  We investigate the dynamical information exchange between a two-state system
and its environment which is measured by 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 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 numerical renormalization group.
