Optimal Gaussian squeezed states for atom interferometry in the presence of phase diffusion

We optimize the signal-to-noise ratio of a Mach-Zehnder atom interferometer with Gaussian squeezed input states in the presence interactions. For weak interactions, our results coincide with those of Huang and Moore [Y. P. Huang and M. G. Moore, Phys. Rev. Lett. 100, 250406 (2008)], with an optimal initial number variance sigma(o) alpha N-1/3 and an optimal signal-to-noise ratio s(o) alpha N-2/3 for the total atom number N. As the interaction strength u increases past unity, phase diffusion becomes dominant, leading to a transition in the optimal squeezing from initial number squeezing to initial phase squeezing with sigma(o) alpha root uN and s(o) alpha root N/u shot-noise scaling. The initial phase squeezing translates into hold-time number squeezing, which is less sensitive to interactions than coherent states and improves so by a factor of root u.