Theory of four-wave-mixing in phonon polaritons

Third order anharmonic scattering in light-matter systems can drive a wide variety of practical and physically interesting processes from lasing to polariton condensation. Motivated by recent experimental results in the nonlinear optics of localised phonon polaritons, in this Letter we develop a quantum theory capable of describing four-wave mixing in arbitrarily inhomogeneous photonic environments. Using it we investigate Kerr self-interaction and parametric scattering of surface and localised phonon polaritons, showing both processes to be within experimental reach.