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本文研究运动光格中非线性作用随时空变化的玻色-爱因斯坦凝聚体的混沌时空动力学. 在运动光格势强度和非线性作用调制强度较小的情况下, 系统满足微扰条件, 将Melnikov函数法应用于理论分析, 得到了系统的Melnikov时空混沌判据. 当系统不满足微扰条件时, 数值模拟表明, 对于原子间呈吸引作用的玻色-爱因斯坦凝聚体, 非线性作用调制强度的增大可以加深系统的时空混沌程度. 在某些参数区域, 非线性作用调制频率对系统时空动力学行为具有重要影响. 进一步的数值研究结果揭示, 较大的化学势不仅可以抑制吸引系统的时空混沌, 还可以抑制排斥系统的时空混沌. 基于以上研究结果, 在实验中可以根据需要规避或引发玻色-爱因斯坦凝聚系统的时空混沌.
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关键词:
- 玻色-爱因斯坦凝聚 /
- 运动光格 /
- Melnikov混沌判据 /
- 混沌
The dynamical behaviors of Bose-Einstein condensates (BECs) depend largely on the nonlinear interactions between BEC atoms. The advancement of experimental techniques enables the rapid and effective modulation of the nonlinear interactions through Feshbach resonance technique. At present, both the nonlinear time-varying interaction and nonlinear space-varying interaction have been realized, respectively, thus making it possible to simultaneously modulate the nonlinear interactions in time and space through the combination of techniques. It will provide more options to conduct various studies by manipulating the BECs. Therefore, BECs with time- and space-varying interactions must possess unique advantages in studying BEC dynamics. This paper focuses on the chaotic spatiotemporal dynamics of BECs with nonlinear time- and space-varying interactions in moving optical lattices. When the intensities of the moving optical lattice potential and the modulation of the nonlinear interaction are small, the system satisfies the perturbation conditions and the Melnikov-function method is used in the theoretical analyses to obtain the Melnikov spatiotemporal chaotic criterion of the system. When the system does not meet the perturbation conditions, numerical simulations show that for a BEC with an attractive atomic interaction, increasing the modulation intensity of the nonlinear interaction can deepen the degree of spatiotemporal chaos in the system. In certain parameter regions, the modulation frequency of the nonlinear interaction can have a significant influence on the spatiotemporal dynamical behavior of the system. Further numerical research results show that larger chemical potentials can suppress the spatiotemporal chaos not only in the attractive BEC but also in the repulsive BEC. Based on the above research results, spatiotemporal chaos in BEC system can be avoided or triggered off in experiments as needed. -
Keywords:
- Bose–Einstein condensates /
- travelling optical lattice /
- Melnikov chaotic criterion /
- chaos
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图 1 改变非线性作用调制强度过程中庞加莱截面的演化, 其他参数和初始条件设置为$ \tilde \mu = 0.8 $, $ \tilde v = 2.0 $, $ \tilde I = 0.23 $, $ {\tilde g_0} = - 0.75 $, $ \omega = 0.4 $, $ A(0) = 0.26 $, $ \dot A(0) = 0 $
Fig. 1. Evolution of Poincaré sections in the process of changing the modulation intensity of the nonlinear interaction. The other parameters and initial conditions are set as $ \tilde \mu = 0.8 $, $ \tilde v = 2.0 $, $ \tilde I = 0.23 $, $ {\tilde g_0} = - 0.75 $, $ \omega = 0.4 $, $ A(0) = 0.26 $, $ \dot A(0) = 0 $.
图 2 增大化学势过程中相图的演化 (a)—(c)图对应的化学势$ \tilde \mu $的值分别为1.7, 2.5, 10; 其他参数和初始条件与图1(d)相同
Fig. 2. Evolution of phase portraits with the increase of chemical potential: The values of the chemical potential $ \tilde \mu $ corresponding to (a)–(c) are 1.7, 2.5, and 10, respectively. The other parameters and initial conditions are the same as Fig. 1(d).
图 3 不同$ \omega $值对应的庞加莱截面 (a), (a') $ \omega = 29.999 $; (b), (b') $ \omega = 30 $; (c), (c') $ \omega = 30.00001 $; 其他参数和初始条件与图1(d)相同
Fig. 3. The Poincaré sections for different values of $ \omega $: (a), (a') $ \omega = 29.999 $; (b), (b') $ \omega = 30 $; (c), (c') $ \omega = 30.00001 $. The other parameters and initial conditions are the same as Fig.1 (d).
图 4 不同$ \omega $值时的相图和对应的庞加莱截面 (a), (a') $ \omega = 1 $; (b), (b') $ \omega = 2.2 $; (a), (b)是相图; (a'), (b')是对应的庞加莱截面; 参数和初始条件设置为$ \tilde \mu = 2.11 $, $ \tilde v = 2.0 $, $ \tilde I = 0.25 $, $ {\tilde g_0} = {\tilde g_1} = - 0.7 $, $ A(0) = 0.28 $, $ \dot A(0) = 0 $
Fig. 4. The phase portraits and corresponding Poincaré sections for different values of $ \omega $: (a), (a') $ \omega = 1 $; (b), (b') $ \omega = 2.2 $; (a), (b) are phase portraits; (a'), (b') are the corresponding Poincaré sections. The parameters and initial conditions are set as $ \tilde \mu = 2.11 $, $ \tilde v = 2.0 $, $ \tilde I = 0.25 $, $ {\tilde g_0} = {\tilde g_1} = - 0.7 $, $ A(0) = 0.28 $, $ \dot A(0) = 0 $.
图 6 原子间呈排斥作用的BEC系统的混沌相图和对应的$ A(\zeta ) $混沌时空演化曲线, 参数和初始条件为$ \tilde \mu = 3 $, $ \tilde v = $$ 1 $, $ \tilde I = 0.9 $, $ {\tilde g_1} = 0.6 $, $ {\tilde g_0} = 0.7 $, $ \omega = 1.2 $, $ A(0) = 0.6 $, $ \dot A(0) = 0 $
Fig. 6. The chaotic phase portrait and corresponding chaotic spatiotemporal evolution curve of $ A(\zeta ) $ of a BEC system with a repulsive interaction between atoms. The parameters and initial conditions are set as $ \tilde \mu = 3 $, $ \tilde v = 1 $, $ \tilde I = 0.9 $, $ {\tilde g_1} = 0.6 $, $ {\tilde g_0} = 0.7 $, $ \omega = 1.2 $, $ A(0) = 0.6 $, $ \dot A(0) = 0 $.
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[1] Anderson M H, Ensher J R, Matthews M R, E Wieman C, Cornell E A 1995 Science 269 198
Google Scholar
[2] Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M, Ketterle W 1995 Phys. Rev. Lett. 75 3969
Google Scholar
[3] Efremidis N K, Sears S, Christodoulides D N, Fleischer J W, Segev M 2002 Phys. Rev. E 66 046602
Google Scholar
[4] Dahan B M, Peik E, Reichel J, Castin Y, Salomon C 1996 Phys. Rev. Lett. 76 4508
Google Scholar
[5] Anderson B P, Kasevich M A 1998 Science 282 1686
Google Scholar
[6] Liu S, Xiong H, Xu Z, Huang G 2003 J. Phys. B 36 2083
Google Scholar
[7] 高吉明, 狄国文, 鱼自发, 唐荣安, 徐红萍, 薛具奎 2024 物理学报 73 130503
Google Scholar
Gao J M, Di G W, Yu Z F, Tang R A, Xu H P, Xue J K 2024 Acta Phys. Sin. 73 130503
Google Scholar
[8] Pachos J K, Knight P L 2003 Phys. Rev. Lett. 91 107902
Google Scholar
[9] Morsch O, Müller J H, Cristiani M, Ciampini D, Arimondo E 2001 Phys. Rev. Lett. 87 140402
Google Scholar
[10] Bhattacherjee A B, Pietrzyk M 2008 Cent. Eur. J. Phys. 6 26
[11] Choi D I, Niu Q 1999 Phys. Rev. Lett. 82 2022
Google Scholar
[12] Wu B, Niu Q 2001 Phys. Rev. A 64 061603(R
Google Scholar
[13] Cristiani M, Morsch O, Müller J H, Ciampini D, Arimondo E 2002 Phys. Rev. A 65 063612
Google Scholar
[14] Liu J, Fu L B, Ou B Y, Chen S G, Choi D I, Wu B, Niu Q 2002 Phys. Rev. A 66 023404
Google Scholar
[15] Choi D I, Niu Q 2003 Phys. Lett. A 318 558
Google Scholar
[16] Trombettoni A, Smerzi A 2001 Phys. Rev. Lett. 86 2353
Google Scholar
[17] Berg-Sørensen K, Mølmer K 1998 Phys. Rev. A 58 1480
Google Scholar
[18] Holthaus M 2000 J. Opt. B: Quantum Semiclassical Opt. 2 589
Google Scholar
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Google Scholar
[20] Scott R G, Martin A M, Fromhold T M, Bujkiewicz S, Sheard F W, Leadbeater M 2003 Phys. Rev. Lett. 90 110404
Google Scholar
[21] Filho V S, Gammal A, Frederico T, Tomio L 2000 Phys. Rev. A 62 033605(R
Google Scholar
[22] Hai W H, Lee C H, Chong G S, Shi L 2002 Phys. Rev. E 66 026202
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
Wang L, Liu J S, Li J, Zhou X L, Chen X R, Liu C F, Liu W M 2020 Acta Phys. Sin. 69 010303
Google Scholar
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Google Scholar
Wang Q Q, Zhou Y S, Wang J, Fan X B, Shao K H, Zhao Y X, Song Y, Shi Y R 2023 Acta Phys. Sin. 69 010308
Google Scholar
[35] Ruprecht P A, Edwards M, Burnett K, Clark C W 1996 Phys. Rev. A 54 4178
Google Scholar
[36] Denschlag J H Simsarian, J E , Häffner H, McKenzie C, Browaeys A, Cho D, Helmerson K, Rolston S L, Phillips W D 2002 J. Phys. B 35 3095
Google Scholar
[37] Mellish A S, Duffy G, McKenzie C, Geursen R, Wilson A C 2003 Phys. Rev. A 68 051601(R
Google Scholar
[38] Fallani L, Cataliotti F S, Catani J, Fort C, Modugno M, Zawada M, Inguscio M 2003 Phys. Rev. Lett. 91 240405
Google Scholar
[39] Kagan Y, Surkov E L, Shlyapnikov G V 1997 Phys. Rev. Lett. 79 2604
Google Scholar
[40] Cornish S L, Claussen N R, Roberts J L, Cornell E A, Wieman C E 2000 Phys. Rev. Lett. 85 1795
Google Scholar
[41] Kevrekidis P G, Theocharis G, Frantzeskakis D J, Malomed B A 2004 Phys. Rev. Lett. 90 230401
Google Scholar
[42] Theocharis G, Schmelcher P, Kevrekidis P G, Frantzeskakis D J 2005 Phys. Rev. A 72 033614
Google Scholar
[43] Abdullaev F K, Garnier J 2005 Phys. Rev. A 72 061605(R
Google Scholar
[44] He J R, Li H M 2011 Phys. Rev. E 83 066607
Google Scholar
[45] Avelar A T, Bazeia D, Cardoso W B 2009 Phys. Rev. A 79 025602(R
Google Scholar
[46] Wang D S, Hu X H, Liu W M 2010 Phys. Rev. A 82 023612
Google Scholar
[47] Beitia J B, García V M P, Vekslerchik V, Konotop V V 2008 Phys. Rev. Lett. 100 164102
Google Scholar
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Google Scholar
[49] Arroyo Meza L E, Souza Dutra A de, Hott M B 2013 Phys. Rev. E 88 053202
Google Scholar
[50] Cardoso W B, Leão S A, Avelar A T, Bazeia D, Hussein M S 2010 Phys. Lett. A 374 4594
Google Scholar
[51] Wang D S, Song S W, Xiong B, Liu W M 2011 Phys. Rev. A 84 053607
Google Scholar
[52] Wang D S, Song S W, Liu W M 2012 Journal of Physics: Conference Series 400 012078
Google Scholar
[53] He J R, Yi L 2014 Phys. Lett. A 378 1085
Google Scholar
[54] Rong S G, Hai W H, Xie Q T, Zhong H H 2012 Chaos 22 033109
Google Scholar
[55] Parker N G, Proukakis N P, Barenghi C F, Adams C S 2004 Phys. Rev. Lett. 92 160403
Google Scholar
[56] Parker N G, Proukakis N P,Barenghi C F, Adams C S 2004 J. Phys. B 37 S175
Google Scholar
[57] Proukakis N P, Parker N G, Barenghi C F, Adams C S 2004 Phys. Rev. Lett. 93 130408
Google Scholar
[58] Gardiner S A, Jaksch D, Dum R, Cirac J I, Zoller P 2000 Phys. Rev. A 62 023612
Google Scholar
[59] Machholm M, Pethick CJ, Smith H 2003 Phys. Rev. A 67 053613
Google Scholar
[60] Goldstein E V, Meystre P 1999 Phys. Rev. A 59 1509
Google Scholar
[61] Goldstein E V, Meystre P 1999 Phys. Rev. A 59 3896
Google Scholar
[62] Ling H Y 2001 Phys. Rev. A 65 013608
Google Scholar
[63] 刘式适, 刘式达 2012 物理学中的非线性方程 (第二版)(北京: 北京大学出版社)第68页
Liu S K, Liu S D 2012 Nonlinear Equations in Physics (2nd edition) (Beijing: Peking University Press) p68
[64] Li F, Zhou B J, Shu W X, Luo H L, Huang Z Y, Tian L 2008 Eur. Phys. J. D 50 75
Google Scholar
[65] Stavans J, Heslot F, Libchaber A 1985 Phys. Rev. Lett. 55 596
Google Scholar
[66] Bishop A R, Forest M G, McLaughlin D W, Overman II E A 1986 Physica D 23 293
Google Scholar
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