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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 (a), (b), (e), (f)自旋向上; (c), (d), (g), (f)自旋向下
Figure 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;
Figure 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
Figure 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
Figure 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
Figure 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)
Figure 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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