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 $(2\gamma)^{1/3}$ 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 $\phi=\chi t\sim 0$ can be detected at the limit even in the presence of the dephasing.