## Abstract

We investigate the possibility of an experimentally feasible cascaded four-wave mixing (FWM) system [Phys. Rev. Lett. **113**, 023602 (2014)] to generate tripartite entanglement. We verify that genuine tripartite entanglement is present in this system by calculating the covariances of three output beams and then considering the violations of the inequalities of the three-mode entanglement criteria, such as two-condition criterion, single-condition criterion, optimal single-condition criterion and the positivity under partial transposition (PPT) criterion. We also consider the possibilities of the bipartite entanglement of any pair of the three output beams using the Duan-Giedke-Cirac-Zoller criterion and PPT criterion. We find that the tripartite entanglement and the bipartite entanglement for the two pairs are present in the whole gain region. The entanglement characteristics under different entanglement criteria are also considered. Our results pave the way for the realization and application of multipartite entanglement based on the cascaded FWM processes.

© 2016 Optical Society of America

## 1. Introduction

Multipartite entanglement is important not only for fundamental tests of quantum effects [1, 2] but also for their numerous possible applications in quantum technologies [3]. A number of different schemes for the generation of multipartite entangled states have been already theoretically proposed and experimentally implemented. The most well established technique consists of mixing several independently generated single mode squeezed states in a linear optical interferometer made of beamsplitters and propagation over well-controlled distances [4–7]. Recent progress towards scalable quantum optical networks includes the experimental generation of ultra-large-scale continuous variable (CV) cluster state multiplexed in both the time domain [8] and the frequency domain [9–11]. Meanwhile, there are many different kinds of criteria for detecting multipartite entanglement [12–16]. Among them, positivity under partial transposition (PPT) criterion is a necessary and sufficient criterion for Gaussian state under certain conditions while most of them are the sufficient criteria.

Four-wave mixing (FWM) is an effective *χ*^{(3)} nonlinear technique to generate twin quantum correlated beams and bipartite entangled beams [17] in the CV domain. Recently, it has been reported that by cascading two FWM processes, tunable delay of Einstein–Podolsky–Rosen entangled states [18], low-noise amplification of an entangled states [19], SU(1,1) interferometer [20, 21], and quantum mutual information [22] have been experimentally realized.

Very recently, our group has theoretically proposed and experimentally demonstrated a cascaded FWM processes to produce multiple quantum correlated beams in hot atomic vapor [23]. In order to experimentally realize the tripartite entanglement in the cascaded FWM system, it is necessary to theoretically investigate its system parameter dependence. There are also some other cascaded systems studied by other groups for generating multipartite entanglement. According to the different ways of cascading, we could classify the related works into three categories, including cascading pump, cascading seed&pump and cascading seed. For the case of cascading pump, in the work [24], by successively employing the pump reflected by the first optical parametric oscillator (OPO) to pump the second one, the cascading pump act as an entanglement distributor among two OPOs, the existence of pentapartite entanglement (two signal beams+two idler beams+one pump beam) in this system is demonstrated using the PPT criterion. For the case of cascading seed&pump, one of signal and idler beams produced by the first nondegenerate optical parametric amplifiers (NOPO) can be used for the pump light of the second NOPO. The three-color entanglement among signal and idler beams produced by the second NOPO and the retained another beam of the first NOPO is theoretically proposed [25] and experimentally demonstrated [26]. The similar works [27, 28] use the cascaded parametric down-conversion processes to generate and observe the photon triplet and the three-photon energy-time entanglement, respectively. For the case of cascading seed, different from the above related works, in our work, one of signal and idler beams produced by the first FWM is used for the seed light of the second FWM. The tripartite entanglement among one signal and two idler beams produced by the two cascaded FWM processes is theoretically demonstrated using the three-mode entanglement criteria. In addition, our system has the following advantages that ensures its scalability. Firstly, as the number of quantum modes increases, so does the total degree of quantum correlations. Secondly, this method has advantage of being spatially separated due to the spatial multimode nature of the FWM system. Finally, this method is phase insensitive without the need of complicated phase locking technique. Due to the above advantages, this cascaded proposal is a promising candidate for tripartite entanglement generation and has many potential applications in the fields of quantum networks. Inspired by this, in this paper we theoretically analyze the dependence of genuine tripartite entanglement on the power gains of the system in the cascaded FWM processes using the three-mode entanglement criteria [13, 15, 16], such as two-condition criterion, single-condition criterion, optimal single-condition criterion and the PPT criterion.

This article is organized as follows. In Sec. 2 we describe the physical system of the cascaded FWM processes under consideration. In Sec. 3 we use the Duan-Giedke-Cirac-Zoller (DGCZ) criterion [12], and PPT criterion [13, 16, 29, 30] to characterize bipartite entanglement potentially existed in this system. In Sec. 4 we use the two-condition criterion, the single-condition criterion, the optimal single-condition criterion [15], and PPT criterion [13, 16, 31] to characterize the tripartite entanglement. In Sec. 5 we give a brief summary of this paper.

## 2. Cascaded FWM processes

Firstly, we give the optical quadrature definitions in our analysis, as the criteria will depend on these quadratures. For the three modes described by the bosonic annihilation oparators *â _{k}* involved in our system, where

*k*=1, 2, 3, we define quadrature operators for each mode as

*X̂*,

_{k}*Ŷ*]=2

_{k}*i*,

*X̂*and

_{k}*Ŷ*are respectively the amplitude and phase quadratures.

_{k}The energy level diagram of a single FWM process is shown in Fig. 1(b), where two pump photons can convert to one signal photon and one idler photon, or vice versa. In the cell_{1} of Fig. 1(a), an intense pump beam and a much weaker signal beam are crossed in the center of the Rb vapor cell with a slight angle. Then the signal beam is amplified and a new beam called idler beam is generated on the other side of the pump beam at the same time. The signal beam and idler beam have different frequencies. We then construct two similar cascaded FWM processes based on the single FWM process in Fig. 1(a) and Fig. 1(c). We take the signal beam from the first FWM process (cell_{1}) as the seed for the second FWM process (cell_{2}) in Fig. 1(a) and the idler beam from the first FWM process (cell_{1}) as the seed for the second FWM process (cell_{2}) in Fig. 1(c). *â*_{1}, *â*_{2} and *â*_{3} are three newly-generated beams in the output stage of the cascaded processes. For convenience, here we will focus on the genuine tripartite entanglement (*â*_{1}, *â*_{2} and *â*_{3}) existed in the Fig. 1(a). A similar analysis can be made for the tripartite entanglement in Fig. 1(c). The input-output relation of the cascaded FWM processes in Fig. 1(a) can be written as

*â*

_{0}is coherent input signal,

*â*

_{ν1}and

*â*

_{ν2}are vacuum inputs.

*G*(

_{j}*j*=1, 2) is the power gain in the FWM process and

*G*−

_{j}*g*=1. Eq. (2) can be rewritten in terms of the quadrature operators using Eq. (1) as follows

_{j}*V*= (〈

_{mn}*X̂*〉 + 〈

_{m}X̂_{n}*X̂*〉)/2 − 〈

_{n}X̂_{m}*X̂*〉〈

_{m}*X̂*〉 and for the case where

_{n}*m*=

*n*, the covariance, denoted

*V*, reduces to the usual variance,

_{mn}*V*(

*X̂*). In fact, the variances are given by the following gain-dependent moments (we assume that

_{m}*â*

_{0}is vacuum input in the following calculations because homodyne detection technique needs three output beams

*â*

_{1},

*â*

_{2}and

*â*

_{3}are all at vacuum level.)

*X̂*〉 = 〈

_{m}X̂_{n}*Ŷ*〉 =

_{m}Ŷ_{n}*δ*(

_{mn}*m*,

*n*= 0,

*ν*1 and

*ν*2). A similar approach can be used to calculate the covariances which are equivalent to the gain-dependent moments 〈

*X̂*〉 and 〈

_{m}X̂_{n}*Ŷ*〉. The covariances are given by the following gain-dependent moments

_{m}Ŷ_{n}## 3. Bipartite entanglement

#### 3.1. DGCZ criterion

In this section, we will focus on the bipartite entanglement potentially existed in the system. Firstly, we use the DGCZ criterion [12] to analyze the bipartite entanglement potentially existed in the cascaded FWM processes. Using our quadrature definitions in Eq. (1), the inequalities of the criteria are written as below

*D*

_{12},

*D*

_{13}and

*D*

_{23}on the gains

*G*

_{1}and

*G*

_{2}as follows

The violation of the first inequality in Eq. (7) can be used to demonstrate the bipartite entanglement between *â*_{1} and *â*_{2}, so also can the second (third) inequality in Eq. (7) be used to demonstrate the bipartite entanglement between *â*_{1} (*â*_{2}) and *â*_{3} (*â*_{3}). The region plot of Eq. (7) is shown in Fig. 2(a). The entanglement region of *â*_{1} and *â*_{2} (*D*_{12}<4) is the region *B* in Fig. 2(a). As shown in Fig. 2(b), if we set *G*_{2} ≈1, the cascaded FWM processes reduce to the single FWM process, i. e., the first FWM process (cell_{1}). Under this condition, *â*_{1} and *â*_{2} is a simple bipartite entangled beams case. As *G*_{1} is getting larger, *D*_{12} is limited by smaller *G*_{2}. This is because only beam *â*_{2} experiences the quantum amplification from the second FWM process (cell_{2}), which leads to their noise unbalance, thus the performance of the entanglement between beams *â*_{1} and *â*_{2} is very sensitive to the *G*_{2}. Similarly, The entanglement region of *â*_{2} and *â*_{3} (*D*_{23}<4) is the region *A* in Fig. 2(a). As shown in Fig. 2(c), if we set *G*_{1} ≈1, the cascaded FWM processes reduce to the single FWM process, i. e., the second FWM process (cell_{2}). In this case, *â*_{2} and *â*_{3} is also a simple bipartite entangled beams case. It should be noted that the value of *D*_{13} is always more than or equal to 4 for any *G*_{1}, *G*_{2}>1.

#### 3.2. PPT criterion

Secondly, compared with the above sufficient criterion (DGCZ criterion), here a necessary and sufficient criterion, i. e., PPT criterion, can also be used to characterize the bipartite entanglement potentially existed in the system. For example, the entanglement between *â*_{1} and *â*_{2} can be completely characterized by their covariance matrix (CM_{12}) which can be written as

*â*

_{1}and

*â*

_{2}is absent if and only if both of the symplectic eigenvalues of the partially transposed (PT) CM

_{12}are greater than or equal to 1 [13,16,29,30]. Following this idea, the entanglement between

*â*

_{1}and

*â*

_{2}can be characterized by the smaller symplectic eigenvalue

*B*

_{1}. If the smaller symplectic eigenvalue

*B*

_{1}is smaller than 1, the bipartite entanglement exists between

*â*

_{1}and

*â*

_{2}. Substituting Eq. (5) and Eq. (6) in Eq. (9), we can get the detailed expression for

*B*

_{1}as shown below

*B*

_{1}is shown in Fig. 3(a). It can be proved that the value of the whole region for any

*G*

_{1},

*G*

_{2}>1 is smaller than 1 meaning that

*â*

_{1}is entangled with

*â*

_{2}. Similarly, the smaller symplectic eigenvalue

*B*

_{2}of PT CM

_{23}of

*â*

_{2}and

*â*

_{3}is given by

*G*

_{1},

*G*

_{2}>1 meaning that

*â*

_{2}is entangled with

*â*

_{3}.

In addition, the smaller symplectic eigenvalue *B*_{3} of PT CM_{13} of *â*_{1} and *â*_{3} is given by

*G*

_{1},

*G*

_{2}>1 as shown in Fig. 3(c) meaning that

*â*

_{1}is not entangled with

*â*

_{3}. This is due to the fact that

*â*

_{1}and

*â*

_{3}never interact with each other directly.

## 4. Tripartite entanglement

#### 4.1. Two-condition criterion

In this section, the tripartite entanglement will be investigated using different criteria, i. e., two-condition criterion, single-condition criterion, the optimal single-condition criterion and the PPT criterion. Firstly, we will consider the genuine tripartite entanglement between the three output modes from the cascaded FWM system using the two-condition criterion [15]. Using our quadrature definitions in Eq. (1), the two-condition criterion gives a set of inequalities

*O*(

_{i}*i*=1, 3) are arbitrary real numbers. The violation of the two inequalities in Eq. (13) is sufficient to demonstrate genuine tripartite entanglement. We will investigate the optimization of the two-condition criterion using the freedom allowed in the choice of the

*O*. The optimal expressions of

_{i}*O*( ${O}_{i}^{\mathit{opt}}$) can be obtained by the direct differentiation for the left-hand side of Eq. (13) with respect to the

_{i}*O*, then we get

_{i}The contour plot of Eq. (13) is shown in Fig. 4. The dependence of *V*_{12} on *G*_{1} and *G*_{2} is shown in Fig. 4(a). The region of *V*_{12}<4 is enlarged compared to the one of *D*_{12} <4 in Fig. 2(b) when we consider the phase quadrature of *â*_{3} (*Ŷ*_{3}). The variance of *Ŷ*_{1}+*Ŷ*_{2}+*Ŷ*_{3} becomes smaller than the one of *Ŷ*_{1}+*Ŷ*_{2} which claims that *â*_{3} has correlation with *â*_{1}+*â*_{2}. The dependence of *V*_{23} on *G*_{1} and *G*_{2} is shown in Fig. 4(b), the region of *V*_{23}<4 is enlarged compared to the one of *D*_{23}<4 in Fig. 2(c) when we consider the phase quadrature of *â*_{1} (*Ŷ*_{1}). The variance of *Ŷ*_{1}+*Ŷ*_{2}+*Ŷ*_{3} becomes smaller than the one of *Ŷ*_{2}+*Ŷ*_{3} which claims that *â*_{1} has correlation with *â*_{2}+*â*_{3}.

The overlapped region of *V*_{12} <4 and *V*_{23} <4 in Fig. 4(c) means that the genuine tripartite entanglement is present in this system.

#### 4.2. Single-condition criterion

Secondly, for simplicity, it is also possible to develop a single-condition criterion to demonstrate genuine tripartite entanglement using the combined quadrature variances [15]. If one of the following formulas in Eq. (15) is less than 2, we could claim that there exists genuine tripartite entanglement in this system

The dependence of *V*_{213} on the gains *G*_{1} and *G*_{2} can be written as follows

The contour plot of *V*_{213} is shown in Fig. 5, the value of most of the region in Fig. 5 is less than 2. We can use this entanglement criterion to directly detect the three-mode entanglement, and it also clearly shows the existence of tripartite entanglement in this cascaded system. But the values of *V*_{312} and *V*_{123} are always bigger than 2 due to the fact that *â*_{2} is correlated with both *â*_{1} and *â*_{3}, whereas *â*_{1} and *â*_{3} are not correlated with each other directly. In fact, they are from two independent FWM processes in serial connection.

#### 4.3. Optimal single-condition criterion

Inspired by above results in Sec. 4.2, we should pay more attention to the optimization of *V*_{213} (
${V}_{213}^{\mathit{opt}}$) through introducing different factors [15] to cancel the impact of noise unbalance between *â*_{1} and *â*_{3}. In this way, we can express the
${V}_{213}^{\mathit{opt}}$ as follows

*F*

_{1}and

*F*

_{3}are arbitrary real numbers whose optimal expressions ( ${F}_{1}^{\mathit{opt}}$ and ${F}_{3}^{\mathit{opt}}$) can be obtained by differentiating either

*X̂*

_{2}−

*F*

_{1}

*X̂*

_{1}−

*F*

_{3}

*X̂*

_{3}or

*Ŷ*

_{2}+

*F*

_{1}

*Ŷ*

_{1}+

*F*

_{3}

*Ŷ*

_{3}with respect to

*F*

_{1}and

*F*

_{3}, then we get

The contour plot of
${V}_{213}^{\mathit{opt}}$ is shown in Fig. 6(a), it depends on *G*_{1} and *G*_{2} symmetrically which is similar to the squeezing degree of the triple beams (1/[2*G*_{1}*G*_{2} − 1]) [23]. Thus in the following discussions we will analyze the tripartite entanglement using the optimal single-condition criterion in the condition *G*=*G*_{1}=*G*_{2}. In this case, Eq. (17) can be reduced to

${F}_{1}^{\mathit{opt}}$ and
${F}_{3}^{\mathit{opt}}$ in Eq. (20) tend to
$\sqrt{1/G}$ and 1 respectively when *G* becomes infinite, resulting that the coefficients of the quadratures from the modes *â*_{2} and *â*_{3} become equal. This makes sense because when *G* becomes infinite, the powers of the modes *â*_{2} and *â*_{3} are equal and it is unnecessary to introduce different factors to cancel the noise unbalance between the modes *â*_{2} and *â*_{3}.

To confirm the existence of the tripartite entanglement in this system, we should consider whether Eq. (17) is smaller than its boundaries according to Ref. [15]. The boundaries of
${V}_{213}^{\mathit{opt}}$ are
$2\left(\left|{F}_{1}^{2}\right|+\left|1-{F}_{3}^{2}\right|\right)$ (*â*_{2} and *â*_{3} may be entangled or not, but *â*_{1} is not entangled with the rest),
$2\left(\left|{F}_{3}^{2}\right|+\left|1-{F}_{1}^{2}\right|\right)$ (*â*_{1} and *â*_{2} may be entangled or not, but *â*_{3} is not entangled with the rest), and
$2\left(1+\left|{F}_{1}^{2}+{F}_{3}^{2}\right|\right)$ (*â*_{1} and *â*_{3} may be entangled or not, but *â*_{2} is not entangled with the rest). To answer this question, we plot the dependence of
${V}_{213}^{\mathit{opt}}$ (Trace *A*),
$2\left(1+\left|{F}_{1}^{2}+{F}_{3}^{2}\right|\right)$ (Trace *B*),
$2\left(\left|{F}_{3}^{2}\right|+\left|1-{F}_{1}^{2}\right|\right)$ (Trace *C*), and
$2\left(1+\left|{F}_{1}^{2}+{F}_{3}^{2}\right|\right)$ (Trace *D*) on *G* in Fig. 6(b). As shown in Fig. 6(b),
$2\left(\left|{F}_{1}^{2}\right|+\left|1-{F}_{3}^{2}\right|\right)$ (Trace *B*) decreases with the increasing of *G*,
$2\left(\left|{F}_{3}^{2}\right|+\left|1-{F}_{1}^{2}\right|\right)$ (Trace *C*) and
$2\left(1+\left|{F}_{1}^{2}+{F}_{3}^{2}\right|\right)$ (trace D) increase with the increasing of *G*.

As shown in Fig. 6(b),
$2\left(\left|{F}_{1}^{2}\right|+\left|1-{F}_{3}^{2}\right|\right)$ (Trace *B*) is the smallest one of all the boundaries which means that tripartite entanglement is present if
${V}_{213}^{\mathit{opt}}$ (Trace *A*) is smaller than
$2\left(\left|{F}_{1}^{2}\right|+\left|1-{F}_{3}^{2}\right|\right)$ (Trace *B*). Thus we define the quantity
$R={V}_{213}^{\mathit{opt}}/2\left(\left|{F}_{1}^{2}\right|+\left|1-{F}_{3}^{2}\right|\right)$ to characterize the dependence of tripartite entanglement on *G* and *R* < 1 means that tripartite entanglement is present. As shown in Fig. 6(c), the tripartite entanglement is present in the whole gain region. Therefore, the optimal single-condition criterion can be used to characterize the tripartite entanglement in this cascaded system.

#### 4.4. PPT criterion

Finally, the PPT criterion used to characterize the bipartite entanglement can also be applied to the tripartite entanglement because the PPT criterion is also a necessary and sufficient criterion for all 1× N decompositions of Gaussian states, where 1+N is the total number of the entangled modes [13, 16, 31]. For the tripartite entanglement condition, the three possible 1× 2 partitions have to be tested. All the partitions of the three-mode state are inseparable when the smallest symplectic eigenvalue for each of the three PT covariance matrices is smaller than 1, i. e., genuine tripartite entanglement is present in this system. Generally, the covariance martix CM_{123} for the three modes can be written as

*â*

_{1}, the entanglement between

*â*

_{1}and the rest beams (

*â*

_{2}and

*â*

_{3}) can be characterized by the smallest symplectic eigenvalue

*T*

_{1}. Substituting Eq. (5) and Eq. (6) in Eq. (21), we can get the detailed expression of

*T*

_{1}which is not listed here due to its complexity. As shown in Fig. 7(a), the value of

*T*

_{1}in the whole gain region is smaller than 1 for any

*G*

_{1},

*G*

_{2}>1 meaning that beam

*â*

_{1}is entangled with the rest beams (

*â*

_{2}and

*â*

_{3}). It should be noted that the behavior of

*T*

_{1}is independent on the gain

*G*

_{2}. This is due to the fact that the entanglement between beam

*â*

_{1}and the rest beams (

*â*

_{2}and

*â*

_{3}) is only decided by the first FWM process (cell

_{1}). Similarly, when the PT operation is applied to the beam

*â*

_{2}(

*â*

_{3}), the smallest symplectic eigenvalue is

*T*

_{2}(

*T*

_{3}). The contour plots of

*T*

_{2}and

*T*

_{3}are shown in Fig. 7(b) and Fig. 7(c) respectively. It can be clearly seen that the values of

*T*

_{2}and

*T*

_{3}are both smaller than 1 for any

*G*

_{1},

*G*

_{2}>1 in the whole gain region, showing the presence of the entanglement for these two partitions. Therefore, the three partitions

*â*

_{1}−(

*â*

_{2},

*â*

_{3}),

*â*

_{2}−(

*â*

_{1},

*â*

_{3}), and

*â*

_{3}−(

*â*

_{1},

*â*

_{2}) are all inseparable and the genuine tripartite entanglement between the three beams

*â*

_{1},

*â*

_{2}and

*â*

_{3}is present for the whole gain region with any

*G*

_{1},

*G*

_{2}>1.

## 5. Conclusion

In summary, we have theoretically predicted that the cascaded FWM processes is a simple system for tripartite entanglement generation. We have used the DGCZ criterion, and the PPT criterion to characterize the bipartite entanglement potentially existed in the cascaded FWM processes. Then we have analyzed the tripartite entanglement using two-condition criterion and gave the region in which the tripartite entanglement is present. Single-condition criterion can also be used to search for a more simpler measurement scheme. We have proposed an optimal single-condition criterion to characterize the tripartite entanglement in this system. Under this optimal single condition criterion, the tripartite entanglement is present in the whole gain gain region. More importantly, the PPT criterion as a necessary and sufficient criterion for Gaussian state under certain conditions has also been used to demonstrate the genuine tripartite entanglement. Under the PPT criterion, the genuine tripartite entanglement is also present in the whole gain region. Our cascaded system for generating triple quantum entangled beams is scalable, simple and phase insensitive, and it can be extended to generate multiple quantum entangled beams via cascading more FWM processes.

The major experimental limitation concerning the maximum number of entangled beams comes from the increasing of the number of the atomic vapor cells. This can be solved by integrating all the pumps into a single cell and crossing them with the probe beam one by one.

## Funding

This work was supported by the National Natural Science Foundation of China (91436211, 11374104, 10974057,11547141,11604297); the SRFDP (20130076110011); the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; the Program for New Century Excellent Talents in University (NCET-10-0383); the Shu Guang project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (11SG26); the Shanghai Pujiang Program (09PJ1404400); the Scientific Research Foundation of the Returned Overseas Chinese Scholars, State Education Ministry; and Program of State Key Laboratory of Advanced Optical Communication Systems and Networks (2016GZKF0JT003).

## Acknowledgments

J. J. would like to thank A. M. Marino for useful discussions.

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