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实现量子中继的关键是克服量子储存器中纠缠态的退相干问题. 目前, 人们常用半导体量子点中的电子自旋来构建纠缠态从而实现量子中继过程. 在该过程中, 两个半导体量子点之间相距很远, 可以认为它们之间没有相互作用. 因此, 量子点内电子自旋与它周围的核自旋之间的超精细相互作用被认为是导致系统退相干的最重要原因之一. 在以前的相关工作中, 人们通常将核自旋对电子自旋的超精细相互作用视为一个大小和方向都是随机的并且满足高斯分布的等效磁场. 本文在考虑核自旋的等效磁场以及外加磁场的情况下, 研究了两个量子点中继系统的退相干问题. 首先利用数值方法分别计算了4组Bell基随时间的演化规律, 发现当外加磁场增大到一定值时, 4组Bell基被分为两类. 体系不可能通过时间演化从一类Bell基跃迁到另一类Bell基, 而只能在同类的两个Bell基之间相互跃迁. 这有效提高了系统的保真度, 并且抑制了核自旋对体系纠缠态的影响, 从而抑制退相干. 其次, 对于给定的较大外加磁场, 采用解析方法研究了核自旋涨落对纠缠态的影响, 给出了初态保真度及退相干时间的解析形式. 发现对于相同的核自旋涨落, 4组Bell基的退相干时间相同, 但是两类Bell基随时间演化的规律不同, 其中一类的保真度在指数衰减的同时伴随快速周期性振荡, 不便于操控. 期望本文的研究能对量子中继过程中纠缠态的选择问题提供理论支持和建议.The decoherence of entanglement states stored in quantum memory is a major obstacle when implementing a quantum repeater. So far, the electron spins in quantum dots are usually utilized to construct entangled states in quantum repeater. In the quantum repeater process, the distance between quantum dots is large, so the interaction between them can be neglected. Thus the hyperfine interaction between the electron spin and its neighbor nuclear spins in the quantum dot is considered to be the main reason for the decoherence of the system. In early researches, the hyperfine interaction between the electron spin and its neighbor nuclear spins was considered as an effective magnetic field whose magnitude and direction are random and the magnitude follows the Gaussian distribution. In this paper, we simultaneously consider an applied magnetic field and the interaction between the electron spin and its neighbor nuclear spins, and investigate the decoherence of the quantum repeater of two quantum dots. We first solve the time evolution of the system by the numerical method, and the result shows that when the applied magnetic field is increased to a certain value, the four Bell states can be divided into two kinds, each with two Bell states. The system cannot transit from the Bell state in one kind to that in the other kind, but can transit between two Bell states with in the same kind. This effectively improves the fidelity of the initial state and suppresses the decoherence of the system. For a given applied magnetic field with large magnitude, we theoretically study the effect of the fluctuation of nuclear spin on the entangled state, and give an analytical expression for each of the fidelity and the decoherence time of the initial state. We show that the decoherence times of the four Bell states are the same, but the time evolutions of the Bell states belonging to different kinds are different obviously. The fidelity of two Bell states not only decays exponentially but also oscillates rapidly, so such two Bell states are difficult to be manipulated and not suggested in quantum repeater process. The results in this paper are expected to provide theoretical suggestions for selecting the entangled states in quantum repeater.
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Keywords:
- quantum teleportation /
- quantum repeater /
- decoherence of entanglement state /
- quantum fidelity
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图 1 量子中继步骤 (a)在每一段量子传输线路上分发纠缠态; (b)在量子中继器中进行纠缠态的转移操作; (c) Alice做Bell基测量, 量子比特1的信息通过量子比特2和3组成的纠缠态量子信息传输通道远程传到Bob处
Fig. 1. The process of quantum repeater: (a) Distributing entanglement states in every segment of quantum channel; (b) quantum swapping in quantum repeaters; (c) quantum teleportation. After Alice taking Bell state measurement, the information of qubit 1 is transferred to Bob via entanglement state of qubit 2 and 3.
图 2 两个相互纠缠的量子中继器, 纠缠态为两个电子自旋组成的4个Bell基.
Fig. 2. Entanglement of the two separated quantum repeaters. Entangled states are the four Bell states constructed by two electron spins.
图 3 3种不同大小的外加磁场下, 体系处于4个不同Bell基上的几率随时间的演化规律 (a), (b), (c), (d)分别对应系统处于
$ |\phi_{12}^+\rangle $ ,$ |\phi_{12}^-\rangle $ ,$ |\psi_{12}^+\rangle $ ,$ |\psi_{12}^-\rangle $ 的平均几率. 体系初态为$ |\phi_{12}^+\rangle $ , 外加磁场参数为$ B_0 = 0 $ (红色虚线),$ B_0 = 3\varDelta_B $ (黑色点划线),$ B_0 = 10\varDelta_B $ (蓝色实线). 时间以$ t_0 = 1/(\mu_0\varDelta_B) $ 为单位Fig. 3. Time evolution of mean probability in four Bell states: (a)
$ |\phi_{12}^+\rangle $ , (b)$ |\phi_{12}^-\rangle $ , (c)$ |\psi_{12}^+\rangle $ , (d)$ |\psi_{12}^-\rangle $ for different applied magnetic fields. The initial state is$ |\phi_{12}^+\rangle $ , and the parameters are$ B_0 = 0 $ (red dash line),$ B_0 = 3\varDelta_B $ (black dash dot line),$ B_0 = 10\varDelta_B $ (blue solid line). Time is in the unit of$ t_0 = 1/(\mu_0\varDelta_B) $ .
图 4 3种不同大小的外加磁场下, 体系处于4个不同Bell基上的几率随时间的演化规律 (a), (b), (c), (d)分别对应系统处于
$ |\phi_{12}^+\rangle $ ,$ |\phi_{12}^-\rangle $ ,$ |\psi_{12}^+\rangle $ ,$ |\psi_{12}^-\rangle $ 的平均几率. 体系的初态为$ |\psi_{12}^+\rangle $ , 外加磁场参数为$ B_0 = 0 $ (红色虚线),$ B_0 = 3\varDelta_B $ (黑色点划线)、$ B_0 = 10\varDelta_B $ (蓝色实线). 时间以$ t_0 = 1/(\mu_0\varDelta_B) $ 为单位Fig. 4. Time evolution of mean probability in four Bell states: (a)
$ |\phi_{12}^+\rangle $ , (b)$ |\phi_{12}^-\rangle $ , (c)$ |\psi_{12}^+\rangle $ , (d)$ |\psi_{12}^-\rangle $ for different applied magnetic fields. The initial state is$ |\psi_{12}^+\rangle $ , and the parameters are$ B_0 = 0 $ (red dash line),$ B_0 = 3\varDelta_B $ (black dash dot line),$ B_0 = 10\varDelta_B $ (blue solid line). Time is in the unit of$ t_0 = 1/(\mu_0\varDelta_B) $ .
图 5 (a)系统仍然处在
$ |\phi_{12}^+\rangle $ 上的平均几率随时间的演化; (b)系统跃迁到$ |\phi_{12}^-\rangle $ 的平均几率随时间的演化. 系统初态为$ |\phi_{12}^+\rangle $ , 时间以$t_0 = 1/(\mu_0\varDelta_B)$ 为单位Fig. 5. (a) Time evolution of the mean probability in
$ |\phi_{12}^+\rangle $ , (b) that in$ |\phi_{12}^{-}\rangle $ . The initial state is$ |\phi_{12}^+\rangle $ , and the time is in the unit of$t_0 = 1/(\mu_0\varDelta_B)$ .
图 6 (a)系统仍然处在
$ |\psi_{12}^+\rangle $ 上的平均几率随时间的演化; (b)系统跃迁到$ |\psi_{12}^-\rangle $ 的平均几率随时间的演化. 系统初态为$ |\psi_{12}^+\rangle $ , 时间以$t_0 = 1/(\mu_0\varDelta_B)$ 为单位Fig. 6. (a) Time evolution of the mean probability in
$ |\psi_{12}^+\rangle $ , (b) that in$ |\psi_{12}^-\rangle $ . The initial state is$ |\psi_{12}^+\rangle $ , and the time is in the unit of$t_0 = 1/(\mu_0\varDelta_B)$ . -
[1] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wotter W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[2] Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter H, Zeilinger A 1997 Nature 390 575
Google Scholar
[3] Ma X S, Herbst T, Scheidl, Wang D, Kropatschek S, Naylor W, Wittmann A, Mech J, Korfler E, Anisimova, Makarov V, Jennewein T, Ursin R, Zeilinger A 2012 Nature 489 269
Google Scholar
[4] Vallone G, Bacco D, Dequal D, Gaiarin S, Luceri V, Bianco G, Villoresi P 2015 Phys. Rev. Lett. 115 040502
Google Scholar
[5] Yin J, Cao Y, Li Y H, Ren J G, Liao S K, Zhang L, Cai W Q, Liu W Y, Li B, Dai H, Li M, Huang Y M, Deng L, Li L, Zhang Q, Liu L N, Chen Y A, Lu C Y, Shu R, Peng C Z, Wang J Y, Pan J W 2017 Phys. Rev. Lett. 119 200501
Google Scholar
[6] Yin J, Cao Y, Li Y H, Liao S K, Zhang L, Ren J G, Cai W Q, Liu W Y, Li B, Dai H, Li G B, Lu Q M, Gong Y H, Xu Y, Li S L, Li F Z, Yin Y Y, Jiang Z Q, Li M, Jia J J, Ren G, He D, Zhou Y L, Zhang X X, Wang N, Chang X, Zhu Z C, Liu N L, Chen Y A, Lu C Y, Shu R, Peng C Z, Wang J Y, Pan J W 2017 Science 356 1140
Google Scholar
[7] Dür W, Briegel H J, Cirac J I, Zoller P 1999 Phys. Rev. A 59 169
Google Scholar
[8] Pfister A D, Salz M, Hettrich M, Poschinger U G, Schmidt-Kaler F 2016 Appl. Phys. B 122 1
Google Scholar
[9] Greve K D 2013 Towards Solid-State Quantum Repeaters: Ultrafast, Coherent Optical Control and Spin-photon Entanglement in Charged InAs Quantum Dots (Heidelberg: Springer) pp21–39
[10] Briegel H J, Dür W, Cirac J I, Zoller P 1998 Phys. Rev. Lett. 81 5932
[11] Freer S, Simmons S, Laucht A, Muhonen J T, Dehollain J P, Kalra R, Mohiyaddin F A, Hudson F E, Itoh K M, McCallum J C, Jamieson D N, Dzurak A S, Morello A 2017 Quantum Sci. Technol. 2 015009
Google Scholar
[12] Kawakami E, Scarlino P, Ward D R, Braakman F R, Savage D E, Lagally M G, Friesen M, Coppersmith S N, Eriksson M A, Vandersypen L M K 2014 Nat. Nanotechnol. 9 666
Google Scholar
[13] Specht H P, Nölleke C, Reiserer A, Uphoff M, Figueroa E, Ritter S, Rempe G 2011 Nature. 473 190
[14] Loock van P, Ladd T D, Sanaka K, Yamaguchi F, Nemoto K, Munro W J, Yamamoto Y 2006 Phys. Rev. Lett. 96 240501
Google Scholar
[15] Duan L M, Lukin M, Cirac J I 2001 Nature 414 413
Google Scholar
[16] Pan J W, Bouwmeester D, Weinfurter H, Zeilinger A 1998 Phys. Rev. Lett. 80 3891
Google Scholar
[17] Lloyd S, Shahriar M S, Shapiro J H, Hemmer P R 2001 Phys. Rev. Lett. 87 167903
Google Scholar
[18] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[19] Malinowski F K, Martins F, Cywiński L, Rudner M S, Nissen P D, Fallahi S, Gardner G C, Manfra M J, Marcus C M, Kuemmeth F 2017 Phys. Rev. Lett. 118 177702
Google Scholar
[20] Liu R B, Yao W, Sham L 2007 New J. Phys. 9 226
Google Scholar
[21] Saito S, Zhu X B, Amsüss R, Matsuzaki Y, Kakuyanagi K, Shimo-Oka T, Mizuochi N, Nemoto K, Munro W J, Semba K 2013 Phys. Rev. Lett. 111 107008
Google Scholar
[22] Xiang Z L, Ashhab S, You J Q, Nori F 2013 Rev. Mod. Phys. 85 623
Google Scholar
[23] Maune B M, Borselli M G, Huang B, Ladd T D, Deelman P W, Holabird K S, Kiselev A A, Alvarado-Rodriguez I, Ross R S, Schmitz A E 2012 Nature 481 344
Google Scholar
[24] Yao W, Liu R B, Sham L J 2007 Phys. Rev. Lett. 98 077602
Google Scholar
[25] Merkulov I A 2002 Phys. Rev. B 65 205309
Google Scholar
[26] Assali L V C, Petrilli H M, Capaz R B, Koiller B, Hu X D, Sarma S D 2011 Phys. Rev. B 83 165301
Google Scholar
[27] Eisenberg B, Sullivan R 2008 Math. Mag. 81 362
Google Scholar
[28] Reilly D J, Taylor J M, Laird E A, Petta J R, Marcus C M, Hanson M P, Gossard A C 2007 Phys. Rev. Lett. 101 236803
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