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

We investigate both theoretically 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, energy diffusion 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 propagation, energy diffusion and quantum scrambling efficiently: when the phase equals to $\pi/2$, we observe a maximum directed current and energy diffusion, while a minimum scrambling phenomenon protected by the $\mathcal{PT}$-symmetry; when the phase is $\pi$, both the directed transport and the energy diffusion are suppressed, in contrast, the quantum scrambling is enhanced by the non-Hermiticity. Possible applications of our findings are discussed.