  We simulate ultracold Sagnac atom interferometers using quantum-mechanical
matter wavepackets, e.g. Bose-Einstein condensates, that counter-propagate
within a rotating ring-trap. We find that the accumulation of the relative
phase difference between wavepackets, i.e. the matter wave Sagnac effect, is
manifested as discrete phase jumps. These plateaus result from three effects;
that the atoms should be initially trapped at rest with respect to the rotating
frame, that they counter-propagate with the same group velocities in the
rotating frame, and that the imaging is performed in the rotating frame. We
show that the plateaus persist in the presence of nonlinear atom-atom
interactions, and in atoms undergoing various rotations, and thus will occur
during matter wavepacket experiments. We also introduce the simplest possible
Sagnac atom interferometry scheme which relies on wavepacket dispersion around
a ring-trap.
