  The ability to interferometrically detect inertial rotations via the Sagnac
effect has been a strong stimulus for the development of atom interferometry
because of the potential 10^{10} enhancement of the rotational phase shift in
comparison to optical Sagnac gyroscopes. Here we analyze ballistic transport of
matter waves in a one dimensional chain of N coherently coupled quantum rings
in the presence of a rotation of angular frequency, \Omega. We show that the
transmission probability, T, exhibits zero transmission stop gaps as a function
of the rotation rate interspersed with regions of rapidly oscillating finite
transmission. With increasing N, the transition from zero transmission to the
oscillatory regime becomes an increasingly sharp function of \Omega with a
slope \partialT/\partial \Omega N^2. The steepness of this slope dramatically
enhances the response to rotations in comparison to conventional single ring
interferometers such as the Mach-Zehnder and leads to a phase sensitivity well
below the standard quantum limit.
