Highly nonlinear optical materials with strong effective photon-photon interactions (Kerr-like nonlinearity) are required in the development of novel quantum sources of light as well as for ultrafast and quantum optical signal processing circuitry. Here we report very large Kerr-like nonlinearities by employing strong optical transitions of charged excitons (trions) observed in semiconducting transition metal dichalcogenides (TMDCs). By hybridising trions in monolayer MoSe$_2$ at low electron densities with a microcavity mode, we realise trion-polaritons exhibiting significant energy shifts at very small photon fluxes due to phase space filling. Most notably, the strong trion-polariton nonlinearity is found to be 10 to 1000 larger than in other polariton systems, including neutral exciton-polaritons in TMDCs. Furthermore it exceeds by factors of $sim 10^3-10^5$ the magnitude of Kerr nonlinearity in bare TMDCs, graphene and other widely used optical materials (e.g. Si, AlGaAs etc) in weak light-matter coupling regimes. The results are in good agreement with a theory which accounts for the composite nature of excitons and trions and deviation of their statistics from that of ideal bosons and fermions. This work opens a new highly nonlinear system for quantum optics applications enabling in principle scalability and control through nano-engineering of van der Waals heterostructures.
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