  Results are presented for an open Floquet topological system represented by
Dirac fermions coupled to a circularly polarized laser and an external
reservoir. It is shown that when the separation between quasi-energy bands
becomes small, and comparable to the coupling strength to the reservoir, the
reduced density matrix in the Floquet basis, even at steady-state, has non-zero
off-diagonal elements, with the magnitude of the off-diagonal elements
increasing with the strength of the coupling to the reservoir. In contrast, the
coupling to the reservoir only weakly affects the diagonal elements, hence
inducing an effective coherence. The steady-state reduced density matrix
synchronizes with the periodic drive, and a Fourier analysis allows the
extraction of the occupation probabilities of the Floquet quasi-energy levels.
The lack of detailed balance at steady-state is quantified in terms of an
entropy production rate, and it is shown that this equals the heat current
flowing out of the system, and into the reservoir. It is also shown that the
entropy production rate mainly depends on the off-diagonal components of the
Floquet density matrix. Thus a stronger coupling to the reservoir leads to an
enhanced entropy production rate, implying a more efficient removal of heat
from the system, which in turn helps the system maintain coherence. Analytic
expressions in the vicinity of the Dirac point are derived which highlights
these results, and also indicates how the reservoir may be engineered to
enhance the coherence of the system.
