  We consider a class of either fermionic or bosonic noninteracting open
quantum chains driven by dissipative interactions at the boundaries and study
the interplay of coherent transport and dissipative processes, such as bulk
dephasing and diffusion. Starting from the microscopic formulation, we show
that the dynamics on large scales can be described in terms of fluctuating
hydrodynamics. This is an important simplification as it allows us to apply the
methods of macroscopic fluctuation theory to compute the large deviation (LD)
statistics of time-integrated currents. In particular, this permits us to show
that fermionic open chains display a third-order dynamical phase transition in
LD functions. We show that this transition is manifested in a singular change
in the structure of trajectories: while typical trajectories are diffusive,
rare trajectories associated with atypical currents are ballistic and
hyperuniform in their spatial structure. We confirm these results by
numerically simulating ensembles of rare trajectories via the cloning method,
and by exact numerical diagonalization of the microscopic quantum generator.
