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Using criteria based on superselection rules, we analyze the quantumcorrelations between the two condensate modes of the Bose-Einstein condensateinterferometer of Egorov et al. [Phys. Rev. A 84, 021605 (2011)]. In order todetermine the two-mode correlations, we develop a multi-mode theory thatdescribes the dynamics of the condensate atoms and the thermal fraction throughthe interferometer sequence, in agreement with the experimentally measuredfringe visibility. We thus present experimental evidence for two-mode entangledstates genuinely involving 40,000 ^{87}Rb atoms, and for two-way steerabilitybetween two groups of 20,000 indistinguishable atoms.

Mesoscopic two-mode entangled and steerable states of 40,000 atoms in a Bose-Einstein condensate interferometer