  Recent research has observed the occurrence of pseudo-mobility edge (ME)
within a modulated mosaic model incorporating the Wannier-Stark potential. This
pseudo-ME, which signifies the critical energy that distinguishes between
ergodic and weakly ergodic, or weakly ergodic and nonergodic states, is a
crucial concept in comprehending the transport and localization phenomena in
Wannier-Stark systems. Here we investigate the influence of dissipation on a
finite-height mosaic Wannier-Stark lattice that features such pseudo-MEs by
computing the steady state density matrix. Our findings indicate that
particular dissipation can steer the system into specific states, regardless of
its initial state, predominantly characterized by either ergodic or nonergodic
states. This suggests that dissipation can be harnessed as a novel method for
inducing transitions between these states and manipulating particle
localization behaviors in disorder-free systems.
