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Enhancement of quantum correlations using correlation injection scheme in a cascaded four-wave mixing processes

Hailong Wang, Kai Zhang, Zhihao Ni, and Jietai Jing

Open Access Open Access

Abstract

Quantum correlations and entanglement shared among multiple quantum beams are important for both fundamental science and the development of quantum technologies. The enhancement for them is necessary and important to implement the specific quantum tasks and goals. Here, we report a correlation injection scheme (CIS) which is an effective method to enhance the quantum correlations and entanglement in the symmetrical cascaded four-wave mixing processes, and the properties of quantum correlations and entanglement can be characterized by the values of the degree of intensity-difference squeezing (DS) and the smallest symplectic eigenvalues, respectively. Our results show that the CIS can enhance the quantum correlations and entanglement under certain conditions, while for other conditions it can only decrease the values of the DS and the smallest symplectic eigenvalues to the level of standard quantum limit, respectively. We believe that our scheme is experimentally accessible and will contribute to a deeper understanding of the manipulations of the quantum correlations and entanglement in various quantum networks.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Multipartite correlations and entanglement have attracted considerable attention due to their fundamental scientific significance [1–5] and potential applications in future quantum technologies [6–14]. A large number of different schemes for generating multiple correlated and entangled beams have already been theoretically proposed and experimentally implemented. In particular, four-wave mixing (FWM) process, as a technique based on atomic ground state coherence, has been demonstrated to be a good candidate for generating twin beams [15–17], triple beams [18,19], and quadruple beams [20,21] owning the strong quantum correlations. Among them, the quadruple beams are produced from a symmetrical cascaded FWM processes, i. e., the twin beams generated from the first FWM process are amplified by two symmetrical individual FWM processes, and the degree of intensity-difference squeezing (DS) between the quadruple beams can reach to the level of $-$8 dB [20]. The limitation of a higher DS is subjected to the fact that the dark ports in this system are seeded by vacuum state.

It is well known that for a FWM process, its two output beams are quantum correlated and entangled with each other. It is this quantum correlation and entanglement that give the possibility of many interesting experiments such as the realization of a SU(1,1) interferometer [8,9] and a two-mode phase-sensitive amplifier (PSA) [22–24]. In these experiments the dark ports are seeded by the twin (coherent) beams instead of vacuum state, and interference-induced quantum squeezing enhancement can be realized. Along this line, in this paper we propose a correlation injection scheme (CIS) which fully exploits the quantum correlation and entanglement produced from the first FWM process to enhance the quantum correlations and entanglement in the system. It should be emphasized that the present scheme is distinctly different from the above mentioned SU(1,1) interferometer and PSA. Firstly, in SU(1,1) interferometer the twin beams generated from the first FWM process to be sent into the one sequential FWM process, while in the present scheme the twin beams generated from the first FWM process to be split and then sent into the two sequential FWM processes, and the adjusted parameters gives us more space to find the optimum working conditions for quantum properties. Secondly, for PSA the two weak coherent input beams are not quantum correlated and entangled, while the input beams for two sequential FWM processes in the present scheme are quantum correlated and entangled. In this sense, when the gains of the FWM processes are unchanged, the enhancement of quantum correlations and entanglement can be realized by using this scheme. Actually we find that the CIS in the symmetrical cascaded FWM processes can effectively enhance the quantum correlations and entanglement.

This paper is organized as follows. In Sec. 2, we study how the CIS can enhance the quantum correlations in the symmetrical cascaded FWM processes. In Sec. 3, the effect of the CIS on the quantum entanglement will also be analyzed. We conclude in Sec. 4 with a discussion.

2. Enhancement of quantum correlations from the CIS

The proposed scheme for enhancing quantum correlations by using the CIS is shown in Fig.  1. As shown in Fig.  1(a), the twin beams generated from the FWM$_{1}$ process are seeded into the dark ports in the FWM$_{3}$ and FWM$_{2}$ processes with the reflectivities $R_{1}$ ($1-T_{1}$) and $R_{2}$ ($1-T_{2}$), respectively. Specifically, the signal (idler) beam with the reflectivity $R_{1}$ ($1-T_{1}$) ($R_{2}$ ($1-T_{2}$)) and the idler (signal) beam with the transmission $T_{2}$ ($T_{1}$) are both seeded into the FWM$_{3}$ (FWM$_{2}$) process, the interference will appear in the FWM$_{2}$ and FWM$_{3}$ processes due to the double seed configuration [8,9,22–24]. For the sake of comparison, the scheme for generating quadruple beams without the CIS is also shown in Fig.  1(b). In order to better describe the enhancement mechanism in Fig.  1(a), we list the input-output relation of the system

$$\begin{aligned}\hat{C}_{1} & =C_{11}\hat{a}_{1}+C_{12}\hat{\nu}_{0}^{\dagger} +C_{13}\hat{\nu}_{1}+C_{14}\hat{\nu}_{2}^{\dagger},\\ \hat{C}_{2} & =C_{21}\hat{a}_{1}^{\dagger}+C_{22}\hat{\nu}_{0} +C_{23}\hat{\nu}_{1}^{\dagger}+C_{24}\hat{\nu}_{2},\\ \hat{C}_{3} & =C_{31}\hat{a}_{1}+C_{32}\hat{\nu}_{0}^{\dagger} +C_{33}\hat{\nu}_{1}+C_{34}\hat{\nu}_{2}^{\dagger},\\ \hat{C}_{4} & =C_{41}\hat{a}_{1}^{\dagger}+C_{42}\hat{\nu}_{0} +C_{43}\hat{\nu}_{1}^{\dagger}+C_{44}\hat{\nu}_{2}, \end{aligned}$$
with
$$\begin{aligned}C_{11} & =\sqrt{G_{1}G_{2}T_{1}}-\sqrt{g_{1}g_{2}R_{2}}e^{i\Phi_{1}}, C_{12} & =\sqrt{G_{2}g_{1}T_{1}}-\sqrt{G_{1}g_{2}R_{2}}e^{i\Phi_{1}},\\ C_{13} & =\sqrt{G_{2}R_{1}},C_{14}=\sqrt{g_{2}T_{2}}e^{i\Phi_{1}}, C_{21} & =\sqrt{G_{3}g_{1}T_{2}}-\sqrt{G_{1}g_{3}R_{1}}e^{i\Phi_{2}},\\ C_{22} & =\sqrt{G_{1}G_{3}T_{2}}-\sqrt{g_{1}g_{3}R_{1}}e^{i\Phi_{2}}, C_{23} & =\sqrt{g_{3}T_{1}}e^{i\Phi_{2}},C_{24}=\sqrt{G_{3}R_{2}},\\ C_{31} & =\sqrt{g_{1}g_{3}T_{2}}e^{i\Phi_{2}}-\sqrt{G_{1}G_{3}R_{1}}, C_{32} & =\sqrt{G_{1}g_{3}T_{2}}e^{i\Phi_{2}}-\sqrt{G_{3}g_{1}R_{1}},\\ C_{33} & =\sqrt{G_{3}T_{1}},C_{34}=\sqrt{g_{3}R_{2}}e^{i\Phi_{2}}, C_{41} & =\sqrt{G_{1}g_{2}T_{1}}e^{i\Phi_{1}}-\sqrt{G_{2}g_{1}R_{2}},\\ C_{42} & =\sqrt{g_{1}g_{2}T_{1}}e^{i\Phi_{1}}-\sqrt{G_{1}G_{2}R_{2}}, C_{43} & =\sqrt{g_{2}R_{1}}e^{i\Phi_{1}},C_{44}=\sqrt{G_{2}T_{2}}, \end{aligned}$$
where $G_{i}$ ($i$=1, 2, 3) is the power gain in the FWM$_{i}$ process, $G_{i}-g_{i}$=1 and $\Phi _{1}$ and $\Phi _{2}$ are the phases of phase-sensitive FWM$_{2}$ and FWM$_{3}$ processes, respectively. For the sake of symmetry, it is assumed that $G_{2}=G_{3}$ in the following discussions. Based on Eqs. (1)–(2), the quantum correlations between the quadruple beams ($\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$) and pairwise beams (PBs) (($\hat {C}_{1}$, $\hat {C}_{4}$), ($\hat {C}_{2}$, $\hat {C}_{3}$), ($\hat {C}_{1}$, $\hat {C}_{2}$), ($\hat {C}_{3}$, $\hat {C}_{4}$), ($\hat {C}_{1}$, $\hat {C}_{3}$), ($\hat {C}_{2}$, $\hat {C}_{4}$)) in Fig.  1(a) can be calculated analytically. In addition, quantum correlations between the multiple quantum beams can be characterized by the DS [25], which is the ratio of the variance of the relative intensity of the multiple quantum beams to the variance of standard quantum limit (SQL), and the variance of SQL can be expected by using a differential measurement for a beam in a coherent state with a power equal to the total power of the multiple quantum beams, i. e., $Var(N_{ic})_{SQL}=\sum N_{ic}$, namely
$$DS_{(N_{ic})}=\frac{Var(N_{ic})_{FWM}}{Var( N_{ic})_{SQL}}.$$
Where $N_{ic}$ is the intensity combination between the multiple quantum correlated beams, and DS$_{(N_{ic})}<1$ means the presence of quantum correlation. Firstly, the DS value of the quadruple beams ($\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$) can be expressed by
$$\begin{aligned}DS_{1234}&= \frac{Var(N_{1}-N_{2}+N_{3}-N_{4})_{FWM}}{Var(N_{1}-N_{2}+N_{3}-N_{4})_{SQL}}\\ &= \frac{1}{(2G_{1}-1)(2G_{2}-1)-\alpha}, \end{aligned}$$
where $\alpha =4\sqrt {G_{1}G_{2}g_{1}g_{2}}(\sqrt {T_{1}R_{2}}\cos \Phi _{1}+\sqrt {T_{2}R_{1}}\cos \Phi _{2})$. Eq. (4) will be reduced to $1/((2G_{1}-1)(2G_{2}-1))$ when we set $T_{1}=T_{2}=1$ and $\Phi _{1}=\Phi _{2}=0$, corresponding to the case of the quadruple beams in Fig.  1(b). In order to show the effect of the CIS on the enhancement of quantum correlation, it is assumed that $G_{1}=G_{2}=3$ in the following analysis, thus the DS value of the quadruple beams in Fig.  1(b) is 0.04. As we can see from Eq. (4), the interference terms at the denominator are exactly introduced by the CIS. Therefore, the quantum correlation between the quadruple beams in Fig.  1(a) can be effectively enhanced by manipulating the parameters $T_{1}$, $T_{2}$, $\Phi _{1}$, and $\Phi _{2}$. Obviously, the minimum value of DS$_{1234}$ can be obtained in such a case of $\Phi _{1}=\Phi _{2}=\pi$, the dependence of DS$_{1234}$ on $T_{1}$ and $T_{2}$ for the case of $\Phi _{1}=\Phi _{2}=\pi$ is shown in Fig.  2(a). As shown in Fig.  2(a), all the values are smaller than 0.04, which is the DS value of the quadruple beams in Fig.  1(b), meaning that the enhancement function can be realized by using the CIS. At the same time, the red dashed line in Fig.  2(a) is $T_{1}+T_{2}=1$ in which the minimum value of DS$_{1234}$ (0.02) can be obtained. The physical meaning can be understood as follows. $T_{1}+T_{2}=1$ means $T_{1}=R_{2}$ or $T_{2}=R_{1}$, in this sense Eq. (4) is not dependent of $T_{1}$ and $T_{2}$ and can be directly written as
$$DS_{1234}^{min}=\frac{1}{(2G_{1}-1)(2G_{2}-1)+4\sqrt{G_{1}G_{2}g_{1}g_{2}}}.$$

The enhancement function of the quantum correlation can also be clearly seen in Eq. (5). To further confirm this result, the dependence of DS$_{1234}$ on $T_{1}$ for the case of $\Phi _{1}=\Phi _{2}=\pi$, $T_{2}=0.5$ (Trace A) is shown in Fig.  2(b) and shows that the minimum value can be obtained under the condition of $T_{1}=0.5$, in this sense it can also be confirmed that the minimum value of DS$_{1234}$ can be obtained under the condition of $T_{1}+T_{2}=1$. From the point view of the phases $\Phi _{1}$ and $\Phi _{2}$, the dependence of DS$_{1234}$ on $\Phi _{1}$ and $\Phi _{2}$ for the case of $T_{1}=T_{2}=0.5$ is shown in Figs.  2(c), most of the values are smaller than 0.04, demonstrating the enhancement function of the CIS. Correspondingly, Fig.  2(d) also shows the dependence of DS$_{1234}$ on $\Phi _{1}$ (Trace A) for the case of $T_{1}=T_{2}=0.5$ and $\Phi _{2}=\pi$.

 figure: Fig. 1.

Fig. 1. The comparison between the schemes of generating the quadruple beams with CIS (a) and without CIS (b). $\hat {a}_{1}$ is coherent input signal, $\hat {\nu }_{0}$, $\hat {\nu }_{1}$, and $\hat {\nu }_{2}$ are vacuum inputs, $\hat {c}_{1}$–$\hat {c}_{4}$ are output signals. ${T}_{1}$ and ${T}_{2}$ are the transmissions for the twin beams generated from the FWM$_{1}$ process.

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 figure: Fig. 2.

Fig. 2. The dependence of Eq. (4) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\Phi _{2}=\pi$), (b) $T_{1}$ ($\Phi _{1}=\Phi _{2}=\pi$ and $T_{2}=0.5$), (c) $\Phi _{1}$ and $\Phi _{2}$ ($T_{1}=T_{2}=0.5$), and (d) $\Phi _{1}$ ($T_{1}=T_{2}=0.5$ and $\Phi _{2}=\pi$). The red dashed line in Fig.  2(a) is $T_{1}+T_{2}=1$, and the magenta dashed line (Trace B) in Figs.  2(b)–(d) is 0.04.

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The enhancement function of the quantum correlation between the quadruple beams due to the CIS has been analyzed, a natural question will be raised as to whether the intensities of the quadruple beams generated from Fig.  1(a) are more amplified simultaneously than the ones in Fig.  1(b) in which the gains of the quadruple beams $\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$ are $G_{1}G_{2}$ (9), $G_{2}g_{1}$ (6), $g_{1}g_{2}$ (4), and $G_{1}g_{2}$ (6) for the case of $G_{1}=G_{2}=3$, respectively. This is because the multiple beams should be bright when using DS criterion to quantify the degree of quantum correlation in which SQL is proportional to the total power of the multiple quantum beams. To answer this question, the gains of the quadruple beams $\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$ in Fig.  1(a) are

$$\begin{aligned}G_{c1} & =G_{1}G_{2}T_{1}+g_{1}g_{2}R_{2}-2\sqrt{G_{1}G_{2}g_{1}g_{2} T_{1}R_{2}}\cos\Phi_{1},\\ G_{c2} & =G_{2}g_{1}T_{2}+G_{1}g_{2}R_{1}-2\sqrt{G_{1}G_{2}g_{1}g_{2} T_{2}R_{1}}\cos\Phi_{2},\\ G_{c3} & =g_{1}g_{2}T_{2}+G_{1}G_{2}R_{1}-2\sqrt{G_{1}G_{2}g_{1}g_{2} T_{2}R_{1}}\cos\Phi_{2},\\ G_{c4} & =G_{1}g_{2}T_{1}+G_{2}g_{1}R_{2}-2\sqrt{G_{1}G_{2}g_{1}g_{2} T_{1}R_{2}}\cos\Phi_{1}, \end{aligned}$$
respectively. The gains of the quadruple beams in Eq. (6) are shown in Fig.  3 in the case of minimizing the value of DS$_{1234}$ ($\Phi _{1}=\Phi _{2}=\pi$, $T_{1}+T_{2}=1$), and the gains of the beams $\hat {C}_{1}$ and $\hat {C}_{4}$ ($\hat {C}_{2}$ and $\hat {C}_{3}$) increases (decreases) with the increasing of $T_{1}$. However, the simultaneous amplification of the quadruple beams $\hat {C}_{1}\sim 12.5$, $\hat {C}_{2}\sim 12$, $\hat {C}_{3}\sim 12.5$, and $\hat {C}_{4}\sim 12$ can be obtained for the case of $T_{1}=0.5$ (the vertical dashed line in Fig.  3) compared with the gains of the quadruple beams ($\hat {C}_{1}\sim 9$, $\hat {C}_{2}\sim 6$, $\hat {C}_{3}\sim 4$, and $\hat {C}_{4}\sim 6$) in Fig.  1(b). Therefore, we can claim that the CIS can realize both the enhancement of the quantum correlation and the power amplification between the quadruple beams.

 figure: Fig. 3.

Fig. 3. The comparison of the gains of the quadruple beams with the CIS (solid line) and without the CIS (dashed line). The black solid (dashed) line, the blue solid (dashed) line, the cyan solid (dashed) line, and the red solid (blue dashed) lines are the gains of the beams $\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$, respectively with the CIS (without the CIS). The vertical dashed line is $T_{1}=0.5$.

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So far, the enhancement mechanism of the quantum correlation between the quadruple beams has been discussed, it is also interesting to analyze the pairwise correlations between the PBs. Firstly, the DS value of the PBs $\hat {C}_{1}$ and $\hat {C}_{4}$ ($\frac {Var(N_{1}-N_{4})_{FWM}}{Var(N_{1}-N_{4})_{SQL}}$) can be written as

$$DS_{14}=\frac{(G_{1}T_{1}-g_{1}R_{2})^{2}+(T_{1}-R_{2})^{2}G_{1}g_{1}+\beta }{\gamma(G_{1}T_{1}+g_{1}R_{2})-4\sqrt {G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1}},$$
where $\beta =G_{1}T_{1}R_{1}+g_{1}T_{2}R_{2}$ and $\gamma =2G_{2}-1$. The above equation will be reduced to $\frac {2G_{1}-1}{2G_{2}-1}$ when we set $T_{1}=T_{2}=1$ and $\Phi _{1}=0$, corresponding to the case of the pairwise correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{4}$ with the seeding of a thermal state in Fig.  1(b). In this sense, the study of DS$_{14}$ (Eq. (7)) is actually the question of how to enhance the quantum correlation between them with the seeding of a thermal state by using the CIS. As shown in Eq. (7), the enhancement can be realized by manipulating the parameters $T_{1}$, $T_{2}$, and $\Phi _{1}$. It is clear that the minimum value of DS$_{14}$ can be obtained in such a case of $\Phi _{1}=\pi$. The dependence of DS$_{14}$ on $T_{1}$ and $T_{2}$ for the case of $\Phi _{1}=\pi$ is shown in Fig.  4(a), in which all the DS values are smaller than 1, in this sense the CIS can quantum correlate the two beams that are not quantum correlated before. Furthermore, the minimum value of DS$_{14}$ (0.018) for the case of $\Phi _{1}=\pi$ can be obtained under the condition of $T_{1}=0.9$ and $T_{2}=0$, which means that the idler beam from the FWM$_{1}$ process is totally seeded into the FWM$_{2}$ process. Under this condition, the minimum value is less than the DS values of the two beams generated from SU(1,1) interferometer and PSA. Similarly, the dependence of DS$_{14}$ on $T_{1}$ for the case of $\Phi _{1}=\pi$ and $T_{2}=0$ (Trace A) is shown in Fig.  4(b), the value of DS$_{14}$ can be minimized under the condition of $T_{1}=0.9$. At the same time, the dependence of DS$_{14}$ on $\Phi _{1}$ for the case of $T_{1}=0.9$ and $T_{2}=0$ (Trace A) is shown in Fig.  4(c), in which the enhancement function can be realized for most of the phase range.

 figure: Fig. 4.

Fig. 4. The dependence of Eq. (7) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\pi$), (b) $T_{1}$ ($\Phi _{1}=\pi$ and $T_{2}=0$), (c) $\Phi _{1}$ ($T_{1}=0.9$ and $T_{2}=0$). The magenta dashed line (Trace B) in Figs.  4(b)–(c) is 1.

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In addition, it should be noted that the gains of the PBs $\hat {C}_{1}$ and $\hat {C}_{4}$ are 23.5 and 22.8 respectively for the case of $\Phi _{1}=\pi$, $T_{1}=0.9$, and $T_{2}=0$, therefore, the CIS can also realize both the enhancement of the quantum correlation and power amplification between the PBs $\hat {C}_{1}$ and $\hat {C}_{4}$.

Similarly, the DS value of the PBs $\hat {C}_{2}$ and $\hat {C}_{3}$ ($\frac {Var(N_{2}-N_{3})_{FWM}}{Var(N_{2}-N_{3})_{SQL}}$) can also be written as

$$DS_{23}=\frac{(g_{1}T_{2}-G_{1}R_{1})^{2}+(T_{2}-R_{1})^{2}G_{1}g_{1} +\beta}{\gamma(g_{1}T_{2}+G_{1}R_{1})-4\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2}}.$$
The dependence of DS$_{23}$ on the parameters $T_{1}$, $T_{2}$, and $\Phi _{2}$ is shown in Fig.  5, which is similar to the case of DS$_{14}$ from the point of view of symmetry. However the same minimum value of DS$_{23}$ (0.018) can be obtained for the case of $\Phi _{2}=\pi$, $T_{1}=0.1$, and $T_{2}=1$, which means that the idler beam from the FWM$_{1}$ process is totally seeded into the FWM$_{3}$ process. In the case of the minimum value of DS$_{23}$, the gains of the PBs $\hat {C}_{2}$ and $\hat {C}_{3}$ are 22.8 and 23.5, respectively. Therefore, it can be claimed that the CIS can also realize both the enhancement of the quantum correlation and power amplification between the PBs $\hat {C}_{2}$ and $\hat {C}_{3}$.

 figure: Fig. 5.

Fig. 5. The dependence of Eq. (8) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{2}=\pi$), (b) $T_{1}$ ($\Phi _{2}=\pi$ and $T_{2}=1$), (c) $\Phi _{2}$ ($T_{1}=0.1$ and $T_{2}=1$). The magenta dashed line (Trace B) in Figs.  5(b)–(c) is 1.

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Secondly, the enhancement of the quantum correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ due to the CIS will be discussed, and the DS of the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ ($\frac {Var(N_{1}-N_{2})_{FWM}}{Var(N_{1}-N_{2})_{SQL}}$) can be expressed by

$$DS_{12}=\frac{DS_{12}\_n}{DS_{12}\_d},$$
where the numerator DS$_{12}\_n$ and denominator DS$_{12}\_d$ of DS$_{12}$ are
$$\begin{aligned}DS_{12}\_n & =(G_{1}G_{2}T_{1}+g_{1}g_{2}R_{2}-G_{2}g_{1}T_{2}-G_{1} g_{2}R_{1}-2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1}\\ & +2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2})^{2} +[\sqrt{G_{1}g_{1}}(G_{2}T_{1}-G_{2}T_{2}-g_{2}R_{1}+g_{2}R_{2})\\ & +(2G_{1}-1)(\sqrt{G_{2}g_{2}T_{2}R_{1}}\cos\Phi_{2}-\sqrt{G_{2}g_{2} T_{1}R_{2}}\cos\Phi_{1})]^{2}+(\sqrt{G_{2}g_{2}T_{2}R_{1}}\sin\Phi_{2}\\ & -\sqrt{G_{2}g_{2}T_{1}R_{2}} \sin\Phi_{1})^{2}+[(2G_{2}-1)\sqrt{G_{1}T_{1}R_{1}}-\sqrt{G_{2}g_{1}g_{2}R_{1}R_{2}}\cos\Phi_{1}\\ & -\sqrt{G_{2}g_{1}g_{2}T_{1}T_{2}}\cos\Phi_{2}]^{2}+(\sqrt{G_{2}g_{1} g_{2}R_{1}R_{2}}\sin\Phi_{1}+\sqrt{G_{2}g_{1}g_{2}T_{1}T_{2}}\sin\Phi_{2})^{2}\\ & +[\sqrt{G_{1}G_{2}g_{2}T_{1}T_{2}}\cos\Phi_{1} +\sqrt{G_{1}G_{2}g_{2}R_{1}R_{2}}\cos\Phi_{2}-(2G_{2}-1)\sqrt{g_{1}T_{2}R_{2}}]^{2}\\ & +(\sqrt{G_{1}G_{2}g_{2}T_{1}T_{2}}\sin\Phi_{1}+\sqrt{G_{1}G_{2}g_{2}R_{1}R_{2}}\sin\Phi_{2})^{2} \end{aligned}$$
and
$$\begin{aligned}DS_{12}\_d & =G_{1}G_{2}T_{1}+g_{1}g_{2}R_{2}+G_{2}g_{1}T_{2}+G_{1}g_{2}R_{1} -2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1}\\ & -2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2}, \end{aligned}$$
respectively. The simplified result, $\frac {2G_{1}G_{2}}{2G_{1}-1}-1$ for Eq. (9) can be obtained when $T_{1}$ and $T_{2}$ are both set to equal to 1, $\Phi _{1}$ and $\Phi _{2}$ are both set to equal to 0, corresponding to the case of the pairwise correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ in Fig.  1(b). In this sense, the study of DS$_{12}$ (Eq. (9)) is actually the question of how to enhance the quantum correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ by using the CIS. The PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ generated from the FWM$_{1}$ process are amplified by the symmetrical individual FWM$_{2}$ and FWM$_{3}$ processes which bring the deterioration effect to the quantum correlation. The enhancement results are shown in Fig.  6, Fig.  6(a) gives the dependence of DS$_{12}$ on $T_{1}$ and $T_{2}$ for the case of $\Phi _{1}=\Phi _{2}=0$, the region bounded by the magenta dashed curve (2.6) is such a region in which the DS value of the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ can be decreased by using the CIS, but there is no quantum correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ since the value is more than or equal to 1; While the region bounded by the red dashed curve (1) is such a region in which the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ can be quantum correlated by using the CIS since the value is less than 1. To see this result more explicitly, the dependence of DS$_{12}$ on $T_{1}$ for the case of $\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0.42$ is shown in Fig.  6(b), the regions of DS$_{12}$ (Trace A) less than 2.6 and 1 (Trace B and Trace C) are also consistent with the regions bounded by the magenta and red dashed curves in Fig.  6(a). When $T_{1}$ and $T_{2}$ are set to 0.91 and 0.42, respectively, the results are shown in Fig.  6(c), similar to the case in Fig.  6(a), the regions bounded by the magenta and red dashed curves represent the phase regions of the reduction of the DS value and the generation of quantum correlation respectively by using the CIS. To support this results, Figs.  6(d)–(e) gives the dependence of DS$_{12}$ on $\Phi _{1}$ for the case of $\Phi _{2}=0$, $T_{1}=0.91$, and $T_{2}=0.42$, the results are in agreement with the ones in Fig.  6(c).

 figure: Fig. 6.

Fig. 6. The dependence of Eq. (9) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\Phi _{2}=0$), (b) $T_{1}$ ($\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0.42$), (c) $\Phi _{1}$ and $\Phi _{2}$ ($T_{1}=0.91$ and $T_{2}=0.42$), (d) $\Phi _{1}$ ($\Phi _{2}=0$, $T_{1}=0.91$, and $T_{2}=0.42$), and (e) $\Phi _{1}$ around $2\pi$ in Fig.  6(d). The magenta dashed line (Trace B) is 2.6, and the red dashed line (Trace C) is 1.

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After the enhancement of the quantum correlation is discussed, the power amplification should be concerned. While in the case of the minimum value of DS$_{12}$, the gains of the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$ are 1.8 and 0.7, respectively, less than the ones of 9 and 6 in Fig.  1(b). Thus, it can be claimed that the CIS can only realize the enhancement of the quantum correlation, but not the power amplification between the PBs $\hat {C}_{1}$ and $\hat {C}_{2}$.

Similarly, the DS value of the PBs $\hat {C}_{3}$ and $\hat {C}_{4}$ ($\frac {Var(N_{3}-N_{4})_{FWM}}{Var(N_{3}-N_{4})_{SQL}}$) can be written in a compact form as

$$DS_{34}=\frac{DS_{12}\_n(G_{2}\rightarrow g_{2},g_{2}\rightarrow G_{2} )}{DS_{12}\_d(G_{2}\rightarrow g_{2},g_{2}\rightarrow G_{2})},$$
Eq. (12) means that the expression of DS$_{34}$ can be obtained by substituting $g_{2}$, $G_{2}$ for $G_{2}$, $g_{2}$, respectively once in Eq. (9). Under the condition of $\Phi _{1}=\Phi _{2}=0$ and $T_{1}=T_{2}=1$, DS$_{34}$ will be reduced to $\frac {2G_{1}G_{2}-1}{2G_{1}-1}$. However, the minimum value of DS$_{34}$ can be obtained for the case of $\Phi _{1}$=$\Phi _{2}=0$, $T_{1}=0.09$, and $T_{2}=0.58$. As shown in Fig.  7, the region bounded by the magenta dashed curve (3.4) is such a region in which the DS value of the PBs $\hat {C}_{3}$ and $\hat {C}_{4}$ can be reduced and the region bounded by the red dashed curve (1) is such a region in which the PBs $\hat {C}_{3}$ and $\hat {C}_{4}$ can be quantum correlated by using the CIS. Similar to the case of DS$_{12}$, the CIS can only realize the enhancement of the quantum correlation, but not the power amplification between the PBs $\hat {C}_{3}$ and $\hat {C}_{4}$.

 figure: Fig. 7.

Fig. 7. The dependence of Eq. (12) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\Phi _{2}=0$), (b) $T_{1}$ ($\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0.58$), (c) $\Phi _{1}$ and $\Phi _{2}$ ($T_{1}=0.09$ and $T_{2}=0.58$), (d) $\Phi _{1}$ ($\Phi _{2}=0$, $T_{1}=0.09$, and $T_{2}=0.58$), and (e) $\Phi _{1}$ around 2$\pi$ in Fig.  6(d). The magenta dashed line (Trace B) is 3.4, and the red dashed line (Trace C) is 1.

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Lastly, the DS value of the PBs $\hat {C}_{1}$ and $\hat {C}_{3}$ ($\frac {Var(N_{1}-N_{3})_{FWM}}{Var(N_{1}-N_{3})_{SQL}}$) can be expressed as

$$DS_{13}=\frac{DS_{13}\_n}{DS_{13}\_d},$$
where the numerator DS$_{13}\_n$ and denominator DS$_{13}\_d$ of DS$_{13}$ are
$$\begin{aligned}DS_{13}\_n & =(G_{1}G_{2}T_{1}+g_{1}g_{2}R_{2}-g_{1}g_{2}T_{2}-G_{1} G_{2}R_{1}-2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1}\\ & +2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2})^{2} +[\sqrt{G_{1}g_{1}}(G_{2}T_{1}-g_{2}T_{2}-G_{2}R_{1}+g_{2}R_{2})\\ & +(2G_{1}-1)(\sqrt{G_{2}g_{2}T_{2}R_{1}}\cos\Phi_{2}-\sqrt{G_{2}g_{2} T_{1}R_{2}}\cos\Phi_{1})]^{2}+(\sqrt{G_{2}g_{2}T_{2}R_{1}}\sin\Phi_{2}\\ & -\sqrt{G_{2}g_{2}T_{1}R_{2}} \sin\Phi_{1})^{2}+(2G_{2}\sqrt{G_{1}T_{1}R_{1}}-\sqrt{G_{2}g_{1}g_{2} R_{1}R_{2}}\cos\Phi_{1}\\ & -\sqrt{G_{2}g_{1}g_{2}T_{1}T_{2}}\cos\Phi_{2})^{2}+(\sqrt{G_{2}g_{1} g_{2}R_{1}R_{2}}\sin\Phi_{1}+\sqrt{G_{2}g_{1}g_{2}T_{1}T_{2}}\sin\Phi_{2})^{2}\\ & +[\sqrt{G_{1}G_{2}g_{2}T_{1}T_{2}}\cos\Phi_{1} +\sqrt{G_{1}G_{2}g_{2}R_{1}R_{2}}\cos\Phi_{2}-2g_{2}\sqrt{g_{1}T_{2}R_{2}}]^{2}\\ & +(\sqrt{G_{1}G_{2}g_{2}T_{1}T_{2}}\sin\Phi_{1}+\sqrt{G_{1}G_{2}g_{2}R_{1}R_{2}}\sin\Phi_{2})^{2} \end{aligned}$$
and
$$\begin{aligned}DS_{13}\_d & =G_{1}G_{2}T_{1}+g_{1}g_{2}R_{2}+g_{1}g_{2}T_{2}+G_{1}G_{2}R_{1} -2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1}\\ & -2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2}, \end{aligned}$$
respectively. The simplified result, $\frac {2G_{1}G_{2}^{2}+2G_{1}^{2}-3G_{1}-G_{2}+1}{2G_{1}G_{2}-G_{1}-G_{2}+1}$ for Eq. (13) can be obtained when $T_{1}$ and $T_{2}$ are both set to equal to 1, $\Phi _{1}$ and $\Phi _{2}$ are both set to equal to 0, corresponding to the case of the pairwise correlation between the PBs $\hat {C}_{1}$ and $\hat {C}_{3}$ in Fig.  1(b). However, the minimum value of DS$_{13}$ (1) can be obtained for the case of $\Phi _{1}=\Phi _{2}=0$, $T_{1}=1~(0)$, and $T_{2}=0~(1)$. For the convenience of discussion, $T_{1}$ and $T_{2}$ are set to 1 and 0 respectively in the following analysis. As shown in Fig.  8(a), the regions bounded by the magenta dashed curve (4.7) is such a region in which the value of DS$_{13}$ can be reduced. At the same time, this point can be also confirmed in Fig.  8(b) which gives the dependence of DS$_{13}$ on $T_{1}$ for the case of $\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0$, the minimum value can be obtained under the condition of $T_{1}=1$. From the point of view of the phase dependence, as shown in Figs.  8(c)–(d), the optimum value, i. e., 1, can be achieved for the case of $\Phi _{1}=0$. In addition, $T_{2}=0$ leads to the independence of DS$_{13}$ on $\Phi _{2}$. It is worth noting that the gains for the PBs $\hat {C}_{1}$ and $\hat {C}_{3}$ are 1 and 0, respectively, so that the CIS can only reduce the DS value of the PBs $\hat {C}_{1}$ and $\hat {C}_{3}$.

 figure: Fig. 8.

Fig. 8. The dependence of Eq. (13) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\Phi _{2}=0$), (b) $T_{1}$ ($\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0$), (c) $\Phi _{1}$ and $\Phi _{2}$ ($T_{1}=1$ and $T_{2}=0$), (d) $\Phi _{1}$ ($\Phi _{2}=0$, $T_{1}=1$, and $T_{2}=0$). The magenta dashed line (Trace B) is 4.7, and the red dashed line (Trace C) is 1.

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The DS value of the PBs $\hat {C}_{2}$ and $\hat {C}_{4}$ ($\frac {Var(N_{2}-N_{4})_{FWM}}{Var(N_{2}-N_{4})_{SQL}}$) has a compact form as

$$DS_{24}=\frac{DS_{13}\_n(G_{2}\rightarrow g_{2},g_{2}\rightarrow G_{2} )}{DS_{13}\_d(G_{2}\rightarrow g_{2},g_{2}\rightarrow G_{2})},$$
the above equation means that DS$_{24}$ can be also obtained by substituting $g_{2}$, $G_{2}$ for $G_{2}$, $g_{2}$, respectively once in Eq. (13). For the case of $\Phi _{1}=\Phi _{2}=0$ and $T_{1}=T_{2}=1$, DS$_{24}$ will be reduced to $\frac {-4G_{1}G_{2}+2G_{1}G_{2}^{2}+2G_{1}^{2}-G_{1}+G_{2}}{2G_{1}G_{2}-G_{1}-G_{2}}$. However, the minimum value of DS$_{24}$ can be obtained for the case of $\Phi _{1}=\Phi _{2}=0$, $T_{1}+T_{2}=1$. As shown in Fig.  9, the red dashed line in Fig.  9(a) and the region bounded by the magenta dashed curve (3) in Fig.  9(b) represent $T_{1}+T_{2}=1$, and the reduction of the DS value, respectively. Similar to the case of DS$_{13}$, the CIS can only decrease the DS value, but not produce the quantum correlation and the power amplification between the PBs $\hat {C}_{2}$ and $\hat {C}_{4}$.

 figure: Fig. 9.

Fig. 9. The dependence of Eq. (16) on (a) $T_{1}$ and $T_{2}$ ($\Phi _{1}=\Phi _{2}=0$), (b) $\Phi _{1}$ and $\Phi _{2}$ ($T_{1}=T_{2}=0.5$). The red dashed line is $T_{1}+T_{2}=1$, and the magenta dashed line is 3.

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Overall, the enhancement mechanism of the quantum correlations between the quadruple beams ($\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$) and the PBs (($\hat {C}_{1}$, $\hat {C}_{4}$), ($\hat {C}_{2}$, $\hat {C}_{3}$), ($\hat {C}_{1}$, $\hat {C}_{2}$), ($\hat {C}_{3}$, $\hat {C}_{4}$), ($\hat {C}_{1}$, $\hat {C}_{3}$), and ($\hat {C}_{2}$, $\hat {C}_{4}$)) has been extensively investigated, it can realize both the enhancement of the quantum correlation and the power amplification between the quadruple beams ($\hat {C}_{1}$, $\hat {C}_{2}$, $\hat {C}_{3}$, and $\hat {C}_{4}$) and the PBs (($\hat {C}_{1}$, $\hat {C}_{4}$) and ($\hat {C}_{2}$, $\hat {C}_{3}$)), but only the enhancement of the quantum correlation for the PBs ($\hat {C}_{1}$, $\hat {C}_{2}$), ($\hat {C}_{3}$, $\hat {C}_{4}$), ($\hat {C}_{1}$, $\hat {C}_{3}$), and ($\hat {C}_{2}$, $\hat {C}_{4}$)). The enhancement mechanism results from the quantum interference in the FWM$_{2}$ and FWM$_{3}$ processes in which the interference beams are quantum correlated, it is this quantum interference that gives us the opportunities to manipulate even enhance the quantum correlations. Therefore, the enhancement of the quantum correlations in this cascaded FWM processes can be realized by using the CIS.

3. Enhancement of quantum entanglement from the CIS

In the previous section, quantum correlations are the non-classical correlations and not necessarily quantum entanglement. The description of quantum correlations only involve the intensity information of the multiple beams which can be obtained by using direct intensity detection, multipartite quantum entanglement requires balanced homodyne detection to construct the full covariance matrix (CM) of the multiple beams, which fully characterizes the quantum properties of the multiple beams. In addition, multipartite quantum entanglement involving the entire quantum properties of the multiple beams has numerous possible applications, for example, the realization of quantum entanglement swapping between two independent multipartite entangled states [26], the verification of quantum secret sharing among four players [27], and the deterministic generation of orbital angular momentum multiplexed tripartite entanglement [28,29] so on. Therefore, in this section, we will focus on the potential enhancement of the quadripartite [30] and the bipartite entanglement by using the CIS. Generally, the entanglement properties of the Gaussian state in Fig.  1(a) can be completely characterized by its corresponding CM in which all the variances and covariances are necessary for the entanglement criterion. For the amplitude covariances, we use the notation $\langle \hat {X}_{m}\hat {X}_{n}\rangle =(\langle \hat {X}_{m}\hat {X}_{n}\rangle +\langle \hat {X}_{n}\hat {X}_{m}\rangle )/2-\langle \hat {X}_{m}\rangle \langle \hat {X}_{n}\rangle$ ($m, n=C_{1}-C_{4}$) and for the case where $m=n$, the covariances, denoted $\langle \hat {X}_{m}\hat {X}_{n}\rangle$, reduce to the usual variances, $\langle \hat {X}_{m}^{2}\rangle$. The phase quadrature operators can be applied by a similar notation, and in our scheme $\langle \hat {X}_{m}\hat {Y}_{n}\rangle$ is zero. Therefore, all the variances and covariances of the quadruple beams in Fig.  1(a) can be given by

$$\begin{aligned}\langle\hat{X}_{c1}^{2}\rangle & =\langle\hat{Y}_{c1}^{2} \rangle=(2G_{1}-1)(G_{2}T_{1}+g_{2}R_{2})+G_{2}R_{1}+g_{2}T_{2} -4\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1},\\ \langle\hat{X}_{c2}^{2}\rangle & =\langle\hat{Y}_{c2}^{2} \rangle=(2G_{1}-1)(G_{2}T_{2}+g_{2}R_{1})+G_{2}R_{2}+g_{2}T_{1} -4\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2},\\ \langle\hat{X}_{c3}^{2}\rangle & =\langle\hat{Y}_{c3}^{2} \rangle=(2G_{1}-1)(G_{2}R_{1}+g_{2}T_{2})+G_{2}T_{1}+g_{2}R_{2} -4\sqrt{G_{1}G_{2}g_{1}g_{2}T_{2}R_{1}}\cos\Phi_{2},\\ \langle\hat{X}_{c4}^{2}\rangle & =\langle\hat{Y}_{c4}^{2} \rangle=(2G_{1}-1)(G_{2}R_{2}+g_{2}T_{1})+G_{2}T_{2}+g_{2}R_{1} -4\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}R_{2}}\cos\Phi_{1},\\ \langle\hat{X}_{c1}\hat{X}_{c2}\rangle & ={-}\langle\hat{Y} _{c1}\hat{Y}_{c2}\rangle=2G_{2}\sqrt{G_{1}g_{1}T_{1}T_{2}}+2g_{2} \sqrt{G_{1}g_{1}R_{1}R_{2}}\cos(\Phi_{1}+\Phi_{2})\\ & -2g_{1}\sqrt{G_{2}g_{2} T_{2}R_{2}}\cos\Phi_{1}-2g_{1}\sqrt{G_{2}g_{2}T_{1}R_{1}}\cos\Phi_{2},\\ \langle\hat{X}_{c1}\hat{X}_{c3}\rangle & =\langle\hat{Y} _{c1}\hat{Y}_{c3}\rangle={-}2G_{2}g_{1}\sqrt{T_{1}R_{1}}-2g_{1}g_{2} \sqrt{T_{2}R_{2}}\cos(\Phi_{1}-\Phi_{2})\\ & +2\sqrt{G_{1}G_{2}g_{1}g_{2}R_{1}R_{2}}\cos\Phi_{1} +2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}T_{2}}\cos\Phi_{2},\\ \langle\hat{X}_{c1}\hat{X}_{c4}\rangle & ={-}\langle\hat{Y} _{c1}\hat{Y}_{c4}\rangle=[(2G_{1}-1)(T_{1}+R_{2})+R_{1}+T_{2}]\sqrt {G_{2}g_{2}}\cos\Phi_{1}\\ & -2\sqrt{G_{1}g_{1}T_{1}R_{2}}(G_{2}+g_{2}\cos2\Phi_{1}),\\ \langle\hat{X}_{c2}\hat{X}_{c3}\rangle & ={-}\langle\hat{Y} _{c2}\hat{Y}_{c3}\rangle=[(2G_{1}-1)(T_{2}+R_{1})+R_{2}+T_{1}]\sqrt {G_{2}g_{2}}\cos\Phi_{2}\\ & -2\sqrt{G_{1}g_{1}T_{2}R_{1}}(G_{2}+g_{2}\cos2\Phi_{2}),\\ \langle\hat{X}_{c2}\hat{X}_{c4}\rangle & =\langle\hat{Y} _{c2}\hat{Y}_{c4}\rangle={-}2G_{2}g_{1}\sqrt{T_{2}R_{2}}-2g_{1}g_{2} \sqrt{T_{1}R_{1}}\cos(\Phi_{1}-\Phi_{2})\\ & +2\sqrt{G_{1}G_{2}g_{1}g_{2}T_{1}T_{2}}\cos\Phi_{1} +2\sqrt{G_{1}G_{2}g_{1}g_{2}R_{1}R_{2}}\cos\Phi_{2},\\ \langle\hat{X}_{c3}\hat{X}_{c4}\rangle & ={-}\langle\hat{Y} _{c3}\hat{Y}_{c4}\rangle=2G_{2}\sqrt{G_{1}g_{1}R_{1}R_{2}}+2g_{2} \sqrt{G_{1}g_{1}T_{1}T_{2}}\cos(\Phi_{1}+\Phi_{2})\\ & -2g_{1}\sqrt{G_{2}g_{2}T_{1}R_{1}}\cos\Phi_{1}-2g_{1}\sqrt{G_{2}g_{2}T_{2}R_{2}}\cos\Phi_{2}. \end{aligned}$$
Based on Eq. (17), compared with van Loock-Furusawa criterion as a sufficient but not necessary criterion, the positivity under partial transposition (PPT) criterion [31–37] as a necessary and sufficient criterion for Gaussian states under the certain conditions can be used to characterize the quantum entanglement enhancement, since the smallest symplectic eigenvalue from the PPT criterion can be used to quantify the degree of entanglement. The smaller the smallest symplectic eigenvalue is, the stronger the entanglement will be. Therefore, the quantum entanglement can be enhanced if and only if the smallest symplectic eigenvalues in Fig.  1(a) are smaller than the ones in Fig.  1(b). Along this line, the enhancement of the quadripartite entanglement will be analyzed. For the quadripartite entanglement scenario, 2$^{(N-1)}-$1=7 possible bipartitions have to be tested. To test the entanglement enhancement of the system, the partially transposed (PT) operation is applied to any 1–3 mode bipartition, then any 2–2 mode bipartition, respectively. Firstly, the PT operation is applied to 1–3 mode bipartition, i. e., $\hat {c}_{1}|$($\hat {c}_{2}$, $\hat {c}_{3}$, $\hat {c}_{4}$), $\hat {c}_{2}|$($\hat {c}_{1}$, $\hat {c}_{3}$, $\hat {c}_{4}$), $\hat {c}_{3}|$($\hat {c}_{1}$, $\hat {c}_{2}$, $\hat {c}_{4}$), and $\hat {c}_{4}|$($\hat {c}_{1}$, $\hat {c}_{2}$, $\hat {c}_{3}$), respectively, and it represents the beams $\hat {c}_{1}$, $\hat {c}_{2}$, $\hat {c}_{3}$, and $\hat {c}_{4}$ are PT, respectively. The generated four smallest symplectic eigenvalues are $\nu _{1234}^{1}$, $\nu _{1234}^{2}$, $\nu _{1234}^{3}$, and $\nu _{1234}^{4}$, respectively. As shown in Fig.  10, all the values are smaller than 1, which means that all the $1\times 3$ bipartitions are fully inseparable. Take $\nu _{1234}^{1}$ in Fig.  10(a) for example, its minimum value (0.01) can be obtained under the conditions of $T_{1}=1$, $T_{2}=0$, and $\Phi _{1}=\pi$, which means that the idler beam from the FWM$_{1}$ process is totally seeded into the FWM$_{2}$ process. The optimum quadripartite entanglement is more and more approaching to the bipartite entanglement existed in the FWM$_{2}$ process, thus $\nu _{1234}^{1}$ is independent on the phase $\Phi _{2}$. And Fig.  10(a) gives the dependence of $\nu _{1234}^{1}$ on $T_{1}$ and $T_{2}$ for the case of $\Phi _{1}=\pi$, confirming the entanglement enhancement since most of the values are smaller than the magenta dashed curve (0.029), which is the value for $\nu _{1234}^{1}$ in Fig.  1(b). The similar behaviors can also be seen in Figs.  10(b)–(d), so it can be claimed that the CIS can enhance the quantum entanglement of 1–3 mode bipartition.

 figure: Fig. 10.

Fig. 10. The dependence of (a) $\nu _{1234}^{1}$ ($\Phi _{1}=\pi$), (b) $\nu _{1234}^{2}$ ($\Phi _{2}=\pi$), (c) $\nu _{1234}^{3}$ ($\Phi _{2}=\pi$), and (d) $\nu _{1234}^{4}$ ($\Phi _{1}=\pi$) on $T_{1}$ and $T_{2}$. The magenta dashed curve is 0.029 and 0.039 in Figs.  10(a)–(b) and Figs.  10(c)–(d), respectively.

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Secondly, the PT operation is applied to 2–2 mode bipartition, i. e., ($\hat {c}_{1}$, $\hat {c}_{2}$)$|$($\hat {c}_{3}$, $\hat {c}_{4}$), ($\hat {c}_{1}$, $\hat {c}_{3}$)$|$($\hat {c}_{2}$, $\hat {c}_{4}$), and ($\hat {c}_{1}$, $\hat {c}_{4}$)$|$($\hat {c}_{2}$, $\hat {c}_{3}$), respectively, and it represents the PBs ($\hat {c}_{1}$, $\hat {c}_{2}$), ($\hat {c}_{1}$, $\hat {c}_{3}$), and ($\hat {c}_{1}$, $\hat {c}_{4}$) are PT, respectively. The generated three smallest symplectic eigenvalues are $\nu _{1234}^{12}$, $\nu _{1234}^{13}$, and $\nu _{1234}^{14}$, respectively. The minimum value of $\nu _{1234}^{12}$ (0.01) can be obtained for the case of $T_{1}=0 (1)$, $T_{2}=1 (0)$, and $\Phi _{2}~(\Phi _{1})=\pi$, and the minimum value of $\nu _{1234}^{13}$ (0.01) can be obtained for the case of $T_{1}+T_{2}=1$, $\Phi _{1}=\pi$, and $\Phi _{2}=\pi$. As shown in Fig.  11, the dependence of $\nu _{1234}^{12}$ on $T_{1}$ and $T_{2}$ for the case of $\Phi _{2}=\pi$ is shown in Fig.  11(a), most of the values are smaller than the magenta dashed curve (0.059) which is the value of $\nu _{1234}^{12}$ in Fig.  1(b), confirming the CIS can enhance the quantum entanglement between the PBs ($\hat {c}_{1}$, $\hat {c}_{2}$) and the PBs ($\hat {c}_{3}$, $\hat {c}_{4}$); The enhancement function can also be confirmed in $\nu _{1234}^{13}$ in Fig.  11(b), all the values are smaller than 0.018 which is the value of $\nu _{1234}^{13}$ in Fig.  1(b), its minimum value (0.01) can be obtained under the case of $T_{1}+T_{2}=1$ (the red dashed line). On the contrary, the value of $\nu _{1234}^{14}$ can not be decreased due to this fact that $\nu _{1234}^{14}$ is only dependent on the gain $G_{1}$ in the FWM$_{1}$ process, thus the manipulations of the parameters $T_{1}$, $T_{2}$, $\Phi _{1}$ and $\Phi _{2}$ after the FWM$_{1}$ process can not alter the value of $\nu _{1234}^{14}$. Therefore, the CIS can partially enhance the quantum entanglement of the 2–2 mode bipartition.

 figure: Fig. 11.

Fig. 11. The dependence of (a) $\nu _{1234}^{12}$ ($\Phi _{2}=\pi$), (b) $\nu _{1234}^{13}$ ($\Phi _{1}=\Phi _{2}=\pi$) on $T_{1}$ and $T_{2}$. The magenta dashed curve is 0.059 in Fig.  11(a), and the red dashed curve is $T_{1}+T_{2}=1$ in Fig.  11(b).

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After the enhancement effect of the CIS on the quadripartite entanglement has been analyzed, the bipartite entanglement enhancement with the CIS should also be discussed correspondingly. In order to test the bipartite entanglement, the PT operation is applied to 1–1 mode bipartition, i. e., $\hat {c}_{1}|$($\hat {c}_{2}$), $\hat {c}_{1}|$($\hat {c}_{3}$), $\hat {c}_{1}|$($\hat {c}_{4}$), $\hat {c}_{2}|$($\hat {c}_{3}$), $\hat {c}_{2}|$($\hat {c}_{4}$), and $\hat {c}_{3}|$($\hat {c}_{4}$), respectively, which represents that only one beam $\hat {c}_{j}$ ($j=1-3$) in bipartite entanglement is PT. The generated six smallest symplectic eigenvalues are $\nu _{12}^{1}$, $\nu _{13}^{1}$, $\nu _{14}^{1}$, $\nu _{23}^{2}$, $\nu _{24}^{2}$, and $\nu _{34}^{3}$, respectively.

As shown in Fig.  12, the value of $\nu _{12}^{1}$ quantifying the bipartite entanglement between the PBs $\hat {c}_{1}$ and $\hat {c}_{2}$ is shown in Figs.  12(a)–(b), its minimum value can be obtained for the case of $\Phi _{1}=\Phi _{2}=0$ and $T_{1}=T_{2}=0.6$, which means that 60% of the signal and idler beam generated from the FWM$_{1}$ process are seeded into the one port of FWM$_{2}$ and FWM$_{3}$ processes, respectively. In addition, most of the values in Fig.  12(a) are smaller than 2.3 which is the value of $\nu _{12}^{1}$ without the CIS, the regions bounded by the red dashed curve (1) is such a region in which the PBs $\hat {c}_{1}$ and $\hat {c}_{2}$ can be entangled by using the CIS, it means that the CIS can quantum entangle the two beams that are not entangled before. Figure  12(b) shows the dependence of $\nu _{12}^{1}$ on the phases $\Phi _{1}$ and $\Phi _{2}$ under the condition of $T_{1}=T_{2}=0.6$. From the point of view of symmetry, $\nu _{34}^{3}$ in Figs.  12(c)–(d) has the similar dependence with $\nu _{12}^{1}$, its minimum value can be obtained for the case of $\Phi _{1}=\Phi _{2}=0$ and $T_{1}=T_{2}=0.4$, it again means that 60% of the signal and idler beam generated from the FWM$_{1}$ process are seeded into the other port of the FWM$_{3}$ and FWM$_{2}$ processes, respectively. It should be noted that $\nu _{14}^{1}$ ($\nu _{13}^{1}$) also has a symmetrical relation with $\nu _{23}^{2}$ ($\nu _{24}^{2}$). Figure  12(e) shows that all the values are smaller than 0.17 which is the entanglement degree with the seeding of a thermal state in Fig.  1(b), meaning that the entanglement degree can be totally enhanced by using the CIS. In contrast, the CIS can only decrease the values of $\nu _{13}^{1}$ and $\nu _{24}^{2}$ (from 7 to 1), but not quantum entangle the PBs ($\hat {c}_{1}$, $\hat {c}_{3}$) and ($\hat {c}_{2}$, $\hat {c}_{4}$). To sum up, the CIS can effectively enhance the entanglement degree of the 1–1 mode bipartition.

 figure: Fig. 12.

Fig. 12. The dependence of (a) $\nu _{12}^{1}$ ($\Phi _{1}=\Phi _{2}=0$), (c) $\nu _{34}^{3}$ ($\Phi _{1}=\Phi _{2}=0$), (e) $\nu _{14}^{1}$ ($\Phi _{1}=\pi$), (f) $\nu _{23}^{2}$ ($\Phi _{2}=\pi$), (g) $\nu _{13}^{1}$ ($\Phi _{2}=0$), and (h) $\nu _{24}^{2}$ ($\Phi _{1}=0$) on $T_{1}$ and $T_{2}$; The dependence of (b) $\nu _{12}^{1}$ ($T_{1}=T_{2}=0.6$) and (d) $\nu _{34}^{3}$ ($T_{1}=T_{2}=0.4$) on $\Phi _{1}$ and $\Phi _{2}$. The red dashed curve is 1 in Figs.  12(a)–(d), the magenta dashed curve in Figs.  12(b), (d), (g), and (h) is 2.3, 3.2, 7, and 7, respectively.

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As the enhancement of the quantum correlations in the previous section can be realized by using the CIS, in this section the enhancement of the quadripartite and bipartite entanglement can also be accomplished, which results from the manipulations of the quantum entanglement by using the CIS.

4. Conclusions

We have shown that the CIS can effectively enhance the quantum correlations and entanglement in the symmetrical cascaded FWM processes. Specifically, in the quantum correlation section, for some specific intensities combinations, the CIS can enhance their correlation degree; While for other combinations, it can only decrease the DS value between them. Similarly, the CIS can also realize the similar function for the quantum entanglement except one intrinsic 2–2 mode bipartition. The results presented here may find potential applications in the manipulations of the quantum correlations and entanglement in various quantum networks.

Funding

National Natural Science Foundation of China (11804323, 10974057, 11374104, 11874155, 61805225, 91436211); Major Scientific Research Project of Zhejiang Lab (2019DE0KF01); Natural Science Foundation of Shanghai (17ZR1442900); Program of Scientific and Technological Innovation of Shanghai (17JC1400401); National Key Research and Development Program of China (2016YFA0302103); 111 project (B12024); Fundamental Research Funds for the Central Universities (2018GZKF03006); State Key Laboratory of Advanced Optical Communication Systems and Networks.

Disclosures

The author declares no conflicts of interest.

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Figures (12)
Fig. 1.
Fig. 1. The comparison between the schemes of generating the quadruple beams with CIS (a) and without CIS (b). $\hat {a}_{1}$ is coherent input signal, $\hat {\nu }_{0}$ , $\hat {\nu }_{1}$ , and $\hat {\nu }_{2}$ are vacuum inputs, $\hat {c}_{1}$ – $\hat {c}_{4}$ are output signals. ${T}_{1}$ and ${T}_{2}$ are the transmissions for the twin beams generated from the FWM $_{1}$ process.

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Fig. 2.
Fig. 2. The dependence of Eq. (4) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\Phi _{2}=\pi$ ), (b) $T_{1}$ ( $\Phi _{1}=\Phi _{2}=\pi$ and $T_{2}=0.5$ ), (c) $\Phi _{1}$ and $\Phi _{2}$ ( $T_{1}=T_{2}=0.5$ ), and (d) $\Phi _{1}$ ( $T_{1}=T_{2}=0.5$ and $\Phi _{2}=\pi$ ). The red dashed line in Fig.  2(a) is $T_{1}+T_{2}=1$ , and the magenta dashed line (Trace B) in Figs.  2(b)–(d) is 0.04.

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Fig. 3.
Fig. 3. The comparison of the gains of the quadruple beams with the CIS (solid line) and without the CIS (dashed line). The black solid (dashed) line, the blue solid (dashed) line, the cyan solid (dashed) line, and the red solid (blue dashed) lines are the gains of the beams $\hat {C}_{1}$ , $\hat {C}_{2}$ , $\hat {C}_{3}$ , and $\hat {C}_{4}$ , respectively with the CIS (without the CIS). The vertical dashed line is $T_{1}=0.5$ .

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Fig. 4.
Fig. 4. The dependence of Eq. (7) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\pi$ ), (b) $T_{1}$ ( $\Phi _{1}=\pi$ and $T_{2}=0$ ), (c) $\Phi _{1}$ ( $T_{1}=0.9$ and $T_{2}=0$ ). The magenta dashed line (Trace B) in Figs.  4(b)–(c) is 1.

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Fig. 5.
Fig. 5. The dependence of Eq. (8) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{2}=\pi$ ), (b) $T_{1}$ ( $\Phi _{2}=\pi$ and $T_{2}=1$ ), (c) $\Phi _{2}$ ( $T_{1}=0.1$ and $T_{2}=1$ ). The magenta dashed line (Trace B) in Figs.  5(b)–(c) is 1.

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Fig. 6.
Fig. 6. The dependence of Eq. (9) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (b) $T_{1}$ ( $\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0.42$ ), (c) $\Phi _{1}$ and $\Phi _{2}$ ( $T_{1}=0.91$ and $T_{2}=0.42$ ), (d) $\Phi _{1}$ ( $\Phi _{2}=0$ , $T_{1}=0.91$ , and $T_{2}=0.42$ ), and (e) $\Phi _{1}$ around $2\pi$ in Fig.  6(d). The magenta dashed line (Trace B) is 2.6, and the red dashed line (Trace C) is 1.

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Fig. 7.
Fig. 7. The dependence of Eq. (12) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (b) $T_{1}$ ( $\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0.58$ ), (c) $\Phi _{1}$ and $\Phi _{2}$ ( $T_{1}=0.09$ and $T_{2}=0.58$ ), (d) $\Phi _{1}$ ( $\Phi _{2}=0$ , $T_{1}=0.09$ , and $T_{2}=0.58$ ), and (e) $\Phi _{1}$ around 2 $\pi$ in Fig.  6(d). The magenta dashed line (Trace B) is 3.4, and the red dashed line (Trace C) is 1.

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Fig. 8.
Fig. 8. The dependence of Eq. (13) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (b) $T_{1}$ ( $\Phi _{1}=\Phi _{2}=0$ and $T_{2}=0$ ), (c) $\Phi _{1}$ and $\Phi _{2}$ ( $T_{1}=1$ and $T_{2}=0$ ), (d) $\Phi _{1}$ ( $\Phi _{2}=0$ , $T_{1}=1$ , and $T_{2}=0$ ). The magenta dashed line (Trace B) is 4.7, and the red dashed line (Trace C) is 1.

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Fig. 9.
Fig. 9. The dependence of Eq. (16) on (a) $T_{1}$ and $T_{2}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (b) $\Phi _{1}$ and $\Phi _{2}$ ( $T_{1}=T_{2}=0.5$ ). The red dashed line is $T_{1}+T_{2}=1$ , and the magenta dashed line is 3.

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Fig. 10.
Fig. 10. The dependence of (a) $\nu _{1234}^{1}$ ( $\Phi _{1}=\pi$ ), (b) $\nu _{1234}^{2}$ ( $\Phi _{2}=\pi$ ), (c) $\nu _{1234}^{3}$ ( $\Phi _{2}=\pi$ ), and (d) $\nu _{1234}^{4}$ ( $\Phi _{1}=\pi$ ) on $T_{1}$ and $T_{2}$ . The magenta dashed curve is 0.029 and 0.039 in Figs.  10(a)–(b) and Figs.  10(c)–(d), respectively.

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Fig. 11.
Fig. 11. The dependence of (a) $\nu _{1234}^{12}$ ( $\Phi _{2}=\pi$ ), (b) $\nu _{1234}^{13}$ ( $\Phi _{1}=\Phi _{2}=\pi$ ) on $T_{1}$ and $T_{2}$ . The magenta dashed curve is 0.059 in Fig.  11(a), and the red dashed curve is $T_{1}+T_{2}=1$ in Fig.  11(b).

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Fig. 12.
Fig. 12. The dependence of (a) $\nu _{12}^{1}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (c) $\nu _{34}^{3}$ ( $\Phi _{1}=\Phi _{2}=0$ ), (e) $\nu _{14}^{1}$ ( $\Phi _{1}=\pi$ ), (f) $\nu _{23}^{2}$ ( $\Phi _{2}=\pi$ ), (g) $\nu _{13}^{1}$ ( $\Phi _{2}=0$ ), and (h) $\nu _{24}^{2}$ ( $\Phi _{1}=0$ ) on $T_{1}$ and $T_{2}$ ; The dependence of (b) $\nu _{12}^{1}$ ( $T_{1}=T_{2}=0.6$ ) and (d) $\nu _{34}^{3}$ ( $T_{1}=T_{2}=0.4$ ) on $\Phi _{1}$ and $\Phi _{2}$ . The red dashed curve is 1 in Figs.  12(a)–(d), the magenta dashed curve in Figs.  12(b), (d), (g), and (h) is 2.3, 3.2, 7, and 7, respectively.

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Equations (17)

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C ^ 1 = C 11 a ^ 1 + C 12 ν ^ 0 † + C 13 ν ^ 1 + C 14 ν ^ 2 † , C ^ 2 = C 21 a ^ 1 † + C 22 ν ^ 0 + C 23 ν ^ 1 † + C 24 ν ^ 2 , C ^ 3 = C 31 a ^ 1 + C 32 ν ^ 0 † + C 33 ν ^ 1 + C 34 ν ^ 2 † , C ^ 4 = C 41 a ^ 1 † + C 42 ν ^ 0 + C 43 ν ^ 1 † + C 44 ν ^ 2 ,
C 11 = G 1 G 2 T 1 − g 1 g 2 R 2 e i Φ 1 , C 12 = G 2 g 1 T 1 − G 1 g 2 R 2 e i Φ 1 , C 13 = G 2 R 1 , C 14 = g 2 T 2 e i Φ 1 , C 21 = G 3 g 1 T 2 − G 1 g 3 R 1 e i Φ 2 , C 22 = G 1 G 3 T 2 − g 1 g 3 R 1 e i Φ 2 , C 23 = g 3 T 1 e i Φ 2 , C 24 = G 3 R 2 , C 31 = g 1 g 3 T 2 e i Φ 2 − G 1 G 3 R 1 , C 32 = G 1 g 3 T 2 e i Φ 2 − G 3 g 1 R 1 , C 33 = G 3 T 1 , C 34 = g 3 R 2 e i Φ 2 , C 41 = G 1 g 2 T 1 e i Φ 1 − G 2 g 1 R 2 , C 42 = g 1 g 2 T 1 e i Φ 1 − G 1 G 2 R 2 , C 43 = g 2 R 1 e i Φ 1 , C 44 = G 2 T 2 ,
D S ( N i c ) = V a r ( N i c ) F W M V a r ( N i c ) S Q L .
D S 1234 = V a r ( N 1 − N 2 + N 3 − N 4 ) F W M V a r ( N 1 − N 2 + N 3 − N 4 ) S Q L = 1 ( 2 G 1 − 1 ) ( 2 G 2 − 1 ) − α ,
D S 1234 m i n = 1 ( 2 G 1 − 1 ) ( 2 G 2 − 1 ) + 4 G 1 G 2 g 1 g 2 .
G c 1 = G 1 G 2 T 1 + g 1 g 2 R 2 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 , G c 2 = G 2 g 1 T 2 + G 1 g 2 R 1 − 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 , G c 3 = g 1 g 2 T 2 + G 1 G 2 R 1 − 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 , G c 4 = G 1 g 2 T 1 + G 2 g 1 R 2 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 ,
D S 14 = ( G 1 T 1 − g 1 R 2 ) 2 + ( T 1 − R 2 ) 2 G 1 g 1 + β γ ( G 1 T 1 + g 1 R 2 ) − 4 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 ,
D S 23 = ( g 1 T 2 − G 1 R 1 ) 2 + ( T 2 − R 1 ) 2 G 1 g 1 + β γ ( g 1 T 2 + G 1 R 1 ) − 4 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 .
D S 12 = D S 12 _ n D S 12 _ d ,
D S 12 _ n = ( G 1 G 2 T 1 + g 1 g 2 R 2 − G 2 g 1 T 2 − G 1 g 2 R 1 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 + 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 ) 2 + [ G 1 g 1 ( G 2 T 1 − G 2 T 2 − g 2 R 1 + g 2 R 2 ) + ( 2 G 1 − 1 ) ( G 2 g 2 T 2 R 1 cos ⁡ Φ 2 − G 2 g 2 T 1 R 2 cos ⁡ Φ 1 ) ] 2 + ( G 2 g 2 T 2 R 1 sin ⁡ Φ 2 − G 2 g 2 T 1 R 2 sin ⁡ Φ 1 ) 2 + [ ( 2 G 2 − 1 ) G 1 T 1 R 1 − G 2 g 1 g 2 R 1 R 2 cos ⁡ Φ 1 − G 2 g 1 g 2 T 1 T 2 cos ⁡ Φ 2 ] 2 + ( G 2 g 1 g 2 R 1 R 2 sin ⁡ Φ 1 + G 2 g 1 g 2 T 1 T 2 sin ⁡ Φ 2 ) 2 + [ G 1 G 2 g 2 T 1 T 2 cos ⁡ Φ 1 + G 1 G 2 g 2 R 1 R 2 cos ⁡ Φ 2 − ( 2 G 2 − 1 ) g 1 T 2 R 2 ] 2 + ( G 1 G 2 g 2 T 1 T 2 sin ⁡ Φ 1 + G 1 G 2 g 2 R 1 R 2 sin ⁡ Φ 2 ) 2
D S 12 _ d = G 1 G 2 T 1 + g 1 g 2 R 2 + G 2 g 1 T 2 + G 1 g 2 R 1 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 − 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 ,
D S 34 = D S 12 _ n ( G 2 → g 2 , g 2 → G 2 ) D S 12 _ d ( G 2 → g 2 , g 2 → G 2 ) ,
D S 13 = D S 13 _ n D S 13 _ d ,
D S 13 _ n = ( G 1 G 2 T 1 + g 1 g 2 R 2 − g 1 g 2 T 2 − G 1 G 2 R 1 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 + 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 ) 2 + [ G 1 g 1 ( G 2 T 1 − g 2 T 2 − G 2 R 1 + g 2 R 2 ) + ( 2 G 1 − 1 ) ( G 2 g 2 T 2 R 1 cos ⁡ Φ 2 − G 2 g 2 T 1 R 2 cos ⁡ Φ 1 ) ] 2 + ( G 2 g 2 T 2 R 1 sin ⁡ Φ 2 − G 2 g 2 T 1 R 2 sin ⁡ Φ 1 ) 2 + ( 2 G 2 G 1 T 1 R 1 − G 2 g 1 g 2 R 1 R 2 cos ⁡ Φ 1 − G 2 g 1 g 2 T 1 T 2 cos ⁡ Φ 2 ) 2 + ( G 2 g 1 g 2 R 1 R 2 sin ⁡ Φ 1 + G 2 g 1 g 2 T 1 T 2 sin ⁡ Φ 2 ) 2 + [ G 1 G 2 g 2 T 1 T 2 cos ⁡ Φ 1 + G 1 G 2 g 2 R 1 R 2 cos ⁡ Φ 2 − 2 g 2 g 1 T 2 R 2 ] 2 + ( G 1 G 2 g 2 T 1 T 2 sin ⁡ Φ 1 + G 1 G 2 g 2 R 1 R 2 sin ⁡ Φ 2 ) 2
D S 13 _ d = G 1 G 2 T 1 + g 1 g 2 R 2 + g 1 g 2 T 2 + G 1 G 2 R 1 − 2 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 − 2 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 ,
D S 24 = D S 13 _ n ( G 2 → g 2 , g 2 → G 2 ) D S 13 _ d ( G 2 → g 2 , g 2 → G 2 ) ,
⟨ X ^ c 1 2 ⟩ = ⟨ Y ^ c 1 2 ⟩ = ( 2 G 1 − 1 ) ( G 2 T 1 + g 2 R 2 ) + G 2 R 1 + g 2 T 2 − 4 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 , ⟨ X ^ c 2 2 ⟩ = ⟨ Y ^ c 2 2 ⟩ = ( 2 G 1 − 1 ) ( G 2 T 2 + g 2 R 1 ) + G 2 R 2 + g 2 T 1 − 4 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 , ⟨ X ^ c 3 2 ⟩ = ⟨ Y ^ c 3 2 ⟩ = ( 2 G 1 − 1 ) ( G 2 R 1 + g 2 T 2 ) + G 2 T 1 + g 2 R 2 − 4 G 1 G 2 g 1 g 2 T 2 R 1 cos ⁡ Φ 2 , ⟨ X ^ c 4 2 ⟩ = ⟨ Y ^ c 4 2 ⟩ = ( 2 G 1 − 1 ) ( G 2 R 2 + g 2 T 1 ) + G 2 T 2 + g 2 R 1 − 4 G 1 G 2 g 1 g 2 T 1 R 2 cos ⁡ Φ 1 , ⟨ X ^ c 1 X ^ c 2 ⟩ = − ⟨ Y ^ c 1 Y ^ c 2 ⟩ = 2 G 2 G 1 g 1 T 1 T 2 + 2 g 2 G 1 g 1 R 1 R 2 cos ⁡ ( Φ 1 + Φ 2 ) − 2 g 1 G 2 g 2 T 2 R 2 cos ⁡ Φ 1 − 2 g 1 G 2 g 2 T 1 R 1 cos ⁡ Φ 2 , ⟨ X ^ c 1 X ^ c 3 ⟩ = ⟨ Y ^ c 1 Y ^ c 3 ⟩ = − 2 G 2 g 1 T 1 R 1 − 2 g 1 g 2 T 2 R 2 cos ⁡ ( Φ 1 − Φ 2 ) + 2 G 1 G 2 g 1 g 2 R 1 R 2 cos ⁡ Φ 1 + 2 G 1 G 2 g 1 g 2 T 1 T 2 cos ⁡ Φ 2 , ⟨ X ^ c 1 X ^ c 4 ⟩ = − ⟨ Y ^ c 1 Y ^ c 4 ⟩ = [ ( 2 G 1 − 1 ) ( T 1 + R 2 ) + R 1 + T 2 ] G 2 g 2 cos ⁡ Φ 1 − 2 G 1 g 1 T 1 R 2 ( G 2 + g 2 cos ⁡ 2 Φ 1 ) , ⟨ X ^ c 2 X ^ c 3 ⟩ = − ⟨ Y ^ c 2 Y ^ c 3 ⟩ = [ ( 2 G 1 − 1 ) ( T 2 + R 1 ) + R 2 + T 1 ] G 2 g 2 cos ⁡ Φ 2 − 2 G 1 g 1 T 2 R 1 ( G 2 + g 2 cos ⁡ 2 Φ 2 ) , ⟨ X ^ c 2 X ^ c 4 ⟩ = ⟨ Y ^ c 2 Y ^ c 4 ⟩ = − 2 G 2 g 1 T 2 R 2 − 2 g 1 g 2 T 1 R 1 cos ⁡ ( Φ 1 − Φ 2 ) + 2 G 1 G 2 g 1 g 2 T 1 T 2 cos ⁡ Φ 1 + 2 G 1 G 2 g 1 g 2 R 1 R 2 cos ⁡ Φ 2 , ⟨ X ^ c 3 X ^ c 4 ⟩ = − ⟨ Y ^ c 3 Y ^ c 4 ⟩ = 2 G 2 G 1 g 1 R 1 R 2 + 2 g 2 G 1 g 1 T 1 T 2 cos ⁡ ( Φ 1 + Φ 2 ) − 2 g 1 G 2 g 2 T 1 R 1 cos ⁡ Φ 1 − 2 g 1 G 2 g 2 T 2 R 2 cos ⁡ Φ 2 .
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