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

TL;DR

This paper introduces a simplified correlation-function expansion for calculating nth order optical susceptibilities using the Keldysh-Schwinger loop, reducing complexity by not fully time-ordering bra and ket interactions, and demonstrates its application to nonlinear optical signals.

Contribution

It presents a novel, compact expansion method that simplifies the calculation of nonlinear optical susceptibilities by reducing the number of terms needed compared to traditional methods.

Findings

Reduced number of terms from 2n to n+1 in susceptibility expressions.

Demonstrated the method's effectiveness in analyzing four-wave mixing signals.

Revealed different interference effects in nonlinear optical pathways.

Abstract

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.