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In non-conservative nonlinear systems, the basic physical mechanics of soliton generation is that the kinetic energy and nonlinear terms of the system, as well as the gain and dissipation terms reach a double dynamic balance. How to generate stable free high-dimensional solitons in such a system is currently a challenging topic in soliton theory. In this article, we propose a theoretical scheme for realizing two-dimensional free bright solitons in exciton-polariton Bose-Einstein condensates, which proposes a physical mechanism for generating stable two-dimensional free space bright solitons through time periodic modulation interactions and a dual balance between gain and dissipation. In this end, firstly, we obtain the dynamic equations of two-dimensional bright soliton parameters through the Lagrange variational method, and obtain its dynamically stable parameter space. Secondly, the evolution of the generalized dissipative Gross-Pitaveskii equation is numerically simulated to verify the stability of two-dimensional bright solitons. Finally, we add Gaussian noise to simulate a real experimental environment and find that two-dimensional bright solitons are also stable within the observable time range of the experiment. Our experimental scheme opens the door to the study of bright solitons in high-dimensional free space in non-conservative systems.
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Keywords:
- bright soliton /
- polariton condensate /
- Gross-Pitaevskii equation
[1] Saito H, Ueda M 2003 Phys. Rev. Lett. 90 040403
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Google Scholar
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Google Scholar
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图 1 峰值密度
$\left|\psi\left(r=0\right)\right|^{2}$ 和单极矩$\left\langle r\right\rangle = \displaystyle\int r\left|\psi\right|^{2}\mathrm{d}\boldsymbol{r}$ 随时间t演化的图像 (a1), (a2)$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.271$ ; (b1), (b2)$\gamma_{{\rm{C}}}=0.70$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.484$ ; (c1), (c2)$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=2.60$ ,$P_{0}=1.142$ . 周期振荡相互作用强度取为$g(t)= $ $ -2\pi+ 8\pi\sin(\varOmega t)$ , 含时演化的前段$t=0$ —20, 逐渐打开热源极化子间的相互作用, 同时缓慢减弱径向约束势到零. 其他参数为$R=2.40$ ,$\omega=2.00$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ Figure 1. Time evolution of the peak density
$\left|\psi\left(r=0\right)\right|^{2}$ and the monopole moment$\left\langle r\right\rangle = \displaystyle\int r\left|\psi\right|^{2}\mathrm{d}\boldsymbol{r}$ : (a1), (a2)$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.271$ ; (b1), (b2)$\gamma_{{\rm{C}}}=0.70$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.484$ ; (c1), (c2)$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=2.60$ ,$P_{0}=1.142$ . The interaction$g(t)=-2\pi+8\pi\sin(\varOmega t)$ is switched on a little bit at a time while the radial confinement potential is switched off from$t=0$ to 20. Values of other parameters are$R=2.40$ ,$\omega=2.00$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ 
图 2 图1(a)所描述系统中, 概率密度分布随时间演化的图像. 图中颜色明暗代表概率密度的大小, 颜色越亮表示对应的概率密度越大
Figure 2. Time evolution of the the odds density distribution for the system depicted in Fig. 1(a). The brighter and darker colors in the figure represent the magnitude of the density, and the brighter color indicates the corresponding higher density
图 3 考虑噪声的快热库极限下, 峰值密度
$\left|\psi\left(r=0\right)\right|^{2}$ (a)随时间演化图像和(b)概率密度分布图像. 周期振荡相互作用强度取为$g(t)=-2\pi+8\pi\sin(\varOmega t)$ , 噪音强度为$D=0.01$ , 其他参数为$R=2.40$ ,$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.274$ ,$\omega=2.00$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ . 图中颜色明暗代表概率密度的大小, 颜色越亮表示对应的概率密度越大Figure 3. (a) Time evolution of the peak density
$\left|\psi\left(r=0\right)\right|^{2}$ and (b) the density distribution in the limit of fast reservoir considering the noise. The interaction$g(t)=-2\pi+8\pi\sin(\varOmega t)$ is switched on a little bit at a time while the radial confinement potential is switched off from$t=0$ to 20. The noise intensity is$D=0.01$ . Values of other parameters are$R=2.40$ ,$\gamma_{{\rm{C}}}=0.60$ ,$\gamma_{{\rm{R}}}=3.00$ ,$P_{0}=1.274$ ,$\omega=2.00$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ . The brighter and darker colors in the figure represent the magnitude of the density, and the brighter color indicates the corresponding higher density
图 4 根据演化耦合方程(4)和方程(5), 峰值密度
$\left|\psi\left(r=0\right)\right|^{2}$ 和概率密度分布图像 (a1)快热库参数空间中峰值密度$\left|\psi\left(r=0\right)\right|^{2}$ 随时间演化的图像,$R=2.40$ ,$\gamma_{{\rm{R}}}=3.00$ ; (b1)在图(a1)基础上再加入强度为$D=0.01$ 的弱噪声; (c1), (d1)考虑噪声的快热库参数空间中的概率密度分布图; (a2)非快热库参数空间中峰值密度$\left|\psi\left(r=0\right)\right|^{2}$ 随时间演化的图像,$R=0.24$ ,$\gamma_{{\rm{R}}}=0.30$ ; (b2)在图(a2)基础上再加入强度为$D=0.01$ 的弱噪声; (c2), (d2)考虑噪声的非快热库参数空间中的分叉率密度分布图. 周期振荡相互作用强度取为$g(t)=-2\pi+8\pi\sin(\varOmega t)$ , 在$t=30$ 时加入高斯白噪声, 其他参数为$\gamma_{{\rm{C}}}=0.60$ ,$P_{0}=1.265$ ,$\omega=2.20$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ . 图中颜色明暗代表概率密度的大小, 颜色越亮表示对应的概率密度越大Figure 4. Time evolution of the peak density
$\left|\psi\left(r=0\right)\right|^{2}$ and the monopole moment$\left\langle r\right\rangle =\int r\left|\psi\right|^{2}\mathrm{d}\boldsymbol{r}$ while evolving the coupled equations (4) and (5): (a1) Image of peak density$\left|\psi\left(r=0\right)\right|^{2}$ in the parameter space of fast reservoir with time evolution,$R=2.40$ ,$\gamma_{{\rm{R}}}=3.00$ ; (b1) adding a weak noise with intensity D=0.01 to panel (a1); (c1), (d1) image of density distribution in the parameter space of fast reservoir considering the noise; (a2) image of peak density$\left|\psi\left(r=0\right)\right|^{2}$ in the parameter space of non-fast reservoir with time evolution,$R=0.24$ ,$\gamma_{{\rm{R}}}=0.30$ ; (b2) adding a weak noise with intensity D = 0.01 to panel (a2); (c2), (d2) image of density distribution in the parameter space of non-fast reservoir considering the noise. The interaction$g(t)=-2\pi+8\pi\sin(\varOmega t)$ is switched on a little bit at a time while the radial confinement potential is switched off from$t=0$ to 20. Gaussian white noise is added at the moment$t=30$ , and the other parameters are$\gamma_{{\rm{C}}}=0.60$ ,$P_{0}=1.265$ ,$\omega=2.20$ ,$\varOmega=30$ ,$g_{{\rm{R}}}=0$ . The brighter and darker colors in the figure represent the magnitude of the density, and the brighter color indicates the corresponding higher density -
[1] Saito H, Ueda M 2003 Phys. Rev. Lett. 90 040403
Google Scholar
[2] Zhang Y C, Zhou Z W, Malomed B A, Pu H 2015 Phys. Rev. Lett. 115 253901
Google Scholar
[3] Dalfovo F, Stringari S 1996 Phys. Rev. A 53 2477
Google Scholar
[4] Carr L D, Clark C W 2006 Phys. Rev. Lett. 97 010403
Google Scholar
[5] Kartashov Y V, Malomed B A, Torner L 2011 Rev. Mod. Phys. 83 247
Google Scholar
[6] Maucher F, Henkel N, Saffman M, Królikowski W, Skupin S, Pohl T 2011 Phys. Rev. Lett. 106 170401
Google Scholar
[7] Tikhonenkov I, Malomed B A, Vardi A 2008 Phys. Rev. Lett. 100 090406
Google Scholar
[8] Kippenberg T J, Gaeta A L, Lipson M, Gorodetsky M L 2018 Science 361 567
Google Scholar
[9] Smirnov L A, Smirnova D A, Ostrovskaya E A, Kivshar Y S 2014 Phys. Rev. B 89 235310
Google Scholar
[10] Xu X R, Chen L, Zhang Z D, Liang Z X 2019 J. Phys. B 52 025303
Google Scholar
[11] Jia C Y, Liang Z X 2020 Chin. Phys. Lett. 37 040502
Google Scholar
[12] Jia C Y, Wu R K, Hu Y, Liu W M, Liang Z X 2021 Front. Phys. 9 805841
Google Scholar
[13] Zhang Y L, Jia C Y, Liang Z X 2022 Chin. Phys. Lett. 39 020501
Google Scholar
[14] Jia C Y, Liang Z X 2022 Entropy 24 1789
Google Scholar
[15] 余超凡, 梁国栋, 曹锡金 2008 57 4402
Google Scholar
Yu C F, Liang G D, Cao X J 2008 Acta Phys. Sin. 57 4402
Google Scholar
[16] 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
[17] Kasprzak J, Richard M, Kundermann S, et al. 2006 Nature 443 409
Google Scholar
[18] Balili R, Hartwell V, Snoke D, Pfeiffer L, West K 2007 Science 316 1007
Google Scholar
[19] Deng H, Weihs G, Santori C, Bloch J, Yamamoto Y 2002 Science 298 199
Google Scholar
[20] Carusotto I, Ciuti C 2013 Rev. Mod. Phys. 85 299
Google Scholar
[21] Byrnes T, Kim N Y, Yamamoto Y 2014 Nat. Phys. 10 803
Google Scholar
[22] Plumhof J D, Stöferle T, Mai L, Scherf U, Mahrt R F 2014 Nat. Mater. 13 247
Google Scholar
[23] Su R, Ghosh S, Wang J, Liu S, Diederichs C, Liew T C H, Xiong Q H 2020 Nat. Phys. 16 301
Google Scholar
[24] 李峰, 熊启华 2022 物理 51 445
Google Scholar
Li F, Xiong Q H 2022 Physics 51 445
Google Scholar
[25] Sieberer L M, Buchhold M, Diehl S 2016 Rep. Prog. Phys. 79 096001
Google Scholar
[26] Ciuti C, Schwendimann P, Quattropani A 2001 Phys. Rev. B 63 041303(R
Google Scholar
[27] Klass M, Flayac H, Amthor M, Savenko I G, Brodbeck S, Ala-Nissila T, Klembt S, Schneider C, Höfling S 2018 Phys. Rev. Lett. 120 017401
Google Scholar
[28] Schneider C, Winkler K, Fraser M D, Kamp M, Yamamoto Y, Ostrovskaya E A, Höfling S 2016 Rep. Prog. Phys. 80 016503
Google Scholar
[29] Shelykh I A, Kavokin A V, Rubo Y G, Liew T C H, Malpuech G 2010 Semicond. Sci. Technol. 25 013001
Google Scholar
[30] Sanvitto D, Kéna-Cohen S 2016 Nat. Mater. 15 1061
Google Scholar
[31] Takemura N, Trebaol S, Wouters M, Portella-Oberli M T, Deveaud B 2014 Nat. Phys. 10 500
Google Scholar
[32] Sun Y, Yoon Y, Steger M, Liu G, Pfeiffer L N, West K, Snoke D W, Nelson K A 2017 Nat. Phys. 13 870
Google Scholar
[33] Navadeh-Toupchi M, Takemura N, Anderson M D, Oberli D Y, Portella-Oberli M T 2019 Phys. Rev. Lett. 122 047402
Google Scholar
[34] Lai C W, Kim N Y, Utsunomiya S, et al. 2007 Nature 450 529
Google Scholar
[35] St-Jean P, Goblot V, Galopin E, Lemaître A, Ozawa T, Le Gratiet T, Sagnes I, Bloch J, Amo A 2017 Nat. Photonics 11 651
Google Scholar
[36] Manni F, Lagoudakis K G, Pietka B, Fontanesi L, Wouters M, Savona V, André R, Deveaud-Plédran B 2011 Phys. Rev. Lett. 106 176401
Google Scholar
[37] Thunert M, Janot A, Franke H, Sturm C, Michalsky T, Martín T, Viña L, Rosenow B, Grundmann M, Schmidt-Grund R 2016 Phys. Rev. B 93 064203
Google Scholar
[38] Sala V G, Solnyshkov D D, Carusotto I, et al. 2015 Phys. Rev. X 5 011034
Google Scholar
[39] Whittaker C E, Cancellieri E, Walker P M, et al. 2018 Phys. Rev. Lett. 120 097401
Google Scholar
[40] Klembt S, Harder T H, Egorov O A, et al. 2018 Nature 562 552
Google Scholar
[41] Lim H T, Togan E, Kroner M, Miguel-Sanchez J, Imamoğlu A 2017 Nat. Commun. 8 14540
Google Scholar
[42] 陈海军, 任元, 王华 2022 71 056701
Google Scholar
Chen H J, Ren Y, Wang H 2022 Acta Phys. Sin. 71 056701
Google Scholar
[43] 郭慧, 王雅君, 王林雪, 张晓斐 2020 69 010302
Google Scholar
Guo H, Wang Y J, Wang L X, Zhang X F 2020 Acta Phys. Sin. 69 010302
Google Scholar
[44] Kivshar Y S, Królikowski W 1995 Opt. Commun. 114 353
Google Scholar
[45] Frantzeskakis D J 2010 J. Phys. A: Math. Theor. 43 213001
Google Scholar
[46] Tran P 1998 Phys. Rev. E 58 8049
Google Scholar
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