We propose to go beyond the usual Hubbard model description of atoms in optical lattices and show how few-body physics can be used to simulate many-body phenomena, e.g., an electron-phonon system.We take one atomic species to be trapped in a deep optical lattice at full-filling and another to be untrapped spin-polarized fermions (which do not see the optical lattice) but to have an s-wave contact interaction with the first species. For large positive scattering length on the order of lattice spacing, the usual two-body bound (dimer) states overlap forming giant orbitals extending over the entire lattice, which can be viewed as an "electronic" band for the untrapped species while the trapped atoms become the "ions" with their own on-site dynamics, thereby simulating an electron-phonon system with renormalization of the phonon frequencies and Peierls transitions. This setup requires large scattering lengths but minimizes losses, does not need higher bands, and adds degrees of freedom which cannot easily be described in terms of lattice variables, thus opening up intriguing possibilities to explore interesting physics at the interface between few-body and many-body systems.
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