  Recent experiments have demonstrated strong light-matter coupling between
electromagnetic nanoresonators and pristine sheets of two-dimensional
semiconductors, and it has been speculated whether these systems can enter the
quantum regime operating at the few-polariton level. To address this question,
we present a first-principles microscopic quantum theory for the interaction
between excitons in an infinite sheet of two-dimensional material and a
localised electromagnetic resonator. We find that the light-matter interaction
breaks the symmetry of the otherwise translation-invariant system and thereby
effectively generates a localised exciton mode, which is coupled to an
environment of residual exciton modes. This dissipative coupling increases with
tighter lateral confinement, and our analysis reveals this to be a potential
challenge in realising nonlinear exciton-exciton interaction. Nonetheless, we
predict that polariton blockade due to nonlinear exciton-exciton interactions
is well within reach for nanoresonators coupled to transition-metal
dichalcogenides, provided that the lateral resonator mode confinement can be
sufficiently small that the nonlinearity overcomes the polariton dephasing
caused by phonon interactions.
