  We consider the prospects for quantum simulation of condensed matter models
exhibiting strong electron-phonon coupling using a hybrid platform of trapped
laser-cooled ions interacting with an ultracold atomic gas. This system
naturally posesses a phonon structure, in contrast to the standard optical
lattice scenarios usually employed with ultracold atoms in which the lattice is
generated by laser light and thus it remains static. We derive the effective
Hamiltonian describing the general system and discuss the arising energy
scales, relating the results to commonly employed extended Hubbard-Holstein
models. Although for a typical experimentally realistic system the coupling to
phonons turns out to be small, we provide the means to enhance its role and
reach interesting regimes with competing orders. Extended Lang-Firsov
transformation reveals the emergence of phonon-induced long-range interactions
between the atoms, which can give rise to both localized and extended bipolaron
states with low effective mass, indicating the possibility of fermion pairing.
