Cauchy-Schwarz characterization of tripartite quantum correlations in an optical parametric oscillator

TL;DR

This paper investigates the quantum correlations in a three-mode optical parametric oscillator below threshold using a Cauchy-Schwarz inequality approach, revealing tripartite entanglement beyond standard criteria.

Contribution

It introduces a novel Cauchy-Schwarz inequality-based method to characterize tripartite quantum correlations in non-Gaussian states of an optical parametric oscillator.

Findings

Triple correlations indicate tripartite entanglement.

The method applies to non-Gaussian states.

Quantum fluctuations are crucial for correlation analysis.

Abstract

We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below threshold. We employ a perturbative expansion of the It\^o equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A {\bf 67}, 052315 (2003)]. Although not being a necessary condition, these triple correlations are a sufficient evidence of tripartite entanglement. Of course, our characterization of the quantum correlations is applicable to non-Gaussian states, which we show to be the case of the optical parametric oscillator below threshold, provided nonlinear quantum fluctuations are properly taken into account.