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arXiv:1107.2671 · Physical Review A

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

K. Dechoum, W. S. Marques, Antonio Zelaquett Khoury

Published Jun 28, 2011• 2 citations
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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 67, 052315 (2003)]. Although not being a necessary condition, these triple correlations are 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.

DOI
10.1103/physreva.83.063843

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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 67, 052315 (2003)]. Although not being a necessary condition, these triple correlations are 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.
We calculate the quantum correlations existing among the three output fields (pump, signal, and idler) of a triply resonant nondegenerate optical parametric oscillator operating above threshold. By applying the standard criteria [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)], we show that strong tripartite continuous-variable entanglement is present in this well-known and simple system. Furthermore, since the entanglement is generated directly from a nonlinear process, the three entangled fields can have very different frequencies, opening the way for multicolored quantum information networks.
The quantum correlations between the output signal, the output idler and the reflected pump fields generated by non-degenerate optical parametric oscillator operating above the oscillation threshold are theoretically calculated with the semi-classical formulae. According to the multipartite entanglement criteria for the continuous variables, proposed by P. van Loock and A. Furusawa, the calculated results prove the existence of the quantum correlations between the amplitude and the phase quadrature for the three optical fields, i. e. they form a tripartite entangled state. We numerically calculate the dependence of the entanglement on the physical parameters of the optical oscillator and find the optimum operating conditions of the oscillator to produce the three-color tripartite entangled state, which provide the direct references for the design of the continuous variable multipartite entanglement generation systems.
It is theoretically shown that tripartite entanglement with different wavelengths can be generated by cascaded nonlinear interaction in an optical parametric oscillator cavity with parametric down conversion and sum-frequency generation. A sufficient inseparability criterion for continuous-variable tripartite entanglement proposed by van Loock and Furusawa was used to evaluate the degree of the quadrature-phase amplitude correlations between the three modes. The dependences of correlation on the cavity parameters and pump intensity are discussed.
Triple-photon states generated by three-mode spontaneous parametric down-conversion are the paradigm of unconditional non-Gaussian states, essential assets for quantum advantage. How to fully characterize their non-Gaussian entanglement remains however elusive. We propose here a set of sufficient and necessary conditions for separability of the broad family of spontaneously generated three-mode non-Gaussian states. We further derive state-of-the-art conditions for genuine tripartite non-Gaussian entanglement, the strongest class of entanglement. We apply our criteria to triple-photon states revealing that they are fully inseparable and genuinely entangled in moments of order 3n. Our results establish a systematic framework for characterizing the entanglement of triple-photon states and thus fostering their application in quantum information protocols.
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