Beyond the semiclassical approximation in atom interferometry

We describe a quantum perturbative approach to evaluating the phase shift of an atom interferometer in a weakly anharmonic trap. This provides a simple way to evaluate quantum corrections to the standard semi-classical approximation. The calculation benefits from the use of generalized coherent states for a basis. We find that the form of the semi-classical approximation remains valid to first order in the anharmonic perturbation, but that phase differences arise because the trajectory of a quantum wave packet will generally deviate from that of a classical particle. The quantum correction to the phase is a factor $\ell^2/A^2$ smaller than the semi-classical perturbation itself, where $\ell$ is the quantum harmonic oscillator length scale and $A$ is the classical amplitude of the motion. We provide analytical results for one-dimensional perturbations of power three through six in the position coordinate.