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一维耦合腔晶格中磁子-光子拓扑相变和拓扑量子态的调制

李锦芳 何东山 王一平

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一维耦合腔晶格中磁子-光子拓扑相变和拓扑量子态的调制

李锦芳, 何东山, 王一平

Modulation of topological phase transition and topological quantum state of magnon-photon in one-dimensional coupled cavity lattices

Li Jin-Fang, He Dong-Shan, Wang Yi-Ping
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  • 本文提出了基于耦合腔的一维晶格理论方案, 其中每个晶胞包含微波腔光子和磁子, 通过调控磁子与微波光子的耦合来研究系统中的拓扑相变和拓扑量子通道. 首先, 分析了在奇偶晶格数情况下, 系统能谱和边缘态的特征, 并发现边缘态分布可以展示反转过程, 可以实现多通道拓扑量子态传输; 其次, 考虑存在缺陷和无序的扰动, 发现它们在较小值范围内, 可以使能带产生波动和翻转现象, 但边缘状态对其是鲁棒的, 这表明边缘态受到系统的拓扑保护. 该研究结果为研究拓扑磁子-光子提供了一条新途径, 将在量子信息处理中有着广阔的应用前景.
    We propose a theoretical scheme to study the topological properties of magnon-photon in a one-dimensional coupled cavity lattice. Each unit cell is composed of the cavity microwave photon and the magnon, where the magnon is placed inside the cavity. The coupling of cavity microwave photon and magnon is controlled by an external magnetic field, and multiple cavities are coupled with each other to form a one-dimensional coupled cavity lattice system. Here, we study the topological phase transition and topological quantum channels of magnon-photon in the system by adjusting the magnon-photon coupling. Firstly, considering odd and even number lattices, we analyze and discuss the energy spectrum and the edge state in one-dimensional coupled cavity lattices. It is found that the energy band of the system is symmetric, and the edge states in the energy gap have time reversal symmetry, which makes the system topologically protected. At the same time, it is also noted that the maximum value, flipping, and period of the energy spectrum have changed, and the region of the edge state has expanded and extended. In addition, the edge state distribution can undergo the flipping process, which can achieve multi-channel topological quantum state transmission. Besides, considering the presence of defects and disorder in the system, it is found that when the random defect potential is small, the edge state of the system is robust to it, but when the random defect potential is large, the fluctuation of the energy band will be enhanced, and the edge state will be submerged in the energy band. However, when the disorder is very small, it can cause band fluctuations and flipping phenomena, and the edge state is robust to it, indicating the topological protection of the edge state. This work offers an effective way to study topological magnon-photon, which will have promising applications in quantum information processing.
      通信作者: 王一平, ypwang2019@nwafu.edu.cn
    • 基金项目: 陕西省教育厅专项科学研究计划(批准号: 23JK0713)、陕西省自然科学基金(批准号: 2023-JC-YB-020)和中央高校基本科研业务费专项资金(批准号: 2452020019, 2452022027)资助的课题.
      Corresponding author: Wang Yi-Ping, ypwang2019@nwafu.edu.cn
    • Funds: Project supported by the Special Scientific Research Project of Shaanxi Provincial Education Department, China (Grant No. 23JK0713), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2023-JC-YB-020), and the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2452020019, 2452022027).
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    Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys 88 035005Google Scholar

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    Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606Google Scholar

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    Li L, Xu Z, Chen S 2014 Phys. Rev. B 89 085111Google Scholar

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    Li L, Chen S 2015 Phys. Rev. B 92 085118Google Scholar

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    Ge L, Wang L, Xiao M, Wen W, Chan C T, Han D 2015 Opt. Express 23 21585Google Scholar

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    Lin Y J, Compton R L, Jiménez-GarcÍa K, Porto J V, Spielman I B 2009 Nature 462 628Google Scholar

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    Jimenéz-GarcÍa K, LeBlanc L J, Williams R A, Beeler M C, Perry A R, Spielman I B 2012 Phys. Rev. Lett. 108 225303Google Scholar

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    Mei F, You J, Zhang D, Yang X C, Fazi R, Zhu S L, Kwek L C 2014 Phys. Rev. A 90 063638Google Scholar

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    Ganeshan S, Sun K, Das Sarma S 2013 Phys. Rev. Lett. 110 180403Google Scholar

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    Wray L A, Xu V, Xia Y, Hsieh D, Fedorov A V, SanHor Y, Cava R J, Bansil A, Lin H, Hasan M Z 2011 Nat. Phys. 7 32Google Scholar

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    Malki M, Uhrig G S 2017 Phys. Rev. B 95 235118Google Scholar

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    Mei F, You J B, Nie W, Fazio R, Zhu S L, Kwek L C 2015 Phys. Rev. A 92 041805Google Scholar

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    Mei F, Xue Z Y, Zhang D W, Tian L, Lee C, Zhu S L 2016 Quantum Sci. Technol. 1 015006Google Scholar

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    Cao J, Yi X, Wang H F 2020 Phys. Rev. A 102 032619Google Scholar

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    Walther H, Varcoe B T, Englert B G, Becker T 2006 Rep. Prog. Phys. 69 1325Google Scholar

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    Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2019 Chin. Phys. B 28 014202Google Scholar

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    Huang S M, Agarwal G S 2010 Phys. Rev. A 81 033830Google Scholar

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    Wang Y P, Wang H 2023 Quantum Inf. Process. 22 386Google Scholar

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    Wang Z Y, Li Z X, Yuan H Y, Zhang Z Z, Cao Y S, Yan P 2023 Acta Phys. Sin. 72 057503Google Scholar

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    Ren Y L, Xie J K, Li X K, Ma S L, Li F L 2022 Phys. Rev. B 105 094422Google Scholar

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  • 图 1  (a)一维耦合腔晶格模型图, $ J_{n} $是晶胞$ a_{n-1} $和$ a_{n} $之间的耦合; (b) $ a_n $和$ m_n $分别表示腔场和磁子的模式, 其中沿腔z 方向施加磁场H, 可以使磁子模式与腔场模式耦合

    Fig. 1.  (a) Schematic of the one-dimensional coupled cavity lattice system, $ J_{n} $ is the coupling strength between cavities $ a_{n-1} $ and $ a_{n} $; (b) $ a_n $ and $ m_n $ represent the modes of the cavity field and the magnon, an uniform bias magnetic field (H along the z direction) that establishes the magnon-photon coupling.

    图 2  n = 36时, 系统能谱与相位$\phi $的关系图, 其中红色和蓝色线条是系统的两个边缘态模式 (a)$ g=0 $; (b)$ g=1 $; (c) $ g=2$; (d) $ g=3 $. 其他参数为$ J=1 $, $ \phi \in $[$ 0 $, $ 2 \pi $]

    Fig. 2.  Energy spectrum of the system via the phase $\phi $, the red and blue lines are the two edge state modes of the system at n = 36: (a) $ g=0 $; (b) $ g=1 $; (c) $ g=2 $; (d) $ g=3 $. Other parameters are set as $ J=1 $ and $ \phi \in $[$ 0 $, $ 2 \pi $].

    图 3  $ n=37 $时, 系统能谱与相位$\phi $的关系图, 其中红色线条是系统零模边缘态 (a) $ g=0 $; (b) $ g=1 $; (c) $ g=2 $; (d) $ g= 3 $. 其他参数为$ J=1 $, $ \phi \in $[$ 0 $, $ 2 \pi $]

    Fig. 3.  Energy spectrum of the system via the phase $\phi $, the red line is the zero mode edge state of the system, n = 37 sites: (a) $ g=0 $; (b) $ g=1 $; (c) $ g=2 $; (d) $ g=3 $. Other parameters are set as $ J=1 $ and $ \phi \in $[$ 0 $, $ 2 \pi $].

    图 4  系统能谱与晶格数关系图 (a)—(d) $ n=36 $, $ g=0, 1, 2, 3 $; (e)—(h) $ n=37 $, $ g=0, 1, 2, 3 $. 其他参数为$ J=1 $, $ \phi = \pi $

    Fig. 4.  Energy spectrum of the system via the lattice numbers: (a)–(d)$ n=36 $, $ g=0, 1, 2, 3 $; (e)–(h)$ n=37 $, $ g=0, 1, 2, 3 $. Other parameters are set as $ J=1 $ and $ \phi = \pi $.

    图 5  n = 37时, 边缘态(第19模式)的概率分布图 (a)$ \phi =0.7\pi, 1.5\pi $, $ g=1 $, $ J=1 $; (b)$ \phi =1.37 \pi $, $ g=0, 1 $, $ J=1 $

    Fig. 5.  State distribution of the 19 mode is plotted at n = 37: (a) $ \phi =0.7\pi, 1.5\pi $, $ g=1 $, $ J=1 $; (b) $ \phi =1.37 \pi $, $ g=0, 1 $, $ J=1 $

    图 6  边缘态概率分布与晶格数和相位$\phi $的关系图 (a) $ g=0 $; (b) $ g=1 $; (c) $ g=2 $; (d) $ g=3 $; (e) $ g=4 $; (f) $ g=5 $. 其他参数为$ n=37 $, $ J=1 $

    Fig. 6.  State distribution of edge state via the lattice numbers and the phase $\phi $: (a) $ g=0 $; (b) $ g=1 $; (c) $ g=2 $; (d) $ g=3 $; (e) $ g=4 $; (f) $ g=5 $. Other parameters are set as $ n=37 $ and $ J=1 $.

    图 7  系统能谱与相位$\phi $和随机缺陷的关系图 (a) $ W_{{\mathrm{r}}}=0.1 $; (b) $ W_{{\mathrm{r}}}=0.5 $; (c) $ W_{{\mathrm{r}}}=1 $; (d) $ W_{{\mathrm{r}}}=3 $; (e) $ W_{{\mathrm{r}}}=5 $; (f) $ W_{{\mathrm{r}}}=7 $. 其他参数参考图2(b)

    Fig. 7.  Energy spectrum as a function of the phase $\phi $ for different degrees of defect: (a) $ W_{{\mathrm{r}}}=0.1 $; (b) $ W_{{\mathrm{r}}}=0.5 $; (c) $ W_{{\mathrm{r}}}=1 $; (d)$ W_{{\mathrm{r}}}=3 $; (e) $ W_{{\mathrm{r}}}=5 $; (f) $ W_{{\mathrm{r}}}=7 $. Other parameters are set as Fig. 2(b).

    图 8  系统能谱与相位$\phi $和无序的关系图 (a) $ \delta=-0.3 $ (蓝色实线), $ \delta=0.3 $ (黑色虚线); (b)$ \delta=0.1 $ (蓝色实线), $ \delta=0.3 $ (黑色虚线); (c)$ \delta=-0.1 $ (蓝色实线), $ \delta=-0.3 $ (黑色虚线). 其他参数参考图2(b)

    Fig. 8.  Energy spectrum as a function of the phase $\phi $ for different degrees of disorder and the dissipation: (a) $ \delta=-0.3 $ (blue line), $ \delta=0.3 $ (black dotted line); (b) $ \delta=0.1 $ (blue line), $ \delta=0.3 $ (black dotted line); (c) $ \delta=-0.1 $ (blue line), $ \delta=-0.3 $ (black dotted line). Other parameters are set as Fig. 2(b).

  • [1]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [2]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [3]

    Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004Google Scholar

    [4]

    Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys 88 035005Google Scholar

    [5]

    Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606Google Scholar

    [6]

    Li L, Xu Z, Chen S 2014 Phys. Rev. B 89 085111Google Scholar

    [7]

    Li L, Chen S 2015 Phys. Rev. B 92 085118Google Scholar

    [8]

    Mei F, Zhu S L, Zhang Z M, Oh C H, Goldman N 2012 Phys. Rev. A 85 013638Google Scholar

    [9]

    Ge L, Wang L, Xiao M, Wen W, Chan C T, Han D 2015 Opt. Express 23 21585Google Scholar

    [10]

    Lin Y J, Compton R L, Jiménez-GarcÍa K, Porto J V, Spielman I B 2009 Nature 462 628Google Scholar

    [11]

    Jimenéz-GarcÍa K, LeBlanc L J, Williams R A, Beeler M C, Perry A R, Spielman I B 2012 Phys. Rev. Lett. 108 225303Google Scholar

    [12]

    Mei F, You J, Zhang D, Yang X C, Fazi R, Zhu S L, Kwek L C 2014 Phys. Rev. A 90 063638Google Scholar

    [13]

    Ganeshan S, Sun K, Das Sarma S 2013 Phys. Rev. Lett. 110 180403Google Scholar

    [14]

    Wray L A, Xu V, Xia Y, Hsieh D, Fedorov A V, SanHor Y, Cava R J, Bansil A, Lin H, Hasan M Z 2011 Nat. Phys. 7 32Google Scholar

    [15]

    Malki M, Uhrig G S 2017 Phys. Rev. B 95 235118Google Scholar

    [16]

    Mei F, You J B, Nie W, Fazio R, Zhu S L, Kwek L C 2015 Phys. Rev. A 92 041805Google Scholar

    [17]

    Mei F, Xue Z Y, Zhang D W, Tian L, Lee C, Zhu S L 2016 Quantum Sci. Technol. 1 015006Google Scholar

    [18]

    Cao J, Yi X, Wang H F 2020 Phys. Rev. A 102 032619Google Scholar

    [19]

    Stehle P 1970 Phys. Rev. A 2 102Google Scholar

    [20]

    Nathan M I, Fowler A B, Burns G 1963 Phys. Rev. Lett. 11 152Google Scholar

    [21]

    Haroche S, Kleppner D 1989 Phys. Today 42 24Google Scholar

    [22]

    Walther H, Varcoe B T, Englert B G, Becker T 2006 Rep. Prog. Phys. 69 1325Google Scholar

    [23]

    Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2019 Chin. Phys. B 28 014202Google Scholar

    [24]

    Huang S M, Agarwal G S 2010 Phys. Rev. A 81 033830Google Scholar

    [25]

    Wang Y P, Wang H 2023 Quantum Inf. Process. 22 386Google Scholar

    [26]

    Sanchez-Mondragon J J, Narozhny N B, Eberly J H 1983 Phys. Rev. Lett. 51 550Google Scholar

    [27]

    McConnell R, Zhang H, Hu J, Cuk S, Vuletic V 2015 Nature 519 439Google Scholar

    [28]

    Lukin M D 2003 Rev. Mod. Phys. 75 457Google Scholar

    [29]

    Zhang X F, Zou C L, Jiang L, Tang H X 2016 Sci. Adv. 2 e1501286Google Scholar

    [30]

    Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Usami K, Nakamura Y 2015 Science 349 405Google Scholar

    [31]

    王振宇, 李志雄, 袁怀洋, 张知之, 曹云姗, 严鹏 2023 物理学报 72 057503Google Scholar

    Wang Z Y, Li Z X, Yuan H Y, Zhang Z Z, Cao Y S, Yan P 2023 Acta Phys. Sin. 72 057503Google Scholar

    [32]

    Zhang X, Zou C L, Jiang L, Tang H X 2014 Phys. Rev. Lett. 113 156401Google Scholar

    [33]

    Wang Y P, Zhang G Q, Zhang D, Luo X Q, Xiong W, Wang S P, Li T F, Hu C M, You J Q 2016 Phys. Rev. B 94 224410Google Scholar

    [34]

    Ren Y L, Xie J K, Li X K, Ma S L, Li F L 2022 Phys. Rev. B 105 094422Google Scholar

    [35]

    Xiao Y, Yan X H, Zhang Y, Grigoryan V L, Hu C M, Guo H, Xia K 2019 Phys. Rev. B 99 094407Google Scholar

    [36]

    Yu T, Yang H, Song L, Yan P, Cao Y 2020 Phys. Rev. B 101 144414Google Scholar

    [37]

    Cao Y, Yan P 2019 Phys. Rev. B 99 214415Google Scholar

    [38]

    Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M, You J Q 2018 Phys. Rev. Lett. 120 057202Google Scholar

    [39]

    Li J, Zhu S Y, Agarwal G S 2018 Phys. Rev. Lett. 121 203601Google Scholar

    [40]

    刘浪, 王一平 2022 物理学报 71 224202Google Scholar

    Liu L, Wang Y P 2022 Acta Phys. Sin. 71 224202Google Scholar

    [41]

    郑智勇, 陈立杰, 向吕, 王鹤, 王一平 2023 物理学报 72 244204Google Scholar

    Zheng Z Y, Chen L J, Xiang L, Wang H, Wang Y P 2023 Acta Phys. Sin. 72 244204Google Scholar

    [42]

    Wang W, Wang Y P 2022 Acta Phys. Sin. 71 194203 (in Chinese) [王伟, 王一平 2022 物理学报 71 194203]Google Scholar

    Wang W, Wang Y P 2022 Acta Phys. Sin. 71 194203 (in Chinese) Google Scholar

    [43]

    Wang Y P, Wang W, Liu L, Zheng Z Y, Du M E 2022 Results Phys. 42 105999Google Scholar

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出版历程
  • 收稿日期:  2023-09-18
  • 修回日期:  2023-10-30
  • 上网日期:  2023-11-29
  • 刊出日期:  2024-02-20

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