Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Dynamics of multipartite quantum coherence in asymmetric spin-orbit coupled system

Shao Ya-Ting Yan Kai Wu Yin-Zhong Hao Xiang

Dynamics of multipartite quantum coherence in asymmetric spin-orbit coupled system

Shao Ya-Ting, Yan Kai, Wu Yin-Zhong, Hao Xiang
PDF
HTML
Get Citation
  • The time evolution of multipartite quantum coherence is studied in a three-body spin system with an asymmetric interaction. The l1 norm measurement is used to estimate the degree of quantum coherence in the spin system. The decoherence of all components of quantum coherence in the three-body spin system is analyzed by the exact diagnolization and numerical method based on quantum master equation. The environmental noise induced by the spontaneous decay can be simulated by the quantum amplitude damping model. It is found that the time evolution of quantum coherence component is closely related to the quantum property of the whole initial state. When the initial state is a separable pure one, the asymmetric interactions can conduce to the enhancement of the degree of multipartite quantum coherence in a short time interval. Under the influence of quantum noise, the degree of quantum coherence decreases gradually. We pay much attention to the spatial distribution of the degree of quantum coherence in a many-body system. The additivity relationship of bipartite component and tripartite coherence can exist if the initial state is chosen to be a Werner-like state. This kind of the coherence additivity between all bipartite components and global coherence can be extended to an arbitrary N-body Werner-like state. But this additivity relationship depends on the l1 norm coherence measurement. Owing to the asymmetric interaction and noise, the degree of tripartite quantum coherence is more than the sum of all degrees of bipartite quantum coherence. The difference between the degree of tripartite coherence and the sum of all degrees of bipartite coherence is increased in a short time interval. The environmental noise can also suppress the difference in the coherence degree. The degree of the nearest neighboring bipartite coherence decreases more quickly than those of other bipartite coherences. The asymmetric interaction gives rise to the improvement in the degree of bipartite coherence and tripartite coherence. The coherence of the next-nearest neighboring two systems can be robust against the environmental noise. These results are helpful in preparing the multipartite quantum resources. We can utilize the system of coupled micro-cavities to realize the quantum spin system with controllable asymmetric interaction. In this way, the global coherence and bipartite coherence can be manipulated effectively by the quantum electromagnetic technology.
      Corresponding author: Hao Xiang, xhao@mail.usts.edu.cn
    [1]

    吕为民 2019 博士学位论文 (北京: 中国科学技术大学)

    Lv W M 2019 Ph. D. Dissertation (Beijing: University of Science and Technology of China) (in Chinese)

    [2]

    Barney R D, Huele J F S V 2019 Phys. Scr. 94 105105

    [3]

    Ding Z Y, Yang H, Yuan H, Wang D, Yang J, Ye L 2019 Phys. Rev. A 100 022308

    [4]

    Kouzakov K A, Chotorlishvili L, Wtzel J, Berakdar J, Ernst A 2019 Phys. Rev. A 100 022311

    [5]

    Guo W J, Wei L F 2017 Chin. Phys. B 26 010303

    [6]

    Liu Z, Qiu L, Pan F 2017 Quantum Inf. Process. 16 109

    [7]

    Khmelinskii I, Makarov V I 2020 Chem. Phys. 532 110671

    [8]

    Afsaneh E, Bagheri H M, Jafari M 2018 J. Phys. B: At. Mol. Opt. Phys. 51 145502

    [9]

    高明 2011 博士学位论文 (湖南: 国防科学技术大学)

    Gao M 2011 Ph. D. Dissertation (Hunan: National University of Defense Technology) (in Chinese)

    [10]

    Chuchem M, Cohen D 2010 Physica E 42 555

    [11]

    Trushechkin A S 2019 Math Notes 106 986

    [12]

    Rodriguez-Rosario C A, Frauenheim T, Aspuru-Guzik A 2019 Open Syst. Inf. Dyn. 26 1950022

    [13]

    Balazadenh L, Najarbashi G, Tavana A 2018 Sci. Rep. 8 17789

    [14]

    Escalera-Moreno L, Gaita-Ariño A, Coronado E 2019 Phys. Rev. B 100 064405

    [15]

    Lin C W, Sels D, Wang Y B 2020 Phys. Rev. A 101 022320

    [16]

    Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401

    [17]

    Zhao M J, Ma T, Quan Q, Fan H, Pereira R 2019 Phys. Rev. A 100 012315

    [18]

    Hu M L, Fan H 2018 Phys. Rev. A 98 022312

    [19]

    Gao D M, Lv T H 2019 Int. J. Theor. Phys. 58 1195

    [20]

    Sargolzahi I, Mirafzali S Y 2019 Phys. Rev. A 100 042121

    [21]

    张志君 2017 硕士学位论文 (南京: 南京邮电大学)

    Zhang Z J 2017 M. S. Thesis (Nanjing: Nanjing University of Posts and Telecommunications) (in Chinese)

    [22]

    范洪义 2014 物理学报 63 020302

    Fan H Y 2014 Acta Phys. Sin. 63 020302

    [23]

    田聪, 鹿翔, 张英杰, 夏云杰 2019 物理学报 68 150301

    Tian C, Lu X, Zhang Y J, Xia Y J 2019 Acta Phys. Sin. 68 150301

    [24]

    Yuan Z G, Zhang P, Li S S, Jian J 2016 Ann. Phys. 365 223

    [25]

    Karpat G, Cakmak B, Fanchini F F 2014 Phys. Rev. B 90 104431

    [26]

    Malvezzi A L, Karpat G, Cakmak B, Fanchini F F, Debarba T, Vianna R O 2016 Phys. Rev. B 93 184428

    [27]

    Korenic A, Perovic S, Cirkovic M M, Miquel P A 2020 Prog. Biophys. Mol. Biol. 150 1

    [28]

    Xu P, Hu Y H, Hou X W 2018 Physica A 491 282

    [29]

    Peng H P, Fang M F, Yu M, Zou H M 2018 Int. J. Theor. Phys. 57 1872

    [30]

    Ming F, Wang D, Shi W N, Huang A J, Du M M, Sun W Y, Ye L 2018 Quantum Inf. Process. 17 267

    [31]

    Xie Y X, Gao Y Y 2019 Laser Phys. Lett. 16 045202

    [32]

    Radhakrishnan C, Ermakov I, Byrnes T 2017 Phys. Rev. A 96 012341

    [33]

    Hui N J, Xu Y Y, Wang J, Zhang Y, Hu Z D 2017 Physica B 510 7

    [34]

    Feldman K E 2006 J. Phys. A: Math. Gen. 39 1039

    [35]

    Levitov L S, Rashba E I 2001 Phys. Rev. B 67 115324

    [36]

    Dzyaloshinskii I 1958 Phys. Chem. Solids 4 241

    [37]

    Moriya T 1960 Phys. Rev. 120 91

    [38]

    De Raedt H, Miyashita S, Michielsen K and Machida M 2004 Phys. Rev. B 70 064401

    [39]

    Hartmann M J, Brand O F G S L, Plenio M B 2007 Phys. Rev. Lett. 99 160501

    [40]

    Chen Z X, Zhou Z W, Zhou X, Zhou X F, Guo G C 2010 Phys. Rev. A 81 022303

    [41]

    Yu T, Eberly J H 2004 Phys. Rev. Lett. 93 140404

  • 图 1  当初态为$ \left| {001} \right\rangle $时, 量子相干组分演化, 参量选取为$ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$, 黑色实线为$ {C_{123}}$, 红色虚线和绿色点划线分别为$ {C_{12}}, {C_{13}}$, 蓝色点线为$ {C_{23}}$

    Figure 1.  The dynamics of all fractions of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be$ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.

    图 2  当初态为$ \left| {001} \right\rangle $时, 量子相干组分差值的演化, 参量选取为$ \varGamma = 0.5, $ $ n = 0.2$ (a) 当$ \gamma = 0.2$时, 黑色实线对应参数$ D = 0.3, $ 红色虚线对应$ D = 0.5;$ (b)当$ D = 0.2$时, 黑色实线对应参数$ \gamma = 0.3, $ 红色虚线对应$ \gamma = 0.7$

    Figure 2.  The dynamics of the difference of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be $ \varGamma = 0.5, $ $ n = 0.2$: (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.

    图 3  当初态为W态时, 量子相干组分演化, 参量$ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$,  黑色实线为$ {C_{123}}$, 红色虚线和绿色点划线分别为$ {C_{12}}, {C_{13}}$, 蓝色点线为$ {C_{23}}$

    Figure 3.  The dynamics of all fractions of quantum coherence for the initial W state. The parameters are chosen to be $ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.

    图 4  当初态为W态时, 量子相干组分差值的演化, 参量选取为$ \varGamma = 0.5, $ $ n = 0.2$ (a) 当$ \gamma = 0.2$时, 黑色实线对应参数$ D = 0.3, $ 红色虚线对应$ D = 0.5;$ (b)当$ D = 0.2$时, 黑色实线对应参数$ \gamma = 0.3, $ 红色虚线对应$ \gamma = 0.7$

    Figure 4.  The dynamics of the difference of quantum coherence for the initial W state. The parameters are chosen to be $ \varGamma = 0.5, $ $ n = 0.2$; (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.

  • [1]

    吕为民 2019 博士学位论文 (北京: 中国科学技术大学)

    Lv W M 2019 Ph. D. Dissertation (Beijing: University of Science and Technology of China) (in Chinese)

    [2]

    Barney R D, Huele J F S V 2019 Phys. Scr. 94 105105

    [3]

    Ding Z Y, Yang H, Yuan H, Wang D, Yang J, Ye L 2019 Phys. Rev. A 100 022308

    [4]

    Kouzakov K A, Chotorlishvili L, Wtzel J, Berakdar J, Ernst A 2019 Phys. Rev. A 100 022311

    [5]

    Guo W J, Wei L F 2017 Chin. Phys. B 26 010303

    [6]

    Liu Z, Qiu L, Pan F 2017 Quantum Inf. Process. 16 109

    [7]

    Khmelinskii I, Makarov V I 2020 Chem. Phys. 532 110671

    [8]

    Afsaneh E, Bagheri H M, Jafari M 2018 J. Phys. B: At. Mol. Opt. Phys. 51 145502

    [9]

    高明 2011 博士学位论文 (湖南: 国防科学技术大学)

    Gao M 2011 Ph. D. Dissertation (Hunan: National University of Defense Technology) (in Chinese)

    [10]

    Chuchem M, Cohen D 2010 Physica E 42 555

    [11]

    Trushechkin A S 2019 Math Notes 106 986

    [12]

    Rodriguez-Rosario C A, Frauenheim T, Aspuru-Guzik A 2019 Open Syst. Inf. Dyn. 26 1950022

    [13]

    Balazadenh L, Najarbashi G, Tavana A 2018 Sci. Rep. 8 17789

    [14]

    Escalera-Moreno L, Gaita-Ariño A, Coronado E 2019 Phys. Rev. B 100 064405

    [15]

    Lin C W, Sels D, Wang Y B 2020 Phys. Rev. A 101 022320

    [16]

    Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401

    [17]

    Zhao M J, Ma T, Quan Q, Fan H, Pereira R 2019 Phys. Rev. A 100 012315

    [18]

    Hu M L, Fan H 2018 Phys. Rev. A 98 022312

    [19]

    Gao D M, Lv T H 2019 Int. J. Theor. Phys. 58 1195

    [20]

    Sargolzahi I, Mirafzali S Y 2019 Phys. Rev. A 100 042121

    [21]

    张志君 2017 硕士学位论文 (南京: 南京邮电大学)

    Zhang Z J 2017 M. S. Thesis (Nanjing: Nanjing University of Posts and Telecommunications) (in Chinese)

    [22]

    范洪义 2014 物理学报 63 020302

    Fan H Y 2014 Acta Phys. Sin. 63 020302

    [23]

    田聪, 鹿翔, 张英杰, 夏云杰 2019 物理学报 68 150301

    Tian C, Lu X, Zhang Y J, Xia Y J 2019 Acta Phys. Sin. 68 150301

    [24]

    Yuan Z G, Zhang P, Li S S, Jian J 2016 Ann. Phys. 365 223

    [25]

    Karpat G, Cakmak B, Fanchini F F 2014 Phys. Rev. B 90 104431

    [26]

    Malvezzi A L, Karpat G, Cakmak B, Fanchini F F, Debarba T, Vianna R O 2016 Phys. Rev. B 93 184428

    [27]

    Korenic A, Perovic S, Cirkovic M M, Miquel P A 2020 Prog. Biophys. Mol. Biol. 150 1

    [28]

    Xu P, Hu Y H, Hou X W 2018 Physica A 491 282

    [29]

    Peng H P, Fang M F, Yu M, Zou H M 2018 Int. J. Theor. Phys. 57 1872

    [30]

    Ming F, Wang D, Shi W N, Huang A J, Du M M, Sun W Y, Ye L 2018 Quantum Inf. Process. 17 267

    [31]

    Xie Y X, Gao Y Y 2019 Laser Phys. Lett. 16 045202

    [32]

    Radhakrishnan C, Ermakov I, Byrnes T 2017 Phys. Rev. A 96 012341

    [33]

    Hui N J, Xu Y Y, Wang J, Zhang Y, Hu Z D 2017 Physica B 510 7

    [34]

    Feldman K E 2006 J. Phys. A: Math. Gen. 39 1039

    [35]

    Levitov L S, Rashba E I 2001 Phys. Rev. B 67 115324

    [36]

    Dzyaloshinskii I 1958 Phys. Chem. Solids 4 241

    [37]

    Moriya T 1960 Phys. Rev. 120 91

    [38]

    De Raedt H, Miyashita S, Michielsen K and Machida M 2004 Phys. Rev. B 70 064401

    [39]

    Hartmann M J, Brand O F G S L, Plenio M B 2007 Phys. Rev. Lett. 99 160501

    [40]

    Chen Z X, Zhou Z W, Zhou X, Zhou X F, Guo G C 2010 Phys. Rev. A 81 022303

    [41]

    Yu T, Eberly J H 2004 Phys. Rev. Lett. 93 140404

  • Citation:
Metrics
  • Abstract views:  547
  • PDF Downloads:  35
  • Cited By: 0
Publishing process
  • Received Date:  26 July 2020
  • Accepted Date:  24 August 2020
  • Available Online:  22 October 2020
  • Published Online:  05 January 2021

Dynamics of multipartite quantum coherence in asymmetric spin-orbit coupled system

    Corresponding author: Hao Xiang, xhao@mail.usts.edu.cn
  • School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, China

Abstract: The time evolution of multipartite quantum coherence is studied in a three-body spin system with an asymmetric interaction. The l1 norm measurement is used to estimate the degree of quantum coherence in the spin system. The decoherence of all components of quantum coherence in the three-body spin system is analyzed by the exact diagnolization and numerical method based on quantum master equation. The environmental noise induced by the spontaneous decay can be simulated by the quantum amplitude damping model. It is found that the time evolution of quantum coherence component is closely related to the quantum property of the whole initial state. When the initial state is a separable pure one, the asymmetric interactions can conduce to the enhancement of the degree of multipartite quantum coherence in a short time interval. Under the influence of quantum noise, the degree of quantum coherence decreases gradually. We pay much attention to the spatial distribution of the degree of quantum coherence in a many-body system. The additivity relationship of bipartite component and tripartite coherence can exist if the initial state is chosen to be a Werner-like state. This kind of the coherence additivity between all bipartite components and global coherence can be extended to an arbitrary N-body Werner-like state. But this additivity relationship depends on the l1 norm coherence measurement. Owing to the asymmetric interaction and noise, the degree of tripartite quantum coherence is more than the sum of all degrees of bipartite quantum coherence. The difference between the degree of tripartite coherence and the sum of all degrees of bipartite coherence is increased in a short time interval. The environmental noise can also suppress the difference in the coherence degree. The degree of the nearest neighboring bipartite coherence decreases more quickly than those of other bipartite coherences. The asymmetric interaction gives rise to the improvement in the degree of bipartite coherence and tripartite coherence. The coherence of the next-nearest neighboring two systems can be robust against the environmental noise. These results are helpful in preparing the multipartite quantum resources. We can utilize the system of coupled micro-cavities to realize the quantum spin system with controllable asymmetric interaction. In this way, the global coherence and bipartite coherence can be manipulated effectively by the quantum electromagnetic technology.

    • 作为重要的量子资源, 量子相干[1-3]已被广泛应用在量子信息处理[4,5]、量子测量[6]和生物物理[7]等领域. 根据量子态叠加原理, 量子相干可以表征物理系统的非经典特性. 一方面, 由于量子相干可以是单个系统中不同能态的叠加, 人们通过实验可以观测到不同能态的振荡现象. 另一方面, 量子相干也包含在多个系统之间, 表现出一种非经典关联, 与量子纠缠和量子失协都存在紧密联系. 在量子光学领域中, 人们通常用光场相空间分布和多点关联函数来描述光场的量子相干性. 利用光子偏振态的相干叠加, 人们可以观测到量子干涉条纹, 为实验测量量子相干性提供了一种有效方法. 随着量子调控技术的发展, 人们对量子相干开展了丰富的理论和实验研究, 研究对象从量子光学系统扩展到固态自旋系统和光学混合量子系统[8,9]等. 这些量子相干系统已经从单体二维封闭系统[10]发展到多体高维开放系统[11]. 同时, 人们将量子相干资源与量子热力学[12]、量子相变[13]、量子退相干[14]和量子耗散[15]等理论紧密相联, 利用量子相干度阐述了一些重要量子现象的产生机理. 为了定量评价物理系统的量子相干性, 2014年Baumgratz等[16]基于量子资源理论提出了一些有效的量子相干度量方法, 和作为量子相干度量应满足的必要条件. 一些常用判据被用于量子相干度量, 如基于${l_1}$范数[17,18]的量子相干度量、基于相对熵[19,20]的量子相干度量和基于其他量子态距离判据[21]的度量方法.

      目前, 人们非常关注多体系统的量子相干分布特性[22]和动力学特性[23]. 一方面, 这些量子相干特性可以用来解释多体系统的集合特性(如量子相变); 另一方面, 不同量子相干组分的含时演化可以帮助人们更好地理解量子资源在时空的分布规律, 为发展量子网络通信提供必要的量子资源. 于是, 一些典型量子多体系统, 如量子自旋系统[24-26]、光学混合量子系统、甚至一些生物系统[27]都成为了多体量子相干性的研究对象. 其中, 海森堡自旋系统[28]的量子相干就是一种具有代表性的多体量子相干. 最近, 人们讨论了量子相干对基于海森堡XXX自旋工作物质的量子热机效率的影响[29]. 在海森堡自旋系统中, 人们分析了不确定性关系及量子相干性的联系[30], 定量评价了海森堡XY自旋系统的量子相干非局域优势及量子相变表征[31]. 在这些研究工作中, 人们都考虑了由于自旋轨道耦合引起的非对称Dzyaloshinsky-Moriya (DM)相互作用[32,33]. 然而, 任何系统都会不可避免地与周围环境发生相互作用, 这种作用将引发量子退相干和量子耗散. 于是, 我们以具有非对称自旋DM相互作用的多体量子系统为研究对象, 分析多体量子相干组分的含时演化规律, 讨论环境噪声对多体量子相干组分的影响, 揭示量子相干的时空分布特性.

      本文借助精确对角化[34]和量子主方程数值计算方法, 定量研究了具有非对称自旋-轨道耦合作用的三体系统量子相干. 研究发现, 存在一类多体量子相干态, 其整体量子相干度与任意两组分量子相干度之和具有一种紧密联系. 首先, 介绍了具有非对称相互作用的量子自旋系统模型, 通过对三体系统的精确对角化, 解析得到了量子相干态的含时演化规律; 基于${l_1}$范数定量分析了多种组分与整体量子相干度之间的关系, 证实了量子相干加和性的存在条件. 其次, 采用量子主方程数值计算方法, 研究了退振幅型量子噪声对多体量子相干组分的影响. 最后, 得到了一些有效结论, 这些研究结果有助于规模化量子相干的制备, 为量子通信网络提供必要的量子资源.

    2.   系统模型
    • 本文考虑一种具有自旋-轨道耦合相互作用的多体非对称量子自旋系统. 在半导体量子点和分子磁体中, 自旋-轨道耦合产生的相互作用是一种典型的非对称作用, 通常描述导带电子的相互作用形式[35-37]. 这种模型可以用来研究固态量子系统的磁性特征和量子临界现象等[38]. 人们也可以利用微腔耦合系统模拟实现该模型, 用于量子态远程传输和量子通信网络[39,40]. 在外磁场中, 系统模型的哈密顿量表示为

      其中J表示自旋之间量子隧穿的相互作用强度, $\gamma $为相邻自旋之间各项异性耦合参数, D为自旋与轨道耦合的非对称参量, h为沿z方向的外磁场强度, $\sigma _i^ \pm, {{{\sigma}} _i}$为第i个自旋的上升(下降)算子和Pauli自旋算符. 其中, 每个自旋态满足${\sigma ^z}$ $\left| {1(0)} \right\rangle = \pm \left| {1(0)} \right\rangle $. 考虑一种简单情况$N = 3$, ${{D}} = D{{{e}}_z}$. 通过对三体自旋系统的研究, 可以得到一些多体量子相干的空间分布特性.

      首先, 通过精确对角化, 在自旋表象${\big\{ }\left| {111} \right\rangle, \left| {110} \right\rangle, \cdots \left| {000} \right\rangle {\big\}}$中, 得到(1)式的能量本征值及其相应本征态:

      其中, 参量

      经过含时演化算符$U(t) = \sum\nolimits_j^{} {\exp ( - {\rm{i}}{E_j}t)} \left| {{\phi _j}} \right\rangle \left\langle {{\phi _j}} \right|$作用, 这个多体系统在任意t时刻的量子态密度矩阵表示为

      这里, ${{\rho}} (0)$表示系统初态.

      选取两种初态: 类Werner态和类GHZ态. 其中, 整个系统的基态符合类Werner态特征, 是一种典型的非经典态. 首先, 以类Werner态为系统初态, 即$\rho (0) = \left| \varphi \right\rangle \left\langle \varphi \right|, \left| \varphi \right\rangle = a\left| {001} \right\rangle \!+\! b\left| {010} \right\rangle \!+\! c\left| {100} \right\rangle$(${\left| a \right|^2} + {\left| b \right|^2} + {\left| c \right|^2} = 1$). 由于系统哈密顿量满足对易关系$\left[H, \displaystyle\sum\nolimits_i {\sigma _i^z} \right] = 0$, 所以此时系统态的演化发生在希尔伯特子空间$\{ \left| {001} \right\rangle, \left| {010} \right\rangle, \left| {100} \right\rangle \} $里, 任意t时刻量子态满足$\rho (t) = \left| {\psi (t)} \right\rangle \left\langle {\psi (t)} \right|$. 利用(2)式和(3)式得到,

      这里的幺正操作S,

      为了研究多体量子相干的空间分布特性, 分别计算了任意两体系统量子态的约化密度矩阵及其量子相干度. 采用${l_1}$相干性范数来计算量子相干,

      即密度矩阵的非对角元素绝对值之和. Baumgratz等[16]基于量子资源理论, 提出了一种便于计算度量的量子${l_1}$相干性范数判据. 对于d维希尔伯特空间, 选取固定基矢$\{ \left| i \right\rangle \} $来表示量子态, 人们将$\rho =\displaystyle \sum\nolimits_i {{p_i}\left| i \right\rangle \left\langle i \right|}$定义为非相干态. 密度矩阵的非对角元素就包含了系统的量子相干性. 相比其他判据, 这种相干性范数不仅更加直观, 而且满足量子相干判据的基本条件, 如非负性、单调性和凸函数性等性质. 利用(5)式, 得到任意两组分的量子相干, ${C_{12}} = 2\left| {{y_2}y_3^ * } \right|$, ${C_{23}} = 2\left| {{y_1}y_2^ * } \right|$, ${C_{13}} = 2\left| {{y_1}y_3^ * } \right|$. 同时, 当系统初态为类Werner态时, 单个自旋的量子相干为零, 并且三体量子相干表示为

      (6)式表明: 对于类Werner态, 系统在非对称性相互作用下, 三体量子相干度与任意两体量子相干度之间满足一种加和性. 对于一般情况$N > 3$, 根据(1)式描述的哈密顿量特性$\left[H, \displaystyle\sum\nolimits_i {\sigma _i^z} \right] \!=\! 0$, 可以得到某个N体类Werner态, $\left| {{\varphi _N}} \right\rangle \!=\! \displaystyle\sum\nolimits_{i = 1}^N {{a_i}| {{1_i}} \rangle }$, 其中$\left| {{1_i}} \right\rangle = \left| {00 \cdots 1 \cdots 0} \right\rangle $表示第i个自旋格点为$\left| 1 \right\rangle $且其余格点为$\left| 0 \right\rangle $. 利用(5)式的相干性范数判据, 可以得到整个多体系统的量子相干度${C_{{\rm{tot}}}} = \displaystyle \sum\nolimits_{i \ne j} {2\left| {{a_i}a_j^ * } \right|}$. 任意两体量子相干度与整体系统量子相干度也满足这种加和性, 即${C_{{\rm{tot}}}} = \displaystyle\sum\nolimits_{i \ne j} {{C_{ij}}}$. 以$N = 4$为例, ${C_{12}} \!=\! 2\left| {{a_3}a_4^ * } \right|, {C_{13}} \!=\! 2\left| {{a_2}a_4^ * } \right|,$ ${C_{14}} = 2 \left| {{a_2}a_3^ * } \right|, {C_{23}} = 2\left| {{a_1}a_4^ * } \right|$, ${C_{34}} = 2\left| {{a_1}a_2^*} \right|$. 然而, 当初态选取类GHZ态, $\left| \varphi \right\rangle = \sqrt q \left| {000} \right\rangle + \sqrt {1 - q} \left| {111} \right\rangle, (0 < q < 1)$三体量子相干度$C_{123}^{{\rm{GHZ}}} = 2\sqrt {q(1 - q)} \cos (6 ht)$, 任意两组分的量子相干度总为零. 这个结论进一步表明: 类GHZ态是一种不可区分的真实多体量子纠缠态, 任意两组分之间量子关联为零. 多体量子相干度的空间分布依赖于不同初态的性质.

    3.   量子退相干
    • 在实际情况下, 系统总是与周围环境存在相互作用, 其量子相干特性往往在环境影响下发生量子退相干. 接下来考虑一种典型的量子退振幅噪声, 采用量子主方程和量子Kraus算符, 讨论噪声因素对多体量子相干度的影响.

      关于非对称自旋系统, 合理考虑每个量子自旋i与其定域性环境${E_i}$存在弱耦合作用, 这种相互作用可以用量子退振幅噪声模型来描述[41]. 为了有效描述这种量子噪声模型, 人们通常采用量子主方程和量子信道来研究噪声对量子系统含时演化的影响. 自旋系统与周围环境初态为${\rho _{SE}} = \rho (0) \otimes ({\left| {000} \right\rangle _E}\left\langle {000} \right|)$, 其中${\left| {000} \right\rangle _E} = {\left| 0 \right\rangle _{{E_1}}} \otimes {\left| 0 \right\rangle _{{E_2}}} \otimes {\left| 0 \right\rangle _{{E_3}}}$表示三个定域性环境初态为真空态, $\rho (0)$表示三个自旋系统初态. 在弱耦合近似条件下, 整个自旋系统状态演化可以通过量子主方程描述:

      其中, 超算符Lindblad算符表示为

      这里, n表示有限温度环境的平均光子数, 当温度越高时, n的数值越大. ${\varGamma _i} = \varGamma $表示每个自旋自发衰减的阻尼系数. 这些噪声因素会影响自旋系统的演化.

      为了说明噪声对多体量子相干度的影响, 首先考虑一种最简单情况, $J = 0, {array}{*{20}{c}} {h = 0} {array}$, 即自旋间相互作用为零. 整个量子态演化可以由量子Kraus算符表示为

      其中, 关于第i个自旋演化的量子Kraus算符分别是:

      利用Kraus算符, 得到任意自旋演化过程,

      这里, 参量$p(t) = 1 - {{\rm e}^{\tfrac{{\varGamma (2 n + 1)}}{2}t}}$与自发衰减系数相关. 对于三体量子态$\rho (0) = \displaystyle\sum\limits_{kls, mnr} {a_{mnr, kls}}({{\left| k \right\rangle }_1} \left\langle m \right| \otimes {{\left| l \right\rangle }_2}\left\langle n \right| \otimes {{\left| s \right\rangle }_3}\left\langle r \right|)$, 任意t时刻的量子态为

      根据量子相干度计算公式(5)式, 选取类Werner态为初态, 得到三体量子相干度和任意两组分量子相干度为:

      如果系统初态选择类GHZ态, 那么在环境噪声影响下, 任意t时刻的量子相干度为$C_{123}^{{\rm{GHZ}}} = 2{(1 - p)^{ \textstyle\frac{3}{2}}}\sqrt {q(1 - q)}$, 其他两体量子相干仍然为零. 当自旋相互作用为零时, 初态处于类Werner态的系统, 其量子相干空间分布仍然满足(6)式所描述的加和性. 而且, 量子相干度随着时间呈现指数衰减规律. 噪声参数p越大, 量子相干数值衰减得越快.

      然而, 当自旋之间存在非对称性相互作用时, 利用量子主方程的数值计算, 发现三体量子相干度与任意两组分量子相干之和存在差异. 当初态为 $\left| {001} \right\rangle $时, 图1描述了三体量子相干度${C_{123}}$、近邻两组分量子相干${C_{12}}$${C_{23}}$、以及次近邻两体量子相干${C_{13}}$的含时演化. 如图1所示, 当时间较短时, 状态演化处于初始阶段, 一些能级组分态$\left| {{\psi _{6, 7}}} \right\rangle $的混合, 使得类Werner态组分在量子态中占有一定比重, 所以量子相干度的数值从零逐渐增大. 其中, 对于相互作用较小的两个自旋, 其量子相干度${C_{13}}$${C_{23}}$在演化初期增长较快. 但是, 随着演化进一步发展, 环境噪声又会导致量子相干度的衰减, 量子退相干现象明显.

      Figure 1.  The dynamics of all fractions of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be$ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.

      为了进一步研究多体量子相干与两体量子相干的关系, 定义一种量子相干组分差值${C_{\rm{r}}} = {C_{123}} - ({C_{12}} + {C_{23}} + {C_{13}})$. 如图2所示, 分析了非均匀自旋相互作用参量D$\gamma $对量子相干性质的影响. 图2(a)的曲线变化表明, 随着自旋与轨道耦合作用D的增加, 量子相干组分差值${C_{\rm{r}}}$也会增大, 但是环境噪声又会大大抑制量子相干, 导致量子退相干. 同样, 图2(b)的曲线变化表明, 各项异性参量$\gamma $的增加也会引起量子相干组分差值${C_{\rm{r}}}$的增加. 这个结论说明, 非均匀相互作用更有利于三体量子相干度的增加.

      Figure 2.  The dynamics of the difference of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be $ \varGamma = 0.5, $ $ n = 0.2$: (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.

      当初态选择W$\left(a = b = c = {1}/{{\sqrt 3 }}\right)$时, 量子相干的各个组分随时间演化行为如图3所示. 所有量子相干组分都呈现出单调衰减现象. 这与(12)式的解析结果相对应.

      Figure 3.  The dynamics of all fractions of quantum coherence for the initial W state. The parameters are chosen to be $ D = 0.2, $ $ \gamma = 0.2, $ $ \varGamma = 0.5, $ $ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.

      但是, 由于自旋相互作用的非对称性, 所以量子相干组分之间存在差异, 相应差值${C_{\rm{r}}}$也会不为零. 图4(a)图4(b)分别表现出, 自旋与轨道耦合作用D和各项异性参量$\gamma $的增加都会引起三体量子相干度的增大, 使得相应量子相干组分差值在演化初期呈现增长现象. 但是, 由于环境噪声的影响, 无论是三体量子相干度, 还是任意两体量子相干度, 都在较长时间里发生衰减, 直至完全消失.

      Figure 4.  The dynamics of the difference of quantum coherence for the initial W state. The parameters are chosen to be $ \varGamma = 0.5, $ $ n = 0.2$; (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.

    4.   结 论
    • 本文以非对称三体自旋系统为研究对象, 重点分析了量子相干的空间分布特性, 及其在环境噪声背景中的量子退相干现象. 发现类Werner态的量子相干度组分之间存在一种特殊加和性, 即三体量子相干度数值等于所有两体量子相干度之和. 在忽略自旋相互作用的极限条件下, 尽管环境噪声会导致系统所有量子相干组分发生衰减, 但是这种量子加和性仍然存在. 对于可分离初态, 自旋之间非均匀相互作用更有利于三体量子相干度的增加. 这些研究在多体系统可能的量子相干特性规律方面提供了一种思路. 同时, 本文采用的精确对角化和量子主方程, 可以推广到任意多体量子自旋系统, 为进一步研究多体量子相干特性提供了有效方法.

Reference (41)

Catalog

    /

    返回文章
    返回