  Using ultrafast resonant pump-probe spectroscopy we investigate A exciton
dynamics 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 our
case both quantities get mixed by the interaction of the excitons. We find that
for short positive delays the spectral lines are blue-shifted and that they
relax to the original energy on a picosecond time scale. For negative delays
pronounced spectral oscillations appear. The comparison between co-circular and
cross-circular excitation schemes further allows us to investigate the
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 can explain all phenomena observed in the
experiment. Our model can be reduced to two levels in the case of a co-circular
excitation, for which we can 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.
