  In condensed matter, it is often difficult to untangle the effects of
competing interactions, and this is especially problematic for superconductors.
Quantum simulators may help: here we show how exploiting the properties of
highly excited Rydberg states of cold fermionic atoms in a bilayer lattice can
simulate electron-phonon interactions in the presence of strong correlation - a
scenario found in many unconventional superconductors. We discuss the core
features of the simulator, and use numerics to compare with condensed matter
analogues. Finally, we illustrate how to achieve a practical, tunable
implementation of the simulation using painted spot potentials.
