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双模纠缠态是量子信息领域一种重要的量子资源, 本文基于四波混频过程从理论上提出了对双模纠缠态的单个模式(单模放大方案)和对双模纠缠态的两个模式(双模放大方案)的放大. 利用光学分束器模型来模拟在光学传输过程中损耗引入的真空场噪声, 利用部分转置正定判据分析了两种不同的放大方案中四波混频过程的增益对初始双模纠缠态的纠缠程度的影响. 结果表明, 在特定的损耗情况下, 两个方案中初始双模纠缠态的纠缠度都随增益的增大而减小, 直至消失, 且双模放大方案中初始双模纠缠态纠缠消失得比单模放大方案中更快. 本文的理论结果为实验上实现基于四波混频过程的双模纠缠态的放大奠定了理论基础.Two-mode entangled state is an important quantum resource for quantum information. In this paper, the amplification of a single mode of two-mode entangled state (single-mode amplification scheme) and two modes of two-mode entangled state (two-mode amplification scheme) are theoretically proposed. Here, the optical beam splitter model is used to simulate the vacuum noise introduced by the loss in the optical transmission process. By utilizing the positivity under partial transpose criterion, we analyze the effect of the gain of the four-wave mixing process on the entanglement degree of the initial two-mode entangled state in two different amplification schemes. In these two schemes, we set the gain of the initial two-mode entangled state generation process to be 1.5, 2.5 and 50.0 respectively, and then change the gain of the amplification process in a certain range. We also set the transmission efficiency of the amplified beams for each of the two schemes to be a definite value. The results show that the entanglement of the initial two-mode entangled state decreases with the gain increasing under the condition of specific transmission loss in two schemes. When the gain does not exceed a certain value, the entanglement of the initial two-mode entangled state can be maintained. Then, with the increase of the gain, the entanglement of the initial two-mode entangled state will disappear. Moreover, the entanglement of the initial two-mode entangled state of the two-mode amplification scheme disappears faster than that of the single-mode amplification scheme. Our theoretical results pave the way for the experimental realization of the amplification of two-mode entangled state based on four-wave mixing process.
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Keywords:
- four-wave mixing /
- quantum entanglement /
- two-mode entangled state /
- optical parametric amplifier
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图 1 一种对EPR光束进行单模放大的方案 (a) 对EPR光束进行单模放大的系统简图; (b) 85Rb D1线的双Λ能级结构
Fig. 1. A scheme for single-mode amplification of EPR beams: (a) Simplified diagram of single-mode amplification of EPR beams; (b) double-Λ energy level structure of 85Rb D1 line.
图 2
${G_1}$ 不同的情况下单模放大方案最小辛本征值与${G_2}$ 的关系Fig. 2. Relationship between the smallest symplectic eigenvalue and
${G_2}$ of the single-mode amplification scheme under different value of${G_1}$ .
图 4
${G_1}$ 不同的情况下双模放大方案最小辛本征值与${G_2}$ 的关系Fig. 4. Relationship between the smallest symplectic eigenvalue and
${G_2}$ of the two-mode amplification scheme under different value of${G_1}$ . -
[1] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
Google Scholar
[2] Braunstein S L, Look P van 2005 Rev. Mod. Phys. 77 513
Google Scholar
[3] 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
[4] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[5] Ralph T C 1999 Phys. Rev. A 61 010303
Google Scholar
[6] Naik D S, Peterson C G, White A G, Berglund A J, Kwiat P G 2000 Phys. Rev. Lett. 84 4733
Google Scholar
[7] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[8] Zhang J, Peng K C 2000 Phys. Rev. A 62 064302
Google Scholar
[9] Heaney L, Vedral V 2009 Phys. Rev. Lett. 103 200502
Google Scholar
[10] Ou Z Y, Pereira S F, Kimble H J, Peng K C 1992 Phys. Rev. Lett. 68 3663
Google Scholar
[11] Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter H, Zeilinger A 1997 Nature 390 575
Google Scholar
[12] Furusawa A, Sørensen J L, Braunstein S L, Fuchs C A, Kimble H J, Polzik E S 1998 Science 282 706
Google Scholar
[13] Li X Y, Pan Q, Jing J T, Zhang J, Xie C D, Peng K C 2002 Phys. Rev. Lett. 88 047904
Google Scholar
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Google Scholar
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Google Scholar
[16] Boyer V, Marino A M, Lett P D 2008 Phys. Rev. Lett. 100 143601
Google Scholar
[17] Kumar P, Kolobov M I 1994 Opt. Commun. 104 374
Google Scholar
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Google Scholar
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Google Scholar
[20] MacRae A, Brannan T, Achal R, Lvovsky A I 2012 Phys. Rev. Lett. 109 033601
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[26] Li Z P, Wang X L, Li C Y, Zhang Y F, Wen F, Ahmed I, Zhang Y P 2016 Laser Phys. Lett. 13 025402
Google Scholar
[27] Abdisa G, Ahmed I, Wang X X, Liu Z C, Wang H X, Zhang Y P 2016 Phys. Rev. A 94 023849
Google Scholar
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Google Scholar
[29] Li C B, Li W, Zhang D, Zhang Z Y, Gu B L, Li K K, Zhang Y P 2019 Laser Phys. Lett. 17 015401
Google Scholar
[30] Pooser R C, Marino A M, Boyer V, Jones K M, Lett P D 2009 Phys. Rev. Lett. 103 010501
Google Scholar
[31] Werner R F, Wolf M M 2001 Phys. Rev. Lett. 86 3658
Google Scholar
[32] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[33] Jasperse M, Turner L D, Scholten R E 2011 Opt. Express 19 3765
Google Scholar
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