  The Cauchy-Schwarz (CS) inequality -- one of the most widely used and
important inequalities in mathematics -- can be formulated as an upper bound to
the strength of correlations between classically fluctuating quantities.
Quantum mechanical correlations can, however, exceed classical bounds.Here we
realize four-wave mixing of atomic matter waves using colliding Bose-Einstein
condensates, and demonstrate the violation of a multimode CS inequality for
atom number correlations in opposite zones of the collision halo. The
correlated atoms have large spatial separations and therefore open new
opportunities for extending fundamental quantum-nonlocality tests to ensembles
of massive particles.
