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In a recent experiment [Hamner C, et al. 2015 Phys. Rev. Lett. 114 070401 ], spin-orbit coupled Bose-Einstein condensates in a translating optical lattice have been successfully prepared into any Bloch band, and directly proved to be the lack of Galilean invariance in the presence of the spin-orbit coupling. The energy band structure of the system becomes complicated because of the lack of Galilean invariance. At present, the energy band structure of the spin-orbit coupled Bose-Einstein condensates in optical lattice is still an open issue, especially the theoretical evidence for the in-depth understanding of the competition mechanism among the spin-orbit coupling, the Raman coupling, the optical lattice and the atomic interactions of the nonlinear energy band structure has not been clear yet.In this paper, based on the two-mode approximation and variational analysis, the nonlinear energy band structure and current density of the spin-orbit coupled Bose-Einstein condensates in the one-dimensional optical lattice are investigated. We find that when the spin-orbit coupling, the Raman coupling, the optical lattice, and the atomic interactions satisfy certain conditions, a loop structure in the Brillouin zone edge will emerge. The critical condition for the loop structure emerging in the Brillouin zone edge is obtained in a parameter space. The Raman coupling and the optical lattice suppress the emergence of the loop structure, while the spin-orbit coupling and the atomic interactions promote the emerging of the loop structure and make the energy band structure more complex. Interestingly, the atomic interactions can make the loop structure occur at both the higher-lying bands and the lowest energy band. The energy band structure is closely related to the current density of the system. The spin-orbit coupling causes the current density to be strongly asymmetric and leads the current density distributions of different spin states to be separated from each other in the momentum space near the boundary of the Brillouin zone. The optical lattice strength and the Raman coupling can weaken the asymmetry. The appearance of loop structure breaks the Bloch oscillation and gives rise to the Landau-Zener tunneling. The separation of the current density distributions of different spin states in the momentum space means the emergence of the spin exchange dynamics. Our results are beneficial to the in-depth understanding of the nonlinear dynamics of the spin-orbit coupled Bose-Einstein condensates in optical lattice. -
Keywords:
- Bose-Einstein condensates /
- optical lattice /
- spin-orbit coupling /
- nonlinear energy band structure
[1] Dai H N, Yang B, Reingruber A, Xu X F, Jiang X, Chen Y A, Yuan Z S, Pan J W 2016 Nat. Phys. 12 783
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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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[13] Zobay O, Garraway B M 1999 Phys. Rev. A 61 033603
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
[22] Chen Yan, Wang H T, Chen Y 2010 J. Phys. B 43 225303
Google Scholar
[23] Seaman B T, Carr L D, Holland M J 2005 Phys. Rev. A 71 033622
Google Scholar
[24] Xie Z W, Liu W M 2004 Phys. Rev. A 70 045602
Google Scholar
[25] Gong B, Li S, Zhang X H, Liu B, Yi W 2019 Phys. Rev. A 99 012703
Google Scholar
[26] Li S, Wang H, Li F, Cui X, Liu B 2020 Phys. Rev. A 102 033328
Google Scholar
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Google Scholar
[28] Mateo A M, Delgado V, Guilleumas M, Mayol R, Brand J 2019 Phys. Rev. A 99 023630
Google Scholar
[29] Kühn S, Judd T E 2013 Phys. Rev. A 87 023608
Google Scholar
[30] Watanabe G, Yoon S, Dalfovo F 2011 Phys. Rev. Lett. 107 270404
Google Scholar
[31] Rosenkranz M, Jaksch D 2008 Phys. Rev. A 77 063607
Google Scholar
[32] Witthaut D, Trimborn F, Kegel V, Korsch H J 2011 Phys. Rev. A 83 013609
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Google Scholar
[37] Hamner C, Zhang Y, Khamehchi M A, Davis M J, Engels P 2015 Phys. Rev. Lett. 114 070401
Google Scholar
[38] Chen Z, Liang Z 2016 Phys. Rev. A 93 013601
Google Scholar
[39] Martone G I 2017 J. Low Temp. Phys. 189 262
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Google Scholar
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Google Scholar
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图 1 不同系统参数下的非线性能谱结构. 每副子图中不同的颜色代表不同的光晶格强度,
$ V_{0} = 0.05 $ (红色),$ 0.2 $ (绿色),$ 0.4 $ (蓝色). 能量最小值处用不同颜色的小球表示. 其他参数:$g = 0.2, \;g_{12} = 0.1$ Figure 1. Nonlinear energy band structure for different system parameter. Different colored curves in every subplots correspond to different optical lattice strength,
$ V_{0} = 0.05 $ (red),$ 0.2 $ (green),$ 0.4 $ (blue). The energy minima is indicated by different colored ball. The other parameters are$ g = 0.2 $ and$ g_{12} = 0.1 $ .
图 2 不同拉曼耦合
$\varOmega$ 下的极化图. 其他参数:$g = 0.2, $ $ g_{12} = 0.1, V_{0} = 0.4$ Figure 2. Spin polarization
$ s $ as a function of Raman coupling$ \varOmega $ for different$ k_{0} $ . The other parameters are$ g = 0.2 $ ,$ g_{12} = 0.1 $ and$ V_{0} = 0.4 $ .
图 3 不同种间原子间相互作用
$ g_{12} $ 下的非线性能谱结构. 其他参数:$\varOmega = 0.1$ ,$ k_{0} = 0.2 $ ,$ V_{0} = 0.1 $ ,$ g = 0.1 $ Figure 3. Nonlinear energy band structure for different interspecies interaction
$ g_ {12} $ . The other parameters are$ \varOmega = 1.0 $ ,$ k_{0} = 0.4 $ ,$ g = 0.2 $ and$ g_{12} = 0.1 $ .
图 4
$ g = 0.2 $ 时最低能带出现loop 的临界$ \varOmega $ , (b) 中不同形状的符号表示相应光晶格强度下(9)式给出的理论值Figure 4. Critical condition for appearing the loop structure in the lowest energy band at
$ g = 0.2 $ . The different symbols in panel (b) represent the theoretical values given by Eq. (9) under the corresponding optical lattice strength.
图 5 不同自旋轨道耦合
$ k_{0} $ 和光晶格强度$ V_{0} $ 下的凝聚体流密度. 其他参数:$ \varOmega = 0.15 $ ,$ g = 0.2 $ ,$ g_{12} = 0.1 $ . 图中红色和黑色的线表示不同自旋态的凝聚体流密度Figure 5. Current density for different spin-orbit coupling
$ k_{0} $ and optical lattice strength$ V_{0} $ . The other parameters are$ \varOmega = 0.15 $ ,$ g = 0.2 $ and$ g_{12} = 0.1 $ . The red and black lines represent the current density of different spin states.
图 6 不同拉曼耦合
$ \varOmega $ 下的能带和相应的凝聚体流密度. 其他参数:$ V_{0} = 0.05 $ ,$ g = 0.2 $ ,$ g_{12} = 0.1 $ . 图中红色和黑色的线表示不同自旋态的流密度Figure 6. Energy band and current density for different Raman coupling
$ \varOmega $ . The other parameters are$ V_{0} = 0.05 $ ,$ g = 0.2 $ and$ g_{12} = 0.1 $ . The red and black lines represent the current density of different spin states. -
[1] Dai H N, Yang B, Reingruber A, Xu X F, Jiang X, Chen Y A, Yuan Z S, Pan J W 2016 Nat. Phys. 12 783
Google Scholar
[2] Dai H N, Yang B, Reingruber A, Sun H, Xu X F, Chen Y A, Yuan Z S, Pan J W 2017 Nat. Phys. 13 1195
Google Scholar
[3] Yang B, Sun H, Huang C J, Wang H Y, Deng Y J, Dai H N, Yuan Z S, Pan J W 2020 Science 369 550
Google Scholar
[4] Yang B, Sun H, Ott R, Wang H Y, Zache T V, Halimeh J C, Yuan Z S, Hauke P, Pan J Wei 2020 Nature 587 392
Google Scholar
[5] Diakonov D, Jensen L M, Pethick C J, Smith H 2002 Phys. Rev. A 66 013604
[6] Wu B, Diener R B, Niu Q 2002 Phys. Rev. A 65 025601
Google Scholar
[7] Dahan M B, Peik E, Reichel J, Castin Y, Salomon C 1996 Phys. Rev. Lett. 76 4508
Google Scholar
[8] Ji W, Zhang K, Zhang W, Zhou L 2019 Phys. Rev. A 99 023604
Google Scholar
[9] Choi D I, Wu B 2003 Phys. Lett. A 318 558
Google Scholar
[10] Ciampini D, Anderlini M, Müller J H, Fuso F, Morsch O, Thomsen J W, Arimondo E 2002 Phys. Rev. A 66 043409
Google Scholar
[11] Wu B, Niu Q 2000 Phys. Rev. A 61 023402
Google Scholar
[12] Konotop V V, Kevrekidis P G, Salerno M 2005 Phys. Rev. A 72 023611
Google Scholar
[13] Zobay O, Garraway B M 1999 Phys. Rev. A 61 033603
Google Scholar
[14] Jona-Lasinio M, Morsch O, Cristiani M, Malossi N, Müller J H, Courtade E, Anderlini M, Arimondo E 2003 Phys. Rev. Lett. 91 230406
Google Scholar
[15] Zhang X, Xu X, Zheng Y, Chen Z, Liu B, Huang C, Malomed B A, Li Y 2019 Phys. Rev. Lett. 123 133901
Google Scholar
[16] Zeng L, Zeng J 2019 Adv. Photonics 1 046004
Google Scholar
[17] Niu L, Jin S, Chen X, Li X, Zhou X 2018 Phys. Rev. Lett. 121 265301
Google Scholar
[18] Stöeferle T, Moritz H, Schori C, Köhl M, Esslinger T 2004 Phys. Rev. Lett. 92 130403
Google Scholar
[19] Machholm M, Pethick C J, Smith H 2003 Phys. Rev. A 67 053613
Google Scholar
[20] Lin Y Y, Lee R K, Kao Y M, Jiang T F 2008 Phys. Rev. A 78 023629
Google Scholar
[21] Hui H Y, Barnett R, Porto J V, Sarma S D 2012 Phys. Rev. A 86 063636
Google Scholar
[22] Chen Yan, Wang H T, Chen Y 2010 J. Phys. B 43 225303
Google Scholar
[23] Seaman B T, Carr L D, Holland M J 2005 Phys. Rev. A 71 033622
Google Scholar
[24] Xie Z W, Liu W M 2004 Phys. Rev. A 70 045602
Google Scholar
[25] Gong B, Li S, Zhang X H, Liu B, Yi W 2019 Phys. Rev. A 99 012703
Google Scholar
[26] Li S, Wang H, Li F, Cui X, Liu B 2020 Phys. Rev. A 102 033328
Google Scholar
[27] Koller S B, Goldschmidt E A, Brown R C, Wyllie R, Wilson R M, Porto J V 2016 Phys. Rev. A 94 063634
Google Scholar
[28] Mateo A M, Delgado V, Guilleumas M, Mayol R, Brand J 2019 Phys. Rev. A 99 023630
Google Scholar
[29] Kühn S, Judd T E 2013 Phys. Rev. A 87 023608
Google Scholar
[30] Watanabe G, Yoon S, Dalfovo F 2011 Phys. Rev. Lett. 107 270404
Google Scholar
[31] Rosenkranz M, Jaksch D 2008 Phys. Rev. A 77 063607
Google Scholar
[32] Witthaut D, Trimborn F, Kegel V, Korsch H J 2011 Phys. Rev. A 83 013609
Google Scholar
[33] Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 83
Google Scholar
[34] Huang L H, Meng Z M, Wang P J, Peng P, Zhang S L, Chen L C, Li D H, Zhou Q, Zhang J 2016 Nat. Phys. 12 540
Google Scholar
[35] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y, Chen S, Liu X J, Pan J W 2016 Science 354 83
Google Scholar
[36] Wang Z Y, Cheng X C, Wang B Z, Zhang J Y, Lu Y H, Yi C R, Niu S, Deng Y, Liu X J, Chen S, Pan J W 2021 Science 372 271
Google Scholar
[37] Hamner C, Zhang Y, Khamehchi M A, Davis M J, Engels P 2015 Phys. Rev. Lett. 114 070401
Google Scholar
[38] Chen Z, Liang Z 2016 Phys. Rev. A 93 013601
Google Scholar
[39] Martone G I 2017 J. Low Temp. Phys. 189 262
Google Scholar
[40] Martone G I, Ozawa T, Qu C, Stringari S 2016 Phys. Rev. A 94 043629
Google Scholar
[41] Zhang Y, Zhang C 2013 Phys. Rev. A 87 023611
Google Scholar
[42] Zhang Y, Gui Z, Chen Y 2019 Phys. Rev. A 99 023616
Google Scholar
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