Gravitational phase shift in matter-wave interferometry ($\Delta \phi =mgA/(\hslash v)$)

For equal-speed nonrelativistic matter waves in an interferometer with signed enclosed vertical area $A$, a uniform gravitational acceleration gives the displayed leading-order phase difference. In a rectangular device, the horizontal paths differ in gravitational potential energy by $mgh$ for a time $\ell /v$, so their phase difference has magnitude $mgh\ell /(\hslash v)$. Assumptions include negligible fractional changes of speed, matched unperturbed optical paths and a consistent treatment of the beam splitters. The Colella–Overhauser–Werner experiment uses neutron interference to test this gravitational phase.