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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.
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Keywords:
- shock wave /
- soliton /
- exciton-polariton condensate /
- incoherent pump
[1] Zhang S, Zhu Z, Du W, Wu X, Ghosh S, Zhang Q, Xiong Q, Liu X 2023 ACS Photonics 10 2414
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[49] Yang J 2011 Nonlinear Waves in Integrable and Nonintegrable Systems (1st Ed.) (SIAM: Philadephia
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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 $, 此时选取(b)图中黑点对应的参数. 图中的插图均是取$ \xi=0 $的截面图
Figure 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 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 $时冲击波的轮廓图, 此时选取(b)图中黑点对应的参数. 图中的插图均是取$ \xi=0 $的截面图
Figure 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 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) 和 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
Figure 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$和$ \sigma_{11}= 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
Figure 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$ \sigma_{10}=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.
Figure 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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[1] Zhang S, Zhu Z, Du W, Wu X, Ghosh S, Zhang Q, Xiong Q, Liu X 2023 ACS Photonics 10 2414
Google Scholar
[2] Kottilil D, Gupta M, Lu S, Babusenan A, Ji W 2023 Adv. Mater. 35 2209094
Google Scholar
[3] Deng H, Weihs G, Santori C, Bloch J, Yamamoto Y 2002 Science 298 199
Google Scholar
[4] Baumberg J J, Kavokin A V, Christopoulos S, Grundy A J D, Butté R, Christmann G, Solnyshkov D D, Malpuech G, Baldassarri Höger von Högersthal G, Feltin E, Carlin J F, Grandjean N 2008 Phys. Rev. Lett. 101 136409
Google Scholar
[5] Zhang Y, Jia C, Liang Z 2022 Chin. Phys. Lett. 39 020501
Google Scholar
[6] Qi W, Guo X G, Dong L W, Zhang X F 2023 Chin. Phys. B 32 030502
Google Scholar
[7] 黄轶凡, 梁兆新 2023 72 100505
Google Scholar
Huang Y F, Liang Z X 2023 Acta Phys. Sin. 72 100505
Google Scholar
[8] 陈礼元, 高超, 林机, 李慧军 2022 71 181101
Google Scholar
Chen L Y, Gao C, Lin J, Li H J 2022 Acta Phys. Sin. 71 181101
Google Scholar
[9] Weisbuch C, Nishioka M, Ishikawa A, Arakawa Y 1992 Phys. Rev. Lett. 69 3314
Google Scholar
[10] Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P, Keeling J M J, Marchetti F M, Szymańska M H, André R, Staehli J L, Savona V, Littlewood P B, Deveaud B, Dang L S 2006 Nature 443 409
Google Scholar
[11] Jia C Y, Liang Z X 2020 Chin. Phys. Lett. 37 040502
Google Scholar
[12] Ye Z, Chen F, Zhou H, Luo S, Sun Z, Xu H, Xu H, Li H, Chen Z, Wu J 2023 Phys. Rev. B 107 L060303
Google Scholar
[13] Balili R, Hartwell V, Snoke D, Pfeiffer L, West K 2007 Science 316 1007
Google Scholar
[14] Szymańska M H, Keeling J, Littlewood P B 2006 Phys. Rev. Lett. 96 230602
Google Scholar
[15] Byrnes T, Kim N Y, Yamamoto Y 2014 Nat. Phys. 10 803
Google Scholar
[16] Zhang K, Wen W, Lin J, Li H J 2021 New J. Phys. 23 033011
Google Scholar
[17] Zhang K, Wen W, Lin J, Li H J 2022 Front. Phys. 10 798562
Google Scholar
[18] Tanese D, Flayac H, Solnyshkov D, Amo A, Lemaître A, Galopin E, Braive R, Senellart P, Sagnes I, Malpuech G, Bloch J 2013 Nat. Commun. 4 1749
Google Scholar
[19] Chen H J, Ren Y, Wang H 2022 Acta Phys. Sin. 71 056701
Google Scholar
[20] Ostrovskaya E A, Abdullaev J, Fraser M D, Desyatnikov A S, Kivshar Y S 2013 Phys. Rev. Lett. 110 170407
Google Scholar
[21] Pinsker F, Flayac H 2014 Phys. Rev. Lett. 112 140405
Google Scholar
[22] El G A, Geogjaev V V, Gurevich A V, Krylov A L 1995 Phys. D 87 186
Google Scholar
[23] El G A, Hoefer M A 2016 Phys. D 333 11
Google Scholar
[24] El G A, Gammal A, Khamis E G, Kraenkel R A, Kamchatnov A M 2007 Phys. Rev. A 76 053813
Google Scholar
[25] 公睿智, 王灯山 2023 72 100503
Google Scholar
Gong R Z, Wang D S 2023 Acta Phys. Sin. 72 100503
Google Scholar
[26] Kamchatnov A M 2021 Phys.-Usp. 64 48
Google Scholar
[27] Bikbaev R, Kudashev V 1994 Phys. Rev. A 190 255
Google Scholar
[28] Wang D S, Xu L, Xuan Z 2022 J. Nonlinear Sci. 32 3
Google Scholar
[29] Kamchatnov A M 2019 Phys. Rev. E 99 012203
Google Scholar
[30] Gong R, Wang D S 2022 Physica D 439 133398
Google Scholar
[31] Gong R, Wang D S 2022 Appl. Math. Lett. 126 107795
Google Scholar
[32] Sagdeev R Z 1962 Sov. Phys. Tech. Phys. 6 867
[33] Wan W, Jia S, Fleischer J 2007 Nat. Phys. 3 46
Google Scholar
[34] Chanson H 2009 Eur. J. Mech. B 28 191
Google Scholar
[35] Hang C, Bai Z, Li W, Kamchatnov A, Huang G 2023 Phys. Rev. A 107 033503
Google Scholar
[36] 贾瑞煜, 方乒乒, 高超, 林机 2021 70 180303
Google Scholar
Jia R Y, Fang P P, Gao C, Lin J 2021 Acta Phys. Sin. 70 180303
Google Scholar
[37] Simplicio C, Purita J, Murrell W, Santos G, Dos Santos R, Lana J 2020 J. Clin. Orthop. Trauma 11 S309
Google Scholar
[38] Takayama K, Saito T 2004 Annu. Rev. Fluid Mech. 36 347
Google Scholar
[39] Yang Z, Zhang B 2023 Combust. Flame 251 112691
Google Scholar
[40] Hu Q, Shen X, Huang Z, Qian X, Jiang J, Yuan M, Pang L 2023 Int. J. Hydrog. Energy 51 830
Google Scholar
[41] Mayilswamy N, Krishnan A, Mundhada M, Deodhar H, Joshi G, Kandasubramanian B 2023 Ind. Eng. Chem. Res. 62 6584
Google Scholar
[42] Ciccarelli G, Johansen C, Parravani M 2010 Combust. Flame 157 2125
Google Scholar
[43] Dominici L, De Giorgi M, Ballarini D, Cancellieri E, Laussy F, Giacobino E, Bramati A, Gigli G, Sanvitto D 2013 IEEE Conference on Lasers and Electro-Optics (CLEO: 2013), June 9–14 2013, San Jose, California, USA, pp1–2
[44] Dominici L, Petrov M, Matuszewski M, Ballarini D, De Giorgi M, Colas D, Cancellieri E, Silva Fernández B, Bramati A, Gigli G, Kavokin A, Laussy F, Sanvitto D 2015 Nat. Commun. 6 8993
Google Scholar
[45] Kamchatnov A M, Kartashov Y V 2012 Europhys. Lett. 97 10006
Google Scholar
[46] Belounis A M, Kessal S 2017 Can. J. Phys. 95 1234
Google Scholar
[47] Wang J L, Wen W, Lin J, Li H J 2023 Chin. Phys. Lett. 40 070302
Google Scholar
[48] Wang Q W, Wang J L, Wen W, Lin J, Li H J 2023 Commun. Theor. Phys. 75 065001
Google Scholar
[49] Yang J 2011 Nonlinear Waves in Integrable and Nonintegrable Systems (1st Ed.) (SIAM: Philadephia
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