Quadripartite continuous-variable entanglement via quadruply concurrent down-conversion

We investigate an experimentally feasible intracavity coupled down-conversion scheme to generate quadripartite entanglement using concurrent nonlinearities. We verify that quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of optimized inequalities of the van Loock–Furusawa type. The entanglement characteristics both above and below the oscillation threshold are considered. We also present analytic solutions for the quadrature operators and the van Loock–Furusawa correlations in the undepleted pump approximation.

Figure 1

Schematic of a ${\chi }^{(2)}$ crystal inside a pumped Fabry-Pérot cavity. Pump lasers drive four intracavity modes with frequencies ${\omega }_1$, ${\omega }_2$, ${\omega }_3$, and ${\omega }_4$ (represented by circles and squares), which are down-converted to four output modes with frequencies ${\omega }_5$, ${\omega }_6$, ${\omega }_7$, and ${\omega }_8$.

Figure 2

Analytic solutions for the optimized van Loock–Furusawa correlations, $V_3$, found by solving the Heisenberg equations of motion in the undepleted pump approximation. A value of less than 4 signifies quadripartite entanglement. All quantities depicted here, and in subsequent graphs, are dimensionless.

Figure 3

Analytic solutions for the optimized van Loock–Furusawa correlations, $V_{56}$, $V_{67}$, and $V_{78}$ with ${\xi }_3=0.5{\xi }_1$, found by solving the Heisenberg equations of motion in the undepleted pump approximation. Having all three of the correlations drop below 4 is sufficient to demonstrate quadripartite entanglement.

Figure 4

The output spectral correlations, $I_{\mathit{ij}}^{\mathrm{out}}(\omega )$, as a function of frequency $\omega$ (units of $\kappa$) corresponding to the quadripartite entanglement criteria given in Eqs. (21, 22, 23). The below-threshold (solid line) and above-threshold (dashed line) cases are shown for cavity pump amplitudes $\epsilon =0.8{\epsilon }_c$ and $\epsilon =1.2{\epsilon }_c$, respectively. The three correlations are equal for each case for the chosen parameters, which are symmetric with ${\chi }_i=\chi$, ${\epsilon }_i=\epsilon$, $\gamma =10$, $\kappa =1$, and $\chi ={10}^{-2}$. All quantities plotted here and in the subsequent graph are dimensionless.

Figure 5

Maximum quadripartite entanglement as a function of the ratio of the cavity pumping to the pumping threshold. The cavity parameters are the same as in Fig. 4, with $\gamma =10$, $\kappa =1$, and $\chi ={10}^{-2}$. Again, all three correlations are equal for these parameters. It should be stressed that at $\epsilon /{\epsilon }_c=1$ the validity of the results is limited as the linearized analysis is no longer valid.