  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.
