We study an imaginary Stark ladder model, and we propose a realization of the model in a dissipative chain with linearly increasing site-dependent dissipation strength. Due to the existence of K symmetry and passive PT symmetry, the model exhibits quite a different feature from its Hermitian counterpart. With the increase of dissipation strength, the system first undergoes a passive PT-symmetry breaking transition, with the shifted eigenvalues changing from real to complex, and then a K-symmetry restoring transition, characterized by the emergence of a pure imaginary spectrum with equal spacing. Accordingly, the eigenstates change from PT-unbroken extended states to PT-broken states, and finally to Stark-localized states. In the framework of the quantum open system governed by the Lindblad equation with linearly increasing site-dependent dissipation, we unveil that the dynamical evolution of a single-particle correlation function is governed by the Hamiltonian of the imaginary Stark ladder model. By studying the dynamical evolution of the density distribution under various initial states, we demonstrate that the damping dynamics displays distinct behaviors in different regions. A localized damping is observed in the strong dissipation limit.
Learning the generator of an open many-body system is more challenging than Hamiltonian learning: local responses, which can directly reveal coherent interaction terms in closed-system dynamics, may also contain dissipative contributions in open-system dynamics. In this paper, we address this challenge by developing an efficient Lindbladian learning framework for a known local candidate generator dictionary with bounded dissipative support and either bounded dual-interaction-graph degree or bounded unweighted local strength. The framework resolves the coherent-dissipative ambiguity by treating local Pauli responses as a linear system over both types of generator terms. Inverting this response system separates their contributions and makes the individual Lindbladian coefficients accessible from local response data in a fixed short-time window. Within this framework, we develop two efficient learning algorithms: Chebyshev--Lobatto response interpolation, which uses logarithmically many short evolution times and has a post-mean cost linear in M, with the stated dependence on ε, and Single-time projected response contraction, which uses a single fixed evolution time and globally inverts a truncated response function. Both procedures estimate M candidate coefficients to entrywise accuracy ε using O(M/ε2) sample and classical post-processing complexity. Our theoretical results establish local response inversion as a scalable paradigm for learning, calibrating, and diagnosing complex quantum systems from experimentally accessible short-time data.