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本文以旋转自旋-轨道角动量耦合玻色-爱因斯坦凝聚体为对象, 通过数值求解单粒子定态Schrödinger方程和具有平均场相互作用的Gross-Pitaevskii方程, 研究了自旋-轨道角动量耦合与旋转对单粒子基态性质和平均场相互作用基态性质的影响, 发现旋转将导致角动量空间中单粒子能谱的二重简并消失, 单粒子能谱不再是关于零点角动量左右对称, 单粒子基态和平均场基态为占据单个角动量的涡旋态, 其角动量大小与旋转频率、激光强度和自旋-轨道角动量耦合有关. 当旋转频率小于临界值时, 基态涡旋的角动量大小不受旋转频率的影响. 当旋转频率超过临界值时, 基态涡旋的角动量大小将随着旋转频率的增加而增加.
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关键词:
- 自旋-轨道角动量耦合 /
- 玻色-爱因斯坦凝聚体 /
- Gross-Pitaevskii方程
By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, we have investigated the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state with a single angular momentum. The angular momentum of the vortex depends on the rotating frequency, the strength of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. While the rotating frequency surpasses the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of the atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.-
Keywords:
- spin-orbital-angular-momentum-coupling /
- Bose-Einstein condensate /
- Gross-Pitaevskii equation
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图 1 $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $ (a)和$ \varOmega_{\text{R}}/\hbar\omega_{\bot}=70 $ (b)时能量最低的两条能级随m的变化, 插图分别为基态的密度和相位分布. 其他参数取值为$ l=1 $, $ \omega_{\bot}=2\pi\times 10 $ Hz, $ \omega_{z}=2\pi\times 200 $ Hz, $ W=25 \text{ μm} $
Fig. 1. Variations of two single-particle lowest energy levels with m for $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $ (a) and $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=70 $ (b), respectively. Other parameters are $ l=1 $, $ \omega_{\bot}=2\pi\times 10 $ Hz, $ \omega_{z}=2\pi\times 200 $ Hz, and $ W=25\text{ μm} $
图 2 (a), (b) 当$ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $时, 最低能级在不同$\varpi $取值下随m的变化和基态角动量$ m_g $随$\varpi $的变化. (c), (d) 当$ \varOmega_{\text{R}}/\hbar\omega_{\bot}=70 $时, 最低能级在不同$\varpi $取值下随m的变化和基态角动量$ m_g $随$\varpi $的变化. (b)和(d)中的插图是$ \varpi=0.5 $和$ \varpi=0.7 $时基态密度和相位分布. (e)是以$\varpi $和$ \varOmega_\text{R}/\hbar\omega_{\bot} $为轴的二维参数空间中基态角动量的变化.其他参数取值分别为$ l=1 $, $ \omega_{\bot}=2\pi\times 10 $ Hz, $ \omega_z=2\pi\times 200 $ Hz, $ W=25{\text{ μm}} $
Fig. 2. When $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $, (a) is the variation of the single-particle lowest energy level with m for different $\varpi $, (b) shows the variation of the angular momentum $ m_g $ of the single-particle ground state with $\varpi $. When $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=70 $, (c) is the variation of the single-particle lowest energy level with m for different $\varpi $, (d) shows the variation of the angular momentum $ m_g $ of the single-particle ground state with $\varpi $. The insets in (b) and (d) are the density and phase of the single-particle ground state for $ \varpi=0.5 $ and $ \varpi=0.7 $, respectively. (e) is the variation of the angular momentum of the single-particle ground state with $\varpi $ and $ \varOmega_{\text{R}}/\hbar\omega_{\bot} $. Other parameters are $ l=1 $, $ \omega_{\bot}=2\pi\times 10 $ Hz, $ \omega_z=2\pi\times 200 $ Hz, and $ W=25{\text{ μm}} $.
图 3 (a) $ \varpi=0 $和$ l=2 $时, $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=100,\; 140,\; 160 $时的基态密度分布. (b), (c) $ l=1 $时, $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $和70的基态角动量$ \left\langle \hat{L}_z\right\rangle $随旋转频率的变化, 插图为$ \varpi=0.5 $和$ \varpi=0.7 $时的基态密度和相位分布. 其他参数为$ a_{\uparrow\uparrow}= a_{\downarrow\downarrow}=100 a_{\text{B}} $, $ a_{\uparrow\downarrow}=50 a_{\text{B}} $, $ \omega_{\bot}=2\pi\times10 $ Hz, $ \omega_{z}=2\pi\times 200 $ Hz, $ N=1000 $和$ W=25{\text{ μm}} $, 其中$ a_{\text{B}} $为玻尔半径
Fig. 3. (a) Densities of the mean-field ground states for $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=100,\; 140,\; 160 $ when $ \varpi=0 $ and $ l=2 $. (b), (c) For $ l=1 $, the variation of the angular momentum $ \left\langle \hat{L}_z\right\rangle $ of the mean-field ground state with the rotating frequency $\varpi $ for $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=30 $ and $ \varOmega_{\text{R}}/\hbar\omega_{\bot}=70 $, respectively. The insets in (b) and (c) are the density and phase of the mean-field ground state for $ \varpi=0.5 $ and $ \varpi=0.7 $, respectively. Other parameters are $ a_{\uparrow\uparrow}=a_{\downarrow\downarrow}=100 a_{\text{B}} $, $ a_{\uparrow\downarrow}=50 a_{\text{B}} $, $ \omega_{\bot}=2\pi\times10 $ Hz, $ \omega_{z}=2\pi\times 200 $ Hz, $ N=1000 $, and $ W=25{\text{ μm}} $, where $ a_{\text{B}} $ is the Bohr radius.
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