  We show that matter-wave interference fringes formed by two overlapping
atomic clouds can yield information about the non-local Bell correlations. To
this end, we consider a simple atomic interferometer, where the clouds are
released from the double-well potential and the relative phase is estimated
from the density fit to this interference pattern. The Bell correlations can be
deduced from the sensitivity of the phase obtained in this way. We examine the
relation between these two quantities for a wide range of ground states of the
double-well, scanning through the attractive and the repulsive interactions.
The presented analysis includes the effects of finite temperature, when excited
states are thermally occupied. We also consider the impact of the spatial
resolution of the single-atom detectors and the fluctuations of the energy
mismatch between the wells. These results establish a link between the
fundamental (non-locality) and application-oriented (quantum metrology) aspects
of entanglement.
