Tripartite entanglement and threshold properties of coupled intracavity down-conversion and sum-frequency generation

The process of cascaded down-conversion and sum-frequency generation inside an optical cavity has been predicted to be a potential source of three-mode continuous-variable entanglement. When the cavity is pumped by two fields, the threshold properties have been analyzed, showing that these are more complicated than in well-known processes such as optical parametric oscillation. When there is only a single pumping field, the entanglement properties have been calculated using a linearized fluctuation analysis, but without any consideration of the threshold properties or critical operating points of the system. In this work we extend this analysis to demonstrate that the singly pumped system demonstrates a rich range of threshold behavior when quantization of the pump field is taken into account and that asymmetric polychromatic entanglement is available over a wide range of operational parameters.

Figure 1

(Color online) Stability of the steady state solutions with ${\chi }_1=0.01$, ${\gamma }_0={\gamma }_1={\gamma }_3=1$, and ${\gamma }_2=3$, as ${\chi }_2$ and the pump amplitude are varied. The dashed line shows the separation between the system with and without threshold.

Figure 2

(Color online) $S_{ijk}$ criteria below threshold for $ϵ=0.5{ϵ}_c$, ${\gamma }_0={\gamma }_1={\gamma }_3=1$, and ${\gamma }_2=3$. In this and all subsequent graphs, the results are dimensionless.

Figure 3

(Color online) $S_{ij}$ criteria below threshold for $ϵ=0.5{ϵ}_c$, ${\gamma }_0={\gamma }_1={\gamma }_3=1$, and ${\gamma }_2=3$.

Figure 4

(Color online) $S_{ijk}$ criteria below threshold with $ϵ=0.9{ϵ}_c$.

Figure 5

(Color online) $S_{ij}$ criteria below threshold with $ϵ=0.9{ϵ}_c$.

Figure 6

(Color online) $S_{ijk}$ criteria for the system without threshold, with $ϵ=1.5{ϵ}_c^o$.

Figure 7

(Color online) $S_{ij}$ criteria for the system without threshold, with $ϵ=1.5{ϵ}_c^o$.

Figure 8

(Color online) Minimum of the $S_{ijk}$ at any frequency between 0 and $10{\gamma }_0$ as the pump varies up to ${ϵ}_c$ in the region with threshold.

Figure 9

(Color online) Tripartite entanglement criteria for the system without threshold, with ${\gamma }_0={\gamma }_1={\gamma }_3=1$, ${\gamma }_2=3$, ${\chi }_1=0.01$, and ${\chi }_2=2{\chi }_1$.

Figure 10

(Color online) Tripartite entanglement criteria for the system without threshold, with ${\gamma }_0={\gamma }_1={\gamma }_3=1$, ${\gamma }_2=3$, ${\chi }_1=0.01$, and ${\chi }_2=3{\chi }_1$.

Figure 11

(Color online) Tripartite entanglement criteria for the system without threshold, with ${\gamma }_0={\gamma }_1={\gamma }_3=1$, ${\gamma }_2=3$, ${\chi }_1=0.01$, and $ϵ=0.5{ϵ}_c^o$. The results are plotted as a function of ${\chi }_2^{\mathrm{crit}}={\chi }_1\sqrt{{\gamma }_2∕{\gamma }_1}$.

Figure 12

(Color online) Tripartite entanglement criteria for the system without threshold, with ${\gamma }_0={\gamma }_1={\gamma }_3=1$, ${\gamma }_2=3$, ${\chi }_1=0.01$, and $ϵ=0.9{ϵ}_c^o$. The results are plotted as a function of ${\chi }_2^{\mathrm{crit}}={\chi }_1\sqrt{{\gamma }_2∕{\gamma }_1}$.

Figure 13

(Color online) Intensity of the output modes $\overline{{\mid {\alpha }_i\mid }^2}$, obtained by stochastic integration of the truncated Wigner equations averaged over $2.65\times {10}^5$ trajectories. Note that the horizontal axis is now time scaled in units of ${\gamma }_1^{-1}$.

Figure 14

(Color online) The $V_{ijk}$ for $ϵ=1.5{ϵ}_c$, obtained by stochastic integration of the truncated Wigner equations averaged over $2.65\times {10}^5$ trajectories. Note that the horizontal axis is now time scaled in units of ${\gamma }_1^{-1}$.

Figure 15

(Color online) Transient behavior of the $V_{ijk}$ with $ϵ=1.5{ϵ}_c$. As seen in Fig. 13, the violations of the inequalities persist for only a short time.