Dynamical speedup of a two-level system induced by coupling in the hierarchical environment

We investigate the dynamics of a two-level system in the presence of an overall environment composed of two layers. The first layer is just one single-mode cavity which decays to memoryless reservoir while the second layer is the two coupled single-mode cavities which decay to memoryless or memory-keeping reservoirs. In the weak-coupling regime between the qubit and the first-layer environment, our attention is focused on the effects of the coupling in the hierarchical environment on the non-Markovian speedup dynamics behavior of the system. We show that, by controlling the coupling in the second-layer environment, the multiple dynamics crossovers from Markovian to non-Markovian and from no-speedup to speedup can be realized. This results hold independently on the nature of the second-layer environment. Differently, we find that how the coupling between the two layers affects the non-Markovian speedup dynamics behavior depends on the nature of the second-layer environment.