Phase modulation of directed transport, energy diffusion, and quantum scrambling in a Floquet non-Hermitian system

We investigate both analytically and numerically the wavepacket's dynamics in momentum space for a Floquet non-Hermitian system with a periodically kicked driven potential. We have deduced the exact expression of a time-evolving wavepacket under the condition of quantum resonance. With this analytical expression, we can investigate thoroughly the temporal behaviors of the directed transport, mean energy, and quantum scrambling. We find interestingly that, by tuning the relative phase between the real part and imaginary part of the kicking potential, one can manipulate the directed transport, mean energy, and quantum scrambling efficiently: When the phase equals to $\pi /2$, we observe a maximum directed transport and mean energy, while a minimum scrambling phenomenon protected by the $\mathcal{PT}$ symmetry; when the phase is $\pi$, both the directed transport and the time dependence of the energy are suppressed; in contrast, the quantum scrambling is enhanced by the non-Hermiticity. For the quantum nonresonance case, we numerically find that the quantum interference effects lead to dynamical localization, characterized by the suppression of the directed transport, the time dependence of the energy, and quantum scrambling. Interestingly, these suppression effects can be adjusted by the phase of the non-Hermitian kicking potential. Possible applications of our findings are discussed.