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在旋量玻色-爱因斯坦凝聚体中, 孤子态作为宏观量子效应的典型状态, 可以通过自旋-轨道耦合进行调控, 这使得对自旋-轨道耦合玻色-爱因斯坦凝聚体中孤子的研究成为近年来超冷原子领域研究的重要课题之一. 本文研究了描述一维自旋-轨道耦合二分量玻色-爱因斯坦凝聚体Gross-Pitaevskii方程的精确求解, 利用直接假设及可积约化方法, 给出了系统多种类型的孤子解, 讨论了相应的孤子动力学以及自旋-轨道耦合效应对系统的量子磁化和自旋-极化态的影响.
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关键词:
- 玻色-爱因斯坦凝聚体 /
- 自旋-轨道耦合 /
- Gross-Pitaevskii方程 /
- 孤子
In a quantum system with spin, spin-orbit coupling is manifested by linking the spin angular momentum of a particle with its orbital angular momentum, which leads to many exotic phenomena. The experimental realization of synthetic spin-orbit coupling effects in ultra-cold atomic systems provides an entirely new platform for exploring quantum simulations. In a spinor Bose-Einstein condensate, the spin-orbit coupling can change the properties of the system significantly, which offers an excellent opportunity to investigate the influence of spin-orbit coupling on the quantum state at the macroscopic level. As typical states of macroscopic quantum effects, solitons in spin-orbit coupled Bose-Einstein condensates can be manipulated by spin-orbit coupling directly, which makes the study on spin-orbit coupled Bose-Einstein condensates become one of the hottest topics in the research of ultracold atomic physics in recent years. This paper investigates exact vector soliton solutions of the Gross-Pitaevskii equation for the one-dimensional spin-orbit coupled binary Bose-Einstein condensates, which has four parameters$\mu$ ,$\delta$ ,$\alpha$ and$\beta$ , where$\mu$ denotes the strength of the spin-orbit coupling,$\delta$ is the detuning parameter,$\alpha$ and$\beta$ are the parameters of the self- and cross-interaction, respectively. For the case$\beta=\alpha$ , by a direct ansatz, two kinds of stripe solitons, namely, the oscillating dark-dark solitons are obtained; meanwhile, a transformation is presented such that from the solutions of the integrable Manakov system, one can get soliton solutions for the spin-orbit coupled Gross-Pitaevskii equation. For the case$\beta=3\alpha$ , a bright-W type soliton for$\alpha>0$ and a kink-antikink type soliton for$\alpha<0$ are presented. It is found that the relation between$\mu$ and$\delta$ can affect the states of the solitons. Based on these solutions, the corresponding dynamics and the impact of the spin-orbit coupling effects on the quantum magnetization and spin-polarized domains are discussed. Our results show that spin-orbit coupling can result in rich kinds of soliton states in the two-component Bose gases, including the stripe solitons as well as the classical non-stripe solitons, and various kinds of multi-solitons. Furthermore, spin-orbit coupling has a remarkable influence on the behaviors of quantum magnetization. In the experiments of Bose-Einstein condensates, there have been many different methods to observe the soliton states of the population distribution, the magnetic solitons, and the spin domains, so our results provide some possible options for the related experiments.-
Keywords:
- Bose-Einstein condensate /
- spin-orbit coupling /
- Gross-Pitaevskii equation /
- soliton
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图 1 (5)式对应的SOC调制的条纹暗孤子. 暗孤子的均匀背景被SOC调制为周期背景, 暗孤子中心与周期背景产生能量交换, 形成了周期背景中的呼吸子传播. 参数取值为
$k= 1$ ,$\delta = 0$ ,$\mu = 3$ ,$\omega = 2$ ,$\alpha = -1$ Fig. 1. Stripe dark soliton given by Eq. (5). The uniform background of the dark soliton is modulated by the SOC into the periodic background, and the energy exchange between the center of the dark soliton and the periodic background results in a breather like soliton propagating in the periodic background. The parameters are chosen as
$k = 1$ ,$\delta = 0$ ,$\mu = 3$ ,$\omega = 2$ ,$\alpha = -1$ 图 2 (5)式对应的不同参数下的赝自旋演化 (a)
$\mu=3/2$ ,$\delta=0$ ,$\omega=-4$ ; (b)$\mu=1/2$ ,$\delta=-1$ ,$\omega=-4$ ; (c)$\delta=1$ ,$\mu=1$ ,$\omega=-4$ . 其他参数为$k=1$ ,$\alpha=-1$ Fig. 2. Evolution of pseudo-spin corresponding to Eq. (5) with different parameters: (a)
$\mu=3/2$ ,$\delta=0$ ,$\omega=-4$ ; (b)$\mu=1/2$ ,$\delta=-1$ ,$\omega=-4$ ; (c)$\delta=1$ ,$\mu=1$ ,$\omega=-4$ . Other parameters are taken as$k=1$ ,$\alpha=-1$ 图 3 (7)式对应的不同参数下的密度分布和磁化分布 (a)—(c)
$\mu = 3/2$ ,$\delta = 0$ ; (d)—(f)$\mu=1$ ,$\delta=-2$ . 其他参数为$G_1 = 2$ ,$G_2 = 1$ ,$k = 1$ ,$\alpha = -1$ Fig. 3. Distributions of the density and magnetization corresponding to Eq. (7) with different parameters: (a)–(c)
$\mu = 3/2$ ,$\delta = 0$ ; (d)–(f)$\mu=1$ ,$\delta=-2$ . Other parameters are taken as$G_1 = 2$ ,$G_2 = 1$ ,$k = 1$ ,$\alpha = -1$ 图 4 (7)式对应的不同参数下的赝自旋演化 (a)
$\mu=3/2$ ,$\delta=0$ ; (b)$\mu=1$ ,$\delta=-2$ ; (c)$\delta=1$ ,$\mu=1$ . 其他参数为$G_1=2$ ,$G_2=1$ ,$k=1$ ,$\alpha=-1$ Fig. 4. Evolution of pseudo-spin corresponding to Eq. (7) with the different parameters: (a)
$\mu=3/2$ ,$\delta=0$ ; (b)$\mu=1$ ,$\delta=-2$ ; (c)$\delta=1$ ,$\mu=1$ . Other parameters are taken as$G_1=2$ ,$G_2=1$ ,$k=1$ ,$\alpha=-1$ -
[1] Pitaevskii L, Stringari S 2016 Bose-Einstein Condensation and Superfluidity (Oxford: Oxford University Press)
[2] Proukakis N P, Snoke D W, Littlewood P B 2017 Universal Themes of Bose-Einstein Condensation (Cambridge: Cambridge University Press)
[3] Zhai H 2020 Ultracold Atomic Physics (Cambridge: Cambridge University Press)
[4] Kawaguchi Y, Ueda M 2012 Phys. Rep. 520 253
Google Scholar
[5] Stamper-Kurn D M, Ueda M 2013 Rev. Mod. Phys. 85 1191
Google Scholar
[6] Liu W M, Kengne E 2019 Schrödinger Equations in Nonlinear Systems (Singapore: Springer Nature Singapore Pte Ltd)
[7] Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 83
Google Scholar
[8] Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan B, Pan G S, Zhao B, Deng Y J, Zhai H, Chen S, Pan J W 2012 Phys. Rev. Lett. 109 115301
Google Scholar
[9] Wang P, Yu Z Q, Fu Z, Miao J, Huang L, Chai S, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301
Google Scholar
[10] Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302
Google Scholar
[11] Zhai H 2015 Rep. Prog. Phys. 78 026001
Google Scholar
[12] Zhang Y P, Mossman M E, Busch T, Engels P, Zhang C W 2016 Front. Phys. 11 118103
Google Scholar
[13] 李吉, 刘伍明 2018 物理学报 67 110302
Google Scholar
Li J, Liu W M 2018 Acta Phys. Sin. 67 110302
Google Scholar
[14] Zhang S C, Jo G B 2019 J. Phys. Chem. Solids 128 75
Google Scholar
[15] 王力, 刘静思, 李吉, 周晓林, 陈向荣, 刘超飞, 刘伍明 2020 物理学报 69 010303
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
[16] 施婷婷, 汪六九, 王璟琨, 张威 2020 物理学报 69 016701
Google Scholar
Shi T T, Wang L J, Wang J K, Zhang W 2020 Acta Phys. Sin. 69 016701
Google Scholar
[17] Achilleos V, Frantzeskakis D J, Kevrekidis P G, Pelinovsky D E 2013 Phys. Rev. Lett. 110 264101
Google Scholar
[18] Achilleos V, Stockhofe J, Frantzeskakis D J, Kevrekidis P G, Schmelcher P 2013 EPL 103 20002
Google Scholar
[19] Lobanov V E, Kartashov Y V, Konotop V V 2014 Phys. Rev. Lett. 112 180403
Google Scholar
[20] Li Y Y, Luo Z H, Liu Y, Chen Z P, Huang C Q, Fu S H, Tan H S, Malomed B A 2017 New. J. Phys. 19 113043
Google Scholar
[21] Kartashov Y V, Konotop V V 2017 Phys. Rev. Lett. 118 190401
Google Scholar
[22] Sakaguchi H, Malomed B A 2018 Phys. Rev. A 97 013607
Google Scholar
[23] Wang C, Gao C, Jian C M, Zhai H 2010 Phys. Rev. Lett. 105 160403
Google Scholar
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Google Scholar
[25] Xu X Q, Han J H 2011 Phys. Rev. Lett. 107 200401
Google Scholar
[26] Radić J, Sedrakyan T A, Spielman I B, Galitski V 2011 Phys. Rev. A 84 063604
Google Scholar
[27] Li J R, Lee J, Huang W, Burchesky S, Shteynas B, Top F Ç, Jamison A O, Ketterle W 2017 Nature 543 91
Google Scholar
[28] Léonard J, Morales A, Zupancic P, Esslinger T, Donner T 2017 Nature 543 87
Google Scholar
[29] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomo C 2002 Science 296 1290
Google Scholar
[30] Nguyen J H V, Luo D, Hulet R G 2017 Science 356 422
Google Scholar
[31] Kengne E, Liu W M, Malomed B A 2021 Phys. Rep. 899 1
Google Scholar
[32] Xu Y, Zhang Y P, Wu B 2013 Phys. Rev. A 87 013614
Google Scholar
[33] Kartashov Y V, Konotop V V, Abdullaev F Kh 2013 Phys. Rev. Lett. 111 060402
Google Scholar
[34] Sakaguchi H, Li B, Malomed B A 2014 Phys. Rev. E 89 032920
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
[40] Liu Y K, Yang S J 2014 EPL 108 30004
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
[45] De S, Campbell D L, Price R M, Putra A, Anderson B M, Spielman I B 2014 Phys. Rev. A 89 033631
Google Scholar
[46] Sakaguchi H, Malomed B A 2019 Symmetry 11 388
Google Scholar
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Google Scholar
[48] Manakov S V 1973 Zh. Eksp. Teor. Fiz. 65 1392
[49] Yang J K 2010 Nonlinear Waves in Integrable and Nonintegrable Systems (Philadelphia: SIAM) pp79–118
[50] Guo B L, Ling L M 2011 Chin. Phys. Lett. 28 110202
Google Scholar
[51] Qin Y H, Zhao L C, Ling L 2019 Phys. Rev. E 100 022212
Google Scholar
[52] Qin Y H, Wu Y, Zhao L C, Yang Z Y 2020 Chin. Phys. B 29 020303
Google Scholar
[53] Tian H Y, Tian B, Yuan Y Q, Zhang C R 2021 Phys. Scr. 96 045213
Google Scholar
[54] Radhakrishnan R, Lakshmanan M 1995 J. Phys. A: Math. Gen. 28 2683
Google Scholar
[55] Lan Y, Zhao L C, Luo X W 2019 Commun. Nonlinear Sci. Numer. Simul. 70 334
Google Scholar
[56] Ohta Y, Wang D S, Yang J K 2011 Stud. Appl. Math. 127 345
Google Scholar
[57] Yan C T 1996 Phys. Lett. A 224 77
Google Scholar
[58] Punya A, Lambrecht W R L 2012 Phys. Rev. B 85 195147
Google Scholar
[59] Sheng X L, Wang Z J, Yu R, Weng H M, Fang Z, Dai X 2014 Phys. Rev. B 90 245308
Google Scholar
[60] Łepkowski S P, Bardyszewski W 2018 Sci. Rep. 8 15403
Google Scholar
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