The generation of genuine quadripartite Einstein–Podolsky–Rosen steering in an optical superlattice

We defined $X_{i}=(\alpha _i+\alpha _i^{\dagger })/2$ and $Y_{i}=(\alpha _i-\alpha _i^{\dagger })/2i$ as the quadrature amplitude and phase component, respectively. Based on the criterion for genuine multipartite EPR steering^4,5,6, a set of inequalities is given as

$$\begin{aligned} V_{1}= & {} \Delta (X_{1}-X_{2})\Delta (Y_{1}+Y_{2}+Y_{3}+Y_{4})<1 \nonumber \\ V_{2}= & {} \Delta (X_{2}-X_{3})\Delta (Y_{1}+Y_{2}+Y_{3}+Y_{4})<1 \nonumber \\ V_{3}= & {} \Delta (X_{3}-X_{4})\Delta (Y_{1}+Y_{2}+Y_{3}+Y_{4})<1 \nonumber \\ V_{4}= & {} \Delta (X_{4}-X_{1})\Delta (Y_{1}+Y_{2}+Y_{3}+Y_{4})<1. \end{aligned}$$

(9)

EPR steering of system i will be confirmed when the values of $V_{i}<1$. Such as $V_{1}<1$ shows the steering exists between the different bipartitions {1,234}, 134,2, {13,24}, or {14,23}. $V_{2}<1$ shows the steering exists between the different bipartitions^5,6 {2,134}, {3, 124}, {12,34}, or {13,24}. $V_{3}<1$ shows the steering exists between the different bipartitions {3,124}, {4, 123}, {13,24}, or {14,23}. $V_{4}<1$ shows the steering exists between the different bipartitions {1,234}, {4,123}, {12,34}, or {13,24}. Remarkably, genuine quadripartite EPR steering will be verified as long as^4,5,6

$$\begin{aligned} V_{t}=V_{1}+V_{2}+V_{3}+V_{4}<1. \end{aligned}$$

(10)

In the following, the quadripartite EPR steering will be discussed both below the threshold and without the threshold, respectively.

Below the threshold

Figure 2 depicts $V_{i}$ and $V_{t}$ versus the normalized analysis frequency $\Omega$ with $\varepsilon =0.08\varepsilon _{c}$, $\gamma _{0}=\gamma _{1}=0.02$, $\gamma _{2}=2\gamma _{1}$, $\gamma _{3}=4\gamma _{1}$, $\gamma _{4}=5\gamma _{1}$, $\kappa _{1}=5\gamma _{1}$, $\kappa _{2}=0.5\kappa _{1}$, and $\ \kappa _{3}=0.2\kappa _{1}$. As shown in Fig. 2, the value of $V_{i}$ is below 1, and most importantly, $V_{t}$ is also below 1 in the whole range of $\Omega$. It shows that the genuine quadripartite EPR steering can be generated in our scheme based on cascaded nonlinear processes.

In Fig. 3, we show the results of $V_{i}$ and $V_{t}$ versus the nonlinear coupling parameter $\kappa _{2}/\kappa _{1}$ with $\varepsilon =0.08\varepsilon _{c}$, $\gamma _{0}=\gamma _{1}=0.02$, $\gamma _{2}=2\gamma _{1}$, $\gamma _{3}=4\gamma _{1}$, $\gamma _{4}=5\gamma _{1}$, $\kappa _{1}=5\gamma _{1}$, $\kappa _{3}=0.2\kappa _{1}$, and $\omega =5\gamma _{0}$. It can be seen that the values of $V_{i}$ and $V_{t}$ increase slowly as the increase of the nonlinear coupling parameter. When $\kappa _{2}>\kappa _{1}$, $V_{i}$ and $V_{t}$ slowly decrease with the increase of $\kappa _2$. However, limited by the threshold condition, $\kappa _2$ cannot continue to increase. Below the threshold, the change of $\kappa _{2}/\kappa _{1}$ has little effect on multipartite quantum steering which is different from the case above the threshold. Nevertheless, $V_{i}$ and $V_{t}$ are all below 1 in the whole range of Fig. 3 which is sufficient to demonstrate that the genuine quadripartite EPR steering can be produced in our scheme.

Figure 2

$V_{i}$ and $V_{t}$ versus the normalized analysis frequency $\Omega$ below the threshold with $\gamma _{0}=\gamma _{1}=0.02$, $\gamma _{2}=2 \gamma _{1}$, $\gamma _{3}=4\gamma _{1}$, $\gamma _{4}=5\gamma _{1}$, $\kappa _{1}=5\gamma _{1}$ , $\kappa _{2}=0.5\kappa _{1}$, and $\kappa _{3}=0.2\kappa _{1}$.

The influences of the damping rate $\gamma _{1}/\gamma _{0}$ on the $V_{i}$ and $V_{t}$ are plotted as a function in Fig. 4 with $\varepsilon =0.08\varepsilon _{c}$, $\gamma _{0}=0.02$, $\gamma _{2}=2\gamma _{1}$, $\gamma _{3}=4\gamma _{1}$, $\gamma _{4}=5\gamma _{1}$, $\kappa _{1}=5\gamma _{1}$, $\kappa _{2}=0.5\kappa _{1}$, and $\ \kappa _{3}=0.2\kappa _{1}$. From Fig. 4, one can see that the values of $V_{i}$ and $V_{t}$ increase with the increase of the damping rates. When $\gamma _{1}/\gamma _{0}>2$, $V_{t}>1$, the quadripartite quantum steering can not be obtained by the cascaded nonlinear process. The damping rate of parametric optical field is smaller than that of pump light, so that all optical fields can resonate in the cavity. A smaller damping rate of parametric optical field can obtain a better multipartite quantum steering correlation.

Figure 5 shows that the values of Vi and $V_{t}$ versus $\varepsilon$ with $\gamma _{0}=\gamma _{1}=0.02$, $\gamma _{2}=2\gamma _{1}$, $\gamma _{3}=4\gamma _{1}$, $\gamma _{4}=5\gamma _{1}$, $\kappa _{1}=5\gamma _{1}$, $\kappa _{2}=0.5\kappa _{1}$, $\kappa _{3}=0.2\kappa _{1}$, and $\omega =5\gamma _{0}$. It can be clearly seen that the value of $V_{i}$ is below 1 and most importantly, $V_{t}$ is also below 1 in the whole range, which demonstrates the success of the quadripartite EPR steering again. With the increase of pump power, the values of Vi and $V_{t}$ increase linearly. When the pump is weak, a better multipartite EPR steering can be obtained. It can be seen from the above analysis, the genuine quadripartite EPR steering can be generated below the threshold by cascaded nonlinear processes in our scheme.

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Without the threshold

In the condition of without oscillation threshold, only when $\varepsilon <\varepsilon _{c}^{^{\prime }}$ the system is stable and the the linearization method is effective. Figure 6 depicts $V_{i}$ and $V_{t}$ versus the normalized analysis frequency $\Omega =\omega /\gamma _{0}$ for $\gamma _{0}=0.1$, $\gamma _{1}=\gamma _{3}=0.2\gamma _{0}$, $\gamma _{2}=0.4\gamma _{0}$, $\gamma _{4}=0.1\gamma _{0}$, $\kappa _{1}=0.1\gamma _{0}$, $\kappa _{2}=4\kappa _{1}$, $\kappa _{3}=2\kappa _{1},$ and $\varepsilon =0.5\varepsilon _{c}^{^{\prime }}$. The curves of $V_{i}$ and $V_{t}$ are below 1 in the whole range of $\Omega$ which shows that the genuine quadripartite EPR steering can be generated in the case of without the threshold.

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Figure 7 shows the effects of the nonlinear coupling parameter $\kappa _{2}/\kappa _{1}$ on $V_{i}$ and $V_{t}$ with $\gamma _{0}=0.1$, $\gamma _{1}=\gamma _{3}=0.2\gamma _{0}$, $\gamma _{2}=0.4\gamma _{0}$, $\gamma _{4}=0.1\gamma _{0}$, $\kappa _{1}=0.1\gamma _{0}$, $\kappa _{3}=2\kappa _{1}$, and $\varepsilon =0.5\varepsilon _{c}^{^{\prime }}$. One can see that when $\kappa _{2}<0.1\kappa _{1}$, the values of $V_{i}$ and $V_{t}$ are all above 1 and there is a sharp decline. In this case, the quadripartite EPR steering can not be obtained. However, when $\kappa _{2}>0.1\kappa _{1}$, it is clearly see that $V_{i}$ and $V_{t}$ are all below 1, and the quadripartite EPR steering are present.

Figure 8 describes $V_{i}$ and $V_{t}$ versus the damping rates $\gamma _{1}/\gamma _{0}$ with $\gamma _{0}=0.1$, $\gamma _{2}=0.4\gamma _{0}$, $\gamma _{3}=0.2\gamma _{0}$, $\gamma _{4}=0.1\gamma _{0}$, $\kappa _{1}=0.1\gamma _{0}$, $\kappa _{2}=4\kappa _{1}$, $\kappa _{3}=2\kappa _{1}$, and $\varepsilon =0.5\varepsilon _{c}^{^{\prime }}$. As shown in Fig. 8, the values of $V_{i}$ and $V_{t}$ are all below 1 in whole range, which also demonstrates the success of the quadripartite EPR steering.

Finally, the effects of changing pump value $\varepsilon$ on the $V_{i}$ and $V_{t}$ with $\gamma _{0}=0.1$, $\gamma _{1}=\gamma _{3}=0.2\gamma _{0}$, $\gamma _{2}=0.4\gamma _{0}$, $\gamma _{4}=0.1\gamma _{0}$, $\kappa _{1}=0.1\gamma _{0}$, $\kappa _{2}=4\kappa _{1}$, $\kappa _{3}=2\kappa _{1}$ is plotted in Fig. 9. It is shown that the values of the $V_{i}$ and $V_{t}$ are all below 1 and a better multipartite EPR steering can be obtained for weaker pump which is same to the case in Fig. 5. This may be because its quantum properties become apparent when the pump power is weak. Generally speaking, the genuine quadripartite EPR steering can be confirmed when the system without oscillation threshold.