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1、NonlinearOptics§IntegralSolutionofDensityMatrixihρ&=[H,ρ]=Hoρ−ρHo+H′ρ−ρH′PerformanUnitary-similaritytransformationwith(Changebase,noteigenenergy)+uu=1,uisanunitarymatrix.+++++uihρ&u==uHoρu−uρHou+uH′ρu−uρH′uWith∂⎛+⎞++ih⎜uρu⎟=ihuρ&u+ihu&+ρu+ihuρu&∂t⎝⎠Weget~⎛+⎞++~~~~ihρ&−ih⎜u&+ρu+uρu&⎟=uHoρu−u
2、ρHou+H′ρ−ρH′⎝⎠Defineusuchthat−ih⎛⎜u&+ρu+u+ρu&⎞⎟=u+Hoρu−u+ρHou⎝⎠+⇒−ihu&+=u+Ho=u+Hoihu&=Houu=e−iHot/h⇒Note:uisatransformationintime!∂ψih=Hoψ∂tFrom,ψ(t)=e−iHo(t−to)/hψ(t)=u(t−t)ψ(t)⇒ooo.Weget~~~ihρ&=[H,ρ]150NonlinearOpticst'ρ˜(tihρ˜=∫t[]H˜,ρ˜dt1+iho)o+t+'ihu(t)ρ(t)u(t)=∫tdt1u(t1)[H(t1),ρ(t1)]u
3、(t1)+ihu+(to)ρ(to)u(to)o1+×u(t)→←u(t)onbothsides,ih1t'ρ(t)=∫tdt1u(t)u+(t1)[H(t1),ρ(t1)]u(t1)u+(t)iho+++u(t)u(to)ρ(to)u(to)u(t)u(t)=e−iHot/hWithu+(t)=eiHot/hu(t)u+(t)=e−iHo(t−t1)/h=u(t−t)11++u(t1)u(t)=u(t−t1)1t'ρ(t)=∫dt1u(t−t1)[H(t1),ρ(t1)]u+(t−t1)ihto++u(t−to)ρ(to)u(t−to).m→←nu(t)n=e−iHot/h
4、n=e−iωntnmu(t)=me−iHot/h=me−iωmtmρ(t)n=1tdte−iωm(t−t1)m[H˜'(t),ρ(t)]iωn(t−t1)∫t111neiho−iω(t−t)iω(t−t)+emomρ(t)nenooρ(t)=1tdte−iωmn(t−t1)[H'(t)ρ(t)−ρ(t)H'(t)]mn∫t1∑ml1ln1ml1ln1ihol−iωmn(t−to)ρ(t)+emno(*)−iωmntAmn(t)≡e"Green'sFunction"151NonlinearOptics1t''ρmn(t)=∫tdt1Amn(t−t1)∑[Hml(t1)ρln(t
5、1)−ρml(t1)Hln(t1)]ihol+Amn(t−to)ρmn(to)Usingperturbationtheory,wecalculate(n+1)thorderρfromnthorderρ,(n+1)'(n)(n)'ρ←∫Hρ−ρH(0)andρmn(t)=Amn(t−to)ρmn(to).Webreaktheintegrationintotwoparts,(n+1)1t'(n)ρln(t)=∫tdt1Aln(t−t1)∑Hlm(t1)ρmn(t1)ihom(H'operateonket)(n+1)1t(n)'ρmp(t)=−∫tdt1Amp(t−t1)∑ρmn(
6、t1)Hnp(t1)ihon(H'operateonbra)(n)Iftherearetotalmstates(fields),inordertogetaspecificρmn,eachinteractionhastocalculate2(integrationhastwoparts)x2(E&E*)xm(eachintegrationsumovermterms)terms.Fornthorder,total2n(2m)nterms.Note:Comparewithasystemwithoutperturbationρ(to)=ψ(to)ψ(to)+ρ(t)=u(t−to)ψ
7、(to)ψ(to)u(t−to)=ψ(t)ψ(t)+ρmn(t)=um(t−to)ψ(to)ψ(to)un(t−to)152NonlinearOptics=Amn(t−to)ρmn(to)Theequationofmotionwithoutperturbationρ&mn+iΩmnρmn=0−iΩmntAmn(t)=eComparewithequation(*),wegettheequationofmotionunderperturbationH'1ρ&+iΩρ=m[]H′,ρnmnmnmnihTwo-
