  Quenching an ultracold bosonic gas in a ring across the Bose-Einstein
condensation phase transition is known, and has been experimentally observed,
to lead to the spontaneous emergence of persistent currents. The present work
examines how these phenomena generalize to a system of two experimentally
accessible explicitly two-dimensional co-planar rings with a common interface,
or to the related lemniscate geometry, and demonstrates an emerging
independence of winding numbers across the rings, which can exhibit flow both
in the same and in opposite directions. The observed persistence of such
findings in the presence of dissipative coupled evolution due to the local
character of the domain formation across the phase transition and topological
protection of the randomly emerging winding numbers should be within current
experimental reach.
