Noninteracting fermionic systems with localized losses: Exact results in the hydrodynamic limit

We investigate the interplay between unitary and nonunitary dynamics after a quantum quench in a noninteracting fermionic chain. In particular, we consider the effect of localized loss processes, for which fermions are added and removed incoherently at the center of the chain. We focus on the hydrodynamic limit of large distances from the localized losses and of long times, with their ratio being fixed. In this limit, the localized losses gives rise to an effective imaginary delta potential (nonunitary 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 losses 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 loss and gain rates the coherent dynamics of the system is arrested, which is a manifestation of the celebrated quantum Zeno effect.