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 Phys. Rev. Lett. {\bf 100}, 250406 (2008), with optimal initial number-variance $\sigma_o\propto N^{1/3}$ and optimal signal-to-noise ratio $s_o\propto N^{2/3}$ for 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 {\it phase}-squeezing with $\sigma_o\propto\sqrt{uN}$ and $s_o\propto\sqrt{N/u}$ shot-noise scaling. The initial phase-squeezing translates into hold-time number-squeezing, which is less sensitive to interactions than coherent states and improves $s_o$ by a factor of $\sqrt{u}$.