Title: Heating Rates under Fast Periodic Driving beyond Linear Response

Abstract: Heating under periodic driving is a generic nonequilibrium phenomenon, and it is a challenging problem in nonequilibrium statistical physics to derive a quantitatively accurate heating rate. In this work, we provide a simple formula on the heating rate under fast and strong periodic driving in classical and quantum many-body systems. The key idea behind the formula is constructing a time-dependent dressed Hamiltonian by moving to a rotating frame, which is found by a truncation of the high-frequency expansion of the micromotion operator, and applying the linear-response theory. It is confirmed for specific classical and quantum models that the second-order truncation of the high-frequency expansion yields quantitatively accurate heating rates beyond the linear-response regime. Our result implies that the information on heating dynamics is encoded in the first few terms of the high-frequency expansion, although heating is often associated with an asymptotically divergent behavior of the high-frequency expansion.