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1、NonlinearOptics§LinearPolarization(withdensitymatrix)•solvedbyperturbationtheory(1)•solvedforρng(forinducedpolarization)(0)(0)I.Assumeinitialpopulationρnn,ρgg.Thereisnoinitialpolarization,(0)(0)ρng=ρgn=0.(1)II.Solveforρngtofirstorderforinducedpolarization.P=Nμ=N()ρngμgn+ρgn
2、μng=N(ρngμgn+c.c.).III.Considersteadystateresponse,ihρ&()1=hΩρ+H′ρ(0)−ρ(0)H′ngngngngggnnngρ&()1+iΩρ=()ih−1H′⎛⎜ρ(gg0)−ρ(nn0)⎞⎟ngngngng⎝⎠ρ&()1+iΩρ=−()ih−11μ⎡Ee−iωt+E*eiωt⎤⎛ρ()0−ρ()0⎞ngngngng⎢⎥⎜⎝ggnn⎟⎠2⎣⎦(1)ρngoscillatesatbothωand-ωfrequencies.∂DefineL≡+iΩng.∂tLρ(1)=−()ih−11μE
3、e−iωt(ρ(0)−ρ(0))ngngggnn2−()ih−11μE*eiωt(ρ(0)−ρ(0))ngggnn2.(1)−1⎧−11−iωt(0)(0)ρng=L⎨−()ihμngEe(ρgg−ρnn)⎩2−11*iωt(0)(0)⎫−()ihμngEe(ρgg−ρnn)⎬2⎭.144NonlinearOptics(1)−1−11−iωt(0)(0)ρng=−()ih()−iω+iΩngμngEe(ρgg−ρnn)2−1−11*iωt(0)(0)−()ih()iω+iΩngμngEe(ρgg−ρnn)2.−1μngE−iωt(0)(0)=
4、Ω()ng−ωe(ρgg−ρnn)2h*−1μngEiωt(0)(0)+Ω()ng+ωe(ρgg−ρnn)2h.−1Neglectanti-resonantterm,recallΩng≡ωng−iT2,(1)−1−1μngE−iωt(0)(0)ρng=(ωng−ω−iT2)e(ρgg−ρnn)2h.(1)⎡1−iωt⎤RecallP=N()ρngμgn+c.c.=⎢εoχEe+c.c.⎥.⎣2⎦N−1−1(0)(0)χ=μngμgn()ωng−ω−iT2(ρgg−ρnn)εoh.2/T2χ"ωngωχ'populationinversion1
5、45NonlinearOptics'"Considerχ=χ+iχ,2'Nμ(0)(0)ωng−ωχ=()ρgg−ρnn22εoh()ωng−ω+()1/T2,2"Nμ(0)(0)1/T2χ=()ρgg−ρnn22εoh()ωng−ω+()1/T2.Inthelimitofχ<<1,1/211"n=()1+χ≅1+χ=1+(χ+iχ)22'"=n+in.'"nωnωnωiziz−zikzccce=e=ee.'χ→refractiveindex"χ→absorption(withpopulationinversion⇒gain)(0)(0)No
6、te:I.Whenρgg=ρnn,there'snoinducedpolarization.II.Forχijtensor,Pi=εoχijEjiiPi=N(ρngμgn+c.c.)N-1ij1(0)(0)χij=μgnμng()ωng−ω−iT2−(ρgg−ρnn)εoh.III.Forisotropicmedium,Duetosymmetryconsideration,146NonlinearOptics⇒PisparalleltoE,⇒χisscalar.Assume1/3atomswillhavetheirdipoletransiti
7、onmomentsparalleltoEfield,1N−1-1(0)(0)χ=μgnμng(ωng−ω−iT2)(ρgg−ρnn)3εoh1N2−1-1≅μ(ωng−ω−iT2)3εoh(0)(0)withρgg=1,ρnn=0.(Eng>>kT)OscillatorStrengthforg→ntransition,22mωngμngfng≡2withdimensionlessunit.3he(withsummationover3dimensions∑fng=1)n2Nfnge−1−1χ=()ωng−ω−iT2.2mωngεoSaturat
8、ioninTwoLevelAtomsIfpopulationdisturbanceistoolarge,wecannotapplyperturbationtheor
