Realising strong photon-photon interactions in a solid-state setting is a major goal with far reaching potential for optoelectronic applications. Using Landaus 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.
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