  We study a 2D system of trion-polaritons at the quantum level and demonstrate
that for monolayer semiconductors they can exhibit a strongly nonlinear optical
response. The effect is due to the composite nature of trion-based excitations
resulting in their nontrivial quantum statistical properties, and enhanced
phase space filling effects. We present the full quantum theory to describe the
statistics of trion-polaritons, and demonstrate that the associated
nonlinearity persists at the level of few quanta, where two qualitatively
different regimes of photon antibunching are present for weak and strong single
photon-trion coupling. We find that single photon emission from
trion-polaritons becomes experimentally feasible in state-of-the-art transition
metal dichalcogenide (TMD) setups. This can foster the development of quantum
polaritonics using 2D monolayers as a material platform.
