Bell correlations in a split two-mode-squeezed Bose-Einstein condensate

We propose and analyze a protocol for observing violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using two spatially separated Bose-Einstein condensates (BECs). To prepare correlated state, spin-changing collisions are used to first two-mode squeezed BEC. This is then split into BECs by controlling spatial wave function, e.g., modifying trapping potential. Finally, also exploited locally, compensate local squeezing terms. The correlators appearing in evaluated three different approaches. In approach, estimated normalized expectation values number operators, similar way evaluating continuous-variable inequalities. An improvement this approach developed sign binning total spin measurements, which allows construction two-outcome measurements violations CHSH without auxiliary assumptions. we show third where maximal CH can be obtained assigning zero vacua outcomes under no-enhancement assumption. effect loss imperfect detection efficiency investigated, observed found robust noise.