Floquet systems coupled to particle reservoirs

Open quantum systems, when driven by a periodic field, can relax to effective statistical ensembles that resemble their equilibrium counterparts. We consider a class of problems in which a periodically driven quantum system is allowed to exchange both energy and particles with a thermal reservoir. We demonstrate that, even for noninteracting systems, effective equilibration to the grand canonical ensemble requires both fine tuning the system-bath coupling and selecting a sufficiently simple driving protocol. We study a tractable subclass of these problems in which the long-time steady state of the system can be determined analytically, and demonstrate that the system effectively thermalizes with fine tuning, but does not thermalize for general values of the system-bath couplings. When the driven system does not thermalize, it supports a tunable persistent current in the steady state without external bias. We compute this current analytically for two examples of interest: (1) a driven double quantum dot, where the current is interpreted as a dc electrical current, and (2) driven Dirac fermions in graphene, where it is interpreted as a valley current.