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Zhao Chao-Ying, Zhang Cheng-Mei. Tripartite continuous-variable entanglement of NOPA system . Chinese Physics B, 2018, 27(8): 084204  
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Tripartite continuous-variable entanglement of NOPA system
Zhao Chao-Ying1, 2, †, Zhang Cheng-Mei3
College of Science, Hangzhou Dianzi University, Hangzhou 310018, China
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
Nokia Solutions and Networks, Hangzhou 310053, China

 

† Corresponding author. E-mail: zchy49@hdu.edu.cn

Projected supported by the National Natural Science Foundation of China (Grant No. 11504074) and the Science Fund from the State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Shanxi, China (Grant No. KF201601).

Abstract

We investigate the fundamental limits to the achievable tripartite continuous-variable (CV) entanglement criterion of a generalized V1 criterion. Our numerical simulation results show that the non-degenerate eigenvalues do effect the performances of the estimated minimum variances. From below the threshold to above the threshold, with the increase of the pump parameter, the tripartite CV entanglement gradually disappears. The different off-diagonal elements seriously distort the weights for entanglement. We can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements εij.

PACS: 42.50.Lc;42.65.Lm
Keyword:continuous-variable entanglement;generalized V1 criterion;non-degenerate eigenvalues
1. Introduction

Optical parametric process is one of the most effective techniques to realize entangled states. All bipartite entangled states are useful for quantum teleportation,[1] dense coding,[2] cryptography,[3] tomography,[4] entanglement swapping,[5] quantum key distribution,[6] quantum computation, and quantum information process.[7,8] Aoki et al. first theoretically proposed that a tripartite continuous-variable (CV) entangled state should be generated by combining three independent squeezed vacuum states.[9] Jing et al. proposed a tripartite entangled state of a bright optical field should be experimentally produced by using an Einstein–Podolsky–Rosen entangled state and linear optics.[10] Pfister et al. showed theoretically a tripartite CV entangled state in a second-order nonlinear medium with concurrent interactions.[11] Guo et al. theoretically produced tripartite entanglement by using a parametric down conversion and sum-frequency process.[12] Furthermore, Pennarun et al. predicted tripartite CV entanglement by using cascaded down-conversion and sum-frequency process.[13] Allevi et al. experimentally demonstrated the nonclassical photon number correlations in tripartite CV states[14] and then investigated the process of entanglement transfer.[15] Weinstein explored the tripartite entanglement witnesses and entanglement sudden death.[16] Wang et al. experimentally investigated tripartite entanglement in a cascaded four-wave-mixing (FWM) system.[17] Wang et al. experimentally demonstrated three quantum correlated beams and quantum steering by using a phase-sensitive cascaded FWM process.[18,19] Zheng et al. investigated different kinds of entanglement in an FWM process with a degenerate pump.[20] Qin et al. experimentally produced multiple quantum correlated beams in a hot rubidium vapor.[21] Furthermore, Daems et al. presented spatial tripartite entanglement.[22]

Several schemes in which the temporal multipartite entanglement was realized[23] were proposed as result of parametric interaction, such as cascaded, concurrent,[24] interlinked,[25] or consecutive parametric interactions,[26] coupled parametric down-conversions.[27] Lv and Jing investigated quadripartite entanglement by using a cascaded FWM process.[28] We investigated theoretically CV multipartite entanglement in a non-degenerate optical parametric amplification (NOPA) system.[29–31] We obtained the generalized criterion V1 for multipartite entanglement. Here we set all off-diagonal elements to be equal, i.e., . In this paper, we will take off-diagonal elements to be different, i.e., , which act as the conditions under which the realistic tripartite entanglement for this system takes place. We can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements εij.

2. Criteria for tripartite entanglement

In the p representation, the Hamiltonian of the system in the rotating frame at the driving frequency is given by the following expression[29–31]

where
where the creation (annihilation) operators (a0), (aj), and (ak) at frequencies ω0, ωj, and ωk refer to the pump, signal, and idler field, respectively; H0 describes the Hamiltonian of the heat bath; V represents the interaction Hamiltonian of the signal, idler field, and heat bath; W represents the interaction Hamiltonian between pump field α0 (included in gain factor εjk = χjkα0) and signal photons , and idler photons .

Assuming perfect phase-matching, ignoring pump depletion, and setting output loss rates to be equal, i.e., kj = k, we will first examine a simplified analytical model for the propagation of fields. Dynamical equations for a non-degenerate optical parametric process in compound modes are

In the positive-p representation, the semi-classical equations of motion for three degenerate parametric modes a1, a2, a3[30] can be obtained by solving the Fokker–Planck equation. Here , , ,
where

Due to the fact that the eigenvalues are different from each other, we can use the eigensolution to construct a transfer matrix uij,[29] which connects the output (b1, b2, b3) and input (a1, a2, a3) and satisfies

Here Δij (−λi) is the cofactor of the determinant.

NOPA with modes a1, a2, a3 can be resolved into a combination of DOPA1 (k, λ1) with mode b1, DOPA2 (k, λ2) with mode b2 and DOPA1 (k, λ3) with mode b3[29]

where L = d/dt + k, in the case of k = 0, , we write out the quantum fluctuations of b1 and b2
Here X1 (0), Y2 (0), and Y3 (0) are the input amplitudes, and we have ⟨[X1 (0)]2⟩ = ⟨[Y2(0)]2⟩ = ⟨[Y3 (0)]2 ⟩ = 1, ⟩X1 (0) ⟩ = ⟨Y2(0)⟩ = ⟨ Y3 (0)⟩ = 0, ⟨ X1 (0)Y2 (0) ⟩ = ⟨ X1 (0)Y3 (0) ⟩ = 0. For the non-degenerate optical fields, the measurements of the idler amplitudes Y2 (l) and Y3(l) will infer the result for the signal amplitude X1 (l).

During the interaction time t = l/c, quantum fluctuation becomes

The output signal amplitude X1 and idler amplitudes Y2 and Y3 satisfy
Equations (7) and (8) correspond to the case of k → 0. Considering the case of loss k > 0 and adding the vacuum fluctuations, the quantum fluctuations are
The correlations between X1 and Y2, and between X1 and Y3 satisfy
We can use the solutions of X1, Y2, and Y3 to find the quadrature variances[29]
The key point is that the entanglement exists among the signal amplitude X1 (located at position 1), the idler phase Y2 (located at position 2), and Y3 (located at position 3). The amplitude and phase components X1, Y2, and Y3 are generated through the NOPA system interaction involved in W of Eq. (1). Now we use Eq. (11) to monitor the evolutions of signal and idler modes of the NOPA system and regard the violation of V1 ≥ 1 as the sufficient condition of entanglement.

3. Numerical calculation

The variations of minimum variance V1 with interaction time t for ε12 = 1.5 ε0, ε13 = 0.6 ε0, ε23 = ε0 are investigated and shown in Fig. 2.

Fig. 1. Input–output relation in NOPA system, the modes a1, a2, a3, and frequencies ω0, ωj, ωk refer to pump, signal, and idler field, respectively. χij represents the effective second-order susceptibility of nonlinear crystal in NOPA system.
Fig. 2. (color online) Plots of variance V1 versus t with (a) ε0 = 0.5 and (b) 2 for k = 0 (solid curve), 0.5 (narrow dashed curve), 0.9 (wide dashed curve), and 1.1 (dotted-dashed curve).

From below the threshold to above the threshold, the curves gradually merge, the position of the minimum variance turns gradually toward the left, and the value of minimum variance V1 decreases. With the increase of loss k, two curves satisfy the condition V1min; 0.82 < 1, and the entanglement disappears gradually. While four curves satisfy the condition V1min; 0.6 < 1, the entanglement always appears.

Given pump parameter ε0 = 2, the value of minimum variance V1 changes with off-diagonal elements εij. In Figs. 3(a)–3(c), four curves satisfy the condition V1min; 0.58, 0.7, 0.76 < 1, respectively, so the entanglement always appears.

Fig. 3. (color online) Plots of variance V1 versus t with ε0 = 2 for k = 0 (solid curve), 0.5 (narrow dashed curve), 0.9 (wide dashed curve), and 1.1 (dashed–dotted curve), for the cases of panel (a): ε12 = 1.5 ε0, ε13 = 0.6 ε0, ε23 = ε0; (b) ε12 = 1.2 ε0, ε13 = 0.4 ε0, ε23 = 0.8 ε0; (c) ε12 = ε0, ε13 = ε0, ε23 = ε0.
4. Conclusions

In this paper, we use the minimum variance criterion to monitor the evolutions of signal and idler modes of a NOPA system with different off-diagonal elements, as these are the conditions under which there exists the realistic tripartite entanglement for this system. Our result is suitable for the whole region from below the threshold to above the threshold. For above the threshold, we can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements εij.

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