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arXiv:quant-ph/0607103 · Physical Review A

Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions

M. K. Olsen, Ashton S. Bradley

Published Dec 11, 2006• 61 citations

Abstract

We examine the tripartite entanglement properties of an optical system using interlinked ${\ensuremath{\chi}}^{(2)}$ interactions, recently studied experimentally in terms of its phase-matching properties by Bondani et al. [Opt. Express 14, 21, 9838 (2006)]. We show that the system produces output modes at three distinct frequencies which are genuinely tripartite entangled, and analyze this entanglement in terms of different measurable correlations. We show that, due to the asymmetry of the process, the detection of this entanglement depends crucially on the correlation functions that are measured. We find that some of the correlations found in the literature fail to register the entanglement, in contrast to symmetric systems, for which the actual choice of correlation to be measured makes little difference.

DOI: 10.1103/physreva.74.063809

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We examine the tripartite entanglement properties of an optical system using interlinked

{\ensuremath{\chi}}^{(2)}

interactions, recently studied experimentally in terms of its phase-matching properties by Bondani et al. [Opt. Express 14, 21, 9838 (2006)]. We show that the system produces output modes at three distinct frequencies which are genuinely tripartite entangled, and analyze this entanglement in terms of different measurable correlations. We show that, due to the asymmetry of the process, the detection of this entanglement depends crucially on the correlation functions that are measured. We find that some of the correlations found in the literature fail to register the entanglement, in contrast to symmetric systems, for which the actual choice of correlation to be measured makes little difference.

We show that the states generated by a three-mode spontaneous parametric down-conversion (SPDC) interaction Hamiltonian possess tripartite entanglement of a different nature to other paradigmatic three-mode entangled states generated by the combination of two-mode SPDC interactions. While two-mode SPDC generates Gaussian states whose entanglement can be characterized by standard criteria based on two-mode quantum correlations, these criteria fail to capture the entanglement generated by three-mode SPDC. We use criteria built from three-mode correlation functions to show that the class of states recently generated in a superconducting-circuit implementation of three-mode SPDC ideally have tripartite entanglement, contrary to recent claims in the literature. These criteria are suitable for triple SPDC but we show that they fail to detect tripartite entanglement in other states which are known to possess it, which illustrates the existence of two fundamentally different notions of tripartite entanglement in three-mode continuous-variable systems.

We address the generation of fully inseparable three-mode entangled states of radiation by interlinked nonlinear interactions in χ(2) media. We show how three-mode entanglement can be used to realize symmetric and asymmetric telecloning machines, which achieve optimal fidelity for coherent states. An experimental implementation involving a single nonlinear crystal in which the two interactions take place simultaneously is suggested. Preliminary experimental results showing the feasibility and the effectiveness of the interaction scheme with a seeded crystal are also presented.

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.