Asymmetric polychromatic tripartite entanglement from interlinked ${\chi }^{\left(2\right)}$ parametric interactions

We examine the tripartite entanglement properties of an optical system using interlinked ${\chi }^{\left(2\right)}$ 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.

Figure 1

(Color online) The analytical solutions of the VLF correlations, with ${\kappa }_1=1.2{\kappa }_2$. Any two of the correlations falling below 4 is sufficient to demonstrate that genuine tripartite entanglement is present. The solid lines use the optimized expressions. All quantities shown in these and subsequent graphs are dimensionless.

Figure 2

(Color online) The analytical solutions of the VLF correlations, with ${\kappa }_2=1.8{\kappa }_1$. Any two of the correlations falling below 4 is sufficient to demonstrate that genuine tripartite entanglement is present. The solid lines use the optimized expressions.

Figure 3

(Color online) The analytical solutions for the $V_{ijk}$ correlations of Eq. (12). On the left-hand side, ${\kappa }_1=1.2{\kappa }_2$, while on the right-hand side ${\kappa }_2=1.8{\kappa }_1$. Although any one of these being below 4 is sufficient to demonstrate genuine tripartite entanglement, we see that only a measurement of $V_{123}$ detects the entanglement, with the other two ($V_{312}$ is the upper line) always remaining above 4.

Figure 4

(Color online) The analytical solutions of the OBR two-mode inference correlations. On the left-hand side, ${\kappa }_1=1.2{\kappa }_2$, while on the right-hand side ${\kappa }_2=1.8{\kappa }_1$. Although all three correlations should be below 1 to demonstrate genuine tripartite entanglement, in each case only one of them $\left(C_1^{\mathit{OBR}}\right)$ goes below this level, while the other two are both exactly equal to 1.

Figure 5

(Color online) The analytical solutions of the OBR correlations which infer combined mode properties from those of a single mode, with ${\kappa }_1=1.2{\kappa }_2$. All three correlations should be below 4 to demonstrate genuine tripartite entanglement.

Figure 6

(Color online) The analytical solutions of the OBR correlations which infer combined mode properties from those of a single mode, with ${\kappa }_2=1.8{\kappa }_1$. All three correlations should be below 4 to demonstrate genuine tripartite entanglement.

Figure 7

(Color online) The best performance of the different criteria as functions of the interaction strength, for ${\kappa }_1=1.2{\kappa }_2$ (left) and ${\kappa }_2=1.8{\kappa }_1$ (right). For any of the correlations shown, a value of less than 4 signifies genuine tripartite entanglement. For $V_{ij}^{\mathit{opt}}$ of Eq. (11), we take the second lowest value as two of the inequalities must be violated, while for $C_{ij}^{\mathit{OBR}}$ we take the highest value as three inequalities must be violated.