  The coupling of internal electronic excitations to vibrational modes of the
external motion of ultracold atoms in an optical lattice is studied here in
using a perturbation expansion in small atomic displacements. In the Mott
insulator case with one atom per site, the resonance dipole-dipole coupling
between neighboring sites can induce emission and absorption of vibrational
quanta. Within a cavity in the strong exciton-photon coupling regime such
coupling results in polariton-vibration interactions, which create a
significant thermalization mechanism for polaritons toward their minimum
energy, and leading to motional heating of the lattice atoms.
