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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.

A neutron interferometer coherently splits and recombines a neutron matter wave, converting a difference of quantum phase between paths into an output intensity change. A sample in one path can therefore be studied through its phase shift, even when its effect on transmitted intensity is small.

A perfect-crystal implementation uses several diffracting silicon blades machined from one crystal with a common base. Bragg diffraction splits the incident wave into paths, redirects them and recombines them in a geometry analogous to a Mach-Zehnder interferometer. The common base helps preserve relative alignment; thermal and vibration isolation protect phase stability. The institutional apparatus description includes the blade geometry and explains the importance of environmental isolation.^[ref]

Recombination measures a relative phase rather than an individual particle trajectory. Electric, magnetic, gravitational or material interactions can all contribute, so a quantitative result requires controls separating the desired phase from drifts and other path differences. The Colella–Overhauser–Werner experiment is a gravitational example.

For a neutron in the Earth's Newtonian gravitational potential ${\Phi }_E$, the Schrödinger equation isNear the surface, ${\Phi }_E(z)=gz$ up to an additive constant.

The Colella–Overhauser–Werner experiment uses crystal slabs as coherent beam splitters and mirrors in a neutron interferometer. The two paths contain horizontal segments of length $s$ separated vertically by $r$, then recombine. The gravitational potential energy difference is $m_{n}gr$. A neutron of speed $v$ spends time $s/v$ on a horizontal segment, so the gravitationally induced relative quantum phase isUsing the de Broglie wavelength $\lambda =h/(m_{n}v)$ and $\hslash =h/(2\pi )$ givesRotating the apparatus changes the vertical projection of its enclosed area from zero to $rs$. The number of full interference oscillations is thereforeHere $rs=(\sqrt{10}\mathrm{cm}{)}^2=10^{-3}{\mathrm{m}}^2$, soThe expected change is therefore $\boxed{9}$ oscillations to the nearest integer.

Yes in both directions. The Page–Geilker experiment tests whether a classical gravitational field is sourced by the expectation value of macroscopically superposed matter configurations. The Colella–Overhauser–Werner experiment uses the Earth's effectively classical field as an external potential and therefore does not distinguish an expectation-value semiclassical source law from a quantized gravitational field.

Conversely, the Colella–Overhauser–Werner experiment directly observes coherent matter-wave interference and a gravitationally generated relative phase in a neutron wavefunction. Page and Geilker did not maintain and recombine coherent branches of the macroscopic source, and their torsion-balance measurement was not an interference experiment. Thus their experiment did not test the neutron gravitational phase effect. The experiments probe opposite sides of the coupling: quantum matter responding coherently to gravity, and gravity responding to branch-dependent quantum matter.