Parametric projection operator technique for second order non-linear response

Highlights: ► Molecular excited state dynamics in a four-wave-mixing experiment is considered. ► Parametric projector is introduced to reduced the description to electronic states. ► Dependence of the dynamics on the delay between the first two pulses is studied. ► 2D coherent spectroscopy is found to see this dependence only as an average. -- Abstract: We demonstrate the application of the recently introduced parametric projector operator technique to a calculation of the second order non-linear optical response of a multilevel molecular system. We derive a parametric quantum master equation (QME) for the time evolution of the excited state of an excitonic system after excitation by the first two pulses in the usual spectroscopic four-wave-mixing scheme. This master equation differs from the usual QME by a correction term which depends on the delay τ between the pulses. In the presence of environmental degrees of freedom with finite bath correlation time and in the presence of intramolecular vibrations we find distinct dynamics of both the excited state populations and the electronic coherence for different delays τ.