Effective dynamics of cold atoms flowing in two ring shaped optical potentials with tunable tunneling

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.