Optimal phase sensitivity of atomic Ramsey interferometers with coherent spin states

We present a detailed analysis of phase sensitivity for a nonlinear Ramsey interferometer, which utilize effective mean-field interaction of a two-component Bose–Einstein condensate in phase accumulation. For large enough particle number N and small phase shift ?, analytical results of the Ramsey signal and the phase sensitivity are derived for a product coherent state |θ,0?. When collisional dephasing is absent, we confirm that the optimal sensitivity scales as 2/N^3/2 for polar angle of the initial state θ = π/4 or 3π/4. The best-sensitivity phase satisfies different transcendental equations, depending upon the initial state and the observable being measured after the phase accumulation. In the presence of the collisional dephasing, we show that the N^-3/2-scaling rule of the sensitivity maintains with spin operators J^x and J^y measurements. A slightly better sensitivity is attainable for optimal coherent state with θ = π/6 or 5π/6.