1. Introduction

Multimode continuous variable quantum entangled source, which is an alternative candidate of discrete variables, has played an important role in quantum information processing and quantum metrology [1–3]. So far, the mature technology to generate the two-mode entangled source is the spontaneous parametric down-conversion process with a $\chi ^{(2)}$ crystal [4,5]. However, due to the weak nonlinearity in the crystal [4], a highly reflective external cavity is generally used to form an optical parametric oscillator (OPO) to enhance the nonlinear effect, thereby improving the conversion efficiency [6]. In general, the preparation of multimode entangled sources requires cascading or paralleling multiple such nonlinear processes [7–10]. However, with the increase in the number of modes, further requirements are imposed on the complexity and scalability of the experimental system. Recently, many source-saving methods based on the OPO have been proposed and demonstrated to generate multimode entangled sources, such as the multipartite entanglement of 60 modes with an optical frequency comb of sidebands [11], on-demand multimode entanglement networks using light pulses of an optical frequency comb [12,13], and the cluster states of 10,000 entangled modes in the timing multiplexing [14], as well as the hexapartite entanglement in the OPO process above the threshold [15,16].

Another method for generating entanglement is using the FWM process in warm atoms. Via employing the double-$\Lambda$ energy level in Rb vapor medium non-degenerate four-wave-mixing (FWM) could generate squeezing and entanglement in between the probe and conjugate light [17–20]. Compared with the OPO, the single-pass FWM process in warm atoms has the following advantages: Firstly because of the strong third-order nonlinearity of hot atoms, it doesn’t require an optical cavity, which greatly simplifies the experimental device; then the nonclassical state of light field generated by FWM naturally matches the atomic transition, which lays a good foundation for the quantum state storage based on the atomic system [21,22]; thirdly, the nonclassical states generated by FWM may have more spatial modes, which greatly improves the information capacity [23,24]. To date, the FWM process has been used to realize many quantum protocols, including entanglement imaging [25,26], four-wave slow light [27], delayed EPR entanglement [27,28], nonlinear interferometer [29,30], quantum noiseless amplification [31], etc.

Furthermore, the FWM process is extended to multimode scenario by using a dual-pump protocol [32]. By adjusting the angle between the two pump beams, it was possible to experimentally produce spatially symmetrical 6, 10, and up to 2N quantum correlated beams [33], and multi-spatial-mode entanglement [34]. Also, the non-seeding dual-pump protocol was demonstrated to directly generate four-mode bright correlated beams [35]. Some other works to generate multimode correlation and entanglement via applying structured pumps are also demonstrated [36–38]. These experiments showed that, compared with other physical systems, such as OPO process, the atomic ensemble has better ability in space expansion due to its isotropy and phase matching property. This provides a scalable and flexible recipe to generate multimode entanglement, without the necessity of cascading or paralleling multiple sources.

It is therefore interesting to analyze the dual-pumped FWM model and the multimode entanglement characteristics. In this work, we apply the linear mode transform to reduce the correlated modes into a set of eigenmodes, and obtain the analytical input-output relation of each mode in the four- and six-mode system. Then we analyze in detail the multimode entanglement characteristics via reconstructing the full covariance matrix of the produced Gaussian states. We find that most of the modes are entangled and there exists obvious competition between single- and dual-pump FWM processes in such a multimode system, which provides a new idea for quantum coherent control in the dual-pump configuration.

2. Theoretical model

In this section, we model the dual-pumped FWM process to produce multiple correlated beams, and a mode transform approach is applied to obtain the input-output relation analytically. The schematic diagram of the non-seeding dual-pumped FWM protocol is shown in Fig. 1(A). Two coherent pump beams ($P_A$ and $P_B$) are employed and simultaneously go through a single rubidium vapour with a slight angle. As the two pumps could satisfy the phase matching condition independently or jointly, it has four co-existing FWM processes. As seen in Fig. 1(B), one possible case is two pump photons of $P_A$ are converted to generate two new photons in the mode of $\hat {a}_{1}$ and $\hat {a}_{4}$ via FWM with phase matching condition, 2$\vec {k}_{A}$=$\vec {k}_{1}$+$\vec {k}_{4}$; meanwhile, another case is two pump photons of $P_B$ are converted into $\hat {a}_ {2}$ and $\hat {a}_ {3}$ via FWM, and the phase matching condition is 2$\vec {k}_{B}$=$\vec {k}_{2}$+$\vec {k}_{3}$. Moreover, when the two pump beams are adjusted to an appropriate angle, the double pump effect can appear, which is, annihilating a pump photon of $P_A$ and $P_B$ will generate the correlated $\hat {a}_ {1}$ and $\hat {a}_{3}$ photons, or another case, $\hat {a}_ {2}$ and $\hat {a}_ {4}$ photons. The phase matching condition of the dual-pump FWM process is $\vec {k}_ {a}$ + $\vec {k}_ {b}$ = $\vec {k}_ {1}$ + $\vec {k}_ {3}$ = $\vec {k}_ {2}$ + $\vec {k}_ {4}$. Due to the coexistence of single- and dual-pumped cases, the interaction Hamiltonian of the system under the parametric approximation can be written as

 

Fig. 1. (A) The schematic diagram of the Rubidium cell (purple) with two pump beams of $P_A$ and $P_B$, and the conical rings (green) represent the spontaneous emission via the FWM processes. (B) and (C) The spatial structures of the output fields, i.e., four- and six-mode, respectively. The red, blue and green circles represent the pump, produced Stokes and anti-Stokes beams, respectively. The black lines corresponds to nonlinear interactions of FWM.

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Figure 1(C) is a spatial diagram for generating six-mode quantum correlated beams using the dual-pumped FWM protocol. Similar to Fig. 1(B), two coherent pump beams ($P_A$ and $P_B$) are used, but with a different angle, to generate six correlated beams. From the perspective of the generation mechanism, this can also be divided into the single-pumped and the dual-pumped FWM processes. The first case is a single-pump FWM process, annihilating two pump photons of $P_A$ and generating the correlated $\hat {b}_ {1}$ and $\hat {b}_ {2}$ modes, or $\hat {b}_ {3}$ and $\hat {b}_ {5 }$ modes, and the corresponding phase matching condition is 2 $\vec {k}_ {A}$ = $\vec {k}_ {1}$ + $\vec {k}_ {2}$ = $\vec {k}_ {3}$ + $\vec {k}_ {5}$. On the other side, annihilation two pump photons of $P_B$ can also generate $\hat {b}_ {1}$ and $\hat {b}_ {4}$ modes, or $\hat {b}_ {3}$ and ${b}_ {6}$ modes with phase matching condition 2 $\vec {k}_ {B}$ = $\vec {k}_ {1}$ + $\vec {k}_ {4}$ = $\vec {k}_ {3}$ + $\vec {k}_ {6}$. The other case is the dual-pumped FWM process, that is, annihilating one pump photon of $P_A$ and $P_B$ while generating $\hat {b}_ {2}$ and $\hat {b}_ {6}$ , $\hat {b}_ {4}$ and $\hat {b}_ {5}$, $\hat {b}_ {1}$ and $\hat {b}_ {3}$ modes. The phase matching conditions are: $\vec {k}_ {A}$ + $\vec {k}_ {B}$ = $\vec {k}_ {2}$ + $\vec {k}_ {6 }$ = $\vec {k}_ {4}$ + $\vec {k}_ {5}$ = $\vec {k}_ {1}$ + $\vec {k}_ {3}$. The interaction Hamiltonian of the six-mode system is,

Next, we apply a mode transform approach to solve the input-output relation of the interaction Hamiltonian defined in Eqs. 1 and 2. In general, the interaction Hamiltonian of the dual-pump system has the form $\hat {H}_I=\sum ^n_{i\neq j} c_{ij}\hat {a}_i\hat {a}_j+H.c$, which can be rewritten as $\hat {H}_I=1/2*\vec {v}^TU_m\vec {v}$, where $\vec {v}=(\hat {a}_{1}^{\dagger }, \hat {a}_{2}^{\dagger }, \ldots \hat {a}_{n}^{\dagger }, \hat {a}_{1}, \hat {a}_{2}, \ldots \hat {a}_{n})^T$, and the coupling matrix is,

Considering the symmetry of the coefficient matrix $U_m$, the upper part of the matrix already contains all the information of the system, so we only consider the $U_1$ part of its diagonal elements. For the symmetrical matrix $U_1$, we can always find a unitary transform $S$ such that $U_1$ could be diagonalized to reach $U_d=SU_1S^T=Diag(\lambda _1,\lambda _2\cdots \lambda _n)$. Therefore, for instance, the Hamiltonian, $\hat {H}_{1}$, can be rewritten in the Schmidt modes basis as: $\hat {H_{1}}=\sum _{i=1}^{4} \lambda _i\hat {\xi }_{i} \hat {\xi }_{i}+h.c.$, where $\hat {\xi }_{i} = S \vec {v}$, $\lambda _i$ is the eigenvalue of the corresponding eigenmode. This also works for the six-mode system, $\hat {H}_ {2}=\sum _{i=1}^{6} \lambda _{i} \hat {\xi }_{i} \hat { \xi }_{i} +h.c.$. After the basis change, under the eigenmodes representation, the modes $\{\hat {\xi }_i\}$ are independently squeezed, without being correlated to each other. At this time, $t$, we can easily get the input-output relationship of the operators under the new representation,

For the four-mode system, as seen in Fig. 1(B), its eigenvalues are $\lambda _{1}=-\varepsilon _a-\varepsilon _c$, $\lambda _{2}=\varepsilon _a-\varepsilon _c$, $\lambda _{3}=\varepsilon _c-\varepsilon _a$, $\lambda _{4}=\varepsilon _a+\varepsilon _c$, respectively. The corresponding unitary operator has the form,

3. Results

3.1 Quantum states

A multimode Gaussian state generated by a quadratic Hamiltonian can be fully characterized by its covariance matrix (CM) [39]. In this section, we will first reconstruct the CMs based on the obtained input-output relation of quadrature operators in Eq. 4. A vector of quadrature operators is defined as, $\vec {R}=(\hat {X}_1,\hat {Y}_1,\ldots , \hat {X}_n, \hat {Y}_n)$, satisfying the commutation relation, $[R_i,R_j]=i\Omega _{ij}$, where $\Omega _{ij}$ is the element of the symplectic matrix,

For the four-mode case, we choose to take $\epsilon _b=0.3$, $\epsilon _d=0.24$ and $t=1$, and the CM of the four-mode state is shown in Fig. 2(A).

 

Fig. 2. (A) The covariance matrix of the four-mode state, when $\epsilon _a=\epsilon _b=0.3$, $\epsilon _c=\epsilon _d=0.24$ and $t=1$. The corresponding gain of the electric filed of $\hat {a}_1$ is about 1.1. Here the shot noise that is unity has been subtracted from all the diagonal terms for clarity. (B) The supermodes (eigenmodes) of the four-mode state and their corresponding squeezing levels. (C) The squeezing levels of the supermodes when increasing the interaction time from 0 to 10.

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It is obvious to find the $\sigma (\hat {X}_i, \hat {Y}_j)$ terms in the CM within the same or different modes are all zero, so it has no amplitude-phase correlation in the state. It is also true in other parametric amplification process, such as multimode optical parametric oscillators [17–20]. Besides, as in between the modes, $\{\hat {a}_{1}$, $\hat {a}_{2}\}$ and $\{\hat {a}_{3}$, $\hat {a}_{4}\}$ , both the amplitude and phase correlations are positive, therefore only classical correlation exists. However, the other modes, $\{\hat {a}_{1}, \hat {a}_{4}\}$ ($\{\hat {a}_{2}, \hat {a}_{3}\}$), $\{\hat {a}_{1}, \hat {a}_{3}\}$ ($\{\hat {a}_{2}, \hat {a}_{4}\}$ ), the negative elements within the phase quadratures part of the CM suggests the presence of quantum correlations, and it satisfies $\sigma (\hat {X}_i, \hat {X}_j)=-\sigma (\hat {Y}_i, \hat {Y}_j)$. Here $\{\hat {a}_{1}, \hat {a}_{4}\}$ ($\{\hat {a}_{2}, \hat {a}_{3}\}$) have stronger quantum correlation, as these are generated via a single-pump FWM process which has stronger nonlinear interaction. All the two modes playing different roles (Stokes or anti-Stokes) can have quantum correlations in the four-mode state. The four eigenvalues of the amplitude quadrature of the CM are $\{0.3396, 0.8869, 1.1275, 2.9447\}$, and their corresponding eigenmodes are shown in Fig. 2(B). Note that, for the four-mode case, because of the special symmetry arising from using two pumps with equal power, the eigenmodes keep constant while modulating the interaction coefficients. As seen in Fig. 2(C), we exhibit the evolution of the corresponding squeezing and antisqueezing of the eigenmodes. Experimentally, this is associated with increasing the pump power. We find all these values change linearly when increasing the interaction time, and this also has been verified by many previous experiments [40,41].

For the six-mode case, considering the symmetry, we choose to take $\epsilon _a=\epsilon _c=\epsilon _d=\epsilon _e=0.348$, $\epsilon _f=\epsilon _g=0.585$, $\epsilon _b=0.992$, the CM is calculated, as seen in Fig. 3(A). We find it does not exist $\hat {X}\hat {Y}$ correlation neither. The most correlated modes are $\hat {b}_1$ and $\hat {b}_3$, where the correlation of its amplitude quadratures is positive, while the phase quadratures part are negative. It suggests a strong quantum correlation existing in between these two modes. Also the modes $\{\hat {b}_1, \hat {b}_2\}$ ($\{\hat {b}_1, \hat {b}_4\}$, $\{\hat {b}_3, \hat {b}_5\}$, $\{\hat {b}_3, \hat {b}_6\}$) have less strong quantum correlations, and the modes $\{\hat {b}_2, \hat {b}_6\}$ ($\{\hat {b}_4, \hat {b}_5\}$) are least quantum correlated. For all the other possibilities of two mode permutation have only classical correlations. This can be explained as the quantum correlation origins from the interaction Hamiltonian of the FWM processes. As seen in Fig. 3(B), the six eigenmodes are shown, and their corresponding eigenvlaues are $\{0.0712, 0.3104, 0.5992, 1.6690, 3.2216, 14.0423\}$. When the interaction strength is increased, the squeezing and antisqueezing values change linearly in Fig. 3(C).

 

Fig. 3. (A) The covariance matrix of the six-mode state, when $\epsilon _a=\epsilon _c=\epsilon _d=\epsilon _e=0.348$, $\epsilon _f=\epsilon _g=0.585$, $\epsilon _b=0.992$ and $t=1$. The corresponding gain of the electric filed of $\hat {b}_1$ is about 1.7. Here the shot noise that is unity has been subtracted from all the diagonal terms for clarity. (B) The supermodes (eigenmodes) of the six-mode state and their corresponding squeezing levels. (C) The squeezing levels of the supermodes when increasing the interaction time from 0 to 10.

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3.2 Entanglement criteria and two-mode entanglement

In this section, we apply the Duan [42] and the positivity under partial transposition (PPT) [43] criteria to investigate the two-mode entanglement of the produced states. Duan criterion is implemented by using the quantum correlations of the associated modes, and is defined as below,

PPT criterion works by partial transposing the CM including two subsystems, and the inseparability can be checked by the symplectic eigenvalues. It is sufficient and necessary for the case of 1 versus n-modes (1-n) and only sufficient for m versus n-modes (m-n) [43,44]. The CM of any bipartition where each part can be composed of one or several modes. For a bipartite system described by the covariance matrix $\sigma _{AB}$, the partial transposition operation on part $A$ is equivalent to the transformation through matrix $\Delta _A=(\bigoplus ^m_{k=1}\mathrm {diag}(1,-1))\bigoplus I_{B}$, where the first factor represents its mirror reflection in phase space, and the second one corresponds to the other subsystems. The symplectic eigenvalues of the partial transposed covariance matrix can be used to show the presence of entanglement.

As seen in Fig. 4(A), Duan criteria are verified for all the possible two-mode permutations with modulating the interaction time. We find $D_{\hat {a}_1, \hat {a}_2}$ and $D_{\hat {a}_3, \hat {a}_4}$ always go up and are larger than 4, this suggests no entanglement existing in between these modes. This is due to these modes do not interact with each other in the Hamitonian in Equ. 1. However, the modes $\{\hat {a}_{1}, \hat {a}_{3}\}$ ($\{\hat {a}_{2}, \hat {a}_{4}\}$), $\{\hat {a}_{1}, \hat {a}_{4}\}$ ($\{\hat {a}_{2}, \hat {a}_{3}\}$) correspond to generated the photon pairs via the FWMs, and thus their Duan values have opposite results. We find $D_{\hat {a}_1, \hat {a}_3}$ ($D_{\hat {a}_2, \hat {a}_4}$), $D_{\hat {a}_1, \hat {a}_4}$ ($D_{\hat {a}_2, \hat {a}_3}$) all decrease from the initial value of 4, so these modes are entangled. Moreover, some differences still exist in between them, e.g., $D_{\hat {a}_1, \hat {a}_4}$ is always smaller than $D_{\hat {a}_1, \hat {a}_3}$. Heuristically speaking, the modes $D_{\hat {a}_1, \hat {a}_4}$ have stronger entanglement than the modes $\hat {a}_1$ and $\hat {a}_3$, as the corresponding interaction parameter, associated with the phase matching condition, are chose differently.

 

Fig. 4. Bipartition entanglement for the two-mode subsystems. Duan (A) and PPT (C) criteria in four-mode system (3 kinds of equivalent modes). (B) and (G) are same as (A) and (C) but in six-mode system (6 kinds of equivalent modes). (D)-(F) and (H) are (A)-(C) and (G) with larger interaction time $t$.

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Interestingly, we find the entanglement of $\{\hat {a}_1, \hat {a}_3\}$ ($\{\hat {a}_2, \hat {a}_4\}$) increases and then decreases when increasing the interaction time, as seen in Fig. 4(D). This is because, when increasing the interaction time, the excess noises from the correlation within $\{\hat {a}_1, \hat {a}_4\}$ and $\{\hat {a}_2, \hat {a}_3\}$ also increase and are dominant. Note that this type competition also occurs in other systems of multimode entanglement, such as in the six-mode system. Besides, for the four-mode case, compared with general single-pumped FWMs, the dual-pumped FWM could enhance the entanglement in the modes $\{\hat {a}_1, \hat {a}_4\}$ and $\{\hat {a}_2, \hat {a}_3\}$, meanwhile also generates more possible entangled modes $\{\hat {a}_1, \hat {a}_3\}$ and $\{\hat {a}_2, \hat {a}_4\}$.

Then the Duan criteria is verified for the six-mode case, as seen in Fig. 4(B). We find $D_{\hat {b}_1, \hat {b}_5}$, $D_{\hat {b}_2, \hat {b}_4}$, $D_{\hat {b}_2, \hat {b}_5}$ are always larger than 4, as the corresponding modes don’t interact with each other via FWM, and thus no entanglement exist in these modes. However, $D_{\hat {b}_1, \hat {b}_3}$, $D_{\hat {b}_2, \hat {b}_6}$ are less than 4 and decrease with increasing the interaction time, which means the corresponding modes are entangled. Note that in this case, the central dual-pumped FWMs have the strongest interaction, and thus it determines the modes $\hat {b}_1$ and $\hat {b}_{3}$ are most entangled, and the modes $\hat {b}_{2}$ and $\hat {b}_{6}$ via the other dual-pumped FWMs are less entangled as the phase matching angle is not optimized. The related Duan values from the dual-pumped FWMs continue decreasing to zero even if the interaction time is quite large, as seen in Fig. 4(E). Besides, as the single pumped FWMs generate the weakest correlation, $D_{\hat {b}_1, \hat {b}_2}$, $D_{\hat {b}_1, \hat {b}_4}$, $D_{\hat {b}_3, \hat {b}_5}$, $D_{\hat {b}_3, \hat {b}_6}$ firstly decrease and then increase, and become lager than 4, as the excess noises get dominant to destroy the initially existed entanglement when the interaction time becomes large.

The PPT criteria are also verified for the four-mode case as seen in Fig. 4(C) and (F), and the six-mode case as seen in Fig. 4(G) and (H). We can find the results of the PPT criteria are similar as the Duan criteria.

3.3 Multimode entanglement

In this section, we study multimode (more than two modes) entanglement of the produced states. As the symmetry existing within dual-pumped FWM processes, we list all the equivalent modes permutations in Table 1 and 2 for the four- and six-mode systems, respectively. Based on the number of the modes, it is categorized into several kinds of equivalent modes in different types of entanglement. For instance, as seen in Table 1, in the two mode subsystem, it has three kinds of equivalent modes and only one type of bipartition, 1 versus 1-mode. In the following, the multimode entanglement of all the kinds and types of modes permutations will be analyzed.

Table 1. Equivalent modes permutations in the four-mode system.

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Table 2. Equivalent modes permutations in the six-mode system.

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3.3.1 Three-mode entanglement

The entanglement of the three-mode subsystem in the four-mode system is all verified, as seen in Fig. 5. For instance, it has three possible bipartition entanglement among the modes, $\{\hat {a}_1, \hat {a}_2, \hat {a}_3\}$. We find $\hat {a}_1-\{\hat {a}_2\hat {a}_3\}$ and $\hat {a}_2-\{\hat {a}_1\hat {a}_3\}$ are always entangled increasedly while modulating the interaction time, and the entanglement of the former is larger than the latter, as the FWM interaction existing in the bipartition is stronger in the former. In the other hand, $\hat {a}_1-\{\hat {a}_2\hat {a}_3\}$, the entanglement increases and then decreases, this is because the excess noise is gradually dominant while increasing the interaction time.

 

Fig. 5. Three-mode entanglement in four-mode case (the type of 1-2 modes, 1 kind of equivalent modes). (A) The PPT values when the interaction time is increased from 0 to 1. (B) Same as (A) but with extended interaction time.

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In the six-mode case, the entanglement of the three-mode subsystem, $\{\hat {b}_1, \hat {b}_2, \hat {b}_3\}$, $\{\hat {b}_1, \hat {b}_2, \hat {b}_4\}$, $\{\hat {b}_1, \hat {b}_2, \hat {b}_5\}$, $\{\hat {b}_1, \hat {b}_2, \hat {b}_6\}$, $\{\hat {b}_1, \hat {b}_5, \hat {b}_6\}$, $\{\hat {b}_2, \hat {b}_4, \hat {b}_5\}$, is also calculated, as seen in Fig. 6. When it does not contain any FWM interaction in between the two sides of the bipartition, such as $\hat {b}_5-\{\hat {b}_1\hat {b}_2\}$ in Fig. 6(F), $\hat {b}_1-\{\hat {b}_5\hat {b}_6\}$, $\hat {b}_5-\{\hat {b}_1\hat {b}_6\}$, $\hat {b}_6-\{\hat {b}_1\hat {b}_5\}$ in Fig. 6(K), and $\hat {b}_2-\{\hat {b}_4\hat {b}_5\}$ in Fig. 6(L), these corresponding modes are not entangled. Oppositely, when FWM interaction exists in between the two parts, the modes are entangled and could be divided into two cases. Firstly when only single-pumped FWM interaction occurs in between the two sides, the PPT values will decease then increase to infinity with modulating the interaction time. For instance, $\hat {b}_2-\{\hat {b}_1\hat {b}_3\}$ in Fig. 6(D), $\hat {b}_1-\{\hat {b}_2\hat {b}_4\}$, $\hat {b}_2-\{\hat {b}_1\hat {b}_4\}$, $\hat {b}_4-\{\hat {b}_1\hat {b}_2\}$ in Fig. 6(E), and $\hat {b}_1-\{\hat {b}_2\hat {b}_5\}$, $\hat {b}_2-\{\hat {b}_1\hat {b}_5\}$ in Fig. 6(F), and $\hat {b}_1-\{\hat {b}_2\hat {b}_6\}$ in Fig. 6(J). The second case is once dual-pumped FWM happens in two sides, the PPT values will decrease continuously to zero, which validates for all the other entangled modes permutations.

 

Fig. 6. Three-mode entanglement in six-mode case (the type of 1-2 modes, 6 kinds of equivalent modes). (A)-(C) and (G)-(I) The PPT values when the interaction time is increased from 0 to 1. (D)-(F) and (J)-(L) Same as (A)-(C) and (G)-(I)but with extended interaction time.

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3.3.2 Four-mode entanglement

The four-mode entanglement of the four-mode system is shown in Fig. 7. We find All the 1-3 modes are entangled in the entire parametric region. And with a high symmetry in the four-mode system, all the amounts of the 1-3 modes entanglement are equal. Besides, the PPT values of the 2-2 mode cases always decrease. This is because for a whole system without tracing modes, it is a pure state and the excess noises appeared in subsystems are correlated within one single side. For $\{\hat {a}_1\hat {a}_2\}-\{\hat {a}_3\hat {a}_4\}$, the PPT values are smaller compared to the other 2-2 mode cases. This could be understood that stronger FWM interactions exist in between the two sides.

 

Fig. 7. Four-mode entanglement in four-mode case (two types of 1-3 and 2-2 modes). The PPT values of 1-3 (A) and 2-2 (C) modes when the interaction time is increased from 0 to 1. (B) and (D) are same as (A) and (C) but with extended interaction time.

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Then the four-mode entanglement of the six-mode system is studied. The PPT values of 1-3 modes permutations are shown in Fig. 8(A-D, I, J). When it has not FWM interactions in two sides, the PPT values are not less than 1, such as $\hat {b}_5-\{\hat {b}_1\hat {b}_2\hat {b}_6\}$ in Fig. 8(I). But when dual-pumped FWM existing in two sides, the PPT values decrease continuously to zero. For instance, $\hat {b}_3-\{\hat {b}_1\hat {b}_2\hat {b}_4\}$, $\hat {b}_1-\{\hat {b}_2\hat {b}_3\hat {b}_4\}$ in Fig. 8(E), and $\hat {b}_4-\{\hat {b}_1\hat {b}_2\hat {b}_5\}$, $\hat {b}_5-\{\hat {b}_1\hat {b}_2\hat {b}_4\}$ in Fig. 8(H). Besides, when only single-pumped FWM existing in two sides, the PPT values decrease then increase back to one, such as, $\hat {b}_2-\{\hat {b}_1\hat {b}_3\hat {b}_4\}$ in Fig. 8(E), and $\hat {b}_2-\{\hat {b}_1\hat {b}_4\hat {b}_5\}$, $\hat {b}_1-\{\hat {b}_2\hat {b}_4\hat {b}_5\}$ in Fig. 8(H). In this case, the dual-pumped FWM in one side causes the excess noises to be dominant with large interaction time. The results of 2-2 modes are shown in Fig. 8(K, L, O, P, and Q-X). As the corresponding rules are similar, here we don’t explain for simplicity.

 

Fig. 8. Four-mode entanglement in six-mode case (two types of 1-3 and 2-2 modes, 6 kinds of equivalent modes). The PPT values of 1-3 (A)-(D), (I) and (J) and 2-2 (K), (L), and (Q)-(T) modes when the interaction time is increased from 0 to 1. (E)-(H), (M), (N), (O), (P), and (U)-(X) are same as (A)-(D), (I), (J), (K), (L) and (Q)-(T) respectively, but with extended interaction time.

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3.3.3 Five- and six-mode entanglement

In the six-mode system, each subsystem that consists of five modes has two categories of bipartition: 1-4 and 2-3 modes, while in the whole system, any bipartition must fall into the 3 categories: 1-5, 2-4, and 3-3 modes. The five-mode entanglement is shown in Fig. 9, and its corresponding rules are similar as in the four-mode entanglement. It has two cases: When only single-pumped FWM existing in two sides, the PPT values decrease then increase to one; and when dual-pumped FWM existing in two sides, the PPT values decrease continuously to zero. The six-mode entanglement is shown in Fig. 10. However, for the six-mode entanglement, we find that, all the 1-5, 2-4 and 3-3 modes are entangled in the entire parameter interval. Moreover, all the PPT values continuously decrease when modulating the interaction time. The reason is similar as the four entanglement in the four-mode case, i.e, when it is a pure state without tracing modes, the excess noises appeared in subsystems are correlated on one single side.

4. Conclusion

In summary, using the linear mode transformation on the quadratic Hamiltonian of the dual-pumped FWM process, we obtain the input-output relation of each mode, and further study the entanglement characteristics of the four- and six-mode systems . We find that, no matter whether it is the 1-n or m-n modes entanglement, the two Gaussian states with four or six modes show the common evolution properties in the corresponding PPT values. Firstly, if it does not contain any FWM interaction in between the two parts, they are not entangled. Secondly, in a subsystem, if there exists a FWM process with the weakest interaction strength between the two parts, the initial entangled modes could become separable. When another strong interaction FWM is evolved in either single side, the PPT values of the bipartition entanglement increases first and then decreases. But note that this doesn’t work for a whole system without tracing modes, where the excess noises appeared in subsystems are correlated, therefore the entanglement in the whole system is always enhanced via increasing the interaction strength. Thirdly, if a FWM process with strong interaction exists in between the two parts, the entanglement will be enhanced throughout the parameter interval. The essential reason for all those phenomena is that the FWM processes with different interaction strengths in the multimode system are competing with each other. It results in the multimode state via dual-pumped FWM containing both quantum correlation and excess noise, and thus in the subsystems, the related entanglement could be reduced and further disappear when the excess noise increases faster.

 

Fig. 9. Five-mode entanglement in six-mode (two types of 1-4 and 2-3 modes, 2 kinds of equivalent modes). (A) and (G) The PPT values of 1-4 modes when increasing the interaction time from 0 to 1. (B), (C),(H) and (I) The PPT values of 2-3 modes. (D)-(F) and (J)-(L) The interaction time from 0 to 5 corresponding to (A)-(C) and (G)-(I) respectively.

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Fig. 10. Six-mode entanglement in six-mode case (three types of 1-5, 2-4 and 3-3 modes). (A) The PPT values of 1-5 modes when increasing the interaction time from 0 to 1. (B), (C) and (G) The PPT values of 2-4 modes. (H) and (I) The PPT values of 3-3 modes. (D)-(F) and (J)-(L) The interaction time from 0 to 5 corresponding to (A)-(C) and (G)-(I) respectively.

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It is worth mentioning that, in our theoretical model, we assume that the power of the two pumps is equal. If the power of the two pumps is not equal, some interesting conclusions may also exist. Moreover, due to the isotropy of the atomic medium, the model can be easily extended to multimode in spatial with adjusting the angles in between the seed and pump beams as well as employing more pump beams. This work, which is tightly connected to experiments, provides a new idea for generating on-demand quantum networks.