  We investigate the interplay between unitary dynamics after a quantum quench
and localized dissipation in a noninteracting fermionic chain. In particular,
we consider the effect of gain and loss processes, for which fermions are added
and removed incoherently. We focus on the hydrodynamic limit of large distances
from the source of dissipation and of long times, with their ratio being fixed.
In this limit, the localized dissipation gives rise to an effective delta
potential (dissipative impurity), and the time-evolution of the local
correlation functions admits a simple hydrodynamic description in terms of the
fermionic occupations in the initial state and the reflection and transmission
amplitudes of the impurity. We derive this hydrodynamic framework from the ab
initio calculation of the microscopic dynamics. This allows us to analytically
characterize the effect of dissipation for several theoretically relevant
initial states, such as a uniform Fermi sea, homogeneous product states, or the
inhomogeneous state obtained by joining two Fermi seas. In this latter setting,
when both gain and loss processes are present, we observe the emergence of
exotic nonequilibrium steady states with stepwise uniform density profiles. In
all instances, for strong dissipation the coherent dynamics of the system is
arrested, which is a manifestation of the celebrated quantum Zeno effect.
