  We develop an open-system dynamical theory of the Casimir interaction between
coherent atomic waves and a material surface. The system --- the external
atomic waves --- disturbs the environment --- the electromagnetic field and the
atomic dipole degrees of freedom --- in a non- local manner by leaving
footprints on distinct paths of the atom interferometer. This induces a
non-local dynamical phase depending simultaneously on two distinct paths,
beyond usual atom-optics methods, and comparable to the local dynamical phase
corrections. Non-local and local atomic phase coherences are thus equally
important to capture the interplay between the external atomic motion and the
Casimir interaction. Such dynamical phases are obtained for finite-width
wavepackets by developing a diagrammatic expansion of the disturbed environment
quantum state.
