Multi-Mode Correlation in a Concurrent Parametric Amplifier

Abstract

A concurrent parametric amplifier consisting of two pump beams is used to investigate the possibility of generating multi-mode correlation and entanglement. The existence of three-mode entanglement is demonstrated by analyzing the violation degree of three-mode entanglement criteria, including the sufficient criterion, i.e., two-condition and optimal single-condition criterion, and necessary and sufficient criterion, i.e., positivity under partial transposition (PPT) criterion. Besides, two-mode entanglement generated from any pair is also studied by using the Duan criterion and PPT criterion. We find that three-mode entanglement and two-mode entanglement of the two pairs are present in the whole parameter region. Our results pave the way for the realization and application of multi-mode correlation and entanglement based on the concurrent parametric amplifiers.

1. Introduction

Parametric amplifier, for example, four-wave mixing (FWM), acts as a nonlinear interaction process that permits the transfer of energy and momentum between multiple optical modes, specifically, two pump modes can be converted into a signal and an idler modes via various nonlinear media [1,2,3]. For the atomic medium, a high-power pump beam intersects a low-power seed beam in a hot rubidium cell, causing them to interact and generate an idler beam, then the amplified seed (signal) and the newly-generated idler beams can be demonstrated to be quantum correlated and entangled [4,5,6,7,8,9,10]. Based on this simple model of only one pump mode and only one seed mode, other variations have been presented. For example, when the FWM process is seeded by the coherent signal and idler beams and only pumped by one pump beam, a scheme of a two-mode phase-sensitive amplifier has been constructed, and its classical and quantum properties have been theoretically analyzed [11] and experimentally measured [12]. Similarly, when the FWM process is seeded by two coherent signal beams at the same angle but in opposite directions on either side of the pump beam, this dual-seed scheme has been allowed to achieve intensity-difference squeezing at ultra-low frequency [13].

On the other hand, with the FWM process pumped by two coherent pump beams with non-degenerate frequency and only seeded by one coherent signal beam, a noiseless optical amplifier has been experimentally realized, and its noise figure is always superior than that obtained with a phase-insensitive amplifier with the same gain [14].

Motivated by the above scientific advances, we propose a concurrent parametric amplifier scheme in which FWM process is pumped by the two coherent pump beams with the degenerate frequency and only seeded by one coherent signal beam. As shown in Figure 1, two pump beams P₁ and P₂ are focused and crossed in the center of a hot rubidium vapor cell. A coherent seed beam, red-shifted from pump beam as shown in the block in Figure 1, is seeded into the vapor cell, and it symmetrically crosses with the two pump beams on one plane with the proper crossing angles [15] to eliminate any other cascaded FWM processes. Under this experimental condition, each pump beam will interact with the seed beam individually by means of FWM process. The seed beam is amplified (S) and two idler beams (I₁ and I₂) are simultaneously generated, therefore the interaction mechanism of the triple output beams S, I₁ and I₂ constitute a concurrent parametric amplification process [16]. In this work, the quantum properties of two-mode and three-mode entanglement existed in the triple output beams generated from the concurrent parametric amplifier will be discussed in detail. Firstly, the Hamiltonian describing this concurrent parametric amplifier can be written as below

$$\begin{array}{c}\hfill \widehat{H}=i\hslash [{\epsilon }_1({\widehat{a}}_s^{†}{\widehat{a}}_1^{†}-{\widehat{a}}_s{\widehat{a}}_1)+{\epsilon }_2({\widehat{a}}_s^{†}{\widehat{a}}_2^{†}-{\widehat{a}}_s{\widehat{a}}_2)],\end{array}$$

(1)

with ${\epsilon }_i$ (i = 1 and 2) representing the interaction strength and ${\widehat{a}}_i$ (i = s, 1, and 2) the bosonic annihilation operators. By applying the Heisenberg equation of motion to Equation (1), the solution for the annihilation operators is found to be

$$\begin{array}{cc}\hfill {\widehat{a}}_s\left(t\right)& =A{\widehat{a}}_s\left(0\right)+B{\widehat{a}}_1^{†}\left(0\right)+C{\widehat{a}}_2^{†}\left(0\right),\\ \hfill {\widehat{a}}_1\left(t\right)& =B{\widehat{a}}_s^{†}\left(0\right)+D{\widehat{a}}_1\left(0\right)+E{\widehat{a}}_2\left(0\right),\\ \hfill {\widehat{a}}_2\left(t\right)& =C{\widehat{a}}_s^{†}\left(0\right)+E{a}_1\left(0\right)+F{\widehat{a}}_2\left(0\right),\end{array}$$

(2)

with

$$\begin{array}{cc}\hfill A& =\sqrt{G},\\ \hfill B& =\frac{\alpha \sqrt{G-1}}{\sqrt{1+{\alpha }^2}},\\ \hfill C& =\frac{\sqrt{G-1}}{\sqrt{1+{\alpha }^2}},\\ \hfill D& =\frac{1+{\alpha }^2\sqrt{G}}{1+{\alpha }^2},\\ \hfill E& =\frac{\alpha (\sqrt{G}-1)}{1+{\alpha }^2},\\ \hfill F& =\frac{{\alpha }^2+\sqrt{G}}{1+{\alpha }^2},\end{array}$$

(3)

where $\mathrm{\Gamma }=\sqrt{{\epsilon }_1^2+{\epsilon }_2^2}$, $cosh\left(\mathrm{\Gamma }t\right)=\sqrt{G}$, and ${\epsilon }_1/{\epsilon }_2=\alpha$. Before investigating the two-mode and three-mode entanglement existing in the system, the optical quadrature definitions should be given firstly due to the requirement of the following criteria. Concerning the three modes characterized by bosonic annihilation operators ${\widehat{a}}_i$ involved in the present system, where i = s, 1, and 2, quadrature operators can be defined as follows:

$$\begin{array}{c}\hfill X_i={\widehat{a}}_i+{\widehat{a}}_i^{†},Y_i=-i({\widehat{a}}_i-{\widehat{a}}_i^{†}),\end{array}$$

(4)

such that $[X_i,Y_i]$ = 2i, and $X_i$ and $Y_i$ are, respectively, position and momentum quadratures. Following the definitions of Equation (4), Equation (2) can be recast in the form of quadrature operators:

$$\begin{array}{cc}\hfill X_s\left(t\right)& =A{X}_s\left(0\right)+B{X}_1\left(0\right)+C{X}_2\left(0\right),\\ \hfill X_1\left(t\right)& =B{X}_s\left(0\right)+D{X}_1\left(0\right)+E{X}_2\left(0\right),\\ \hfill X_2\left(t\right)& =C{X}_s\left(0\right)+E{X}_1\left(0\right)+F{X}_2\left(0\right),\end{array}$$

(5)

and

$$\begin{array}{cc}\hfill Y_s\left(t\right)& =A{Y}_s\left(0\right)-B{Y}_1\left(0\right)-C{Y}_2\left(0\right),\\ \hfill Y_1\left(t\right)& =-B{Y}_s\left(0\right)+D{Y}_1\left(0\right)+E{Y}_2\left(0\right),\\ \hfill Y_2\left(t\right)& =-C{Y}_s\left(0\right)+E{Y}_1\left(0\right)+F{Y}_2\left(0\right).\end{array}$$

(6)

Based on the above relations about quadrature operators, the variances and covariances of the position and momentum quadratures can be obtained to analyze the violation degree of different entanglement criteria. $V\left(X_i\right)=\langle X_i^2\rangle -{\langle X_i\rangle }^2$ represents the position quadrature variance. For the covariance, it can be defined as $V_{ij}=(\langle X_i{X}_j\rangle +\langle X_j{X}_i\rangle )/2-\langle X_i\rangle \langle X_j\rangle$. It should be noted that the covariance V${}_{ij}$ will reduce to the usual variance V($X_i$) under the condition of $i=j$. In reality, the variances of the three modes can be expressed as

$$\begin{array}{cc}\hfill 〈X_s^2\left(t\right)〉& =〈Y_s^2\left(t\right)〉=A^2+B^2+C^2=2G-1,\hfill \\ \hfill 〈X_1^2\left(t\right)〉& =〈Y_1^2\left(t\right)〉=B^2+D^2+E^2=\frac{1+{\alpha }^2(2G-1)}{1+{\alpha }^2},\hfill \\ \hfill 〈X_2^2\left(t\right)〉& =〈Y_2^2\left(t\right)〉=C^2+E^2+F^2=\frac{{\alpha }^2+(2G-1)}{1+{\alpha }^2},\hfill \end{array}$$

(7)

and here, we used the fact that the mean values of quadrature operators are all equal to 0 and $\langle X_i\left(0\right)X_j\left(0\right)\rangle =\langle Y_i\left(0\right)Y_j\left(0\right)\rangle ={\delta }_{ij}$ (i, j = s, 1, and 2). Similarly, the covariances can be expressed by

$$\begin{array}{cc}\hfill 〈X_s\left(t\right)X_1\left(t\right)〉& =-〈Y_s\left(t\right)Y_1\left(t\right)〉=AB+BD+CE\hfill \\ & =\frac{2\alpha \sqrt{G(G-1)}}{\sqrt{1+{\alpha }^2}},\hfill \\ \hfill 〈X_s\left(t\right)X_2\left(t\right)〉& =-〈Y_s\left(t\right)Y_2\left(t\right)〉=AC+BE+CF\hfill \\ & =\frac{2\sqrt{G(G-1)}}{\sqrt{1+{\alpha }^2}},\hfill \\ \hfill 〈X_1\left(t\right)X_2\left(t\right)〉& =〈Y_1\left(t\right)Y_2\left(t\right)〉=BC+DE+EF\hfill \\ & =\frac{2\alpha (G-1)}{1+{\alpha }^2},\hfill \end{array}$$

(8)

and the above expressions can be used to discuss the entanglement properties of both two-mode and three-mode cases; this is due to the fact that the triple beams in the concurrent parametric amplifier are Gaussian states, which can be fully quantified by their corresponding covariance matrix (CM).

2. Two-Mode Entanglement

2.1. Duan Criterion

On the one hand, two-mode entanglement in this concurrent parametric amplifier is analyzed using a sufficient criterion, i.e., Duan criterion [17], which is based on the total variance of a pair of Einstein–Podolsky–Rosen-type operators, $X_i-X_j$ and $Y_i+Y_j$. For physical entangled continuous variable states, this variance will rapidly reduce to zero by increasing the correlation degree. Thus, if any inequality in Equation (9) is violated, there will exist two-mode entanglement between any pair. Based on the quadrature definitions in Equation (4), the inequalities can be expressed asnumerical order.

$$\begin{array}{ccc}\hfill D_{s1}& =& V(X_s-X_1)+V(Y_s+Y_1)\ge 4,\hfill \\ \hfill D_{s2}& =& V(X_s-X_2)+V(Y_s+Y_2)\ge 4,\hfill \\ \hfill D_{12}& =& V(X_1-X_2)+V(Y_1+Y_2)\ge 4.\hfill \end{array}$$

(9)

We calculate the dependence of $D_{s1}$, $D_{s2}$, and $D_{12}$ on the gain G and interaction strength ratio $\alpha$ as

$$\begin{array}{cc}\hfill D_{s1}& =2[2G-1-\frac{4\alpha \sqrt{G(G-1)}}{\sqrt{1+{\alpha }^2}}+\frac{1+{\alpha }^2(2G-1)}{1+{\alpha }^2}],\hfill \\ \hfill D_{s2}& =4[G-\frac{2\sqrt{G(G-1)}}{\sqrt{1+{\alpha }^2}}+\frac{G-1}{1+{\alpha }^2}],\hfill \\ \hfill D_{12}& =4G.\hfill \end{array}$$

(10)

The violation of the first, second, and third inequalities in Equation (9) can be used to claim the existence of two-mode entanglement between ${\widehat{a}}_s$ and ${\widehat{a}}_1$, ${\widehat{a}}_s$ and ${\widehat{a}}_2$, and ${\widehat{a}}_1$ and ${\widehat{a}}_2$, respectively. As depicted in Figure 2a, as G and α get larger, the value of $D_{s1}$ becomes smaller and smaller. This is because, under this condition, the concurrent parametric amplifier will reduce to a simple single pump amplifier only pumped by P${}_1$, and two-mode entanglement between the modes ${\widehat{a}}_s$ and ${\widehat{a}}_1$ will dominate. Contrary to the dependence of $D_{s1}$ on G and α, the two-mode entanglement between ${\widehat{a}}_s$ and ${\widehat{a}}_2$ depicted in Figure 2b can be improved by means of a larger value of G and a smaller value of α, meaning that, in this situation, the concurrent parametric amplifier will reduce to the simple single-pump amplifier only pumped by P${}_2$, and two-mode entanglement between the modes ${\widehat{a}}_s$ and ${\widehat{a}}_2$ will dominate. This also explains the opposite behaviors between Figure 2a,b. More interestingly, as depicted in Figure 2c, two-mode entanglement between ${\widehat{a}}_1$ and ${\widehat{a}}_2$ is absent. The reason for this phenomenon is that the two modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ are both generated from the seed mode ${\widehat{a}}_s$ and compete with each other.

2.2. Positivity under Partial Transposition Criterion

On the other hand, the PPT criterion as a necessary and sufficient criterion can be used to quantify two-mode entanglement in the concurrent parametric amplifier. Generally, two-mode entanglement shared by ${\widehat{a}}_s$ and ${\widehat{a}}_1$ can be fully quantified by the following CM${}_{s1}$ [18,19]:

$$\begin{array}{c}\hfill C{M}_{s1}=\left[\begin{array}{cccc}〈{\widehat{X}}_s^2\left(t\right)〉\hfill & 0\hfill & 〈{\widehat{X}}_s\left(t\right){\widehat{X}}_1\left(t\right)〉\hfill & 0\hfill \\ 0\hfill & 〈{\widehat{Y}}_s^2\left(t\right)〉\hfill & 0\hfill & 〈{\widehat{Y}}_s\left(t\right){\widehat{Y}}_1\left(t\right)〉\hfill \\ 〈{\widehat{X}}_s\left(t\right){\widehat{X}}_1\left(t\right)〉\hfill & 0\hfill & 〈{\widehat{X}}_1^2\left(t\right)〉\hfill & 0\hfill \\ 0\hfill & 〈{\widehat{Y}}_s\left(t\right){\widehat{Y}}_1\left(t\right)〉\hfill & 0\hfill & 〈{\widehat{Y}}_1^2\left(t\right)〉\hfill \end{array}\right].\end{array}$$

(11)

Only when both of the symplectic eigenvalues of the partially transposed (PT) CM${}_{s1}$ are no less than 1, this indicates the absence of two-mode entanglement between them [18,19,20,21]. In this way, the smaller symplectic eigenvalue $E_{s1}$ can be used to quantify two-mode entanglement between ${\widehat{a}}_s$ and ${\widehat{a}}_1$, i.e., if $E_{s1}$ is smaller than 1, two-mode entanglement will exist between them. Substituting Equations (7) and (8) into Equation (11), we can obtain the detailed result for $E_{s1}$ as below:

$$\begin{array}{c}\hfill E_{s1}=\frac{1}{{(1+{\alpha }^2)}^{3/2}}\sqrt{\begin{array}{c}1+{\alpha }^2+(1+{\alpha }^2)[{\alpha }^4+2{(G+2G{\alpha }^2)}^2-2G(1+3{\alpha }^2+4{\alpha }^4)]-\\ 2(1+{\alpha }^2)[G+(2G-1){\alpha }^2]\sqrt{(G-1)[G{(1+2{\alpha }^2)}^2-1]}\end{array}},\end{array}$$

(12)

and the dependence of $E_{s1}$ is depicted in Figure 3a. Due to the value of $E_{s1}$ being smaller than 1, thus ${\widehat{a}}_s$ is quantum entangled with ${\widehat{a}}_1$ in the whole parametric region (G > 1 and $\alpha$ > 0). Similarly, the smaller symplectic eigenvalue $E_{s2}$ of ${\widehat{a}}_s$ and ${\widehat{a}}_2$ can be calculated as

$$\begin{array}{c}\hfill E_{s2}=\frac{1}{{(1+{\alpha }^2)}^{3/2}}\sqrt{\begin{array}{c}1+{\alpha }^2+(1+{\alpha }^2)[{\alpha }^4+2G^2{(2+{\alpha }^2)}^2-2G(4+3{\alpha }^2+{\alpha }^4)]\\ -2(1+{\alpha }^2)[G(2+{\alpha }^2)-1]\sqrt{(G-1)[G{(2+{\alpha }^2)}^2-{\alpha }^4]}\end{array},}\end{array}$$

(13)

and its value is smaller than 1 in the whole parametric region as depicted in Figure 3b, meaning that ${\widehat{a}}_s$ is also quantum entangled with ${\widehat{a}}_2$.

Besides, the smaller symplectic eigenvalue $E_{12}$ of ${\widehat{a}}_1$ and ${\widehat{a}}_2$ can be given by

$$\begin{array}{c}\hfill E_{12}=\frac{1-G-{\alpha }^2+G{\alpha }^2+\sqrt{G^2-4{\alpha }^2+8G{\alpha }^2-2G^2{\alpha }^2+G^2{\alpha }^4}}{1+{\alpha }^2},\end{array}$$

(14)

and its value is larger than 1 in the whole parametric region, as depicted in Figure 3c, indicating that ${\widehat{a}}_1$ is not quantum entangled with ${\widehat{a}}_2$. This is determined by the following fact: ${\widehat{a}}_1$ and ${\widehat{a}}_2$ are both generated from ${\widehat{a}}_s$ and in a competitive relationship.

3. Three-Mode Entanglement

3.1. Two-Condition Criterion

In the following, we analyze three-mode entanglement by using different criteria, i.e., two-condition, optimal single-condition, and PPT. First of all, a set of inequalities based on the two-condition criterion [22] is given by

$$\begin{array}{ccc}\hfill V_{s1}& =& V(X_s-X_1)+V(Y_s+Y_1+O_2{Y}_2)\ge 4,\hfill \\ \hfill V_{12}& =& V(X_1-X_2)+V(Y_1+Y_2+O_s{Y}_s)\ge 4,\hfill \end{array}$$

(15)

where $O_i$ (i = s, 2), as arbitrary real numbers can be used to minimize the values in Equation (15). If the two inequalities in Equation (15) are both violated, then it can be deemed as a sufficient criterion to claim the presence of genuine three-mode entanglement. Following this idea, via direct differentiation of Equation (15) with regard to $O_i$, the optimal results of $O_i$ ($O_i^{opt}$) can be given by

$$\begin{array}{ccc}\hfill O_s& =& \frac{-(〈Y_s{Y}_1〉+〈Y_s{Y}_2〉)}{〈Y_s^2〉}\hfill \\ & =& \frac{2\sqrt{G(G-1)}(1+\alpha )}{(2G-1)\sqrt{1+{\alpha }^2}},\hfill \\ \hfill O_2& =& \frac{-(〈Y_s{Y}_2〉+〈Y_1{Y}_2〉)}{〈Y_2^2〉}\hfill \\ & =& \frac{2[\alpha -G\alpha +\sqrt{G(G-1)(1+{\alpha }^2)}]}{2G-1+{\alpha }^2},\hfill \end{array}$$

(16)

and substituting Equation (16) into Equation (15), the detailed expression of Equation (15) can be expressed as

$$\begin{array}{cc}\hfill V_{s1}& =\frac{\begin{array}{c}4(G+{\alpha }^2)[G+2G{\alpha }^2-\alpha \end{array}(\alpha +2\sqrt{G(G-1)(1+{\alpha }^2)})]}{(1+{\alpha }^2)(2G-1+{\alpha }^2)}\hfill \\ \hfill V_{12}& =\frac{4G[G{(\alpha -1)}^2+2\alpha ]}{(1+{\alpha }^2)(2G-1)}\hfill \end{array}$$

(17)

The contour plot of Equation (17) is shown in Figure 4. The dependence of $V_{s1}$ on G and $\alpha$ is shown in Figure 4a. The region of $V_{s1}$ < 4 is enlarged compared to the one of $D_{s1}$ < 4 in Figure 2a when we consider the phase quadrature of ${\widehat{a}}_2$ ($Y_2$). The variance of $Y_s$ + $Y_1$ + $Y_2$ becomes smaller than the one of $Y_s$ + $Y_1$, which claims that ${\widehat{a}}_2$ has a correlation with ${\widehat{a}}_s$ + ${\widehat{a}}_1$. The dependence of $V_{12}$ on G and $\alpha$ is shown in Figure 4b, and the region of $V_{12}$ < 4 is also enlarged compared to the one of $D_{12}$ < 4 in Figure 2c due to the same reason.

The overlapped light blue region between $V_{s1}$ < 4 and $V_{12}$ < 4 in Figure 4c means that genuine three-mode entanglement is present in this system.

3.2. Optimal Single-Condition Criterion

Secondly, the single-condition criterion using the combined quadrature variances [22] can be used to test and verify the presence of genuine three-mode entanglement. If its value in Equation (18) is no more than 2, genuine three-mode entanglement can be verified.

$$\begin{array}{ccc}\hfill V_{s12}& =& V[X_s-\frac{1}{\sqrt{2}}(X_1+X_2)]+V[Y_s+\frac{1}{\sqrt{2}}(Y_1+Y_2)]\hfill \\ & =& 2[3G-1+(G-1)\frac{2\alpha }{1+{\alpha }^2}-\frac{2(1+\alpha )\sqrt{2G(G-1)}}{\sqrt{1+{\alpha }^2}}].\hfill \end{array}$$

(18)

The dependence of $V_{s12}$ is depicted in Figure 5, and its value in most of the region is no more than 2, which clearly shows the presence of genuine three-mode entanglement. Inspired by the above results, by introducing different factors [22] instead of $1/\sqrt{2}$, the noise unbalance between ${\widehat{a}}_1$ and ${\widehat{a}}_2$ in Equation (18) can be effectively canceled; this criterion can be called the optimal single-condition criterion. In this way, $V_{s12}^{opt}$ can be expressed as follows:

$$V_{s12}^{opt}=V[X_s-F_1{X}_1-F_2{X}_2]+V[Y_s+F_1{Y}_1+F_2{Y}_2],$$

(19)

where the optimal expressions (F${}_1^{opt}$ and F${}_2^{opt}$) of F${}_1$ and F${}_2$ can be calculated as

$$\begin{array}{ccc}\hfill F_1^{opt}& =& \frac{2\alpha \sqrt{G(G-1)}}{(2G-1)\sqrt{1+{\alpha }^2}},\hfill \\ \hfill F_2^{opt}& =& \frac{2\sqrt{G(G-1)}}{(2G-1)\sqrt{1+{\alpha }^2}},\hfill \end{array}$$

(20)

respectively. Substituting Equation (20) into Equation (19), $V_{s12}^{opt}$ can be simply written as

$$\begin{array}{c}\hfill V_{s12}^{opt}=\frac{2}{2G-1}.\end{array}$$

(21)

The contour plot of $V_{s12}^{opt}$ is depicted in Figure 6a, and it is not dependent on interaction strength ratio $\alpha$, meaning that the generation of the two modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ is the mode $X_s$; thus, $F_1{X}_1+F_2{X}_2$ is a combined idler mode with respect to the signal mode $X_s$.

To verify the presence of genuine three-mode entanglement, all the boundaries should be calculated according to [22]. If the value of Equation (21) is smaller than the smallest boundary, there will exist genuine three-mode entanglement. In this sense, the boundaries of $V_{s12}^{opt}$ are $B_1=2(1+\left|F_1^2+F_2^2\right|)$, $B_2=2(\left|F_1^2\right|+\left|1-F_2^2\right|)$, and $B_3=2(\left|F_2^2\right|+\left|1-F_1^2\right|)$ with the detailed expressions of

$$\begin{array}{ccc}\hfill B_1& =& 4-\frac{2}{{(2G-1)}^2},\hfill \\ \hfill B_2& =& \frac{2+2[1+8G(G-1)]{\alpha }^2}{{(2G-1)}^2(1+{\alpha }^2)},\hfill \\ \hfill B_3& =& \frac{2[1+8G(G-1)+{\alpha }^2]}{{(2G-1)}^2(1+{\alpha }^2)},\hfill \end{array}$$

(22)

respectively. To find the smallest value between $B_1$, $B_2$, and $B_3$, we should obtain their extreme points when the value of $\alpha$ is fixed. Thus, the smallest values of $B_2$ and $B_3$ can be obtained:

$$\begin{array}{ccc}\hfill {\left(B_2\right)}_{min}& =& \underset{\alpha \to 0}{lim}\frac{2+2[1+8G(G-1)]{\alpha }^2}{{(2G-1)}^2(1+{\alpha }^2)}=\frac{2}{{(2G-1)}^2},\hfill \\ \hfill {\left(B_3\right)}_{min}& =& \underset{\alpha \to \infty }{lim}\frac{2[1+8G(G-1)+{\alpha }^2]}{{(2G-1)}^2(1+{\alpha }^2)}=\frac{2}{{(2G-1)}^2}.\hfill \end{array}$$

(23)

As Equation (23) shows, the value of $B_2$ in the limit of $\alpha =0$ is equivalent to the one of $B_3$ in the limit of $\alpha =\infty$, and this is because under both two conditions, three-mode entanglement will reduce to two-mode entanglement only pumped by one pump beam; thus, ${\left(B_2\right)}_{min}$ and ${\left(B_3\right)}_{min}$ cannot be viewed as the boundary of three-mode entanglement. The other effective boundary $B_1$ (trace B) is depicted in Figure 6b, and its value is always larger than the one of $V_{s12}^{opt}$ (trace A); thus, genuine three-mode entanglement is present in the whole gain range. Thus, the optimal single-condition criterion can be used to quantify genuine three-mode entanglement more efficiently than the single-condition criterion.

3.3. PPT Criterion

Finally, the PPT criterion can also be used to quantify genuine three-mode entanglement [18,19,23]. For three-mode entanglement, the three possible 1 × 2 partitions (${\widehat{a}}_s$− (${\widehat{a}}_1$, ${\widehat{a}}_2$), ${\widehat{a}}_1$− (${\widehat{a}}_s$, ${\widehat{a}}_2$), and ${\widehat{a}}_2$− (${\widehat{a}}_s$, ${\widehat{a}}_1$)) have to be tested. When the smallest symplectic eigenvalue for each of the three PT CMs is smaller than 1, all the partitions are inseparable, and genuine three-mode entanglement will exist.

When the PT operation is applied to the mode ${\widehat{a}}_s$, the entanglement between ${\widehat{a}}_s$ and the rest of the modes (${\widehat{a}}_1$ and ${\widehat{a}}_2$) can be quantified by the smallest symplectic eigenvalue $T_{s-12}$:

$$T_{s-12}=\frac{\sqrt{1+3{\alpha }^2+3{\alpha }^4+{\alpha }^6-8G{(1+{\alpha }^2)}^3+8G^2{(1+{\alpha }^2)}^3-4(2G-1){(1+{\alpha }^2)}^3\sqrt{G(G-1)}}}{{(1+{\alpha }^2)}^{3/2}}.$$

(24)

As depicted in Figure 7a, the value of $T_{s-12}$ in the whole region is smaller than 1, meaning that the mode ${\widehat{a}}_s$ is quantum entangled with the rest of the modes (${\widehat{a}}_1$ and ${\widehat{a}}_2$). It should be emphasized that the value of $T_{s-12}$ is independent of the interaction strength ratio $\alpha$, and this is because the combination of the two idler modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ can be viewed as a combined idler mode. Similarly, when the PT operation is applied to the modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$, the smallest symplectic eigenvalues are $T_{1-s2}$ and $T_{2-s1}$, and their detailed results can be written as

$$T_{1-s2}=\frac{\sqrt{\begin{array}{c}1+(8G-5){\alpha }^2+(8G^2-5){\alpha }^4+[1+8G(G-1)]{\alpha }^6\\ -4[(2G-1){\alpha }^2-1](\alpha +{\alpha }^3)\sqrt{(G-1)(1+G{\alpha }^2)}\end{array}}}{{(1+{\alpha }^2)}^{3/2}},$$

(25)

and

$$\begin{array}{c}\hfill T_{2-s1}=\frac{\sqrt{\begin{array}{c}1-5{\alpha }^2-5{\alpha }^4+{\alpha }^6+8G^2(1+{\alpha }^2)+8G({\alpha }^4-1)\\ -4({\alpha }^2+1)(2G-1+{\alpha }^2)\sqrt{(G-1)(G+{\alpha }^2)}\end{array}}}{{(1+{\alpha }^2)}^{3/2}},\end{array}$$

(26)

respectively. The contour plots of $T_{1-s2}$ and $T_{2-s1}$ are depicted in Figure 7b,c, respectively. It can be clearly seen that the values of $T_{1-s2}$ and $T_{2-s1}$ are both smaller than 1 in the whole parametric region, showing that the three partitions are all inseparable and the existence of genuine three-mode entanglement in the whole parameter region.

4. Conclusions

In conclusion, we theoretically predicted that the concurrent parametric amplifier as a simple system can be used to generate two-mode and three-mode entanglement. The Duan criterion and PPT criterion can be used to quantify two-mode entanglement, which is present in the two pairs. The two-condition and optimal single-condition criterion were both analyzed to search for the entanglement region. In the case of the optimal single-condition criterion, genuine three-mode entanglement is present in the whole parameter region. More importantly, the PPT criterion was also used to claim the existence of genuine three-mode entanglement in the whole parameter region. Our concurrent parametric amplifier for generating multi-mode correlation and entanglement is integrated and phase-insensitive.