Quantum-limited metrology in the presence of collisional dephasing

Including collisional decoherence explicitly, phase sensitivity for estimating effective scattering strength $chi$ of a two-component Bose-Einstein condensate is derived analytically. With a measurement of spin operator $hat{J}_{x}$, we find that the optimal sensitivity depends on initial coherent spin state. It degrades by a factor of $(2gamma)^{1/3}$ below super-Heisenberg limit $propto 1/N^{3/2}$ for particle number $N$ and the dephasing rate $1<!<gamma<N^{3/4}$. With a $hat{J}_y$ measurement, our analytical results confirm that the phase $phi=chi tsim 0$ can be detected at the limit even in the presence of the dephasing.