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二维激子-极化子凝聚体中冲击波的产生与调控

王金玲 张昆 林机 李慧军

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二维激子-极化子凝聚体中冲击波的产生与调控

王金玲, 张昆, 林机, 李慧军

Generation and modulation of shock waves in two-dimensional polariton condensates

Wang Jin-ling, Zhang Kun, Lin Ji, Li Hui-jun
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  • 由于半导体微腔中形成的激子-极化子凝聚体能在室温实现, 且具有非平衡、强相互作用等特性, 使其成为研究非平衡量子系统非线性特性的一个理想平台. 本文采用谱方法与四阶龙格库塔法, 探究二维极化子凝聚体中产生和调控冲击波的方案. 发现, 若在高凝聚率时淬火凝聚体与热库之间的交叉相互作用, 可将初始制备的亮孤子调制成两种波速不同的旋转对称型冲击波, 而初始的类暗孤子只能转变成单一波速的旋转对称型冲击波; 若淬火外势, 则可将类暗孤子转化成各向异性的超声速冲击波, 并给出冲击波对外势宽度的依赖关系. 若在低凝聚率时调控外势和非相干泵浦, 可在均匀凝聚体中激发出多种各向异性冲击波, 还可通过它们的振幅调控冲击波的波数和振幅, 并展示了激发冲击波所需外势或非相干泵浦的宽度范围. 文中方案不仅为激子-极化子凝聚体中产生和调控冲击波提供理论指导, 找到了与实验相似的对称型冲击波, 而且为非平衡或不可积系统中激发冲击波开辟了一条普适捷径, 可能成为调控孤子向冲击波转变的一种范式.
    Due to the ability of exciton-polariton condensates formed in semiconductor microcavities to be achieved at room temperature and their characteristics such as non-equilibrium and strong interactions, they have become an ideal platform for studying the nonlinear properties of non-equilibrium quantum systems. In 2013, the research group led by L. Dominici observed two-dimensional symmetric shock waves in the polariton condensate driven by coherent pump. However, owing to the characteristics of this system, theoretical researches have lagged behind. In one-dimensional polariton condensates, disregarding cross-interaction of the system, a type of asymmetric shock wave was respectively discovered by A. M. Kamchatnov in 2012 and A. M. Belounis in 2017. In 2023, utilizing the adiabatic approximation, our research team not only uncovered sparse wave, symmetric, and asymmetric shock waves in the system, but also revealed that the symmetric shock waves are triggered by cross-interaction. At present, there is no theoretical research on shock waves in two-dimensional polariton condensate. In this paper, spectral methods and fourth-order Runge-Kutta methods are used to explore the generation and control of shock waves in two-dimensional polariton condensates. It is found that when the cross-interaction between the condensate and the polariton thermal reservoir is quenched at high condensation rates, the initially prepared bright solitons can be modulated into two types of rotationally symmetric shock waves with different velocities, while the initial dark-like solitons can only transform into a single velocity rotationally symmetric shock wave. If quenching the external potential, the dark-like solitons can be transformed into anisotropic supersonic shock waves, and the dependence of shock wave on the width of the external potential is also shown. When the external potential and incoherent pumping are controlled at low condensation rates, multiple anisotropic shock waves can be excited in a uniform condensate, and their amplitudes can be used to control the wave number and amplitude of the shock waves and the range of widths for the external potential or incoherent pumping to excite shock waves is also demonstrated. The proposed methods in this paper not only provide theoretical guidance for the generation and control of shock waves in exciton-polariton condensates, but also find symmetric shock waves similar to experiments (Nat. Commun. 6 , 8993) without adopting any approximation, and open up a universal pathway for exciting shock waves in non-equilibrium or non-integrable systems, which may become a paradigm for transforming solitons into shock waves and significantly propel the rapid development of shock wave theory in different domains.
  • 图 1  取亮孤子为初态, 淬火交叉相互作用 $ \sigma_{2} $ 产生的冲击波. (a) 参数取$ \sigma_{2}=1 $时, 亮孤子的轮廓图; (b) 冲击波波速 $ v_{\eta} $ 与声速 $ C_{\eta} $ 随交叉相互作用强度 $ \sigma_{2} $ 的变化曲线; (c)和 (d) 分别为$ \sigma_{2}=4, s=18 $ 和$ \sigma_{2}=5.8, s=16.5 $ 时冲击波的轮廓图, 二者波速分别为$ v_{\eta}=1.57 $和$ v_{\eta}=1.63 $, 此时选取 图1 (b) 中黑点对应的参数. 图1中的插图均是取$ \xi=0 $ 的截面图

    Fig. 1.  A bright soliton is chosen as the initial incident wave, shock waves are generated through quenching the cross-interaction strength $ \sigma_{2} $. (a) Profile of the bright soliton with $ \sigma_{2}=1 $; (b) Curves illustrating the variation of shock wave velocity and sound speed as a function of $ \sigma_{2} $; (c) and (d) represent the profiles of shock waves at $ \sigma_{2}=4, s = 18, v_{\eta}=1.57 $ and $ \sigma_{2}=5.8, s = 16.5, v_{\eta}=1.63 $, respectively, with corresponding parameters marked by the black dots in panel (b). The insets in figure 1 are cross-sections of the bright soliton and shock waves at $ \xi=0 $.

    图 2  取类暗孤子为初态, 淬火交叉相互作用强度 $ \sigma_{2} $ 产生的冲击波. (a) 参数取$ \sigma_{2}=1 $时, 类暗孤子的轮廓图; (b) 冲击波波速 $ v_{\eta} $ 与声速 $ C_{\eta} $ 随交叉相互作用强度 $ \sigma_{2} $ 的变化曲线; (c) 和 (d) 分别为 $ \sigma_{2}=4.5, s=18, v_{\eta}=1.5 $ 和 $ \sigma_{2}=6.5, s=18, $$ v_{\eta}=1.5 $ 时冲击波的轮廓图, 此时选取 图2 (b) 中黑点对应的参数. 图2中的插图均是取 $ \xi=0 $ 的截面图

    Fig. 2.  A dark-like soliton is chosen as the initial incident wave, shock waves are generated through quenching the cross-interaction strength $ \sigma_{2} $. (a) Profile of the dark-like soliton with $ \sigma_{2}=1 $; (b) Curves illustrating the variation of shock wave velocity and sound speed as a function of $ \sigma_{2} $; (c) and (d) represent the profiles of shock waves at $ \sigma_{2}=4.5, s=18, v_{\eta}=1.5 $ and $ \sigma_{2}=6.5, s=18, v_{\eta}=1.5 $, respectively, with corresponding parameters marked by the black dots in panel (b). The insets in figure 2 are cross-sections of the dark-like soliton and shock waves at $ \xi=0 $.

    图 3  以类暗孤子为初态, 淬火外势产生的冲击波. (a) 冲击波的存在区域图. (b)—(c) 分别为 $ \sigma_{11}=3, v_{\eta}=1.42, s=11 $ 和$ \sigma_{11}=7, v_{\eta}=1.42, s=11 $的冲击波轮廓图, 其中 $ d_{\xi}=8, d_{\eta}=1 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示. (d) $ \sigma_{11}=3 $, $ d_{\xi}=\infty $, $ d_{\eta}=1, s=8 $ 时冲击波的投影图, 此时, 冲击波速度为$ v_{\eta}=1.42 $, 图3(e)3(f) 分别给出 3(d) 中 $ \xi=-16.16 $ 与$ \xi=0 $ 处两条黑色虚线对应的截面图

    Fig. 3.  A dark-like soliton is chosen as the initial incident wave, shock waves are produced by quenching the external potential. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{11}=3, \, 7, s=11 $, here, $ d_{\xi}=8, d_{\eta}=1, v_{\eta}=1.42 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line)directions are displayed on the right side of (b) and (c); (d) Projection of the shock wave, here, $ \sigma_{11}=3, d_{\xi}=\infty, d_{\eta}=1, s=8, v_{\eta}=1.42 $; The cross-sections corresponding to the two black dashed lines for $ \xi=-16.16 $ and $ \xi=0 $ are illustrated in (e)and (f), respectively.

    图 4  通过调控外势在均匀凝聚体中激发的冲击波. (a) 冲击波的存在区域图. (b)—(c) $ \sigma_{11}= $ 3和7时对应的冲击波轮廓图, 其中 $ d_{\xi}=8, d_{\eta}=1, v_{\eta}=1.72, s=11 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示

    Fig. 4.  Shock waves induced by potential in a homogeneous condensate. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{11}=3, \, 7 $, here, $ d_{\xi}=8, d_{\eta}=1, v_{\eta}=1.72, s=11 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line)directions are displayed on the right side of (b) and (c).

    图 5  通过调控泵浦在均匀凝聚体中激发的冲击波. (a) 冲击波的存在区域图. (b)—(c) $ \sigma_{10}= $ 3和7时对应的冲击波轮廓图, 其中 $ d_{\xi}=10, d_{\eta}=2, v_{\eta}=1.57, s=17 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示.

    Fig. 5.  Shock waves induced by incoherent pump in a homogeneous condensate. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{10}=3, \, 7 $, here, $ d_{\xi}=10, d_{\eta}=2, v_{\eta}=1.57, s=17 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line)directions are displayed on the right side of (b) and (c).

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  • 收稿日期:  2024-02-02
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  • 上网日期:  2024-04-11

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