William Angerer - Department of Physics and Astronomy - University ...

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where p(O) is the density operator in thermal equilibrium, and p(i) represent the ith
order perturbation to p(O). Once p is determined, the nonlinear polarization is calculated
as
(5.9)
vVe simplify the rules for Feynman diagrams by observing that the three virtual
transitions for second-harmonic generation leave the state unaltered. Although each
photon process induces a phase shift in p(2), the net phase shift is zero for SHG.
Therefore, we will disregard the net phase shift caused by each photon process. The
rules for double sided Feynman diagrams are as follows.
1. The state starts with Il)p~?)(ll. For SHG, the state also ends with Il)p}?l(ll.
2. For the ket side, a vertex that brings the state from Il) to 1m) by absorption of a
photon with angular frequency WI corresponds to the matrix element
(ml - erjE(wdll)
iii
(5.10)
where -erj is the dipole moment operator along the direction i and E(;.;d is the
electric field. For an emission process, the matrbc element is
(ml - eriE*(wdll)
iii
(5.11)
Similarly, the matrix element that represents absorption of a photon on the bra side
is
-(ll - eriE(wdlm)
iii
(5.12)
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