- We investigate how to estimate from atom-position measurements the relative phase of two Bose-Einstein condensates released from a double-well potential. We demonstrate that the phase-estimation sensitivity via the fit of the average density to the interference pattern is fundamentally bounded by shot noise. The shot noise can be overcome by estimating the phase from the measurement of order √N (or higher) correlation functions. The optimal estimation strategy, saturating the quantum Fisher information, requires the measurement of the Nth-order correlation function. We demonstrate that a different estimation method—based on the detection of the center of mass of the interference pattern—also provides sub-shot-noise sensitivity. The implementation of both protocols, however, might be experimentally challenging.
