Exciton-exciton interaction beyond the hydrogenic picture in a MoSe$_2$ monolayer in the strong light-matter coupling regime

In transition metal dichalcogenides layers of atomic scale thickness, the electron-hole Coulomb interaction potential is strongly influenced by the sharp discontinuity of the dielectric function across the layer plane. This feature results in peculiar non-hydrogenic excitonic states, in which exciton-mediated optical nonlinearities are predicted to be enhanced as compared to their hydrogenic counterpart. To demonstrate this enhancement, we performed optical transmission spectroscopy of a MoSe$_2$ monolayer placed in the strong coupling regime with the mode of an optical microcavity, and analyzed the results quantitatively with a nonlinear input-output theory. We find an enhancement of both the exciton-exciton interaction and of the excitonic fermionic saturation with respect to realistic values expected in the hydrogenic picture. Such results demonstrate that unconventional excitons in MoSe$_2$ are highly favourable for the implementation of large exciton-mediated optical nonlinearities, potentially working up to room temperature.