Phase estimation from atom position measurements

We study the measurement of the positions of atoms as a means of estimating the relative phase between two Bose-Einstein condensates. We consider N bosonic atoms released from a double-well trap, which form an interference pattern; we show that the measurement of the position of N atoms has a sensitivity that saturates the bound set by the quantum Fisher information, and allows for estimation at the Heisenberg limit of precision. Phase estimation through the measurement of the center of mass of the interference pattern can also provide sub-shot-noise sensitivity. Finally, we study the effect of an overlap of the two clouds on the estimation precision when Mach-Zehnder interferometry is performed in a double well. We find that a nonzero overlap of the clouds strongly reduces the phase sensitivity.