  We are interested in dynamics of quantum many-body systems under continuous
observation, and its physical realizations involving cold atoms in lattices. In
the present work we focus on continuous measurement of atomic currents in
lattice models, including the Hubbard model. We describe a Cavity QED setup,
where measurement of a homodyne current provides a faithful representation of
the atomic current as a function of time. We employ the quantum optical
description in terms of a diffusive stochastic Schr\"odinger equation to follow
the time evolution of the atomic system conditional to observing a given
homodyne current trajectory, thus accounting for the competition between the
Hamiltonian evolution and measurement back-action. As an illustration, we
discuss minimal models of atomic dynamics and continuous current measurement on
rings with synthetic gauge fields, involving both real space and synthetic
dimension lattices (represented by internal atomic states). Finally, by `not
reading' the current measurements the time evolution of the atomic system is
governed by a master equation, where - depending on the microscopic details of
our CQED setups - we effectively engineer a current coupling of our system to a
quantum reservoir. This provides novel scenarios of dissipative dynamics
generating `dark' pure quantum many-body states.
