Quantum-limited metrology in the presence of collisional dephasing

Including collisional decoherence explicitly, phase sensitivity for estimating effective scattering strength χ of a two-component Bose-Einstein condensate is derived analytically. With a measurement of spin operator Ĵ_x, we find that the optimal sensitivity depends on initial coherent spin state. It degrades by a factor of (2γ)^1/3 below super-Heisenberg limit ∝1/N^3/2 for particle number N and the dephasing rate 1≪γ<N^3/4. With a Ĵ_y measurement, our analytical results confirm that the phase ϕ=χt~0 can be detected at the limit even in the presence of the dephasing.