The optimal frequency window for Floquet engineering in optical lattices

The concept of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting novel properties. It has been employed, for instance, for the realization of artificial magnetic fluxes in optical lattices and, typically, it is based on two approximations. First, the driving frequency is assumed to be low enough to suppress resonant excitations to high-lying states above some energy gap separating a low energy subspace from excited states. Second, the driving frequency is still assumed to be large compared to the energy scales of the low-energy subspace, so that also resonant excitations within this space are negligible. Eventually, however, deviations from both approximations will lead to unwanted heating on a time scale $\tau$. Using the example of a one-dimensional system of repulsively interacting bosons in a shaken optical lattice, we investigate the optimal frequency (window) that maximizes $\tau$. As a main result, we find that, when increasing the lattice depth, $\tau$ increases faster than the experimentally relevant time scale given by the tunneling time $\hbar/J$, so that Floquet heating becomes suppressed.