  We examine the dynamics of circulating modes of a Bose-Einstein condensate
confined in toroidal lattice. Nonlinearity due to interactions leads to
criticality that separates oscillatory and self-trapped phases among
counter-propagating modes which however share the same physical space. In the
mean-field limit, the criticality is found to substantially enhance sensitivity
to rotation of the system. Analysis of the quantum dynamics reveals the
fluctuations near criticality are significant, that we explain using
spin-squeezing formalism visualized on a Bloch sphere. We utilize the squeezing
to propose a Ramsey interferometric scheme that suppresses fluctuation in the
relevant quadrature sensitive to rotation.
