Non-Markovian dynamical effects and time evolution of the entanglement entropy of a dissipative two-state system

We investigate the dynamical information exchange between a two-state system and its environment which is measured by the von Neumann entropy. It is found that in the underdamping regime, the entropy dynamics exhibits an extremely non-Markovian oscillation-hump feature, in which oscillations manifest quantum coherence and a hump of envelop demonstrates temporal memory of bath. It indicates that the process of entropy exchange is bidirectional. When the coupling strength increases up to a certain threshold, the hump along with ripple disappears, which is indicative of the coherent-incoherent dynamical crossover. The long-time limit of entropy evolution reaches the ground-state value which agrees with that of the numerical renormalization group.