Atom interferometry is a sensitive tool for measuring relativistic effects, but there are no known non-trivial exact solutions for relativistic atom interferometer phase shifts. Here we derive relativistically exact expressions within the usual semiclassical approximation for a wide range of experimentally interesting atom interferometer pulse sequences in flat spacetime, including Mach-Zehnder, resonant, and large momentum transfer interferometer geometries. As an example, the leading order phase shift
ωagT2/c for a Mach-Zehnder clock atom interferometer is found to become
ωa(1+2mωa)(e−gT/c−1)2c/g when all relativistic kinematics are included. We calculate exact phase shifts for both clock (single-photon) interferometers and Raman or Bragg (two-photon) interferometers.
We reanalyzed our atom interferometer measurement of the electric polarizability of lithium, now accounting for the Sagnac effect due to the Earth's rotation. The resulting correction to the polarizability is very small, but the visibility as a function of the applied phase shift is now better explained. The fact that the Sagnac and polarizability phase shifts are both proportional to ${v}^{\ensuremath{-}1}$, where
v is the atom velocity, suggests that a phase shift of the Sagnac type could be used as a counterphase to compensate for the electric polarizability phase shift. This exact compensation opens the way to higher-accuracy measurements of atomic polarizabilities, and we discuss how this can be done in practice and the final limitations of the proposed technique.