5.74 Introductory Quantum Mechanics II - MIT OpenCourseWare

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p. 10-25rephasing.” Analysis of R 1and R 3reveals that these terms are non-rephasing and rephasing,respectively.phase acquiredφ0−iωba τ1eτ 1−iω ba τ 3eτ 3+iωba τ3enon-rephasingrephasingP()tt 1 t 2t 3tFor the present case of a third-order spectroscopy applied to a two-level system, weobserve that the two rephasing functions R 2and R 3have the same emission frequency andwavevector, and would therefore both contribute equally to a given detection geometry. The twoterms differ in which population state they propagate during the τ 2 variable. Similarly, the nonrephasingfunctions R 1and R 4each have the same emission frequency and wavevector, but differby the τ 2 population. For transitions between more than two system states, these terms could beseparated by frequency or wavevector (see appendix). Since the rephasing pair R 2and R 3bothcontribute equally to a signal scattered in the −k + k +k1 2 3direction, they are also referred to asS I . The nonrephasing pair R1and R4both scatter in the + k1 − k2+k3direction and are labeled asS <strong>II</strong> .Our findings for the four independent correlation functions are summarized below.S IS <strong>II</strong>rephasingnon-rephasingωsigR2ω1ω2ω3R3ω1ω2ω3R1ω1ω2ω3R4ω1ω2ω3ksigτ 2 population− + + − k1+ k2+ k3excited state− + + − k1+ k2+ k3ground state+ − + + k1 − k2+ k3ground state+ − + + k1 − k2+ k3excited state
p. 10-26Frequency Domain Representation 4A Fourier-Laplace transform of P (3) (t) with respect to the time intervals allows us to obtain an(3)expression for the third order nonlinear susceptibility, χ ( ω , ω , ω ) :P (3) ( )1 2 3ω = χ(3) ( ω ; ω , ω , ω ) E E E (10.16)sig sig 1 2 3 1 2 3where χ ( n ) ∞= ∫ d τ0eiΩ τ∞d e R(τ)( τ , τ ,Kτ ) . (10.17)n n iΩ11τ nnL∫ 01 1 2 nHere the Fourier transform conjugate variables Ω mto the time-interval τ mare the sum over allfrequencies for the incident field interactions up to the period for which you are evolving:m∑ ii=1For instance, the conjugate variable for the third time-interval of a +ksum over the three preceding incident frequencies Ω3 = ω1 − ω2+ ω3.Ω = ω (10.18)m− k1 2 3+k experiment is theIn general, χ (3) is a sum over many correlation functions and includes a sum over states:χ( 3 )1 i 3 4(ω , ω , ω = ⎛ ⎞ p ⎡6 ⎝h ⎠ abcd α = 1*1 2 3 ) ⎜ ⎟ ∑ a ∑ ⎣ χ χα −α ⎦⎤(10.19)Here a is the initial state and the sum is over all possible intermediate states. Also, to describefrequency domain experiments, we have to permute over all possible time orderings. Mostgenerally, the eight terms in R (3)of the time-ordering of the input fields. 5lead to 48 terms for χ (3) , as a result of the 3!=6 permutationsGiven a set of diagrams, we can write the nonlinear susceptibility directly as follows:1) Read off products of light-matter interaction factors.2) Multiply by resonance denominator terms that describe the propagation under H 0. In thefrequency domain, if we apply eq. (10.17) to response functions that usephenomenological time-propagators of the form eq. (10.1), we obtain1τm ρab⇒(Ωm −ωba ) − iΓ baĜ ( )Ω m is defined in eq. (10.18).. (10.20)3) As for the time domain, multiply by a factor of ( −1 ) nfor n bra side interactions.4) The radiated signal will have frequency ω sig= ∑ω iand wavevector k sig= ∑ k i.ii
Page 3 and 4: p. 10-16Feynman Diagrams 1Feynman d
Page 6: p. 10-19( ) = Tr ⎡⎣ μ (t) μ (
Page 10: p. 10-23Third-Order Nonlinear Spect
Page 17: p. 10-30Note that for the S I and S

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