  Using a novel approach to ultrafast resonant pump-probe spectroscopy we
investigate the spectral shape and dynamics of absorption features related to
the A exciton in an hBN/MoSe$_2$/hBN van der Waals heterostructure. While in a
pure two-level system a pump-probe experiment measures the occupation or the
polarization dynamics, depending on the time ordering of the pulse pair, in the
transition metal dichalcogenide (TMD) system both quantities get thoroughly
mixed by strong exciton-exciton interaction. We find that for short positive
delays the spectral lines experience pronounced changes in their shape and
energy and they relax to the original situation on a picosecond time scale. For
negative delays distinctive spectral oscillations appear indicating the
first-time observation of perturbed free induction decay for a TMD system. The
comparison between co-circular and cross-circular excitation schemes further
allows us to investigate the rapid inter-valley scattering. By considering a
three-level system as a minimal model including the local field effect,
excitation induced dephasing and scattering between the excited states we
explain all phenomena observed in the experiment with excellent consistency.
Our handy model can be even further reduced to two levels in the case of a
co-circular excitation, for which we derive analytic expressions to describe
the detected signals. This allows us to trace back the spectral shapes and
shifts to the impact of local field effect and excitation induced dephasing
thus fully reproducing the complex behavior of the observed effects.
