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