Generation of tripartite entanglement from cascaded four-wave mixing processes

Hailong Wang; Zhan Zheng; Yaxian Wang; Jietai Jing

doi:10.1364/OE.24.023459

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

{\hat{X}}_{k} = {\hat{a}}_{k} + {\hat{a}}_{k}^{†}, {\hat{Y}}_{k} = - i ({\hat{a}}_{k} - {\hat{a}}_{k}^{†}),

such that [X̂_k, Ŷ_k]=2i, X̂_k and Ŷ_k are respectively the amplitude and phase quadratures.

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 cell₁ 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 (cell₁) as the seed for the second FWM process (cell₂) in Fig. 1(a) and the idler beam from the first FWM process (cell₁) as the seed for the second FWM process (cell₂) in Fig. 1(c). â₁, â₂ and â₃ are three newly-generated beams in the output stage of the cascaded processes. For convenience, here we will focus on the genuine tripartite entanglement (â₁, â₂ and â₃) 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

\begin{array}{l} {\hat{a}}_{1} & = & \sqrt{G_{1}} {\hat{a}}_{ν 1} + \sqrt{g_{1}} {\hat{a}}_{0}^{†}, \\ {\hat{a}}_{2} & = & \sqrt{G_{1} G_{2}} {\hat{a}}_{0} + \sqrt{G_{2} g_{1}} {\hat{a}}_{ν 1}^{†} + \sqrt{g_{2}} {\hat{a}}_{ν 2}^{†}, \\ {\hat{a}}_{3} & = & \sqrt{G_{2}} {\hat{a}}_{ν 2} + \sqrt{G_{1} g_{2}} {\hat{a}}_{0}^{†} + \sqrt{g_{1} g_{2}} {\hat{a}}_{ν 1}, \end{array}

where â₀ 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_j=1. Eq. (2) can be rewritten in terms of the quadrature operators using Eq. (1) as follows

(\begin{array}{l} {\hat{X}}_{1} \\ {\hat{X}}_{2} \\ {\hat{X}}_{3} \end{array}) = (\begin{matrix} \sqrt{G_{1}} & \sqrt{g_{1}} & 0 \\ \sqrt{G_{2} g_{1}} & \sqrt{G_{1} G_{2}} & \sqrt{g_{2}} \\ \sqrt{g_{1} g_{2}} & \sqrt{G_{1} g_{2}} & \sqrt{G_{2}} \end{matrix}) (\begin{matrix} {\hat{X}}_{ν 1} \\ {\hat{X}}_{0} \\ {\hat{X}}_{ν 2} \end{matrix}),

and

(\begin{array}{l} {\hat{Y}}_{1} \\ {\hat{Y}}_{2} \\ {\hat{Y}}_{3} \end{array}) = (\begin{matrix} \sqrt{G_{1}} & - \sqrt{g_{1}} & 0 \\ - \sqrt{G_{2} g_{1}} & \sqrt{G_{1} G_{2}} & - \sqrt{g_{2}} \\ \sqrt{g_{1} g_{2}} & - \sqrt{G_{1} g_{2}} & \sqrt{G_{2}} \end{matrix}) (\begin{matrix} {\hat{Y}}_{ν 1} \\ {\hat{Y}}_{0} \\ {\hat{Y}}_{ν 2} \end{matrix}) .

From the above expressions for the quadrature operators, it is possible to get the variances and covariances of the amplitude and phase quadratures necessary to calculate the entanglement criteria.

V ({\hat{X}}_{m}) = 〈 {\hat{X}}_{m}^{2} 〉 - {〈 {\hat{X}}_{m} 〉}^{2}

denotes the amplitude quadrature variance. Whereas for the covariance, we use the notation V_mn = (〈X̂_mX̂_n〉 + 〈X̂_nX̂_m〉)/2 − 〈X̂_m〉〈X̂_n〉 and for the case where m=n, the covariance, denoted V_mn, reduces to the usual variance, V(X̂_m). In fact, the variances are given by the following gain-dependent moments (we assume that â₀ is vacuum input in the following calculations because homodyne detection technique needs three output beams â₁, â₂ and â₃ are all at vacuum level.)

\begin{array}{l} 〈 {\hat{X}}_{1}^{2} 〉 & = & 〈 {\hat{Y}}_{1}^{2} 〉 = 2 G - 1, \\ 〈 {\hat{X}}_{2}^{2} 〉 & = & 〈 {\hat{Y}}_{2}^{2} 〉 = 2 G_{1} G_{2} - 1, \\ 〈 {\hat{X}}_{3}^{2} 〉 & = & 〈 {\hat{Y}}_{3}^{2} 〉 = 2 G_{1} G_{2} - 2 G_{1} + 1, \end{array}

since the expectation values of quadratures are all zero. Here we have used the fact that 〈X̂_mX̂_n〉 = 〈Ŷ_mŶ_n〉 = δ_mn (m, n = 0, ν1 and ν2). A similar approach can be used to calculate the covariances which are equivalent to the gain-dependent moments 〈X̂_mX̂_n〉 and 〈Ŷ_mŶ_n〉. The covariances are given by the following gain-dependent moments

\begin{array}{l} 〈 {\hat{X}}_{1} {\hat{X}}_{2} 〉 & = & - 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 = 2 \sqrt{G_{1} G_{2} (G_{1} - 1)}, \\ 〈 {\hat{X}}_{1} {\hat{X}}_{3} 〉 & = & 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 = 2 \sqrt{G_{1} (G_{1} - 1) (G_{2} - 1)}, \\ 〈 {\hat{X}}_{2} {\hat{X}}_{3} 〉 & = & - 〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉 = 2 G_{1} \sqrt{G_{2} (G_{2} - 1)}, \end{array}

we will use the above expressions to analyze the following entanglement criteria, because the three beams output from the cascaded FWM system are Gaussian states, and the entanglement properties of Gaussian states can be completely characterized by their covariance matrix.

Fig. 1 Proposed schemes for generating tripartite entanglement based on the cascaded FWM processes. (a) Cascaded FWM processes, â₀ is coherent input signal, â_ν1 and â_ν2 are vacuum inputs, G₁ and G₂ are the power gains of cell₁ and cell₂, respectively. â₁, â₂ and â₃ are three output beams. (b) Energy level diagram of ⁸⁵Rb D1 line for the single FWM process. 0.8 GHz is one-photon detuning, 4 MHz is two-photon detuning. (c) Another cascaded FWM processes.

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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

\begin{array}{l} D_{12} & = & V ({\hat{X}}_{1} - {\hat{X}}_{2}) + V ({\hat{Y}}_{1} + {\hat{Y}}_{2}) \geq 4, \\ D_{13} & = & V ({\hat{X}}_{1} - {\hat{X}}_{3}) + V ({\hat{Y}}_{1} + {\hat{Y}}_{3}) \geq 4, \\ D_{23} & = & V ({\hat{X}}_{2} - {\hat{X}}_{3}) + V ({\hat{Y}}_{2} + {\hat{Y}}_{3}) \geq 4 . \end{array}

We calculate the dependence of D₁₂, D₁₃ and D₂₃ on the gains G₁ and G₂ as follows

\begin{array}{l} D_{12} & = & 4 [G_{1} (G_{2} + 1) - 2 \sqrt{G_{1} G_{2} (G_{1} - 1)} - 1], \\ D_{13} & = & 4 G_{1} G_{2}, \\ D_{23} & = & 4 [2 G_{1} G_{2} - 2 G_{1} \sqrt{G_{2} (G_{2} - 1)} - G_{1}] . \end{array}

The violation of the first inequality in Eq. (7) can be used to demonstrate the bipartite entanglement between â₁ and â₂, so also can the second (third) inequality in Eq. (7) be used to demonstrate the bipartite entanglement between â₁ (â₂) and â₃ (â₃). The region plot of Eq. (7) is shown in Fig. 2(a). The entanglement region of â₁ and â₂ (D₁₂<4) is the region B in Fig. 2(a). As shown in Fig. 2(b), if we set G₂ ≈1, the cascaded FWM processes reduce to the single FWM process, i. e., the first FWM process (cell₁). Under this condition, â₁ and â₂ is a simple bipartite entangled beams case. As G₁ is getting larger, D₁₂ is limited by smaller G₂. This is because only beam â₂ experiences the quantum amplification from the second FWM process (cell₂), which leads to their noise unbalance, thus the performance of the entanglement between beams â₁ and â₂ is very sensitive to the G₂. Similarly, The entanglement region of â₂ and â₃ (D₂₃<4) is the region A in Fig. 2(a). As shown in Fig. 2(c), if we set G₁ ≈1, the cascaded FWM processes reduce to the single FWM process, i. e., the second FWM process (cell₂). In this case, â₂ and â₃ is also a simple bipartite entangled beams case. It should be noted that the value of D₁₃ is always more than or equal to 4 for any G₁, G₂>1.

Fig. 2 (a) Region plot of Eq. (7). Region A is a region where D₂₃ <4, region B is a region where D₁₂ <4, region C is a region where D₁₂, D₁₃ and D₂₃ in Eq. (7) are all falling above 4; (b) The contour plot of D₁₂; and (c) The contour plot of D₂₃.

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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 â₁ and â₂ can be completely characterized by their covariance matrix (CM₁₂) which can be written as

{CM}_{12} = [\begin{array}{l} 〈 {\hat{X}}_{1}^{2} 〉 & 0 & 〈 {\hat{X}}_{1} {\hat{X}}_{2} 〉 & 0 \\ 0 & 〈 {\hat{Y}}_{1}^{2} 〉 & 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 \\ 〈 {\hat{X}}_{1} {\hat{X}}_{2} 〉 & 0 & 〈 {\hat{X}}_{2}^{2} 〉 & 0 \\ 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 & 0 & 〈 {\hat{Y}}_{2}^{2} 〉 \end{array}] .

The bipartite entanglement between â₁ and â₂ is absent if and only if both of the symplectic eigenvalues of the partially transposed (PT) CM₁₂ are greater than or equal to 1 [13,16,29,30]. Following this idea, the entanglement between â₁ and â₂ can be characterized by the smaller symplectic eigenvalue B₁. If the smaller symplectic eigenvalue B₁ is smaller than 1, the bipartite entanglement exists between â₁ and â₂. Substituting Eq. (5) and Eq. (6) in Eq. (9), we can get the detailed expression for B₁ as shown below

B_{1} = - 1 + G_{1} + G_{1} G_{2} - \sqrt{G_{1}^{2} - 4 G_{1} G_{2} + 2 G_{1}^{2} G_{2} + G_{1}^{2} G_{2}^{2}},

the contour plot of B₁ is shown in Fig. 3(a). It can be proved that the value of the whole region for any G₁, G₂ >1 is smaller than 1 meaning that â₁ is entangled with â₂. Similarly, the smaller symplectic eigenvalue B₂ of PT CM₂₃ of â₂ and â₃ is given by

B_{2} = - G_{1} + 2 G_{1} G_{2} - \sqrt{(1 - 2 G_{1}) + {(- G_{1} + 2 G_{1} G_{2})}^{2}},

its contour plot is shown in Fig. 3(b), its value is smaller than 1 for any G₁, G₂ >1 meaning that â₂ is entangled with â₃.

Fig. 3 The contour plot of (a) B₁; and (b) B₂; and (c) B₃.

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In addition, the smaller symplectic eigenvalue B₃ of PT CM₁₃ of â₁ and â₃ is given by

B_{3} = - 1 + 2 G_{1} - G_{1} G_{2} + \sqrt{{(1 - 2 G_{1} + G_{1} G_{2})}^{2} - (1 - 2 G_{1} G_{2})},

its value is always larger than 1 for any G₁, G₂ >1 as shown in Fig. 3(c) meaning that â₁ is not entangled with â₃. This is due to the fact that â₁ and â₃ 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

\begin{array}{l} V_{12} & = & V ({\hat{X}}_{1} - {\hat{X}}_{2}) + V ({\hat{Y}}_{1} + {\hat{Y}}_{2} + O_{3} {\hat{Y}}_{3}) \geq 4, \\ V_{23} & = & V ({\hat{X}}_{2} - {\hat{X}}_{3}) + V (O_{1} {\hat{Y}}_{1} + {\hat{Y}}_{2} + {\hat{Y}}_{3}) \geq 4, \end{array}

where 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_i. The optimal expressions of O_i (

O_{i}^{opt}

) can be obtained by the direct differentiation for the left-hand side of Eq. (13) with respect to the O_i, then we get

\begin{array}{l} O_{1}^{opt} & = & \frac{- (〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 + 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉)}{〈 {\hat{Y}}_{1}^{2} 〉} \\ = & \frac{2 \sqrt{G_{1} G_{2} (G_{1} - 1)} - 2 \sqrt{G_{1} (G_{1} - 1) (G_{2} - 1)}}{2 G_{1} - 1}, \\ O_{3}^{opt} & = & \frac{- (〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 + 〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉)}{〈 {\hat{Y}}_{3}^{2} 〉} \\ = & \frac{2 G_{1} \sqrt{G_{2} (G_{2} - 1)} - 2 \sqrt{G_{1} (G_{1} - 1) (G_{2} - 1)}}{2 G_{1} G_{2} - 2 G_{1} + 1}, \end{array}

The contour plot of Eq. (13) is shown in Fig. 4. The dependence of V₁₂ on G₁ and G₂ is shown in Fig. 4(a). The region of V₁₂<4 is enlarged compared to the one of D₁₂ <4 in Fig. 2(b) when we consider the phase quadrature of â₃ (Ŷ₃). The variance of Ŷ₁+Ŷ₂+Ŷ₃ becomes smaller than the one of Ŷ₁+Ŷ₂ which claims that â₃ has correlation with â₁+â₂. The dependence of V₂₃ on G₁ and G₂ is shown in Fig. 4(b), the region of V₂₃<4 is enlarged compared to the one of D₂₃<4 in Fig. 2(c) when we consider the phase quadrature of â₁ (Ŷ₁). The variance of Ŷ₁+Ŷ₂+Ŷ₃ becomes smaller than the one of Ŷ₂+Ŷ₃ which claims that â₁ has correlation with â₂+â₃.

Fig. 4 (a) The contour plot of V₁₂; (b) The contour plot of V₂₃; and (c) The light blue region is the region of V₂₃ <4, the light orange region is the region of V₁₂<4.

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The overlapped region of V₁₂ <4 and V₂₃ <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

\begin{array}{l} V_{213} & = & V [{\hat{X}}_{2} - \frac{1}{\sqrt{2}} ({\hat{X}}_{1} + {\hat{X}}_{3})] + V [{\hat{Y}}_{2} + \frac{1}{\sqrt{2}} ({\hat{Y}}_{1} + {\hat{Y}}_{3})], \\ V_{312} & = & V [{\hat{X}}_{3} - \frac{1}{\sqrt{2}} ({\hat{X}}_{1} + {\hat{X}}_{2})] + V [{\hat{Y}}_{3} + \frac{1}{\sqrt{2}} ({\hat{Y}}_{1} + {\hat{Y}}_{2})], \\ V_{123} & = & V [{\hat{X}}_{1} - \frac{1}{\sqrt{2}} ({\hat{X}}_{2} + {\hat{X}}_{3})] + V [{\hat{Y}}_{1} + \frac{1}{\sqrt{2}} ({\hat{Y}}_{2} + {\hat{Y}}_{3})] . \end{array}

The dependence of V₂₁₃ on the gains G₁ and G₂ can be written as follows

\begin{array}{l} V_{213} & = & 2 {[\sqrt{G_{2} - 1} - \sqrt{\frac{G_{2}}{2}}]}^{2} + \\ 2 {[\sqrt{(G_{1} - 1) G_{2}} - \sqrt{\frac{G_{1}}{2}} - \sqrt{\frac{(G_{1} - 1) (G_{2} - 1)}{2}}]}^{2} \\ + 2 {[\sqrt{G_{1} G_{2}} - \sqrt{\frac{G_{1} - 1}{2}} - \sqrt{\frac{G_{1} (G_{2} - 1)}{2}}]}^{2} . \end{array}

The contour plot of V₂₁₃ 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 V₃₁₂ and V₁₂₃ are always bigger than 2 due to the fact that â₂ is correlated with both â₁ and â₃, whereas â₁ and â₃ are not correlated with each other directly. In fact, they are from two independent FWM processes in serial connection.

Fig. 5 The contour plot for V₂₁₃ in Eq. (16).

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4.3. Optimal single-condition criterion

Inspired by above results in Sec. 4.2, we should pay more attention to the optimization of V₂₁₃ ( $V_{213}^{opt}$ ) through introducing different factors [15] to cancel the impact of noise unbalance between â₁ and â₃. In this way, we can express the $V_{213}^{opt}$ as follows

V_{213}^{opt} = V [{\hat{X}}_{2} - F_{1} {\hat{X}}_{1} - F_{3} {\hat{X}}_{3}] + V [{\hat{Y}}_{2} + F_{1} {\hat{Y}}_{1} + F_{3} {\hat{Y}}_{3}],

where F₁ and F₃ are arbitrary real numbers whose optimal expressions (

F_{1}^{opt}

and

F_{3}^{opt}

) can be obtained by differentiating either X̂₂ − F₁X̂₁ − F₃X̂₃ or Ŷ₂ + F₁Ŷ₁ + F₃Ŷ₃ with respect to F₁ and F₃, then we get

\begin{array}{l} F_{1}^{opt} & = & \frac{〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 〈 {\hat{Y}}_{3}^{2} 〉 - 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉}{{〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉}^{2} - 〈 {\hat{Y}}_{1}^{2} 〉 〈 {\hat{Y}}_{3}^{2} 〉}, \\ F_{3}^{opt} & = & \frac{〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉 〈 {\hat{Y}}_{1}^{2} 〉 - 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉}{{〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉}^{2} - 〈 {\hat{Y}}_{1}^{2} 〉 〈 {\hat{Y}}_{3}^{2} 〉} . \end{array}

The contour plot of $V_{213}^{opt}$ is shown in Fig. 6(a), it depends on G₁ and G₂ symmetrically which is similar to the squeezing degree of the triple beams (1/[2G₁G₂ − 1]) [23]. Thus in the following discussions we will analyze the tripartite entanglement using the optimal single-condition criterion in the condition G=G₁=G₂. In this case, Eq. (17) can be reduced to

V_{213}^{opt} = \frac{2}{2 G^{2} - 1},

and Eq. (18) can be reduced to

\begin{array}{l} F_{1}^{opt} & = & \frac{2 G \sqrt{G - 1}}{2 G^{2} - 1}, \\ F_{3}^{opt} & = & \frac{2 G \sqrt{G (G - 1)}}{2 G^{2} - 1}, \end{array}

Fig. 6 (a) The contour plot of $V_{213}^{opt}$ in Eq. (17); (b) The dependence of $V_{213}^{opt}$ (trace A); $2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ (trace B); $2 (| F_{3}^{2} | + | 1 - F_{1}^{2} |)$ (trace C); and $2 (1 + | F_{1}^{2} + F_{3}^{2} |)$ (trace D) on gain G; and (c) The dependence of R on gain G. 1 for trace A, $R = V_{213}^{opt} / 2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ for trace B.

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$F_{1}^{opt}$ and $F_{3}^{opt}$ 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 â₂ and â₃ become equal. This makes sense because when G becomes infinite, the powers of the modes â₂ and â₃ are equal and it is unnecessary to introduce different factors to cancel the noise unbalance between the modes â₂ and â₃.

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 $V_{213}^{opt}$ are $2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ (â₂ and â₃ may be entangled or not, but â₁ is not entangled with the rest), $2 (| F_{3}^{2} | + | 1 - F_{1}^{2} |)$ (â₁ and â₂ may be entangled or not, but â₃ is not entangled with the rest), and $2 (1 + | F_{1}^{2} + F_{3}^{2} |)$ (â₁ and â₃ may be entangled or not, but â₂ is not entangled with the rest). To answer this question, we plot the dependence of $V_{213}^{opt}$ (Trace A), $2 (1 + | F_{1}^{2} + F_{3}^{2} |)$ (Trace B), $2 (| F_{3}^{2} | + | 1 - F_{1}^{2} |)$ (Trace C), and $2 (1 + | F_{1}^{2} + F_{3}^{2} |)$ (Trace D) on G in Fig. 6(b). As shown in Fig. 6(b), $2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ (Trace B) decreases with the increasing of G, $2 (| F_{3}^{2} | + | 1 - F_{1}^{2} |)$ (Trace C) and $2 (1 + | F_{1}^{2} + F_{3}^{2} |)$ (trace D) increase with the increasing of G.

As shown in Fig. 6(b), $2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ (Trace B) is the smallest one of all the boundaries which means that tripartite entanglement is present if $V_{213}^{opt}$ (Trace A) is smaller than $2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ (Trace B). Thus we define the quantity $R = V_{213}^{opt} / 2 (| F_{1}^{2} | + | 1 - F_{3}^{2} |)$ 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.

Fig. 7 The contour plot of (a) T₁; (b) T₂; and (c) T₃.

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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 CM₁₂₃ for the three modes can be written as

{CM}_{123} = [\begin{array}{l} 〈 {\hat{X}}_{1}^{2} 〉 & 0 & 〈 {\hat{X}}_{1} {\hat{X}}_{2} 〉 & 0 & 〈 {\hat{X}}_{1} {\hat{X}}_{3} 〉 & 0 \\ 0 & 〈 {\hat{Y}}_{1}^{2} 〉 & 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 & 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 \\ 〈 {\hat{X}}_{1} {\hat{X}}_{2} 〉 & 0 & 〈 {\hat{X}}_{2}^{2} 〉 & 0 & 〈 {\hat{X}}_{2} {\hat{X}}_{3} 〉 & 0 \\ 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{2} 〉 & 0 & 〈 {\hat{Y}}_{2}^{2} 〉 & 0 & 〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉 \\ 〈 {\hat{X}}_{1} {\hat{X}}_{3} 〉 & 0 & 〈 {\hat{X}}_{2} {\hat{X}}_{3} 〉 & 0 & 〈 {\hat{X}}_{3}^{2} 〉 & 0 \\ 0 & 〈 {\hat{Y}}_{1} {\hat{Y}}_{3} 〉 & 0 & 〈 {\hat{Y}}_{2} {\hat{Y}}_{3} 〉 & 0 & 〈 {\hat{Y}}_{3}^{2} 〉 \end{array}] .

When the PT operation is applied to beam â₁, the entanglement between â₁ and the rest beams (â₂ and â₃) can be characterized by the smallest symplectic eigenvalue T₁. Substituting Eq. (5) and Eq. (6) in Eq. (21), we can get the detailed expression of T₁ which is not listed here due to its complexity. As shown in Fig. 7(a), the value of T₁ in the whole gain region is smaller than 1 for any G₁, G₂ >1 meaning that beam â₁ is entangled with the rest beams (â₂ and â₃). It should be noted that the behavior of T₁ is independent on the gain G₂. This is due to the fact that the entanglement between beam â₁ and the rest beams (â₂ and â₃) is only decided by the first FWM process (cell₁). Similarly, when the PT operation is applied to the beam â₂ (â₃), the smallest symplectic eigenvalue is T₂ (T₃). The contour plots of T₂ and T₃ are shown in Fig. 7(b) and Fig. 7(c) respectively. It can be clearly seen that the values of T₂ and T₃ are both smaller than 1 for any G₁, G₂ >1 in the whole gain region, showing the presence of the entanglement for these two partitions. Therefore, the three partitions â₁−(â₂, â₃), â₂−(â₁, â₃), and â₃−(â₁, â₂) are all inseparable and the genuine tripartite entanglement between the three beams â₁, â₂ and â₃ is present for the whole gain region with any G₁, G₂ >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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Generation of tripartite entanglement from cascaded four-wave mixing processes

Abstract

1. Introduction

2. Cascaded FWM processes

3. Bipartite entanglement

3.1. DGCZ criterion

3.2. PPT criterion

4. Tripartite entanglement

4.1. Two-condition criterion

4.2. Single-condition criterion

4.3. Optimal single-condition criterion

4.4. PPT criterion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Equations (21)

Optics Express