Structure-dynamics relation in shaken optical lattices

Shaken optical lattices permit to coherently modify the tunneling of particles in a controllable manner. We introduce a general relation between the geometry of shaken lattices and their admissible effective dynamics. Using three different examples, we illustrate the symmetries of the emerging tunneling rates. The results provide a clear framework to understand the relation between lattice geometry and accessible dynamics, and a tool to straightforwardly derive truncated effective Hamiltonians on arbitrary geometries.