We introduce and solve a model of fermions hopping between neighbouring sites
on a line with random Brownian amplitudes and open boundary conditions driving
the system out of equilibrium. The average dynamics reduces to that of the
symmetric simple exclusion process. However, the full distribution encodes for
a richer behaviour entailing fluctuating quantum coherences which survive in
the steady limit. We determine exactly the system state steady distribution. We
show that these out of equilibrium quantum fluctuations fulfil a large
deviation principle and we present a method to recursively compute exactly the
large deviation function. On the way, our approach gives a solution of the
classical symmetric simple exclusion process using fermion technology. Our
results open the route towards the extension of the macroscopic fluctuation
theory to many body quantum systems.Comment: 5 pages + SM, 2 figure
