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Floquet topological insulators are noninteracting quantum systems that, whendriven by a time-periodic field, are described by effective Hamiltonians whosebands carry nontrivial topological invariants. A longstanding question concernsthe possibility of selectively populating one of these effective bands, therebymaximizing the system's resemblance to a static topological insulator. We studysuch Floquet systems coupled to a zero-temperature thermal reservoir thatprovides dissipation. We find that the resulting electronic steady states aregenerically characterized by a finite density of excitations above theeffective ground state, even when the driving has a small amplitude and/orlarge frequency. We discuss the role of reservoir engineering in mitigatingthis problem.

Occupation of topological Floquet bands in open systems