Perturbative Page Curve Induced by External Impulse

In this work, we extend the recent study of entropy dynamics induced by an external impulse in open quantum systems, where the entropy response follows the Page curve. For small system-bath coupling $\kappa$, we expect that the entropy first increases exponentially ${\kappa }^2{e}^{\mathrm{\varkappa }t}$ in the early-time regime $t\lesssim \mathrm{\mid }\log \kappa \mathrm{\mid }$ due to quantum many-body chaos, and then decreases as $e^{-{\lambda }_{0}t}$ with ${\lambda }_0\propto {\kappa }^2$ due to the energy relaxation. These results are confirmed through explicit calculations using two methods: 1) generalized Boltzmann equation for systems with quasi-particles; 2) scramblon effective theory in the early-time regime and perturbation theory in the late-time regime for 0+1-d systems. We also prove that in the second stage, the entropy of the system is equal to the coarse-grained entropy.