  There now exists in the literature two different expressions for the phase
shift of a matter-wave interferometer caused by the passage of a gravitation
wave. The first, a commonly accepted expression that was first derived in the
1970s, is based on the traditional geodesic equation of motion (EOM) for a test
particle. The second, a more recently derived expression, is based on the
geodesic deviation EOM. The power-law dependence on the frequency of the
gravitational wave for both expressions for the phase shift is different, which
indicates fundamental differences in the physics on which these calculations
are based. Here we compare the two approaches by presenting a series of
side-by-side calculations of the phase shift for one specific
matter-wave-interferometer configuration that uses atoms as the interfering
particle. By looking at the low-frequency limit of the different expressions
for the phase shift obtained, we find that the phase shift calculated via the
geodesic deviation EOM is correct, and the ones calculated via the geodesic EOM
are not.
