  Using criteria based on superselection rules, we analyze the quantum
correlations between the two condensate modes of the Bose-Einstein condensate
interferometer of Egorov et al. [Phys. Rev. A 84, 021605 (2011)]. In order to
determine the two-mode correlations, we develop a multi-mode theory that
describes the dynamics of the condensate atoms and the thermal fraction through
the interferometer sequence, in agreement with the experimentally measured
fringe visibility. We thus present experimental evidence for two-mode entangled
states genuinely involving 40,000 ^{87}Rb atoms, and for two-way steerability
between two groups of 20,000 indistinguishable atoms.
