Exact Semiclassical Phase Shifts for Relativistic Atom Interferometers in Flat Spacetime

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 $\omega_a g T^2/c$ for a Mach-Zehnder clock atom interferometer is found to become $\omega_a \left(1 + \frac{\omega_a}{2m}\right)(e^{-gT/c}-1)^2 c/g$ when all relativistic kinematics are included. We calculate exact phase shifts for both clock (single-photon) interferometers and Raman or Bragg (two-photon) interferometers.