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We study a system represented by a Bose-Einstein condensate interacting witha cavity field in presence of a strong off-resonant pumping laser. This systemcan be described by a three-mode Gaussian state, where two are the atomic modescorresponding to atoms populating upper and lower momentum sidebands and thethird mode describes the scattered cavity field light. We show that, as aconsequence of the collective atomic recoil instability, these modes possess agenuine tripartite entanglement that increases unboundedly with the evolutiontime and is larger than the bipartite entanglement in any reduced two-modebipartition. We further show that the state of the system exhibits genuinetripartite nonlocality, which can be revealed by a robust violation of theSvetlichny inequality when performing displaced parity measurements. Our exactresults are obtained by exploiting the powerful machinery of phase-spaceinformational measures for Gaussian states, which we briefly review in theopening sections of the paper.

Genuine tripartite entanglement and nonlocality in Bose-Einstein condensates by collective atomic recoil