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采用虚时演化法求解拉曼光晶格的平均场Gross-Pitaevskii方程, 基于其基态波函数, 研究该模型中超冷原子的拓扑性质. 研究发现光晶格深度和原子间相互作用强度的竞争导致了丰富的基态结构相图. 当只有标量光晶格存在时, 超冷原子在实空间没有涡旋产生, 在动量空间表现为拓扑平庸的密度峰; 当只有拉曼光晶格存在时, 超冷原子在实空间出现大小相同的涡旋; 当标量光晶格和拉曼光晶格共同作用时, 超冷原子在实空间出现大涡旋和小涡旋, 正反涡旋交错排列, 在动量空间出现具有拓扑非平庸相位的衍射峰, 在自旋表象中产生半量子化的斯格明子晶格.
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关键词:
- 拉曼光晶格 /
- 超冷原子 /
- 平均场Gross-Pitaevskii方程 /
- 虚时演化法
The ground-state topological properties of ultracold atoms in composite scalar-Raman optical lattices are systematically investigated by solving the two-component Gross-Pitaevskii equation through the imaginary time evolution method. Our study focuses on the interplay between scalar and Raman optical lattice potentials and the role of interatomic interactions in shaping real-space and momentum-space structures. The competition between lattice depth and interaction strength gives rise to a rich phase diagram of ground-state configurations. In the absence of Raman coupling, atoms in scalar optical lattices exhibit topologically trivial periodic density distributions without forming vortices. When only Raman coupling exists, a regular array of vortices of equal size will appear in one spin component, while the other spin component will remain free of vortices. Strikingly, when scalar and Raman lattices coexist, the system develops complex vortex lattices with alternating large and small vortices of opposite circulation, forming a staggered vortex configuration in real space. In momentum space, the condensate wave function displays nontrivial diffraction peaks carrying a well-defined topological phase structure, whose complexity increases with the depth of the optical potentials increasing. In spin space, we observe the emergence of a lattice of half-quantized skyrmions (half-skyrmions), each carrying a topological charge of ±1/2. These topological textures are confirmed by calculating the spin vector field and integrating the topological charge density. Our results demonstrate how the combination of scalar and Raman optical lattices, together with tunable interactions, can induce nontrivial real-space spin textures and momentum-space topological features. These findings offers new insights into the controllable realization of topological quantum states in cold atom systems.
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Keywords:
- Raman optical lattice /
- ultracold atom /
- mean field Gross-Pitaevskii equation /
- imaginary time evolution method
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图 1 当只有(a)—(d)标量势(V0 = 7, M0 = 0)或(e)—(h)拉曼势(V0 = 7, M0 = 5)时, 实空间((a), (c), (e), (g))密度分布图和((b), (d), (f), (h))相位分布图, 其中(a), (b), (e), (f)自旋向上; (c), (d), (g), (f)自旋向下
Fig. 1. Distribution of density ((a), (c), (e), (g)) and phase diagram ((b), (d), (f), (h)) in real space when there is only (a)–(d) scalar potential (V0 = 7, M0 = 0) or (e)–(h) Raman potential (V0 = 7, M0 = 5): (a), (b), (e), (f) Spin-up; (c), (d), (g), (f) spin-down.
图 2 当标量-拉曼势并存(V0 = 7, M0 = 5, g22 = 400)时, 实空间密度分布图(左)和相位分布图(右) (a), (b)自旋向上; (c), (d)自旋向下
Fig. 2. Distribution of density (left) and phase diagram (right) in real space when scalar-Raman potential coexists (V0 = 7, M0 = 5, g22 = 400): (a), (b) Spin-up; (c), (d) spin-down.
图 3 (a)—(d) V0 = 7, M0 = 5以及(e)—(h) V0 = 15, M0 = 9情况下动量空间波函数分布图 (a), (e)自旋向上的波函数实部分布图; (b), (f)自旋向上的波函数虚部分布图; (c), (g)自旋向下的波函数实部分布图; (d), (h)自旋向下的波函数虚部分布图
Fig. 3. Distribution diagram of momentum space wave function when (a)–(d) V0 = 7, M0 = 5 and (e)–(h) V0 = 15, M0 = 9: (a), (e) Spin-up distribution of the real part of the wave function; (b), (f) spin-up imaginary distribution of wave functions; (c), (g) spin-down distribution of the real part of the wave function; (d), (h) spin-down imaginary distribution of wave functions.
图 4 标量-拉曼势并存(V0 = 7, M0 = 5, g22 = 600)时, 实空间密度分布图(左)和相位分布图(右) (a), (b)自旋向上; (c), (d)自旋向下
Fig. 4. Distribution of density (left) and phase diagram (right) in real space when scalar-Raman potential coexists (V0 = 7, M0 = 5, g22 = 600): (a), (b) Spin-up; (c), (d) spin-down.
图 5 (a)自旋结构分布图; (b)拓扑结构分布图
Fig. 5. (a) Spin structure distribution diagram; (b) topological structure distribution map.
图 6 (a)实空间密度分布图和(b)能量随时间演化图(其中V0 = 7, M0 = 5, g22 = 400)
Fig. 6. Distribution of density (a) and energy evolution diagram over time (b) in real space (V0 = 7, M0 = 5, g22 = 400).
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[1] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198
Google Scholar
[2] Bradley C C, Sackett C A, Tollett J J, Hulet R G 1995 Phys. Rev. Lett. 75 1687
Google Scholar
[3] Gaubatz U, Rudecki P, Schiemann S, Bergmann K 1990 J. Chem. Phys. 92 5363
Google Scholar
[4] Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 83
Google Scholar
[5] Aidelsburger M 2015 Artificial Gauge Fields with Ultracold Atoms in Optical Lattices (New York: Springer) pp27–47
[6] Struck J, Simonet J, Sengstock K 2014 Phys. Rev. A 90 031601
Google Scholar
[7] Xu Z F, You L, Ueda M 2013 Phys. Rev. A 87 063634
Google Scholar
[8] Anderson B M, Spielman I B, Juzeliūnas G 2013 Phys. Rev. Lett. 111 125301
Google Scholar
[9] Wall M L, Koller A P, Li S, Zhang X, Cooper N R, Rey A M 2016 Phys. Rev. Lett. 116 035301
Google Scholar
[10] Liu X J, Borunda M F, Liu X, Sinova J 2009 Phys. Rev. Lett. 102 046402
Google Scholar
[11] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y J, Chen S, Liu X J, Pan J W 2016 Science 354 83
Google Scholar
[12] Wang Z Y, Cheng X C, Wang B Z, Zhang J Y, Lu Y H, Yi C R, Niu S, Deng Y J, Liu X J, Chen S, Pan J W 2021 Science 372 271
Google Scholar
[13] Liu S, Hua W, Zhang D J 2020 Rep. Math. Phys. 86 271
Google Scholar
[14] Han W, Zhang S Y, Jin J J, Liu W M 2012 Phys. Rev. A 85 043626
Google Scholar
[15] Han W, Juzeliūnas G, Zhang W, Liu W M 2015 Phys. Rev. A 91 013607
Google Scholar
[16] 郭慧, 王雅君, 王林雪, 张晓斐 2008 69 010302
Google Scholar
Guo H, Wang Y J, Wang L X, Zhang X F 2008 Acta Phys. Sin. 69 010302
Google Scholar
[17] Kasamatsu K, Tsubota M, Ueda M 2005 Phys. Rev. A 71 043611
Google Scholar
[18] Wu C J, Mondragon-Shem I, Zhou X F 2011 Chin. Phys. Lett. 28 097102
Google Scholar
[19] Xu Z F, Lü R, You L 2011 Phys. Rev. A 83 053602
Google Scholar
[20] Sinha S, Nath R, Santos L 2011 Phys. Rev. Lett. 107 270401
Google Scholar
[21] Hu H, Ramachandhran B, Pu H, Liu X J 2012 Phys. Rev. Lett. 108 010402
Google Scholar
[22] Yang S J, Wu Q S, Zhang S N, Feng S 2008 Phys. Rev. A 77 033621
Google Scholar
[23] Heinze S, Von Bergmann K, Menzel M, Brede J, Kubetzka A, Wiesendanger R, Blügel S 2011 Nat. Phys. 7 713
Google Scholar
[24] Leslie L S, Hansen A, Wright K C, Deutsch B M, Bigelow N P 2009 Phys. Rev. Lett. 103 250401
Google Scholar
[25] 丁贝, 王文洪 2018 物理 47 15
Google Scholar
Ding B, Wang W H 2018 Physics 47 15
Google Scholar
[26] Nagaosa N, Tokura Y 2013 Nat. Nanotechnol. 8 899
Google Scholar
[27] Fert A, Reyren N, Cros V 2017 Nat. Rev. Mater. 2 17031
Google Scholar
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