Rotation Sensing via Josephson-frequency Splitting in a Toroidal Superfluid

We show that a toroidal superfluid interrupted by $n$ tunneling barriers realizes a compact Josephson gyroscope with an $n$-enhanced response to rotation. In the small-amplitude regime, we derive analytically the normal mode spectrum of the coupled population-phase oscillations. In the absence of rotation, pairs of modes are degenerate: a finite angular velocity $\mathrm{\Omega }$ lifts this degeneracy through a Doppler shift, producing a frequency splitting that grows linearly with both $\mathrm{\Omega }$ and $n$. Full numerical simulations confirm this prediction and reveal long-lived two-frequency beatings, in sharp contrast with the monochromatic Josephson oscillations of the nonrotating system. These beatings provide a direct rotation signal with estimation uncertainty scaling as $\mathrm{\Delta }\mathrm{\Omega }\sim n^{-3\mathrm{/}2}$, while remaining robust against imperfections and dynamical excitations. These results identify multi-junction toroidal superfluids as scalable, micrometer-size rotation sensors compatible with current experimental platforms.