We show that the well known geometric phase, the Gouy phase in optics can be defined for matter waves in vacuum as well. In particular we show that the underlying physics for the "matter waves" Gouy phase is the generalized Schrödinger-Robertson uncertainty principle, more specifically, the off diagonal elements of the covariance matrix. Recent experiments involving the diffraction of fullerene molecules and the uncertainty principle are shown to be quantitatively consistent with the existence of a Gouy phase for matter waves.

The Schrödinger equation for an atomic beam predicts that it must have a phase anomaly near the beam waist analogous to the Gouy phase of an electromagnetic beam. We propose here a feasible experiment which allows for the direct determination of this anomalous phase using Ramsey interferometry with Rydberg atoms. Possible experimental limitations are discussed and shown to be completely under control within the present day technology. We also discuss how this finding can open the possibility to use the spatial mode wavefunctions of atoms as q-dits, since the Gouy phase is an essential ingredient for making rotations in the quantum states.

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