  We propose a mechanism to use nonlinearity arising from inter-particle
interactions to significantly enhance rotation sensitivity of matter-wave
interferometers. The method relies on modifying Sagnac interferometers by
introducing a weak circular lattice potential that couples modes with opposite
orbital angular momenta (OAM). The primary observable comprises of the modal
population distributions measured at particular times. This provides an
alternate mechanism for rotation sensing that requires substantially smaller
ring size, even in the linear non-interacting regime. Nonlinearity can improve
the sensitivity, as well as operation timescales, by several orders of
magnitude.
