  We propose an approach for quantum simulation of electron-phonon interactions
using Rydberg states of cold atoms and ions. We show how systems of cold atoms
and ions can be mapped onto electron-phonon systems of the Su-Schrieffer-Heeger
type. We discuss how properties of the simulated Hamiltonian can be tuned and
how to read physically relevant properties from the simulator. In particular,
use of painted spot potentials offers a high level of tunability, enabling all
physically relevant regimes of the electron-phonon Hamiltonian to be accessed.
