Matter Waves and Quantum Correlations

Abstract

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 Cauchy-Schwarz (CS) inequality is ubiquitous in mathematics and physics [1]. Its utility ranges from proofs of basic theorems in linear algebra to the derivation of the Heisenberg uncertainty principle. In its basic form, the CS inequality simply states that the absolute value of the inner product of two vectors cannot be larger than the product of their lengths. In probability theory and classical physics, the CS inequality can be applied to fluctuating quantities and states that the expectation value of the cross correlation hI 1 I 2 i between two quantities I 1 and I 2 is bounded from above by the autocorrelations in each quantity,This inequality is satisfied, for example, by two classical currents emanating from a common source. In quantum mechanics, correlations can, however, be stronger than those allowed by the CS inequality [2][3][4]. Such correlations have been demonstrated in quantum optics using, for example, antibunched photons produced via spontaneous emission [5], or twin photon beams generated in a radiative cascade [6], parametric down conversion [7], and optical four-wave mixing [8]. Here the discrete nature of the light and the strong correlation (or anticorrelation in antibunching) between photons is responsible for the violation of the CS inequality. The violation has even been demonstrated for two light beams detected as continuous variables [8].In this work we demonstrate a violation of the CS inequality in matter-wave optics using pair-correlated atoms formed in a collision of two Bose-Einstein condensates (BECs) of metastable helium [9-12] (see Fig. 1). The CS inequality which we study is a multimode inequality, involving integrated atomic densities, and therefore is different from the typical two-mode situation studied in quantum optics. Our results demonstrate the potential of atom optics experiments to extend the fundamental tests of quantum mechanics to ensembles of massive particles. Indeed, violation of the CS inequality implies the possibility of (but is not equivalent to) formation of quantum states that exhibit the Einstein-Podolsky-Rosen (EPR) correlations or violate a Bell's inequality [3]. The EPR and Bell-state correlations are of course of wider significance FIG. 1 (color online). Diagram of the collision geometry. (a) Two cigar-shaped condensates moving in opposite directions along the axial direction z shortly after their creation by a Bragg laser pulse (the anisotropy and spatial separation are not to scale). (b) Spherical halo of scattered atoms produced by fou...