Formations of generalized Wannier-Stark ladders: Theorem and applications

The Wannier-Stark ladder (WSL) is a basic concept, supporting periodic oscillation, widely used in many areas of physics. In this paper, we investigate the formations of WSL in generalized systems, including strongly correlated and non-Hermitian systems. We present a theorem on the existence of WSL for a set of general systems that are translationally symmetric before the addition of a linear potential. For a non-Hermitian system, the WSL becomes complex but maintains a real energy level spacing. We illustrate the theorem using 1D extended Bose-Hubbard models with both real and imaginary hopping strengths. It is shown that the Bloch-Zener oscillations of correlated bosons are particularly remarkable under resonant conditions. Numerical simulations for cases with boson numbers $n=2$, $3$, and $4$ are presented. Analytical and numerical results for the time evolution of the $n$-boson-occupied initial state indicate that all evolved states exhibit quasi periodic oscillations, but with different profiles, depending on the Hermiticity and interaction strength.