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Coherent dynamics and mapping of excitons in single-layer MoSe$_2$ and WSe$_2$ at the homogeneous limit

We perform coherent nonlinear spectroscopy of excitons in single-layers of MoSe$_2$ and WSe$_2$ encapsulated between thin films of hexagonal boron nitride. Employing four-wave mixing microscopy we identify virtually disorder free areas, generating exciton optical response at the homogeneous limit. Focussing on such areas, we measure exciton homogeneous broadening as a function of environmental factors, namely temperature and exciton density. Exploiting FWM imaging, we find that at such locations, nonlinear absorption of the exciton excited states and their coherent couplings can be observed. Using the WSe$_2$ heterostructure, we infer coherence and density dynamics of the exciton 2S state. Owing to its increased radiative lifetime, at low temperatures, the dephasing of the 2S state is longer than of the 1S transition. While scanning various heterostructures across tens of micrometers, we conclude that the disorder, principally induced by strain variations, remain to be present creating spatially varying inhomogeneous broadening.