A free-oscillation interferometer uses atoms confined in a harmonic trap. Bragg scattering from an off-resonant laser is used to split an atomic wave function into two separated packets. After one or more oscillations in the trap, the wave packets are recombined by a second application of the Bragg laser to close the interferometer. Anharmonicity in the trap potential can lead to a phase shift in the interferometer output. In this paper, analytical expressions for the anharmonic phase are derived at leading order for perturbations of arbitrary power in the position coordinate. The phase generally depends on the initial position and velocity of the atom, which are themselves typically uncertain. This leads to degradation in the interferometer performance, and can be expected to limit the use of a cm-scale device to interaction times of about 0.1 s. Methods to improve performance are discussed.
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