  We study an open quantum system of free fermions on an infinite lattice
coupled to a localized particle source. In the long time limit, the total
number of fermions in the system increases linearly with growth rate dependent
on the lattice geometry and dimensionality. We express the growth rate in terms
of lattice Green functions and derive explicit formulae in one dimension and
for the square lattice. The interplay between the dynamics and the coupling to
the environment leads, in contrast to classical systems, to a non-monotonic
dependence of the particle growth rate on the input rate. We show that for all
lattices the particle growth rate is inversely proportional to the input rate
when the latter becomes large. This is a manifestation of the quantum Zeno
effect.
