Atom Interferometer Phase Shear and Spacetime Sectional Curvature

Atom interferometry is a natural laboratory for precision tests of general relativity, but there is no simple relationship between atom interferometer phase and geometric properties of spacetime. Here we show that a different atom interferometer observable, the phase shear, can be expressed directly as the integrated sectional curvature over a spacetime surface enclosed by the interferometer arms and final beamsplitter. This is a consequence of a generalized Gauss-Bonnet theorem, which also explicitly computes small correction terms arising from gravitational redshift of atom optics pulses. This synthesis of quantum mechanics, relativity, and differential geometry affords a manifestly coordinate-free and representation-free means of measuring spacetime properties. Additionally, it provides a convenient computational tool for predicting atom interferometer properties in arbitrary background spacetimes.