Partially-Time-Ordered Schwinger-Keldysh Loop Expa

arXiv:0711.2813v1[quant-ph]18Nov2007Partially-Time-OrderedSchwinger-KeldyshLoopExpansionofCoherentNonlinearOpticalSusceptibilitiesShaulMukamelDepartmentofChemistry,UniversityofCalifornia,Irvine,CA92697(Dated:February2,2008)AbstractAcompactcorrelation-functionexpansionisdevelopedforn’thorderopticalsusceptibilitiesinthefrequencydomainusingtheKeldysh-Schwingerloop.Bynotkeepingtrackoftherelativetimeorderingofbraandketinteractionsatthetwobranchesoftheloop,theresultingexpressionscontainonlyn+1basicterms,comparedtothe2ntermsrequiredforafullytime-ordereddensitymatrixdescription.SuperoperatorGreen’sfunctionexpressionsforχ(n)derivedusingbothexpansionsreflectdifferenttypesofinterferencesbetweenpathways.Thesearedemonstratedforcorrelation-inducedresonancesinfourwavemixingsignals.1I.INTRODUCTIONTimeorderedexpansionsformthebasisfortheperturbativecalculationofstaticanddy-namicalpropertiesofinteractingmany-bodysystems.Thenonlinearresponsetoasequenceofnshort(impulsive)pulsesismostnaturallycalculatedinreal(physical)time.Theresult-ingresponsefunctionscontain2nbasicterms,stemmingfromthefactthateachinteractioncanoccureitherwiththeketorwiththebraofthesystemdensitymatrix.Thisfullytimeorderedexpansionisroutinelyusedforcomputingultrafast(femtosecond)opticalsignalsinmolecules,semiconductorsandothermaterials.ThephysicalpictureisrecastintermsofthedensitymatrixinLiouvillespace.Many-bodytheoryofexternallydrivensystemsisincontrastcommonlyformulatedusingnonequilibriumGreen’sfunctionswhichactinHilbertspace[1-5].Time-orderingisthenmaintainedonanartificialKeldysh-Schwingerloop,[6,7]whichcorrespondstobothforwardandbackwardevolutioninphysicaltimeandformsthebasisforpeturbativediagrammatictechniques.Theloopprovidesaformalbookkeepingdeviceforvariousinteractions.Weonlykeeptrackofthenumberofinteractionswiththeketandthebrabutnotoftheirrelativetimeordering.Thenonlinearresponsefunctionrecastusingtheseartificial(loop)timevariableshasthenaconsiderablyreducednumberofterms,n+1.Time-domainopticalexperimentsperformedusingimpulsiveultrashortpulsesmaybedescribedontheloop,buttherequiredtransformationfromloop-toreal-timevariablesmakesithardtoattributephysicalmeaningtothevariousterms[8].Inthispaperweshowthatthelooptimeorderingismostsuitableforcomputingnonlinearsusceptibilitiesinthefrequencydomain,wherereal-timeorderingisnotmaintainedinanycase.Thefrequencyvariablesaredirectlyconjugatedtothevariousdelayperiodsalongtheloop.InSec.IIwederivethecorrelationfunctionloopexpressionsforthethirdordersusceptibility.Sincetheloopexpansionismuchmorecompact,itmaybeadvantageoustoperformmany-bodycalculationsinthefrequencydomainontheloopandthenswitchtothetimedomainbyaFouriertransform.ThiswayonemayexploitthefullpowerofmanybodyGreen’sfunctiontechniques.TheseexpressionsarethenrecastinSec.IIIusingadiagrammaticrepresentationintermsofsuperoperatorsinLiouvillespace.Theloopandthetime-orderedexpressionsarecomparedinSec.IVandshowntocontainadifferentstructureofresonances.Asuperficiallookatthetwotypesofexpressionsmaysuggestthattheypredictdifferenttypesofresonances.Thisishowevermisleadingsincethevarioustermsinterfere.Consequentlysomeapparentresonancesmaycancelandothersmaybeinducedby2dephasingprocesses.Simplediagrammaticrulesareprovidedwhichallowtocomputethepartiallytime-orderedexpressions.ThesesubtleeffectsareillustratedinSec.Vbyapplyingthisformalismtostudycorrelation-inducedresonancesinfourwavemixing.ThefourpointdipolecorrelationfunctioniscalculatedforamultilevelsystemwhoseenergylevelsfluctuatebycouplingtoaBrownianoscillatorbath.Themodelallowsforanarbitrarydegreeofcorrelationbetweenthesefluctuations.Theexpressionsmaynotbegenerallyfactorizedintoproductsofeitherreal-timedelaysorloop-delaysandtheresultingcomplexpatternofresonancesmaynotbeattributedtospecifictimedelays.Whenthesefluctuationsarenegligibletheloopexpressionsbestrevealtheresonances.Inthelimitoffastfluctuations(homogeneousdephasing)therealtimeexpressionsshowtheseresonances.ThesesubtleeffectsaredemonstratedinSec.Vwhereweillustratethedifferentroleofinterferenceinthetwotypesofexpansion.WeconcludebyadiscussionoftheseresultsinSec.VI.II.NONLINEARSUSCEPTIBILITIESONTHEKELDYSHLOOPWeconsiderasysteminteractingwithanexternalelectricopticalfieldE(t).ThecouplingHamiltonianisHint=−E(t)V,whereVisthedipoleoperator.ThenonlinearpolarizationP(n)hasn+1terms[9,10]P(n)(t)=nXm=0hψ(n−m)(t)|V|ψ(m)(t)i.(1)Here|ψ(m)istheperturbedwavefuntiontom′thorderintheexternalfield.Weshallcarryoutthecalculationforthethirdorderresponse,n=3.Thegeneralizationton’thorderisstraightforward.ThefieldconsistsofthreemodesandexpandedasE(t)=3Xj=1Ej(t)exp(−iωjt)+c.c.(2)Eq.(1)nowhasfourtermswhichcorrespondtom=3,2,1,0andarerepresentedbytheFeynmandiagrams(a),(b),(c),(d)showninFig.1respectively.ThesysteminteractswiththefieldsE1,E2andE3attimesτ1,τ2andτ3respectively,andthepolarizationiscalculatedatτ4byintegratingoverthetimevariablesτj.Eachdiagramrepresentsadifferentorderingofτjalongtheloop.FouriertransformofEq.(1)tothefrequencydomaingivesP(3)(ωs)≡Z∞−∞dtexp(iωst)P(3)(t)=Pa+Pb+Pc+Pd(3)3wherePa(ωs)=Z∞−∞dτ4Zτ4−∞dτ3Zτ3−