  We investigate polariton quantum correlations in a coherently driven box
cavity in the low driving regime, with a particular focus on accounting for the
polarization degree of freedom. The possibility of having different interaction
strengths between co- and cross-circularly polarized polaritons as well as a
realistic linear-polarization splitting allows one to model the system as two
coupled nonlinear resonators with both self- and cross-Kerr-like
nonlinearities, thus making our results potentially relevant for other
experimental platforms. Within an effective wave-function approach, we obtain
analytical expressions for the steady-state polarization-resolved polariton
populations and second-order correlation functions, which agree very well with
our numerical results obtained from a Lindblad master equation. Notably, we
highlight that depending on the excitation polarization (circular or linear),
both the unconventional (interference-mediated) and conventional (mediated by
nonlinearities) antibunchings can be investigated in a single cavity. Moreover,
using our results, we argue that recent experiments on confined fiber-cavity
polaritons are likely to have probed a regime where the dominant interaction is
between cross-polarized polaritons, which is characteristic of the polariton
Feshbach resonance. We furthermore investigate the regime close to resonance
using a two-channel model, and we show that systems with large biexciton
binding energies, such as atomically thin semiconductors, are promising
platforms for realizing strong polariton antibunching.
