Long-term evolution of non-Hermitian systems in quantum-walk dynamics

We present a study on the long-term evolution of a non-Hermitian one-dimensional system with on-site dissipations in quantum-walk dynamics. By investigating various cases, including different evolution operations, lattice sizes, initial positions, and dissipations, we uncover remarkable and unconventional features. They include the emergence of distinct final states after a sufficiently long evolution time, the persistence of metastable states for extended periods, the survival of extended states only within the inner region of a domain-wall structure, and the occurrence of instantaneous ultrafar jumps during the evolution process. To provide insight into the underlying physical mechanisms behind these observations, we undertake a comprehensive investigation accompanied by detailed analysis, clarify the relation between time-evolved (meta)stable states and system eigenstates, and analyze the symmetry of final states. Moreover, our research reveals some phenomena that differ from those seen in previous experimental results, which can be attributed to the limited evolution time in the experiments. These theoretical findings underscore the significance of long-term evolution in non-Hermitian systems based on quantum walks.