The Lindblad master equation is one of the main approaches to open quantum systems. While it has been widely applied in the context of condensed matter systems to study properties of steady states in the limit of long times, the actual route to such steady states has attracted less attention yet. Here, we investigate the nonequilibrium dynamics of spin chains with a local coupling to a single Lindblad bath and analyze the transport properties of the induced magnetization. Combining typicality and equilibration arguments with stochastic unraveling, we unveil for the case of weak driving that the dynamics in the open system can be constructed on the basis of correlation functions in the closed system, which establishes a connection between the Lindblad approach and linear response theory at finite times. In this way, we provide a particular example where closed and open approaches to quantum transport agree strictly. We demonstrate this fact numerically for the spin-1/2 XXZ chain at the isotropic point and in the easy-axis regime, where superdiffusive and diffusive scaling is observed, respectively.

Funding: Deutsche Forschungsgemeinschaft (DFG), under Grants No. 397107022 (GE 1657/3-2) and No. 397067869 (STE 2243/3-2), within the DFG Research Unit FOR 2692, under Grant No. 355031190. Moreover, DFG Grant No. 456666331. European Union's Horizon Europe research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101060162.