  Strong coupling between light and the fundamental excitations of a
two-dimensional electron gas (2DEG) are of foundational importance both to pure
physics and to the understanding and development of future photonic
nanotechnologies. Here we study the relationship between spin polarization of a
2DEG in a monolayer semiconductor, MoSe$_2$, and light-matter interactions
modified by a zero-dimensional optical microcavity. We find robust
spin-susceptibility of the 2DEG to simultaneously enhance and suppress
trion-polariton formation in opposite photon helicities. This leads to
observation of a giant effective valley Zeeman splitting for trion-polaritons
(g-factor >20), exceeding the purely trionic splitting by over five times.
Going further, we observe robust effective optical non-linearity arising from
the highly non-linear behaviour of the valley-specific strong light-matter
coupling regime, and allowing all-optical tuning of the polaritonic Zeeman
splitting from 4 to >10 meV. Our experiments lay the groundwork for engineering
quantum-Hall-like phases with true unidirectionality in monolayer
semiconductors, accompanied by giant effective photonic non-linearities rooted
in many-body exciton-electron correlations.
