mit非线性光学讲义nonlinear optics ix

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1、NonlinearOptics§DiagrammaticRepresentationAllowimmediatetranslationintobothmultiplesolution+steadystates.Equations:(n+1)1t'(n)(n)'ρmn(t)=∫dt1Amn(t−t1)∑[Hml(t1)ρln(t1)−ρml(t1)Hln(t1)](1)ihtolWecanbreakaboveequationintotwoparts,ρ(n+1)1t'(n)ln(t)=∫tdt1Aln(t−t1)∑Hlm(t1)ρmn(t1)(2)ihomH'operatesonk

2、etstateand−1tρ(n+1)(n)'mp(t)=dt1Amp(t−t1)∑ρmn(t1)Hnp(t1)(3)ih∫tonH'operatesonbrastate(n)(t)=A(t−t)ρ(n)−iΩmntwithρmn1mn1omn&Amn=e.Werepresent(1)asρmnn+1ρlnρmlnWerepresent(2)and(3)asρlnρmpn+1ρmnn163NonlinearOpticsFeynmannDiagramTechnique1.Timeisplottedvertically2.Representketandbracomponentsofd

3、ensitymatrixbyverticallinesketbra3.RepresentperturbationmatrixelementsbyverticesForequation(2)lntt1tomnForequation(3)mptt1tomn164NonlinearOpticsDefinelnt≡Aln(t−t1)t1lnl't1≡Hlm(t1)mp't1≡Hnp(t1)nRules:(0)1.Includeρmn(to)factortoaccountforinitialcondition.2.Includeafactorof-1foreachbrainteractio

4、n.3.Foreachinteractionattimetj,integratefromtouptotj+1,−1includeafactorof(ih).(n)(n)4.ρmn(t1)=Amn(t1−to)ρmn(to).5.Frequencycomponentofperturbation165NonlinearOpticsForketstate:rtj+ω'+ω1−iωtj≡Hrl(tj)=−μrlEe2lr'−ω1*iωtj−ωt≡Hrl(tj)=−μrlEej2lForbrastate:k+ω'+ω1−iωtj≡Hmk(tj)=−μmkEet2jmkt−ωj'−ω1*iω

5、tj≡Hmk(tj)=−μmkEe2m166NonlinearOpticsLinearPolarizationProcessng(0)(1)ρgg=1,wantlinearpolarization~ρng(t)ngtt1⇒toggTherearetwochoicesforinteraction.AngtFromgroundtoexcitedstateswilltabsorbphotonforresonance.1+ωtoggBngt−ωt1togg167NonlinearOpticsTranslatediagramA(1)−1t'+ω(0)ρng(t)=(ih)∫tdt1Ang(

6、t−t1)Hng(t1)ρggowheretimeevolutionforzeroorderisneglected.(0)Taketo→−∞,assumeρggconstant,wegetsteadystatesolution,(1)−1−1⎛1−iωt⎞(0)ρng(t)=h(ω−Ωng)⎜−μngEe⎟ρgg⎝2⎠"Resonantterm"→absorbaphoton.TranslatediagramB(1)−1t'−ω(0)ρng(t)=(ih)∫tdt1Ang(t−t1)Hng(t1)ρggoForsteadystate,(1)−1−1⎛1*iωt⎞(0)ρng(t)=

7、h(−ω−Ωng)⎜−μngEe⎟ρgg⎝2⎠"Antiresonantterm"(0)Ifρnn≠0(populationinexcitedstate),wehavetoconsiderngtt1tonnTherearetwochoiceforinteraction168NonlinearOpticsCngtt1−ωtonnDngtt1+ωFromexcitedtogroundstateswillemitphotonforresonance.tonnTranslatediagr