Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator

We examine the feasibility of generating continuous-variable multipartite entanglement in an intracavity concurrent downconversion scheme that has been proposed for the generation of cluster states by Menicucci et al. [Phys. Rev. Lett. 101, 130501 (2008)]. By calculating optimized versions of the van Loock-Furusawa correlations we demonstrate genuine quadripartite entanglement and investigate the degree of entanglement present. Above the oscillation threshold the basic cluster state geometry under consideration suffers from phase diffusion. We alleviate this problem by incorporating a small injected signal into our analysis. Finally, we investigate squeezed joint operators. While the squeezed joint operators approach zero in the undepleted regime, we find that this is not the case when we consider the full interaction Hamiltonian and the presence of a cavity. In fact, we find that the decay of these operators is minimal in a cavity, and even depletion alone inhibits cluster state formation.

Figure 1

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

Figure 2

Optimized van Loock-Furusawa correlations, $I_{36}$ and $I_{45}$ (black solid line) and $I_{56}$ (red dashed-dot line), 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. All quantities depicted here and in subsequent graphs are dimensionless.

Figure 3

Optimized van Loock-Furusawa correlations, $I_{36}$ and $I_{45}$ (black solid line) and $I_{56}$ (red dashed-dot line) with ${\xi }_2=0.5{\xi }_1$.

Figure 4

Intensities of the high (red dashed-dot and blue solid lines) and low-frequency modes (dashed green and small dashed purple lines) calculated using the positive-$P$ equations without a cavity present. The number of trajectories is 300 000. The parameters used are ${\chi }_{1,2}=0.01$, ${\alpha }_{1,2}(0)=1\times {10}^3$ and ${\alpha }_{3,4,5,6}(0)=0$. The black solid lines show the low-frequency modes in the undepleted pump approximation.

Figure 5

The region of stability (the plateau) and the transition to instability for a range of pump amplitudes, ${\epsilon }_1$ and ${\epsilon }_2$, found by investigating the behavior of the negative real part of the eigenvalue ${\lambda }_3$, denoted Re[${\lambda }_3^{-}$].

Figure 6

The spectral correlations below threshold, $I_{\mathit{ij}}^{\mathrm{out}}(\omega )$, for the intracavity concurrent scheme. The parameter values are $\chi =0.01$, $\gamma =1$, and $\epsilon =0.987{\epsilon }_c$. This value of $\epsilon$ gives best violation of the inequalities. The correlation $I_{56}^{\mathrm{out}}$ is given by the red dashed-dot line, while $I_{36}^{\mathrm{out}}$ and $I_{45}^{\mathrm{out}}$ are equal and given by the black solid line.

Figure 7

The spectral correlations above threshold for the intracavity concurrent scheme. The parameter values are $\chi =0.01$, $\gamma =1$, and $\epsilon =1.49{\epsilon }_c$. The other parameters are the same as in the below threshold case. The correlation $I_{56}^{\mathrm{out}}$ is repesented by the red dashed-dot line. The correlations $I_{36}^{\mathrm{out}}$ and $I_{45}^{\mathrm{out}}$ are equal and represented by the black solid line.

Figure 8

Maximum quadripartite entanglement as a function of the ratio of the cavity pumping to the pumping threshold. All other cavity parameters are the same as in Figs. 6 and 7. Again, the correlations $I_{36}^{\mathrm{out}}$ and $I_{45}^{\mathrm{out}}$ are equal for these parameters and hence cannot be differentiated on the graph. Note that for $\epsilon /{\epsilon }_c=1$ the validity of the results is limited as the linearized analysis is no longer valid.

Figure 9

Plot of the spectral correlations $I_{36}^{\mathrm{out}}$ and $I_{45}^{\mathrm{out}}$ as a function of the pumping rate $\epsilon /{\epsilon }_c$ and frequency $\omega$ (in units of $\gamma$). All other cavity parameters are the same as in Fig. 8. Note that we crop the peak of the spectra for visualization purposes.

Figure 10

Plot of the spectral correlation $I_{56}^{\mathrm{out}}$ as a function of the pumping rate $\epsilon /{\epsilon }_c$ and frequency $\omega$ (in units of $\gamma$). All other cavity parameters are the same as in Fig. 8.

Figure 11

Squeezed joint operators, $O_i$, in the undepleted pump approximation (black solid lines) and positive-$P$ results without a cavity (colored dashed lines). The number of trajectories for the stochastic simulation is 300,000 with $\chi =0.01$ and ${\alpha }_{1,2}(0)=1\times {10}^3$.

Figure 12

Squeezed joint operators, $O_i$, calculated using the positive-$P$ equations in the presence of a cavity. The operators are calculated for the below threshold case using 50,000 trajectories, with $\epsilon =0.6472{\epsilon }_c$, $\chi =0.01$, and $\gamma =1$.