  The quantum dynamics of population-balanced fractional vortices and
population-imbalanced vortices in an effective two-state bosonic system, made
of two coupled discrete circuits with few sites, is addressed within the
Bose-Hubbard model. % We show that for low on-site interaction, the tunneling
of quantized vortices between the rings performs a coherent, oscillating
dynamics connecting current states with chiral symmetry. The vortex-flux
transfer dually follows the usual sinusoidal particle current of the Josephson
effect, in good agreement with a mean-field approximation. Within such regime,
the switch of persistent currents in the rings resembles flux-qubit features,
and is feasible to experimental realization. On the contrary, strong
interatomic interactions suppress the chiral current and lead the system into
fragmented condensation.
