By forming hybrid light-matter states, optical cavities offer a route for engineering material properties, however, unambiguously probing the effects of light-matter coupling remains difficult. Here, we show that the polarization-resolved statistics of photons transmitted through a cavity, measurable via $g^{\left(2\right)}$, provide one such diagnostic. By relating $g^{\left(2\right)}$ to matter correlation functions such as the Raman structure factor, we link photon bunching and antibunching to material properties. By applying this method to the stripy-to-antiferromagnetic transition in the Kitaev-Heisenberg spin model, we find that polarization-dependent patterns of bunching and antibunching encode the magnetic point-group symmetries of each phase and characterize the behavior at the phase boundary. Finally, we predict measuring $g^{\left(2\right)}$ for output photon pairs polarized orthogonal to the input field will isolate higher-order light-matter scattering processes that probe higher-order material correlations.

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• Cavity quantum electrodynamics
• Crystal symmetry
• Photon statistics
• Kitaev-Heisenberg model