Effects of time on the evolution of a wave packet in the tunneling dynamics

We investigate the time-dependent electron wave packet in a one-dimensional geometry with the potential bent by a homogeneous external field. Based on the behaviors of the wave packet over time, we observe a crossover time. After this crossover time, the temporal evolution of the wave packet comes into a new regime, where the wave packet evolves in a self-similar structure. To establish the time scale of this crossover quantitatively, we utilize the Loschmidt Echo function, through which the time at which the crossover occurs can be extracted. We also find the time of the maximum ionization velocity can be comparable with the semi-classical tunneling delay time.