Violation of the Cauchy-Schwarz inequality with matter waves. K. V. Kheruntsyan, J.-C. Jaskula, P. Deuar, M. Bonneau, G. B. Partridge, J. Ruaudel, R. Lopes, D. Boiron and C. I Westbrook in Phys. Rev. Lett. 108:260401 (2012). What the paper says!?
The authors «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»
Basically, they split a (cigar-shaped) atomic condensate in two with a Bragg pulse, pass the halves one through the other and measure correlations of the atoms scattered in this encounter, either back-to-back (BB) or together (CL, for collinear). While they see clear correlations:
they actually fail to violate CSI as such (even finding this "counterintuitive"):
The situation we analyze here is counterintuitive in that we observe a peak cross correlation (for pairs of atoms scattered in opposite directions) that is smaller than the peak autocorrelation (for pairs of atoms propagating in the same direction).
This is because the back-to-back correlation covers about 300 times the volume of the collinear one. Integrated over a zone, the lower but much wider BB peak gives more counts than the CL peak. So they introduce a new quantity to average and weight over such effects:
Most importantly, this is multimode physics, not two-modes, i.e., they
generalize the CS inequality to a multimode situation, which takes into account the fact that the cross correlations and autocorrelations in matter-wave optics are usually functions (in our case of momentum).
Without this, they don't violate CSI:
The various correlation functions can have different widths and peak heights, and one must define an appropriate integration domain over multiple momentum modes to recover an inequality that plays the same role as that in the two-mode case and is actually violated
In their case, there is a «6 cm spatial separation between the scattered, correlated atoms.» This is a macroscopic separation. They integrate over the sphere and in momentum space (so not spatial per se):
Their violation however remains small:
Nice comment on how CSI stand as compared to other quantities:
The EPR and Bell-state correlations are of course of wider significance to foundational principles of quantum mechanics than those that violate a CS inequality. Nevertheless, the importance of understanding the CS inequality in new physical regimes lies in the fact that: (i) they are the simplest possible tests of stronger-than-classical correlations, and (ii) they can be viewed as precursors, or necessary conditions, for the stricter tests of quantum mechanics.
And indeed the Bell inequalities are seen as the culmination of this:
Our experiment is an important step toward the demonstrations of increasingly restrictive types of nonlocal quantum correlations with matter waves, which we hope will one day culminate in the violation of a Bell inequality as well.
They describe Marino et al.[1] as «The violation has even been demonstrated for two light beams detected as continuous variables».