EE 232 Lightwave DevicesLecture 4: Time-Dependent Perturbation Th F i’ G ld R lTheory, Fermi’s Golden RuleInstructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 4-1©2008. University of CaliforniaTime-Dependent PerturbationConsider a quantum mechanical system:†Consider a single-frequency, time-varying stimulus '()'()'() for 0iwt iwtHrt Hre H re t−+>GGGE†0(,)() () for 0'( , )() ()Hrt Hre H re tHH HrtHti t=+>=+∂GGG=2EnE…() ()φφ∂GG0(,)(,)Assuming 'Th f ti b dHrtirttHHψψ=∂=1Ehv0(,)(,)(,)()nnniE tnnHrt i rttrt reφφφφ−∂=∂===GGThe new wavefunction can be expressed asa linear combination of original eigenstateswith time-varying coefficients:() ()( ) an orthonormal set of eigenstatesnnnrnφφφ=G/2yg(,) () ()( ) : probability of electron at stateniE tnnnrt a t reat nψφ−=∑=GGEE232 Lecture 4-2©2008. University of California*() ()mn mnmn r rdrφφδ==∫GGG() : probability of electron at state at time nat ntTime-Dependent Perturbation (cont’d)(,) (,)Hrt i rttψψ∂=∂GG=// /0//()( ') ()() () ()()()nn niE t iE t iE tnnn n nnnn niE t iE tda tiEHH at re i te i at r edtda tφφφ−− −−⎛⎞+=+⎜⎟⎝⎠∑∑ ∑∑∑== ===GG===//*()'()Multiply both sides by (i.e., multiply by (nniE t iE tnnnnmda tHatne i nedtmrφ−−=∑∑===) and integrate over )rGG///'() ()() '()1nnmiE t iE t iE tnmnnnida t da ta t mH ne i mne i edt dtda t−−−==∑∑∑====='()1() ()mnitmnmnnmnmnda tatH tedt iEEωω=−=∑=EE232 Lecture 4-3©2008. University of Californiamn=First-Order Perturbation0To track the order of perturbation, let'HH Hλ=+(0) (1) 2 (2)(0)() () () () ...Group terms with the same order of :()nn n nat a t a t a tda tλλλ=++ +(0)(1)(0) '()0 ( ) constant ()1() ()mnmmitmda tatdtda tatHteω=⇒==∑(2)() ()(nmnnmatHtedt ida∑=(1) ')1() ()mnitnmnntatHtedt iω=∑=ndt i=EE232 Lecture 4-4©2008. University of CaliforniaFirst-Order Perturbation (Cont’d)(0)(0)Initial state at t=0 and final state () 1iifat⎧=⎪⎨⎪()(0)(1)'''†() 0 if ()11()mi mimfit itit itfi fi fiat mida tHte He He edt i iωωωω−⎨=≠⎪⎩==+==()() ()''†(1 mi miit itfi fiidt i iHe H eiωω ωωω−+=+===)()titωωω+⎛⎞((1) '1()miiffieat Hω−−==)()'†11We are only interested at frequencies near resonance:mititfimi mieHωωωωω ωω+⎛⎞−−+⎜⎟−+⎝⎠()()2222''†2(1)2222We are only interested at frequencies near resonance:sin sin4422()mi mifi fifttHHatωωωω−+⎛⎞ ⎛⎞⎜⎟ ⎜⎟⎝⎠ ⎝⎠=+==EE232 Lecture 4-5©2008. University of California()()2222()fmi miωωωω−+==Fermi’s Golden Rule⎛⎞⎛⎞()2222sin2sinc42fifitttωωωωωω−⎛⎞⎜⎟−⎛⎞⎝⎠=⎜⎟⎝⎠−0.60.81yi2sinc2fitωω−⎛⎞⎜⎟⎝⎠() ( ) as 2fifittωωπδω ω⎝⎠→− →∞60− 40− 20− 0 20 40 6000.20.4ti()fitωω−22''†2(1)2222() ( ) ( )Transition Rate:fi fiffifitH tHatππδω ω δω ω=−+ +==()fi22''†2(1)2Transition Rate:22() ( )fi fiif f fiHHdWatdtππδω ω→== −+=2()fiδωω+=dt=22''†1Note: ( ) ( )fi fiEEδωδωωω−− = −−===EE232 Lecture 4-6©2008. University of California22''†22() ()fi fiif fi fiHHWEE EEππδωδω→=−−+ −+====Physical Interpretation22''†22() ()fi fiif f i f iHHWEE EEππδωδω→=−−+ −+====Absorption of a photonfiEEω=+=Emission of a photonfiEEω=− =fEiEEhvhvE• Conservation of energyiEfEEE232 Lecture 4-7©2008. University of California• Transition rate is proportional to the square of the “matrix element”Distributed Final States• If the final state is a distribution of states, the transition rate is proportional to the density of 22''†22() ()fifiHHWEE EEππδδ++==states of the final state:() ()ffif f f i f f iWEE EEρδωρδω→=−−+−+====fiEEω=+=fiEEω=− =Absorption of a photonEmission of a photonfEhvhviEEE232 Lecture 4-8©2008. University of