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 an 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 the two condensates is thoroughly analyzed analytically, as a 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 nonvanishing difference.
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