  Interferometers, which are built using spatially propagating light or matter
waves, are commonly used to measure physical quantities. These measurements are
made possible by exploiting the interference between waves traveling along
different paths. This study introduces a novel approach to sensing of the
Aharonov--Bohm phase, an ion matter-wave interferometer operating within a
two-dimensional circular trajectory in a trap potential. The ion orbitals in
the potential form Lissajous curves, causing the direction of ion rotation to
reverse. This reversal results in a corresponding change in the interference
phase. Our study is groundbreaking as it is the first attempt to utilize
propagating matter waves of an ion in constructing an interferometer for the
measurement of physical quantities. Given that the scale factor of the
interferometer to the cyclotron motion and the rotation of the system is
common, the sensitivity to the Aharonov--Bohm phase in this study corresponds
to a rotation sensitivity of approximately 300~rad/s. Besides advancing
interferometry, our work also lays the foundation for future research into the
use of ion matter waves in gyroscopic applications.
