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回音壁式耦合光力学系统中的混沌现象

华志豪 郭琴 樊碧璇 谢旻

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回音壁式耦合光力学系统中的混沌现象

华志豪, 郭琴, 樊碧璇, 谢旻

Route to chaos in whispering gallery mode coupled opto-mechanical systems

Hua Zhi-Hao, Guo Qin, Fan Bi-Xuan, Xie Min
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  • 在光力学系统中, 辐射压力引起的非线性可导致系统产生混沌等多种丰富的动力学行为. 从有序转变到混沌有各种各样的途径, 本文研究了回音壁式耦合光力系统中两种不同途径通往混沌的现象——倍周期分岔道路和准周期道路. 两个全同的光力学子系统通过光学倏逝波进行耦合, 调控子系统间的耦合强度, 可以有效地改变系统的动力学行为. 采用分岔图、李雅普诺夫指数以及相空间轨迹的变化分析系统从有序到混沌的转变. 研究表明, 较强的耦合强度会迫使两个子系统的振荡产生完全同步, 子系统间相互耦合的作用可等效于子系统光学模频率与外加控制场频率失谐量的增加, 耦合光力学系统的动力学行为等效于单个光力学系统的动力学, 可实现倍周期分岔道路通往混沌的转变. 而对于相对较弱的耦合, 耦合系统的动力学将在高维相空间展开, 极限环发生Hopf分岔, 形成环面吸引子. 选取合适的失谐量, 耦合系统可实现从准周期道路进入混沌的现象.
    In opto-mechanical systems, the nonlinearity caused by radiation pressure can lead to various abundant dynamical phenomena such as chaos. Chaos is an important branch of nonlinear dynamics, and researchers focus on understanding the transitions from order to chaos in different systems. In this paper, we investigate the chaotic dynamics in a system consisting of two evanescently coupled identical cavity opto-mechanical subsystems, where the optical fields are in whispering gallery modes. To thoroughly analyze the transition from order to chaos in our system, we utilize the bifurcation diagrams, the Lyapunov exponents, and phase space trajectories to characterize the system properties. It is found that the coupling strength between the two opto-mechanical subsystems plays a crucial role in determining the systemic dynamic behaviors. There are two routes to chaos in our system i.e. the period-doubling transition and the quasiperiodic transition. These routes correspond to strong coupling and weak coupling between the two opto-mechanical subsystems, respectively. Furthermore, the results show that the synchronization between the oscillations in the two opto-mechanical subsystems can occur under strong coupling. In this situation, the dynamic behaviors of the two opto-mechanical subsystems are exactly identical and the manipulation of the coupling strength is equivalent to the tuning of the frequency detuning between the cavity fields and their corresponding driving fields. Consequently, the coupled system behaves as a single opto-mechanical system, enabling a period-doubling transition to chaos through increasing the coupling strength. In the case of weak coupling, the dynamics of the two opto-mechanical subsystems are no longer synchronized, and the coupled system dynamic behaviors unfold in an eight-dimensional phase space. The limit cycles experience the Hopf bifurcation, resulting in the emergence of a toric attractor. Within a certain range of parameters, i.e. appropriate frequency detunings, the two-dimensional torus becomes unstable as coupling strength increases, leading to a quasiperiodic transition into chaos in our coupled opto-mechanical system.
      通信作者: 谢旻, xiemin@jxnu.edu.cn
    • 基金项目: 国家自然科学基金地区科学基金(批准号: 12064018, 11964014)、江西省教育厅科学研究项目(批准号: GJJ191683)和江西省主要学科学术和技术带头人培养计划——青年人才项目(批准号: 20204BCJ23026)资助的课题
      Corresponding author: Xie Min, xiemin@jxnu.edu.cn
    • Funds: Project supported by the Fund for Less Developed Regions of the National Natural Science Foundation of China (Grant Nos. 12064018, 11964014), the Scientific Research Foundation of the Education Department of Jiangxi Province, China (Grant No. GJJ191683), and the the Major Discipline Academic and Technical Leader Training Program of Jiangxi Province, China (Grant No. 20204BCJ23026)
    [1]

    Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391Google Scholar

    [2]

    Marquardt F, Chen J P, Clerk A A, Girvin S M 2007 Phys. Rev. Lett. 99 093902Google Scholar

    [3]

    Wilson-Rae I, Nooshi N, Zwerger W, Kippenberg T J 2007 Phys. Rev. Lett. 99 093901Google Scholar

    [4]

    Vitali D, Gigan S, Ferreira A, Böhm H R, Tombesi P, Guerreiro A, Vedral V, Zeilinger A, Aspelmeyer M 2007 Phys. Rev. Lett. 98 030405Google Scholar

    [5]

    张秀龙, 鲍倩倩, 杨明珠, 田雪松 2018 物理学报 67 104203Google Scholar

    Zhang X L, Bao Q Q, Yang M Z, Tian X S 2018 Acta Phys. Sin. 67 104203Google Scholar

    [6]

    Weis S, Rivière R, Deléglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520Google Scholar

    [7]

    Guo Y, Kai L, Nie W, Yong L 2014 Phys. Rev. A 90 053841Google Scholar

    [8]

    Dorsel A, Mccullen J D, Meystre P, Vignes E, Walther H 1983 Phys. Rev. Lett. 51 1550Google Scholar

    [9]

    Marquardt F, Harris J, Girvin S M 2006 Phys. Rev. Lett. 96 103901Google Scholar

    [10]

    Metzger C, Ludwig M, Neuenhahn C, Ortlieb A, Favero I, Karrai K, Marquardt F 2008 Phys. Rev. Lett. 101 133903Google Scholar

    [11]

    Xie M, Fan B X, He X L, Chen Q Q 2018 Phys. Rev. E 98 052202Google Scholar

    [12]

    Fan B X, Xie M 2017 Phys. Rev. A 95 023808Google Scholar

    [13]

    Zhang M, Wiederhecker G S, Manipatruni S, Barnard A, Mceuen P, Lipson M 2012 Phys. Rev. Lett. 109 233906Google Scholar

    [14]

    Wu Y, Li G, He B, Lin Q 2022 Phys. Rev. A 105 013521Google Scholar

    [15]

    Wang H, Dhayalan Y, Buks E 2016 Phys. Rev. E 93 023007Google Scholar

    [16]

    Heinrich G, Ludwig M, Qian J, Kubala B, Marquardt F 2011 Phys. Rev. Lett. 107 043603Google Scholar

    [17]

    Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar

    [18]

    Rai V, Sreenivasan R 1993 Ecol. Modell. 69 63Google Scholar

    [19]

    Ding W X, Huang W, Wang X D, Yu C X 1993 Phys. Rev. Lett. 70 170Google Scholar

    [20]

    Stavans J, Heslot F, Libchaber A 1985 Phys. Rev. Lett. 55 596Google Scholar

    [21]

    Pomeaub Y, Manneville P 1980 Commun. Math. Phys. 74 189Google Scholar

    [22]

    Gwinn E G, Westervelt R M 1985 Phys. Rev. Lett. 54 1613Google Scholar

    [23]

    Doebeli M 1994 J. Theor. Biol. 166 325Google Scholar

    [24]

    Carmon T, Cross M C, Vahala K J 2007 Phys. Rev. Lett. 98 167203Google Scholar

    [25]

    Bakemeier L, Alvermann A, Fehske H 2015 Phys. Rev. Lett. 114 013601Google Scholar

    [26]

    Lü X Y, Jing H, Ma J Y, Wu Y 2015 Phys. Rev. Lett. 114 253601Google Scholar

    [27]

    Christou S, Kovanis V, Giannakopoulos A E, Kominis Y 2021 Phys. Rev. A 103 053513Google Scholar

    [28]

    Madiot G, Correia F, Barbay S, Braive R 2021 Phys. Rev. A 104 023525Google Scholar

    [29]

    Wurl C, Alvermann A, Fehske H 2016 Phys. Rev. A 94 063860Google Scholar

    [30]

    Roque T F, Marquardt F, Yevtushenko O M 2020 New J. Phys. 22 013049Google Scholar

    [31]

    Buskirk R V, Jeffries C 1985 Phys. Rev. A 31 3332Google Scholar

    [32]

    Sánchez E, Pazó D, Matías M A 2006 Chaos 16 033122Google Scholar

    [33]

    Yang J Z 2000 Phys. Rev. E 61 6521Google Scholar

    [34]

    Borkowski L, Perlikowski P, Kapitaniak T, Stefanski A 2015 Phys. Rev. E 91 062906Google Scholar

    [35]

    Reick C, Mosekilde E 1995 Phys. Rev. E 52 1418Google Scholar

    [36]

    Lindner J F, Meadows B K, Ditto W L, Inchiosa M E, Bulsara A R 1995 Phys. Rev. Lett. 75 3Google Scholar

    [37]

    Kapitaniak T 1994 Phys. Rev. E 50 1642Google Scholar

    [38]

    Gu Y, Bandy D K, Yuan J M, Narducci L M 1985 Phys. Rev. A 31 354Google Scholar

  • 图 1  回音壁式耦合光力学系统示意图

    Fig. 1.  Schematic diagram of the coupled whispering gallery modes cavity opto-mechanical system

    图 2  两个子系统无耦合情形下, 力学模的稳态位置$ x_{1} $随失谐$ \varDelta_{1} $的变化曲线图. 蓝色表示稳定解, 绿色表示参量不稳解, 红色表示不稳定解. 小框内为雅可比矩阵的一对共轭特征值虚根变化的复平面图. 系统参数分别为$ \kappa _{1} = 1.0\omega_{m1}, $ $\gamma_{1} = 0.26\omega_{m1}, $ $g_{1}=-0.0006\omega_{m1}, $ $E_{1} = 2980\omega_{m1} $

    Fig. 2.  System stability diagram of $ x_{1} $ with the varying of detuning $ \varDelta_{1} $ under uncoupling between the two subsystems. Blue stands for the stable branches, green stands for parametric instability, and red stands for unstable branch. The virtual rosots of a pair of conengenvalues for the Jacobi matrix are presented in the small box. The parameters are $\kappa _{1} = 1.0\omega_{m1},$ $\gamma_{1} = 0.26\omega_{m1},$ $g_{1} = -0.0006\omega_{m1},$ $E_{1} = $$ 2980\omega_{m1}$

    图 3  不同耦合强度下两个子系统力学模位置的输出曲线, 蓝实线对应$ x_{1} $, 红虚线对应$ x_{2} $ (a)$ G = 1.0\omega_{m1} $; (b)$ G = 1.5\omega_{m1} $; (c)$ G = 1.7\omega_{m1} $; (d)$ G = 2.3\omega_{m1} $; (e)$ G = 2.8\omega_{m1} $; (f)$ G = 3.0\omega_{m1} $. 两个子系统参数完全相同, 初始条件随机, $\varDelta_{1}=\varDelta_{2}= $$ -1.0\omega_{m1}$, 其余参数和图2相同

    Fig. 3.  Output curves of the two mechanical modes under different coupling strengthes. The blue solid line and the red dashed line correspond to $ x_{1} $ and $ x_{2} $, respectively: (a) $ G = 1.0\omega_{m1} $; (b) $ G = 1.5\omega_{m1} $; (c) $ G = 1.7\omega_{m1} $; (d) $ G = 2.3\omega_{m1} $; (e) $ G = 2.8\omega_{m1} $; (f) $ G = 3.0\omega_{m1} $. The parameters for the two subsystems are exactly the same, and the initial conditions are arbitrary. All parameters are the same as those in Fig. 2 except for $ \varDelta_{1}=\varDelta_{2}=-1.0\omega_{m1} $

    图 4  (a)系统倍周期分岔图; (b)最大李雅普诺夫指数图. 所有参数和图3相同

    Fig. 4.  (a) Schematic period-doubling bifurcation diagram; (b) the curve for the maximum of Lyapunov exponents. All parameters are the same as those in Fig. 3.

    图 5  不同耦合强度下系统达到稳定时的状态图 (a) $ G = 1.47\omega_{m1} $; (b) $ G = 1.53\omega _{m1} $; (c) $ G = 1.60\omega_{m1} $; (d) $ G = 1.87\omega_{m1} $. 其中每张子图中包含4个分图, 左上图对应$I_{1}\text{-}x_{1}\text{-}p_{1}$三维相空间轨迹, 两种颜色表征两组初始条件下的轨迹; 左下图对应$x_{1}\text{-}x_{2}$二维相空间轨迹; 右上图对应动态李雅普诺夫指数谱(前4个李雅普诺夫指数); 右下图对应$ x_{1} $的频率谱. $\varDelta_{1}=\varDelta_{2} = $$ 0.5\omega _{m1}$, 其余参数和图3相同

    Fig. 5.  Stability diagrams of the system under different coupling strengthes: (a) $ G = 1.47\omega_{m1} $; (b) $ G = 1.53\omega _{m1} $; (c) $ G = 1.60\omega_{m1} $; (d) $ G = 1.87\omega_{m1} $. Each subgraph includes four charts, the top left one corresponding to the three-dimensional phase space of $I_{1}\text-x_{1}\text-p_{1}$, and the two colors represented the traces for two sets of initial conditions; the bottom left one corresponding to the two-dimensional phase space of $x_{1}\text{-}x_{2}$; the top right one corresponding to Lyapunov exponents (the top four Lyapunov exponents); and the bottom right one corresponding to the frequency spectrum of $ x_{1} $. The other parameters are the same as those in Fig. 3 except for $ \varDelta_{1}=\varDelta_{2} = 0.5\omega _{m1} $

    图 6  (a)准周期路径分岔图; (b)李雅普诺夫指数图(前4个李雅普诺夫指数). 所有参数与图5相同

    Fig. 6.  (a) Schematic quasiperiodic bifurcation diagram; (b) curves of Lyapunov exponents (the top four Lyapunov exponents). All parameters are the same as those in Fig. 5.

  • [1]

    Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391Google Scholar

    [2]

    Marquardt F, Chen J P, Clerk A A, Girvin S M 2007 Phys. Rev. Lett. 99 093902Google Scholar

    [3]

    Wilson-Rae I, Nooshi N, Zwerger W, Kippenberg T J 2007 Phys. Rev. Lett. 99 093901Google Scholar

    [4]

    Vitali D, Gigan S, Ferreira A, Böhm H R, Tombesi P, Guerreiro A, Vedral V, Zeilinger A, Aspelmeyer M 2007 Phys. Rev. Lett. 98 030405Google Scholar

    [5]

    张秀龙, 鲍倩倩, 杨明珠, 田雪松 2018 物理学报 67 104203Google Scholar

    Zhang X L, Bao Q Q, Yang M Z, Tian X S 2018 Acta Phys. Sin. 67 104203Google Scholar

    [6]

    Weis S, Rivière R, Deléglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520Google Scholar

    [7]

    Guo Y, Kai L, Nie W, Yong L 2014 Phys. Rev. A 90 053841Google Scholar

    [8]

    Dorsel A, Mccullen J D, Meystre P, Vignes E, Walther H 1983 Phys. Rev. Lett. 51 1550Google Scholar

    [9]

    Marquardt F, Harris J, Girvin S M 2006 Phys. Rev. Lett. 96 103901Google Scholar

    [10]

    Metzger C, Ludwig M, Neuenhahn C, Ortlieb A, Favero I, Karrai K, Marquardt F 2008 Phys. Rev. Lett. 101 133903Google Scholar

    [11]

    Xie M, Fan B X, He X L, Chen Q Q 2018 Phys. Rev. E 98 052202Google Scholar

    [12]

    Fan B X, Xie M 2017 Phys. Rev. A 95 023808Google Scholar

    [13]

    Zhang M, Wiederhecker G S, Manipatruni S, Barnard A, Mceuen P, Lipson M 2012 Phys. Rev. Lett. 109 233906Google Scholar

    [14]

    Wu Y, Li G, He B, Lin Q 2022 Phys. Rev. A 105 013521Google Scholar

    [15]

    Wang H, Dhayalan Y, Buks E 2016 Phys. Rev. E 93 023007Google Scholar

    [16]

    Heinrich G, Ludwig M, Qian J, Kubala B, Marquardt F 2011 Phys. Rev. Lett. 107 043603Google Scholar

    [17]

    Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar

    [18]

    Rai V, Sreenivasan R 1993 Ecol. Modell. 69 63Google Scholar

    [19]

    Ding W X, Huang W, Wang X D, Yu C X 1993 Phys. Rev. Lett. 70 170Google Scholar

    [20]

    Stavans J, Heslot F, Libchaber A 1985 Phys. Rev. Lett. 55 596Google Scholar

    [21]

    Pomeaub Y, Manneville P 1980 Commun. Math. Phys. 74 189Google Scholar

    [22]

    Gwinn E G, Westervelt R M 1985 Phys. Rev. Lett. 54 1613Google Scholar

    [23]

    Doebeli M 1994 J. Theor. Biol. 166 325Google Scholar

    [24]

    Carmon T, Cross M C, Vahala K J 2007 Phys. Rev. Lett. 98 167203Google Scholar

    [25]

    Bakemeier L, Alvermann A, Fehske H 2015 Phys. Rev. Lett. 114 013601Google Scholar

    [26]

    Lü X Y, Jing H, Ma J Y, Wu Y 2015 Phys. Rev. Lett. 114 253601Google Scholar

    [27]

    Christou S, Kovanis V, Giannakopoulos A E, Kominis Y 2021 Phys. Rev. A 103 053513Google Scholar

    [28]

    Madiot G, Correia F, Barbay S, Braive R 2021 Phys. Rev. A 104 023525Google Scholar

    [29]

    Wurl C, Alvermann A, Fehske H 2016 Phys. Rev. A 94 063860Google Scholar

    [30]

    Roque T F, Marquardt F, Yevtushenko O M 2020 New J. Phys. 22 013049Google Scholar

    [31]

    Buskirk R V, Jeffries C 1985 Phys. Rev. A 31 3332Google Scholar

    [32]

    Sánchez E, Pazó D, Matías M A 2006 Chaos 16 033122Google Scholar

    [33]

    Yang J Z 2000 Phys. Rev. E 61 6521Google Scholar

    [34]

    Borkowski L, Perlikowski P, Kapitaniak T, Stefanski A 2015 Phys. Rev. E 91 062906Google Scholar

    [35]

    Reick C, Mosekilde E 1995 Phys. Rev. E 52 1418Google Scholar

    [36]

    Lindner J F, Meadows B K, Ditto W L, Inchiosa M E, Bulsara A R 1995 Phys. Rev. Lett. 75 3Google Scholar

    [37]

    Kapitaniak T 1994 Phys. Rev. E 50 1642Google Scholar

    [38]

    Gu Y, Bandy D K, Yuan J M, Narducci L M 1985 Phys. Rev. A 31 354Google Scholar

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出版历程
  • 收稿日期:  2022-12-18
  • 修回日期:  2023-04-05
  • 上网日期:  2023-05-17
  • 刊出日期:  2023-07-20

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