Partially-Time-Ordered Schwinger-Keldysh Loop Expansion of Coherent Nonlinear Optical Susceptibilities

A compact correlation-function expansion is developed for nth order optical susceptibilities in the frequency domain using the Keldysh-Schwinger loop. By not keeping track of the relative time ordering of bra and ket interactions at the two branches of the loop, the resulting expressions contain only n+1 basic terms, compared to the 2n terms required for a fully time-ordered density matrix description. Superoperator Green's function expressions for the nth order suscpeptibility derived using both expansions reflect different types of interferences between pathways .These are demonstrated for correlation-induced resonances in four wave mixing signals.