Periodic driving has emerged as a powerful experimental tool to engineer physical properties of isolated, synthetic quantum systems. However, due to the lack of energy conservation and heating effects, non-trivial (e.g., topological) many-body states in periodically driven (Floquet) systems are generally metastable. Therefore it is necessary to find strategies for preparing long-lived many-body states in Floquet systems. We develop a theoretical framework for describing the dynamical preparation of states in Floquet systems by a slow turn-on of the drive. We find that the dynamics of the system is well approximated by the initial state evolving under a slowly varying effective Hamiltonian $H_{rm eff}^{(s)}(t)$, provided the ramp speed $s gg t_*^{-1} sim e^{-{mathcal{C} frac{omega}{J}}}$, the inverse of the characteristic heating time-scale in the Floquet system. At such ramp speeds, the heating effects due to the drive are exponentially suppressed. We compute the slowly varying effective Hamiltonian $H_{rm eff}^{(s)}(t)$, and show that at the end of the ramp it is identical to the effective Hamiltonian of the unramped Floquet system, up to small corrections of the order $O(s)$. Therefore, the system effectively undergoes a slow quench from $H_0$ to $H_{rm eff}$. As an application, we consider the passage of the slow quench through a quantum critical point (QCP), and estimate the energy absorbed due to the non-adiabatic passage through the QCP via a Kibble-Zurek mechanism. By minimizing the energy absorbed due to both the drive and the ramp, we find an optimal ramp speed $s_* sim t_*^{-z/({d+2z})}$ for which both heating effects are exponentially suppressed. Our results bridge the gap between the numerous proposals to obtain interesting systems via Floquet engineering, and the actual preparation of such systems in their effective ground states.
Download