量子扩散通道中Wigner算符的演化规律

张科; 李兰兰; 任刚; 杜建明; 范洪义

doi:10.7498/aps.69.20200106

摘要

众所周知, 量子态的演化可用与其相应的Wigner函数演化来代替. 因为量子态的Wigner函数和量子态的密度矩阵一样, 都包含了概率分布和相位等信息, 因此对量子态的Wigner函数进行研究, 可以更加快速有效地获取量子态在演化过程的重要信息. 本文从经典扩散方程出发, 利用密度算符的P表示, 导出了量子态密度算符的扩散方程. 进一步通过引入量子算符的Weyl编序记号, 给出了其对应的Weyl量子化方案. 另外, 借助于密度算符的另一相空间表示—Wigner函数, 建立了Wigner算符在扩散通道中演化方程, 并给出了其Wigner算符解的形式. 本文推导出了Wigner算符在量子扩散通道中的演化规律, 即演化过程中任意时刻Wigner算符的形式. 在此结论的基础上, 讨论了相干态经过量子扩散通道的演化情况.

关键词:

Abstract

As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the P representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.

Keywords:

作者及机构信息

1.
淮南师范学院电子工程学院, 淮南　232038

2.
中国科学技术大学研究生院科学岛分院, 合肥　230031

通信作者: 范洪义, fhym@ustc.edu.cn

基金项目: 国家自然科学基金(批准号: 11775208), 安徽省教育厅自然科学重点项目(批准号: KJ2019A0688)和淮南师范学院重点研究项目(批准号: 2019XJZD04)资助的课题

Authors and contacts

1.
School of Electronic Engineering, Huainan Normal University, Huainan 232038, China

2.
Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230031, China

Corresponding author: Fan Hong-Yi, fhym@ustc.edu.cn

Funds: the National Natural Science Foundation of China (Grant No.11775208), the Natural Science Foundation of the Anhui Higher Education Institutions of China (Grant No. KJ2019A0688), and the Key Projects of Huainan Normal University, China(Grant No. 2019XJZD04)

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参考文献

[1]	Chakrabarti R, Yogesh V 2018 Physica A 490 886 Google Scholar
[2]	Wei C P, Xie F S, Zhang H L, Hu L Y 2013 Int. J. Theor. Phys. 52 798 Google Scholar
[3]	Agarwal G S 1971 Phys. Rev. A 3 828 Google Scholar
[4]	Hu L Y, Fan H Y 2009 Opt. Commun. 282 4379 Google Scholar
[5]	Takahashi K 1986 J. Phys. Soc. Jpn. 55 762 Google Scholar
[6]	Fan H Y 1991 Phys. Lett. A 161 1 Google Scholar
[7]	Meng X G, Wang J S, Liang B L 2009 Chin. Phys. B 18 01534 Google Scholar
[8]	Fan H Y 2010 Commun. Theor. Phys. 53 344 Google Scholar
[9]	Fan H Y, Hu L Y 2009 Commun. Theor. Phys. 51 729 Google Scholar
[10]	Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480 Google Scholar
[11]	Kurchan J, Leboeuf P, Saraceno M 1989 Phys. Rev. A 40 6800 Google Scholar
[12]	Fan H Y, Zaidi H R 1987 Phys. Lett. A 124 303 Google Scholar
[13]	Fan H Y, Wang J S 2007 Commun. Theor. Phys. 47 431 Google Scholar
[14]	Fan H Y, Fan Y 1997 Commun. Theor. Phys. 27 105 Google Scholar
[15]	颜森林 2007 物理学报 56 1994 Google Scholar Yan S L 2007 Acta Phys. Sin. 56 1994 Google Scholar
[16]	兰豆豆, 郭晓敏, 彭春生, 姬玉林, 刘香莲, 李璞, 郭龑强 2017 物理学报 66 120502 Google Scholar Lan D D, Guo X M, Peng C S, Ji Y L, Liu X L, Li P, Guo Y Q 2017 Acta Phys. Sin. 66 120502 Google Scholar
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[18]	范洪义, 梁祖峰 2015 物理学报 64 050301 Google Scholar Fan H Y, Liang Z F 2015 Acta Phys. Sin. 64 050301 Google Scholar
[19]	Wigner E P 1932 Phys. Rev. 40 749 Google Scholar
[20]	Fan H Y 1992 J. Phys. A 25 3443 Google Scholar
[21]	袁洪春, 徐学翔 2012 物理学报 61 064205 Google Scholar Yuan H C, Xu X X 2012 Acta Phys. Sin. 61 064205 Google Scholar
[22]	Fan H Y, Yang Y L 2006 Phys. Lett. A 353 439 Google Scholar
[23]	Zhang K, Fan C Y, Fan H Y 2019 Int. J. Theor. Phys. 58 1687 Google Scholar
[24]	Buot F, Jensen K 1990 Phys. Rev. B 42 9429 Google Scholar
[25]	Chountasis S, Vourdas A 1998 Phys. Rev. A 58 1794 Google Scholar
[26]	Fan H Y, Hu L Y, Yuan H C 2010 Chin. Phys. B 19 060305 Google Scholar

施引文献

图 1 (a) $kt = 0$, (b) $kt = 0.8$时, 相干态下Wigner算符的演化规律($\alpha = 1/2\left( {1 + i} \right)$)

Fig. 1. Evolution law of Wigner operator for the coherent state with $\alpha = 1/2\left( {1 + i} \right)$ for (a) $kt = 0$; (b) $kt = 0.8$

下载: 全尺寸图片幻灯片

[1]	Chakrabarti R, Yogesh V 2018 Physica A 490 886 Google Scholar
[2]	Wei C P, Xie F S, Zhang H L, Hu L Y 2013 Int. J. Theor. Phys. 52 798 Google Scholar
[3]	Agarwal G S 1971 Phys. Rev. A 3 828 Google Scholar
[4]	Hu L Y, Fan H Y 2009 Opt. Commun. 282 4379 Google Scholar
[5]	Takahashi K 1986 J. Phys. Soc. Jpn. 55 762 Google Scholar
[6]	Fan H Y 1991 Phys. Lett. A 161 1 Google Scholar
[7]	Meng X G, Wang J S, Liang B L 2009 Chin. Phys. B 18 01534 Google Scholar
[8]	Fan H Y 2010 Commun. Theor. Phys. 53 344 Google Scholar
[9]	Fan H Y, Hu L Y 2009 Commun. Theor. Phys. 51 729 Google Scholar
[10]	Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480 Google Scholar
[11]	Kurchan J, Leboeuf P, Saraceno M 1989 Phys. Rev. A 40 6800 Google Scholar
[12]	Fan H Y, Zaidi H R 1987 Phys. Lett. A 124 303 Google Scholar
[13]	Fan H Y, Wang J S 2007 Commun. Theor. Phys. 47 431 Google Scholar
[14]	Fan H Y, Fan Y 1997 Commun. Theor. Phys. 27 105 Google Scholar
[15]	颜森林 2007 物理学报 56 1994 Google Scholar Yan S L 2007 Acta Phys. Sin. 56 1994 Google Scholar
[16]	兰豆豆, 郭晓敏, 彭春生, 姬玉林, 刘香莲, 李璞, 郭龑强 2017 物理学报 66 120502 Google Scholar Lan D D, Guo X M, Peng C S, Ji Y L, Liu X L, Li P, Guo Y Q 2017 Acta Phys. Sin. 66 120502 Google Scholar
[17]	Weinbub J, Ferry D K 2018 Appl. Phys. Rev. 5 041104 Google Scholar
[18]	范洪义, 梁祖峰 2015 物理学报 64 050301 Google Scholar Fan H Y, Liang Z F 2015 Acta Phys. Sin. 64 050301 Google Scholar
[19]	Wigner E P 1932 Phys. Rev. 40 749 Google Scholar
[20]	Fan H Y 1992 J. Phys. A 25 3443 Google Scholar
[21]	袁洪春, 徐学翔 2012 物理学报 61 064205 Google Scholar Yuan H C, Xu X X 2012 Acta Phys. Sin. 61 064205 Google Scholar
[22]	Fan H Y, Yang Y L 2006 Phys. Lett. A 353 439 Google Scholar
[23]	Zhang K, Fan C Y, Fan H Y 2019 Int. J. Theor. Phys. 58 1687 Google Scholar
[24]	Buot F, Jensen K 1990 Phys. Rev. B 42 9429 Google Scholar
[25]	Chountasis S, Vourdas A 1998 Phys. Rev. A 58 1794 Google Scholar
[26]	Fan H Y, Hu L Y, Yuan H C 2010 Chin. Phys. B 19 060305 Google Scholar

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量子扩散通道中Wigner算符的演化规律