Continuous Measurement of an Atomic Current

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.