Coherence and density dynamics of excitons in a single-layer MoS$_2$ reaching the homogeneous limit

We measure the coherent nonlinear response of excitons in a single-layer of molybdenum disulphide embedded in hexagonal boron nitride, forming a $h$-BN/MoS$_2$/$h$-BN heterostructure. Using four-wave mixing microscopy and imaging, we correlate the exciton homogeneous and inhomogeneous broadenings. We find that the exciton dynamics is governed by microscopic disorder on top of the ideal crystal properties. Analyzing the exciton ultra-fast density dynamics using amplitude and phase of the response, we investigate the relaxation pathways of the resonantly driven exciton population. The surface protection via encapsulation provides stable monolayer samples with low disorder, avoiding surface contaminations and the resulting exciton broadening and modifications of the dynamics. We identify areas localized to a few microns where the optical response is totally dominated by homogeneous broadening. Across the sample of tens of micrometers, weak inhomogeneous broadening and strain effects are observed, attributed to the remaining interaction with the $h$-BN and imperfections in the encapsulation process.