Condition for emergence of the Floquet-Gibbs state in periodically driven open systems

We study probability distribution of a steady state of a periodically driven system coupled to a thermal bath by using a quantum master equation in the weak-coupling limit. It is proved that, even when the external field is strong, the probability distribution is independent of the detailed nature of the thermal bath under the following conditions: (i) the Hamiltonian of the relevant system is bounded and the period of the driving field is short, (ii) the Hamiltonians for the driving field at different times commute, and (iii) the Hamiltonians of the driving field and of the interaction between the relevant system and the thermal bath commute. It is shown that the steady state is described by the Gibbs distribution of the Floquet states of the relevant system at the temperature of the thermal bath.

Figure 1

(a) Schematic illustration of our model. (b) $\Delta$ Prob versus the period of the driving field $T$ where the relevant system is coupled to the thermal baths via $X_1={\sum }_{k=\{x,y,z\}}{\alpha }^k{S}_1^k$ and $X_5={\sum }_{k=\{x,y,z\}}{\alpha }^k{S}_5^k$ for various realizations of ${\alpha }^y/{\alpha }^x=\{0$ (filled circle), $0.01$ (up triangle), $0.1$ (square), 1 (down triangle), $\infty$ (cross)$\}$, and ${\alpha }^z=0$. When $[H^{\mathrm{ex}}\left(t\right),H_{\mathrm{I}}]=0$ (filled circle), the steady state approaches the Floquet-Gibbs state. The larger ${\alpha }^y/{\alpha }^x$ is, the larger $\Delta$ Prob is. (c) Like (b), but the driven spins and the spins coupling to the thermal bath are separated. We study the set $({\alpha }^x,{\alpha }^y,{\alpha }^z)=\{(0.1,0.1,1)\left(\text{filled}\text{circle}\right),(0.1,1,0.1)\left(\text{triangle}\right),\text{and}(1,0.1,0.1)\}$ (square). The steady state approaches the Floquet-Gibbs state independently of the set.