  We show that the general quantum state of synthetically spin-orbit coupled
ultra cold bosonic atom whose condensate was experimentally created recently (
Y. J. Lin {\it et al.}, Nature, {\bf 471}, 83, (2011)), shows entanglement
between motional degrees of freedom ( momentum) and internal degrees of freedom
(hyperfine spin). We demonstrate the violation of Bell-like inequality (CHSH)
for such states that provides a unique opportunity to verify fundamental
principle like quantum non-contextuality for commutating observables which are
not spatially separated. We analyze in detail the Rabi oscillation executed by
such atom-laser system and how that influneces quantities like entanglement
entropy, violation of Bell like Inequality etc. We also discuss the implication
of our result in testing the quantum non-contextuality and Bell's Inequality
vioaltion by macroscopic quantum object like Bose-Einstein Condensate of ultra
cold atoms.
