Fermi's golden rule for heating in strongly driven Floquet systems

We study heating dynamics in isolated quantum many-body systems driven periodically at high frequency and large amplitude. Combining the high-frequency expansion for the Floquet Hamiltonian with Fermi's golden rule (FGR), we develop a master equation termed the Floquet FGR. Unlike the conventional one, the Floquet FGR correctly describes heating dynamics, including the prethermalization regime, even for strong drives, under which the Floquet Hamiltonian is significantly dressed, and nontrivial Floquet engineering is present. The Floquet FGR depends on system size only weakly, enabling us to analyze the thermodynamic limit with small-system calculations. Our results also indicate that, during heating, the system approximately stays in the thermal state for the Floquet Hamiltonian with a gradually rising temperature.

Figure 1

Typical heating dynamics for ${\epsilon }_k=\langle {\mathrm{\Psi }}_k\left|H_0\right|{\mathrm{\Psi }}_k\rangle /L$ under (a) weak ($\omega =13$ and $h_x=0.37$) and (b) strong ($\omega =16$ and $h_x=3.0$) drives. The horizontal axis denotes the Floquet cycle $k$. Solid curves are the results obtained by numerically simulating the unitary evolution. Dashed (dotted) curves show that obtained by the Floquet FGR with the sixth-order Magnus expansion at $L=14$ ($L=16$) and dash-dotted ones by the bare FGR at $L=16$.

Figure 2

Heating rates extracted from unitary dynamics at energy density $\epsilon =-0.48$ by the linear least squares. The solid (dashed) curves show the heating rates obtained by the Floquet (bare) FGR, and the Floquet FGR's order from zeroth to sixth in the legend shows that of the Magnus expansion for $H_F$ (see also text). All the FGR calculations are for $L=14$.

Figure 3

(a) Heating rate $d\epsilon /dt$ calculated by the Floquet FGR with the sixth-order $H_F$ at temperature $\beta =0.1$. (b) Bures distance $D_B({\rho }_{\mathrm{th}}^F,{\rho }_{\mathrm{th}}^0)$ (13) between thermal states for $H_F$ and $H_0$ at temperature $\beta =0.1$. In each panel we used $L=14$.

Figure 4

Typical heating dynamics in the open Fermi-Hubbard chain for a strong drive ($E_0=4$ and $\omega =10$). Solid curves are the results obtained by numerically simulating the unitary evolution. Dotted (dash-dotted) curves show that obtained by the sixth-order-Floquet (bare) FGR at $L=10$.

Figure 5

Heating dynamics for various driving amplitude $h_x$ in (a) $0.25\le h_x\le 3.25$ and (b) $3.5\le h_x\le 6.25$ calculated at frequency $\omega =16$ and system size $L=20$. The inset shows the first 20 Floquet cycles in which ${\epsilon }_k>-0.48$ for $h_x=1.5$ in panel (a) and $h_x=5.5$ in panel (b). The inset also shows the linear least-squares fit.