## 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

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 $\Phi _{2}=\pi$.

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

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

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

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

where the numerator DS$_{12}\_n$ and denominator DS$_{12}\_d$ of DS$_{12}$ areAfter 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

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

where the numerator DS$_{13}\_n$ and denominator DS$_{13}\_d$ of DS$_{13}$ areThe 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

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

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.

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.

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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