从宏观量子电动力学分析色散力毕业论文外文翻译.docx

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