  We investigate the current flow of microcavity polariton condensates loaded
into concentric ring-shaped potentials. The tunneling of the condensates
between different potential rings results in different phase-locked states,
depending on the separation of the potential rings. As a consequence, the
condensate currents in different rings can flow either in the same or opposite
direction depending on the specific configuration of the ring-shaped
potentials. In two concentric standard ring-shaped potentials, the condensates
always circulate in the same direction (co-flowing current) and the vortices
formed in the two rings share the same topological charge because of the
azimuthally uniform distribution of their phase difference. In this case,
increasing the number of the potential rings enables the excitation of
Bessel-like solutions. If the two ring-shaped potentials are engineered into an
eye shape with the inner ring being standard ring-shaped and the outer ring
being elliptically ring-shaped, the phase differences of the condensates in the
two rings along the major and minor axes of the ellipse can be opposite, which
gives rise to a counter-flowing condensate currents.
