High sensitivity quantum interferometry requires more than just access to entangled states. It is achieved through deep understanding of quantum correlations in a system. Integrable models offer the framework to develop this understanding. We communicate the design of interferometric protocols for an integrable model that describes the interaction of bosons in a four-site configuration. Analytic formulae for the quantum dynamics of certain observables are computed. These expose the system's functionality as both an interferometric identifier, and producer, of NOON states. Being equivalent to a controlled-phase gate acting on two hybrid qudits, this system also highlights an equivalence between Heisenberg-limited interferometry and quantum information. These results are expected to open new avenues for integrability-enhanced atomtronic technologies.

6 pages, 4 figures, 1 table