  Atom interferometers provide exquisite measurements of the properties of
non-inertial frames. While atomic interactions are typically detrimental to
good sensing, efforts to harness entanglement to improve sensitivity remain
tantalizing. Here we explore the role of interactions in an analogy between
atomic gyroscopes and SQUIDs, motivated by recent experiments realizing ring
shaped traps for ultracold atoms. We explore the one-dimensional limit of these
ring systems with a moving weak barrier, such as that provided by a
blue-detuned laser beam. In this limit, we employ Luttinger liquid theory and
find an analogy with the superconducting phase-slip qubit, in which the
topological charge associated with persistent currents can be put into
superposition. In particular, we find that strongly-interacting atoms in such a
system could be used for precision rotation sensing. We compare the performance
of this new sensor to an equivalent non-interacting atom interferometer, and
find improvements in sensitivity and bandwidth beyond the atomic shot-noise
limit.
