Entanglement-enhanced Synchronous differential comparison

The quantum entanglement enables the precision measurement and frequency metrology beyond the standard quantum limit that is imposed by the quantum projection noise and photon shot noise. Here we propose employing the entangled atoms in the synchronous differential measurement to enhance the sensitivity of the spatial-shift detection. Two ways of engineering the entangled atoms are studied. The synchronous comparison between two pixels within an entangled atomic cloud leads to a sensitivity enhancement factor of 1.4 over the standard quantum limit. Increasing the atom number hardly further improves the sensitivity. In contrast, the synchronous comparison between two independent pixels that are individually composed of entangled atoms allows for a strong sensitivity enhancement by a factor of, for example, 9.7 with $10^{3}$ entangled atoms in each pixel, corresponding to a reduction of the averaging time by a factor of about $10^{2}$. A large atom number may further elevate the sensitivity. Our work paves the way towards the entanglement-enhanced detection of the gravitational redshift by means of the \emph{in situ} imaging spectroscopy.