Dynamics of non-Hermitian Floquet Wannier-Stark system

We study the dynamics of the non-Hermitian Floquet Wannier-Stark system in the framework of the tight-binding approximation, where the hopping strength is a periodic function of time with Floquet frequency $\omega$. It is shown that the energy level of the instantaneous Hamiltonian is still equally spaced and independent of time $t$ and the Hermiticity of the hopping term. In the case of off resonance, the dynamics are still periodic, while the occupied energy levels spread out at the resonance, exhibiting $t^z$ behavior. Analytic analysis and numerical simulation show that the level-spreading dynamics for real and complex hopping strengths exhibit distinct behaviors and are well described by the dynamical exponents $z=1$ and $z=1/2$, respectively.