  Reservoir engineering lattice states using only localized engineered
dissipation is extremely attractive from a resource point of view, but can
suffer from long relaxation times. Here, we study the relaxation dynamics of
bosonic lattice systems locally coupled to a single squeezed reservoir. Such
systems can relax into a highly non-trivial pure states with long-range
entanglement. In the limit of large system size, analytic expressions for the
dissipation spectrum can be found by making an analogy to scattering from a
localized impurity. This allows us to study the cross-over from perturbative
relaxation to a slow, quantum-Zeno regime. We also find the possibility of
regimes of accelerated relaxation due to a surprising impedance matching
phenomena. We also study intermediate time behaviors, identifying a long-lived
"prethermalized" state associated that exists within a light cone like area.
This intermediate state can be quasi-stationary, and can very different
entanglement properties from the ultimate dissipative steady state.
