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The spinor Bose-Einstein condensate (BEC) provides an ideal platform for observing and manipulating topological structures, which arise from the spin degrees of freedom and the superfluid nature of the gas. Artificial helicoidal spin-orbit coupling (SOC) in the spinor BEC, owing to the spatially varying gauge potential and the more flexible adjustability, provides possibly an unprecedented opportunity to search for novel quantum states. The previous studies of the BEC with helicoidal SOC mainly focus on the two-component case. However, there are few reports on the studies of helicoidal SOC in three-component BEC. Especially considering one-dimensional three-component BEC, whether the helicoidal SOC can generate previously unknown types of topological excitations and phase diagrams is still an unsolved problem. In this work, by solving quasi one-dimensional Gross-Pitaevskii equations, we study the ground state structure of one-dimensional helicoidal spin-orbit coupled three-component BEC. The numerical results show that the helicoidal SOC can induce a phase separation among the components in ferromagnetic BEC. Through numerical calculations of the system, a phase diagram is obtained as a function of the helicoidal SOC strength and gauge potential, which shows the critical conditions for phase separation and phase miscibility in ferromagnetic BEC. Meanwhile, we also study the influences of the helicoidal SOC and the gauge potential on the antiferromagnetic BEC ground state. The numerical results show that the helicoidal SOC is beneficial for the miscibility in antiferromagnetic BEC. When the helicoidal SOC strength or gauge potential increases, the ground state of antiferromagnetic BEC exhibits a stripe soliton structure. Adjusting the strength of helicoidal SOC or gauge potential can control the transitions between a plane-wave soliton and a stripe soliton. In addition, we show the changes of the particle number density maximum and the number of peaks of stripe solitons for adjusting the helicoidal SOC strength or gauge potential. Our results show that helicoidal spin-orbit coupled BEC not only provides a controlled platform for investigating the exotic topological structures, but also is crucial for the transitions between different ground states. This work paves the way for exploring the topological defect and the corresponding dynamical stability in quantum systems subjected to the helicoidal SOC in future.
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Keywords:
- three-component Bose-Einstein condensate /
- helicoidal spin-orbit coupling /
- phase separation /
- phase miscibility
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图 1 不同螺旋自旋-轨道耦合强度下铁磁BEC基态, 3个自旋分量的粒子数密度$ {\rho _1} $, $ {\rho _0} $和$ {\rho _{ - 1}} $分别用黑线、红线和蓝线表示 (a) $ \alpha = 0.15 $; (b) $ \alpha = 0.8 $; 其他模拟参数$ \beta = 0.4 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $
Figure 1. Ground state of the ferromagnetic BEC for different helicoidal spin-orbit coupling strengths, the particle number densities $ {\rho _1} $, $ {\rho _0} $ and $ {\rho _{ - 1}} $ for the three spin components are depicted by black, red, and blue lines, respectively: (a) $ \alpha = 0.15 $; (b) $ \alpha = 0.8 $; the rest of parameters are $ \beta = 0.4 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $.
图 2 不同螺旋规范势下铁磁BEC基态, 3个自旋分量的粒子数密度$ {\rho _1} $, $ {\rho _0} $和$ {\rho _{ - 1}} $分别用黑线、红线和蓝线表示 (a) $ \beta = 0.2 $; (b) $ \beta = 0.8 $; 其余模拟参数$ \alpha = 1 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $
Figure 2. Ground state of the ferromagnetic BEC for different helicoidal gauge potentials, the particle number densities $ {\rho _1} $, $ {\rho _0} $and $ {\rho _{ - 1}} $ for the three spin components are depicted by black, red, and blue lines, respectively: (a) $ \beta = 0.2 $; (b) $ \beta = 0.8 $; the rest of parameters are $ \alpha = 1 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $.
图 3 螺旋自旋-轨道耦合铁磁BEC基态相图, 包括相混合和相分离, 其余模拟参数选为$ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $
Figure 3. Phase diagram of the ground state of the ferromagnetic BEC with the helicoidal spin-orbit coupling, the phase diagram includes phase miscibility and phase separation, the other parameters are $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $.
图 4 (a) 当螺旋自旋-轨道耦合强度$ \alpha $确定时, 粒子数密度$ {\rho _0} $最大值随螺旋规范势$ \beta $的变化; (b) 当螺旋规范势$ \beta $确定时, 粒子数密度$ {\rho _0} $最大值随螺旋自旋-轨道耦合强度$ \alpha $的变化; 其余模拟参数$ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $
Figure 4. (a) The particle number density $ {\rho _0} $maximum as a function of helicoidal gauge potential for a given helicoidal spin-orbit coupling strength; (b) the particle number density $ {\rho _0} $ maximum as a function of helicoidal spin-orbit coupling strength for a given helicoidal gauge potential parameter; the other parameters are $ {\lambda _0} = 600 $, $ {\lambda _2} = - 2.76 $.
图 5 不同螺旋自旋-轨道耦合强度下反铁磁BEC基态, 3个自旋分量的粒子数密度$ {\rho _1} $, $ {\rho _0} $和$ {\rho _{ - 1}} $分别用红线、蓝线和黑线表示 (a) $ \alpha = 0.2 $; (b) $ \alpha = 1.2 $; (c) $ \alpha = 2 $; (d) $ \alpha = 4 $; 其余模拟参数$ \beta = 3 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $
Figure 5. Ground state of the antiferromagnetic BEC for different helicoidal spin-orbit coupling strengths, the particle number densities $ {\rho _1} $, $ {\rho _0} $and $ {\rho _{ - 1}} $for the three spin components are depicted by red, blue, and black lines, respectively: (a) $ \alpha = 0.2 $; (b) $ \alpha = 1.2 $; (c) $ \alpha = 2 $; (d) $ \alpha = 4 $; and the other parameters are $ \beta = 3 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $.
图 6 (a) 当螺旋规范势$ \beta $确定时, 粒子数密度$ {\rho _0} $最大值随螺旋自旋-轨道耦合强度$ \alpha $的变化; (b) 当螺旋规范势$ \beta $确定时, 粒子数密度$ {\rho _0} $波峰数目随螺旋自旋-轨道耦合强度$ \alpha $的变化; 其余模拟参数$ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $
Figure 6. (a) The particle number density $ {\rho _0} $maximum as a function of helicoidal spin-orbit coupling strength for a given helicoidal gauge potential; (b) numbers of wave peaks of the particle number density $ {\rho _0} $ as a function of helicoidal spin-orbit coupling strength for a given helicoidal gauge potential; the other parameters are $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $.
图 7 不同螺旋规范势下反铁磁BEC基态, 3个自旋分量的粒子数密度$ {\rho _1} $, $ {\rho _0} $和$ {\rho _{ - 1}} $分别用红线、蓝线和黑线表示 (a) $ \beta = 0.4 $; (b) $ \beta = 1.2 $; (c) $ \beta = 2 $; (d) $ \beta = 4 $; 其余模拟参数选为$ \alpha = 2 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $
Figure 7. Ground state of the antiferromagnetic BEC for different helicoidal gauge potentials, the particle number densities $ {\rho _1} $, $ {\rho _0} $and $ {\rho _{ - 1}} $ for the three spin components are depicted by red, blue, and black lines, respectively: (a) $ \beta = 0.4 $; (b) $ \beta = 1.2 $; (c) $ \beta = 2 $; (d) $ \beta = 4 $; and the other parameters are $ \alpha = 2 $, $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $.
图 8 (a) 当螺旋自旋-轨道耦合强度$ \alpha $确定时, 粒子数密度$ {\rho _0} $最大值随螺旋规范势$ \beta $的变化; (b) 当螺旋自旋-轨道耦合强度$ \alpha $确定时, 粒子数密度$ {\rho _0} $波峰数目随螺旋自旋-轨道耦合强度$ \beta $的变化; 其余模拟参数选为$ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $
Figure 8. (a) The particle number density $ {\rho _0} $maximum as a function of helicoidal gauge potential for a given helicoidal spin-orbit coupling strength; (b) numbers of wave peaks of the particle number density $ {\rho _0} $as a function of helicoidal gauge potential for a given helicoidal spin-orbit coupling strength; the other parameters are $ {\lambda _0} = 600 $, $ {\lambda _2} = 24 $.
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