  Two-dimensional correlation spectroscopy (2DCS) based on the nonlinear
optical response of excitons to sequences of ultrafast pulses, has the
potential to provide some unique insights into carrier dynamics in
semiconductors. The most prominent feature of 2DCS, cross peaks, can best be
understood using a sum-over-states picture involving the many-body eigenstates.
However, the optical response of semiconductors is usually calculated by
solving truncated equations of motion for dynamical variables, which result in
a quasiparticle picture. In this work we derive Green's function expressions
for the four wave mixing signals generated in various phase-matching directions
and use them to establish the connection between the two pictures. The formal
connection with Frenkel excitons (hard-core bosons) and vibrational excitons
(soft-core bosons) is pointed out.
