We discuss the effect of dissipation on heating which occurs in periodically driven quantum many-body systems. We especially focus on a periodically driven Bose-Hubbard model coupled to an energy and particle reservoir. Without dissipation, this model is known to undergo parametric instabilities which can be considered as an initial stage of heating. By taking the weak on-site interaction limit as well as the weak system-reservoir coupling limit, we find that parametric instabilities are suppressed if the dissipation is stronger than the on-site interaction strength and stable steady states appear. Our results demonstrate that periodically driven systems can emit energy, which is absorbed from external drivings, to the reservoir so that they can avoid heating.

• Received 12 February 2017

DOI:https://doi.org/10.1103/PhysRevA.95.043621