Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems

We discuss the universal nature of relaxation in isolated many-body quantum systems subjected to global and strong periodic driving. Our rigorous Floquet analysis shows that the energy of the system remains almost constant up to an exponentially long time in frequency for arbitrary initial states and that an effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion is a quasiconserved quantity in a long time scale. These two general properties lead to an intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon in nearly integrable systems.

Figure 1

Numerical demonstration of prethermalizationlike phenomenon. (a) Relaxation in the long time scale. (b) Initial relaxation. (c) Transient time evolution after the initial one. Parameters are $(J,B_x,B_z)=(1,0.9045,0.8090)$ and $N=24$. The dotted line in (b) is the expectation value in the equilibrium state of $H_F^{(0)}$ at the inverse temperature $\beta =0.85$, which is determined from the expectation value of $H_F^{(0)}$ at $t=10$.