  Using matter-wave interference, we have investigated thermal phase
fluctuations in narrow coplanar, concentric rings of ultracold fermionic
superfluids. We found that the correlation length decreases with number
density, consistent with theoretical expectations. We also observed that
increasing the coupling between the rings leads to greater overall coherence in
the system. The phase fluctuations increased with a change from periodic to
closed boundary conditions as we applied a potential barrier at one point in a
ring. These results are relevant for the implementation of proposals to utilize
ultracold quantum gases in large and elongated circuit-like geometries,
especially those that require deterministic preparation and control of
quantized circulation states.
