  We consider the mechanical coupling between a two-dimensional Bose-Einstein
condensate with a graphene sheet via the vacuum fluctuations of the
electromagnetic field which are at the origin of the so-called Casimir-Polder
potential. By deriving a self-consistent set of equations governing the
dynamics of the condensate and the flexural (out-of-plane) modes of the
graphene, we can show the formation of a new type of purely acoustic
quasi-particle excitation, a phonon-polariton resulting from the coherent
superposition of quanta of flexural and Bogoliubov modes.
