)
(P,principalvalue;Wba==Wb-Wa),whereG(r,r',w)istheclassical(retarded)Green
tensor(inthefrequencydomain)fortheelectricfield,whichtakesthepresenceof
themacroscopicbodiesintoaccount.Itcanthenbearguedthat,inordertoobtain
theCPpotentialUa(rA)astheposition-dependentpartoftheenergyshift,one
mayreplaceG(rA,rA,w)inEq.
(1)withG(S)(rA,rA,w),whereG(S)(r,r',w)isthe
scatteringpartoftheGreentensor.Hence,
(2)
(3)
(4)
whereUa(rA)hasbeendecomposedintoanoff-resonantpartU~f(rA)andaresonantpartU~(rA),bytakingintoaccounttheanalyticpropertiesoftheGreentensorasafunctionofcomplexw,andconsideringexplicitlythesingularitiesexcludedbytheprincipal-valneintegrationinEq.
(1).
Letusrestrictourattentiontoground-stateat0111S.(Forcesonexcitedatomsleadtodynamicalproblemsingeneral[2]).Inthiscase,thereareofcoursenoresonantcontributions,asonlyupwardtransitionsarepossible[Wab<0inEq.(4)].Thus,onidentifyingthe(isotropic)ground-statepolarizabilityofanatomas
wemaywritetheCPpotentialofaground-stateatomintheformof(see,e.g.
Refs.[1-6])
fromwhichtheforceactingontheat0111followsas
(7)
Nowconsider,insteadoftheforceonasingleground-stateatom,theforceonacollectionofground-stateat0111Sdistributedwitha(coarse-grained)nUInberdensity''7(r)insideaspaceregionofvolumeVr-iI.Whenthemutualinteractionoftheatomscanbedisregarded,itispermissibletosimplyadduptheCPforcesontheindividualatomstoobtaintheforceactingonthecollectionofatomsduetotheirinteractionwiththebodiesoutsidethevolumeVm,Le.
SincethecollectionofatomscanberegardedasconstitutingaweaklydielectricbodyofsusceptibilityXNI(r,i~),
Eq.(8)givestheCasimirforceactingonsuchabody.NotethatspecialcasesofthisformulawerealreadyusedbyLifshitz[7]inthestudyofCasimirforcesbetweendielectricplates.ThequestionishowEq.(8)canbegeneralizedtoanarbitraryground-statebodywhosesusceptibilityXrvI(r,i~)isnotnecessarilysmall.AnanswertothisandrelatedquestionscanbegivenbymeansoftheLorentz-forceapproachtodispersionforces,asdevelopedinRefs.[8,9].
2.LorentzForce
LetusconsidermacroscopicQEDinalinearly,locallyandcausallyrespondingmediumwithgiven(complex)permittivityc(r,w)andperrneabilityp(r,w).Then,ifthecurrentdensitythatentersthemacroscopicMaxwellequationsis
thesource-quantityrepresentationsoftheelectricandinductionfields
readas
wheretheretardedGreentensorG(r,r',w)correspondstotheprescribedmedium.InEqs.(12)and(13),itisassumedthatthemediumcoverstheentirespacesothatsolutionsofthehomogeneousMaxwellequationsdonotappear.Free-spaceregionscanbeintroducedbyperformingthelimitsE~1andJ-L~1,butnotbeforetheendoftheactualcalculations.
Becauseofthepolarizationand/ormagnetizationcurrentsattributedtothemedium,thetotalchargeandcurrentdensitiesaregivenby
where
AswehavenotyetspecifiedthecurrentdensityIN(r)inanyway,theaboveformu-lasaregenerallyvalidsofar,andtheyarevalidbothinclassicalandinquantumelectrodynamics,Inanycase,itisclearthatknowledgeofthecorrelationfunc-tion(IN(r,w)l~(r''w')),wheretheanglebracketsdenoteclassicaland/orquan-turnaveraging,issufficienttoC0111putethecorrelationfunctions(e(r,w)Et(r',w')),
(t(r,w),E(r',w')),(l(r,w)Bt(r',w'))and(t(r,w)B(r',w')),fromwhichthe
(slowlyvaryingpartofthe)Lorentzforcedensityfollowsas
Wherethelimitr'~rmustbeunderstoodinsuchawaythatdivergentself-forces,whichwouldbeformallypresenteveninauniform(bulk)medium,areomitted.TheforceonthematterinavolurneVf\,listhengivenbythevolumeintegral
whichcanberewrittenasthesurfaceintegral
whereT(r)is(theexpectationvalueof)Maxwell'sstresstensor(asopposedtoMinkowski'sstresstensor),whichis(formally)identicalwiththestresstensorinmicroscopicelectrodynamics.NotethatingoingfromEq.(18)toEq.(19),atermresultingfromthe(slowlyvaryingpartofthe)Poyntingvectorhasbeenomitted,whichisvalidunderstationaryconditions.IfIN(r)canberegardedasbeingaclassicalcurrentdensityproducingclassicalradiation,IN(r)~jclass(r,t),thentheLorentzforcecomputedinthiswaygivestheclassicalradiationforcethatactsonthematerialinsidethechosenspaceregionofvolumeVm(seealsoRef.[10]).
3.DispersionForce
AsalreadymentionedinSec.1,thedispersionforceisobtainedifIN(r)isidentifiedwiththenoisecurrentdensityattributedtothepolarizationandmagnetizationofthematerial.Letusrestrictourattentiontothezero-temperaturelimit,Le.letusassumethattheoverallsystemisinitsgroundstate.(Thegeneralizationtothermalstatesisstraightforward.)FrommacroscopicQEDindispersingandabsorbinglinearmedia[11,12]itcanbeshownthattherelevantcurrentcorrelationfunctionreadsas
(I,unittensor).CombiningEqs.(12),(13),(15),(16)and(20),andmakinguseof
standardpropertiesoftheGreentensor,onecanthenshowthat
and
Letusconsider,forinstance,anisolateddielectricbodyofvolumeVl\rIand
susceptibilityXrvI(r,w)inthepresenceofarbitraryrnagnetodielectricbodies,which
arewellseparatedfromthedielectricbody.Inthiscase,furtherevaluationof
Eq.(18)leadstothefollowingformulaforthedispersionforceonthedielectric
body:
whereGrvI(r,r',i~)istheGreentensorofthesystemthatincludesthedielectric
body.Whenthedielectricbodyisnotanisolatedbodybutapartofsomelarger
body(againinthepresenceofarbitrarymagnetodielectricbodies),Eq.(23)must
besupplernentedwithasurfaceintegral,
whichmayberegardedasreflectingthescreeningeffectduetotheresidualpartof
thebody.
AtthispointitshouldbementionedthatifMinkowski'sstresstensorwereused
tocalculatetheforceonadielectricbody,Eq.(24)wouldbereplacedwith
AlthoughbothEq.(24)and(25)properlyreducetoEq.(23)whenthedielectricbodyisanisolatedone,theydifferbyasurfaceintegralinthecasewherethebodyisS0111epartofalargerbody.Inthelattercase,Minkowski'stensorishenceexpectedtoleadtoincorrectandevenself-contradictoryresults[9,13].ItshouldbepointedoutthatthedifferencesbetweentheLorentz-forceapproachtodispersionforcesandapproachesbasedonMinkowski'stensororrelatedquantitiesarenotnecessarilysmall,Forinstance,theground-stateLorentzforce(perunitarea)thatactsonanalmostperfectlyreflectingplanarplate