For the Bose–Einstein condensate in the non-Abelian gauge field, the atomic wavefunction obtains helical modulation, which would lead to nontrivial objects, such as three-dimensional Skyrmion. The spin–orbit coupled condensate is usually capable of supporting Skyrmion ground states, which can be well understood through the wavefunction modulation vector concept. In the present manuscript, the modulation vector is characterized both analytically and numerically with respect to the spin–orbit coupling strength. For the SU(2) symmetric condensate, the modulation vector of the wavefunction is linearly proportional to the spin–orbit coupling strength. In the presence of spin-dependent interaction, the modulus of the modulation vector is predicted to increase or decrease according to the sign of the interaction parameter, resulting in a shrunken or expanded Skyrmion. However, the Skyrmion is unstable for a certain area of the parameter space of the spin–orbit coupling and spin-dependent interaction. The condensate favors a single Skyrmion state or the Skyrmion lattice state according to the strength of the spin–orbit coupling.

中文翻译：

---

自旋轨道耦合凝聚物中三维 Skyrmions 的表征螺旋调制

对于非阿贝尔规范场中的玻色-爱因斯坦凝聚体，原子波函数获得螺旋调制，这将导致非平凡物体，例如三维Skyrmion。自旋轨道耦合凝聚态通常能够支持 Skyrmion 基态，这可以通过波函数调制矢量概念很好地理解。在本手稿中，调制矢量在自旋轨道耦合强度方面进行了分析和数值表征。对于 SU(2) 对称凝聚体，波函数的调制矢量与自旋轨道耦合强度成线性比例。在存在自旋相关相互作用的情况下，调制矢量的模量根据相互作用参数的符号预测增加或减少，导致 Skyrmion 缩小或扩大。然而，对于自旋轨道耦合和自旋相关相互作用的参数空间的某个区域，Skyrmion 是不稳定的。根据自旋轨道耦合的强度，冷凝物有利于单一的 Skyrmion 状态或 Skyrmion 晶格状态。