  We propose a novel type of a Bose-Hubbard ladder model based on an open
quantum-gas--cavity-QED setup to study the physics of dynamical gauge
potentials. Atomic tunneling along opposite directions in the two legs of the
ladder is mediated by photon scattering from transverse pump lasers to two
distinct cavity modes. The resulting interplay between cavity photon
dissipation and the optomechanical atomic back-action then induces an
average-density-dependent dynamical gauge field. The dissipation-stabilized
steady-state atomic motion along the legs of the ladder leads either to a pure
chiral current, screening the induced dynamical magnetic field as in the
Meissner effect, or generates simultaneously chiral and particle currents. For
sufficiently strong pump the system enters into a dynamically unstable regime
exhibiting limit-cycle and period-doubled oscillations. Intriguingly, an
electromotive force is induced in this dynamical regime as expected from an
interpretation based on Faraday's law of induction for the time-dependent
synthetic magnetic flux.
