  We present a theoretical description of the coupling between longitudinal
optical phonons and collective excitations of a two-dimensional electron gas.
By diagonalizing the Hamiltonian of the system, including Coulomb
electron-electron and Fr\"ohlich interactions, we observe the formation of
multisubband polarons, mixed states partially phonon and partially multisubband
plasmon, characterized by a coupling energy which is a significant fraction, up
to ~40%, of the phonon energy. We demonstrate that multisubband plasmons and
longitudinal optical phonons are in the ultra-strong coupling regime in several
III-V and II-VI material systems.
