  In recent years the physics of two-dimensional semiconductors was revived by
the discovery of the class of transition metal dichalcogenides. In these
systems excitons dominate the optical response in the visible range and open
many perspectives for nonlinear spectroscopy. To describe the coherence and
polarization dynamics of excitons after ultrafast excitation in these systems,
we employ the Bloch equation model of a two-level system extended by a local
field describing the exciton-exciton interaction. We calculate four-wave mixing
signals and analyze the dependence of the temporal and spectral signals as a
function of the delay between the exciting pulses. Exact analytical results
obtained for the case of ultrafast ($\delta$-shaped) pulses are compared to
numerical solutions obtained for finite pulse durations. If two pulses are used
to generate the nonlinear signal, characteristic spectral line splittings are
restricted to short delays. When considering a three-pulse excitation the line
splittings, induced by the local field effect, persist for long delays. All of
the found features are instructively explained within the Bloch vector picture
and we show how the exciton occupation dynamics govern the different four-wave
mixing signals.
