Coherent imaging and dynamics of excitons in MoSe$_2$ monolayers epitaxially grown on hexagonal boron nitride

Using four-wave mixing microscopy, we measure the coherent response and ultrafast dynamics of excitons and trions in MoSe$_2$ monolayers grown by molecular beam epitaxy on thin films of hexagonal boron nitride. We assess inhomogeneous and homogeneous broadenings in the transition spectral lineshape. The impact of phonons on the homogeneous dephasing is inferred via the temperature dependence of the dephasing. Four-wave mixing mapping, combined with the atomic force microscopy, reveals spatial correlations between exciton oscillator strength, inhomogeneous broadening and the sample morphology. The quality of coherent optical response of the epitaxially grown transition metal dichalcogenides becomes now comparable with the samples produced by mechanical exfoliation, enabling coherent nonlinear spectroscopy of innovative materials, like magnetic layers or Janus semiconductors.