Energy spectrum and the thermal Mandel Q-parameter of a quadratically coupled phonon-photon polariton

► Energy spectrum of a polariton with quadratic coupling (λ) is determined. ► For phonons, Qth is shown to increase with T and linear coupling κ. ► With λ, increases monotonically. ► Phonons in a polariton show a super-Poissonian statistics for all κ, λ and T. ► nph increases with κ, λ and T but decreases with phonon frequency.