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Based on the cascaded four wave mixing processes, a coherent-feedback control system is constructed by utilizing a linear beam splitter as the feedback controller. Considering the loss of optical propagation in the coherent feedback loop and the absorption effect of Rb vapor cells to beams, we theoretically investigate the entanglement properties of this system under different feedback ratio, gain and phase by calculating the covariance matrix of system and applying the positivity under partial transpose (PPT) criterion to all possible bipartitions. The result shows that the genuine tripartite entanglement exists in the coherent feedback control system, but the entanglement structure of system will be destroyed by the excessive feedback. In addition, when the phase is π, we find that the tripartite entanglement can be enhanced by changing the gains and the reflectivity of the beam splitter in the range of 0.1 to 0.4. The results pave the way for manipulating multipartite entanglement by coherent feedback control and have potential application in quantum communication.
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Keywords:
- four wave mixing /
- coherent feedback /
- quantum entanglement
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图 1 (a)级联四波混频相干反馈控制系统简图; (b)85Rb原子D1线的双Λ型跃迁能级结构图能级图. ∆对应单光子失谐, δ对应双光子失谐
Figure 1. (a) The scheme of coherent feedback control system based on the cascade four wave mixing processes; (b) The Double-Λ type transition energy-level diagram of 85Rb D1 line. ∆ corresponds to one-photon detuning, δ corresponds to two-photon detuning.
图 2 级联四波混频相干反馈控制系统. 绿色虚线框为反馈回路中的光传输损耗模型, 紫红色虚线框为原子吸收损耗模型
Figure 2. The coherent feedback system based on the cascaded four wave mixing processes. The green dashed frame is the optical transmission loss model in the feedback loop, and the pink dashed frames are the loss model of atomic absorption.
图 3 (a)(b)(c)分别对应三种二分情况的最小辛本征值与相位和反射率的关系
Figure 3. (a) (b) (c) correspond to the smallest symplectic eigenvalues of the three bipartitions varying with the value of phase and reflectivity, respectively.
图 4 (a)(b)(c)(d)分别为四种铷池增益情形下系统的三个输出光场强度随相位ϕ的变化. 这里, k = 0.5,
$ {\hat N_1} = \hat a_3^{'\dagger }\hat a_3^{'}$ ,${\hat N_2} = \hat b_2^{'\dagger }\hat b_2^{'} $ ,${\hat N_3} = \hat c_1^{\dagger} {\hat c_1} $ Figure 4. (a)(b)(c)(d) show the relationship between the intensity of three output fields and the phase ϕ under different gains condition, respectively. Here, k = 0.5,
$ {\hat N_1} = \hat a_3^{'\dagger }\hat a_3^{'}, \;{\hat N_2} = \hat b_2^{'\dagger }\hat b_2^{'}, \;{\hat N_3} = \hat c_1^{\dagger} {\hat c_1}$ .
图 5 (a)(b)(c)分别对应三种二分情况的最小辛本征值与增益和反射率的关系
Figure 5. (a) (b) (c) correspond to the smallest symplectic eigenvalues of the three bipartitions varying with the value of gain and reflectivity, respectively.
图 6 当ϕ取180°及G = 3和5时, 三种二分情况的最小辛本征值与反射率的关系
Figure 6. The smallest symplectic eigenvalues of the three bipartitions vary with reflectivity when ϕ is taken the value of 180° and G = 3, 5.
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[1] DiVincenzo D P 1995 Science 270 255
Google Scholar
[2] Lloyd S, Braunstein S L 1999 Phys. Rev. Lett. 82 1784
Google Scholar
[3] Braunstein S L, Loock P van 2005 Rev. Mod. Phys. 77 513
Google Scholar
[4] Weedbrook C, Pirandola S, García-Patrón R, Cerf N J, Ralph T C, Shapiro J H, Lloyd S 2012 Rev. Mod. Phys. 84 621
Google Scholar
[5] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[6] Jing J, Zhang J, Yan Y, Zhao F, Xie C, Peng K 2003 Phys. Rev. Lett. 90 167903
Google Scholar
[7] Yonezawa H, Braunstein S L, Furusawa A 2007 Phys. Rev. Lett. 99 110503
Google Scholar
[8] Jia X, Su X, Pan Q, Gao J, Xie C, Peng K 2004 Phys. Rev. Lett. 93 250503
Google Scholar
[9] Yonezawa H, Aoki T, Furasawa A 2004 Nature 431 430
Google Scholar
[10] Loock P van, Braunstein S L 2000 Phys. Rev. Lett. 84 3482
Google Scholar
[11] Coelho A S, Barbosa F A S, Cassemiro K N, Villar A S, Martinelli M, Nussenzveig P 2009 Science 326 823
Google Scholar
[12] Armstrong S, Wang M, Teh R Y, Gong Q, He Q, Janousek J, Bachor H A, Reid M D, Lam P K 2015 Nat. Phys. 11 167
Google Scholar
[13] Cassemiro K N, Villar A S 2008 Phys. Rev. A 77 022311
Google Scholar
[14] Daems D, Cerf N J 2010 Phys. Rev. A 82 032303
Google Scholar
[15] Yokoyama S, Ukai R, Armstrong S C, et al. 2013 Nat. Photonics 7 982
Google Scholar
[16] Roslund J, de Araújo R M, Jiang S, Fabre C, Treps N 2014 Nat. Photonics 8 109
Google Scholar
[17] McCormick C F, Boyer V, Arimondo E, Lett P D 2007 Opt. Lett. 32 178
Google Scholar
[18] Wang W, Cao L, Lou Y, Du J, Jing J 2018 Appl. Phys. Lett. 112 034101
Google Scholar
[19] Lv S, Jing J 2018 Opt. Commun. 424 63
Google Scholar
[20] Lv S, Jing J 2017 Phys. Rev. A 96 043873
Google Scholar
[21] Wang H, Zheng Z, Wang Y, Jing J 2016 Opt. Express 24 23459
Google Scholar
[22] Zhang K, Wang W, Liu S, Pan X, Du J, Lou Y, Yu S, Lv S, Treps N, Fabre C, Jing J 2020 Phys. Rev. Lett. 124 090501
Google Scholar
[23] Zhou Y, Jia X, Li F, Xie C, Peng K, 2015 Opt. Express 23 4952
Google Scholar
[24] Bechhoefer J 2005 Rev. Mod. Phys. 77 783
Google Scholar
[25] Wiseman H M, Milburn G J 1994 Phys. Rev. A 49 4110
Google Scholar
[26] Nelson R J, Weinstein Y, Cory D, Lloyd S 2000 Phys. Rev. Lett. 85 3045
Google Scholar
[27] Lloyd S 2000 Phys. Rev. A 62 022108
Google Scholar
[28] Kerckhoff J, Nurdin H I, Pavlichin D S, Mabuchi H 2010 Phys. Rev. Lett. 105 040502
Google Scholar
[29] Iida S, Yukawa M, Yonezawa H, Yamamoto N, Furasawa A 2012 IEEE Trans. Autom. Control 57 2045
Google Scholar
[30] Pan X, Chen H, Wei T, Zhang J, Marino A M, Treps N, Glasser R T, Jing J 2018 Phys. Rev. B 97 161115
Google Scholar
[31] Jasperse M, Turner L D, Scholten R E 2011 Opt. Express 19 3765
Google Scholar
[32] Fox M 2006 Quantum Optics: An Introduction (New York: Oxford University Press)
[33] Wiseman H M, Jones S J, Doherty A C 2007 Phys. Rev. Lett. 98 140402
Google Scholar
[34] Jasperse M 2010 M. S. Thesis (Melbourne: The University of Melbourne)
[35] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[36] Duan L M, Giedke G, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 84 2722
Google Scholar
[37] Werner R F, Wolf M M 2001 Phys. Rev. Lett. 86 3658
Google Scholar
[38] Barbosa F A S, Coelho A S, Muñoz-Martínez L F, et al. 2018 Phys. Rev. Lett. 121 073601
Google Scholar
[39] Qin Z, Cao L, Wang H, Marino A M, Zhang W, Jing J 2014 Phys. Rev. Lett. 113 023602
Google Scholar
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