  Realising strong photon-photon interactions in a solid-state setting is a
major goal with far reaching potential for optoelectronic applications. Using
Landau's quasiparticle framework combined with a microscopic many-body theory,
we explore the interactions between exciton-polaritons and trions in a
two-dimensional semiconductor injected with an electron gas inside a
microcavity. We show that particle-hole excitations in the electron gas mediate
an attractive interaction between the polaritons, whereas a trion-polariton
interaction mediated by the exchange of an electron is either repulsive or
attractive depending on the specific polariton branch. These mediated
interactions are intrinsic to the quasiparticles and are also present in the
absence of light. Importantly, they can be tuned to be more than an order of
magnitude stronger than the direct polariton-polariton interaction in the
absence of the electron gas, thereby providing a promising outlook for
non-linear optical components. Finally, we compare our theoretical predictions
with two recent experiments.
