Exploring the effects of phase modulation on the dynamics of the kicked rotor systems

Inspired by the recent experimental progress in the time-driven phase transition in quantum chaos, we investigate comprehensively the energy diffusion of a kicked rotor in the presence of phase modulation. In the classical case, we found that there always exists anomalous diffusion as long as the phase is modulated periodically and changes by 0 or π from kick to kick. On the contrary, for quasiperiodic and random phase modulation, anomalous diffusion is suppressed. On the other hand, in the quantum case, there exist only ballistic energy diffusion and dynamical localization in the standard and periodically shifted cases, while random phase modulation destroys the quantum coherence and totally suppresses the dynamical localization. Furthermore, the quasiperiodic phase modulation is an intermediate phase between the standard case and the random one. In both the classical and quantum cases, quasiperiodic phase modulation is inequivalent to random phase modulation at large kicking times (>10^{3}), thus caution has to be taken when dealing with these two kinds of phase modulation in experiments.