  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.
