  We study the current dynamics of coupled atomic condensates flowing in two
ring-shaped optical potentials. We provide a specific setup where the ring-ring
coupling can be tuned in experimentally feasible way. It is demonstrated that
the imaginary time effective action of the system in a weak coupling regime
provides a two-level-system-dynamics for the phase slip across the two rings.
Through two-mode Gross- Pitaevskii mean field equations, the real time dynamics
of the population imbalance and the phase difference between of the two
condensates is thoroughly analyzed analytically, as function of the relevant
physical parameters of the system. In particular, we find that the macroscopic
quantum self trapping phenomenon is induced in the system if the flowing
currents assume a non vanishing difference.
