  The unavoidable coupling of quantum systems to external degrees of freedom
leads to dissipative (non-unitary) dynamics, which can be radically different
from closed-system scenarios. Such open quantum system dynamics is generally
described by Lindblad master equations, whose dynamical and steady-state
properties are challenging to obtain, especially in the many-particle regime.
Here, we introduce a method to deal with these systems based on the calculation
of (dissipative) lattice Green's function with a real-space decimation
technique. Compared to other methods, such technique enables obtaining compact
analytical expressions for the dynamics and steady-state properties, such as
asymptotic decays or correlation lengths. We illustrate the power of this
method with several examples of driven-dissipative bosonic chains of increasing
complexity, including the Hatano-Nelson model. The latter is especially
illustrative because its surface and bulk dissipative behavior are linked due
to its non-trivial topology, which manifests in directional amplification.
