  We investigate light-matter coupling in metallic crystals where plasmons
coexist with phonons exhibiting large oscillator strength. We demonstrate
theoretically that this coexistence can lead to strong light-matter
interactions without external resonators. When the frequencies of plasmons and
phonons are comparable, hybridization of these collective matter modes occurs
in the crystal. We show that the coupling of these modes to photonic degrees of
freedom gives rise to intrinsic surface plasmon-phonon polaritons, which offer
the unique possibility to control the phonon properties by tuning the electron
density and the crystal thickness. In particular, dressed phonons with reduced
frequency and large wave vectors arise in the case of quasi-2D crystals, which
leads to large enhancements of the electron-phonon scattering in the
vibrational ultrastrong coupling regime. This suggests that photons can play a
key role in determining the quantum properties of certain materials. A
non-perturbative self-consistent Hamiltonian method is presented that is valid
for arbitrarily large coupling strengths.
