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By numerically solving the static and time-dependent Gross-Pitaevskii equations, we systematically investigate the ground-state properties and collective excitations of a weakly interacting Bose gas with the Raman-type spin-orbit coupling in one dimension. Our analysis focuses on three distinct quantum phases—the stripe phase, plane-wave phase, and zero-momentum phase—characterizing their key static properties, such as condensate momentum, spin polarization, and ground-state energy. Using time-dependent simulations, we explore the dynamics of total-density collective modes, including the dipole mode, which drives harmonic oscillations of the atomic cloud's center of mass, and the breathing mode, responsible for periodic expansion and contraction of the density profile. The modes' frequencies exhibit a non-monotonic dependence on the Rabi frequency across the three phases and are significantly suppressed at the transition point between the plane-wave and the zero-momentum phases. Additionally, we study spin-dependent collective excitations, particularly the spin-dipole and spin-breathing modes, governed by the time-dependent spin density distribution $(\delta n(x, t) \equiv n_\uparrow(x, t)-n_\downarrow(x, t)$) as shown in the following figure. Our results reveal that two spin oscillation modes exist only in the stripe and zero-momentum phases, with frequencies remarkably higher in the latter. Notably, in the stripe phase, mode frequencies decrease monotonically with increasing Rabi frequency, whereas they rise linearly in the zero-momentum phase. The spin-dipole mode induces rigid, out-of-phase oscillations of the two spin components, while the spin-breathing mode modulates the spin density distribution periodically. These findings offer fundamental theoretical insights into the dynamic behavior of spin-orbit-coupled quantum gases, particularly regarding spin-related collective excitations, and provide valuable guidance for future cold-atom experiments.
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图 1 拉曼型自旋-轨道耦合玻色气体三种基态物相 ST、PW 以及 ZM 中 (a)凝聚体动量$ P_x $; (b) 自旋极化率$ \langle\sigma_z\rangle $; (c)能量E, 随拉比频率Ω的变化行为. (a)中插图表示各物相中典型密度分布, 其中蓝色实线、红色虚线分别表示自旋向上和自旋向下组分的密度分布. (c)中插图为帮助判断相变类型的能量关于拉比频率的导数$ dE/d\Omega $. 图中竖划线和竖点线分别表示本文理论框架确定的ST-PW和PW-ZM相变点处拉比频率$ 2.08 E_{\rm{r}} $和$ 3.95 E_{\rm{r}} $
Figure 1. (a) The condensate momentum $ P_x $, (b) the spin polarization $ \langle\sigma_z\rangle $, and (c) the energy E in the ST, PW, and ZM phases, as functions of the Rabi frequency Ω in the Raman-type spin-orbit coupled Bose gases. The insets in (a) denote the typical density distribution in respective phases, where the density distributions of the spin-up and spin-down components are indicated by the blue-solid and red-dashed lines, respectively. The insets in (c) denote the derivative of the energy with respect to the Rabi frequency $ dE/d\Omega $ to determine the type of phase transitions. The vertical dashed and dotted lines indicate the critical Rabi frequencies of the ST-PW and PW-ZM phase transitions, respectively, determined by our numerical calculations.
图 2 条纹相中偶极, 呼吸, 自旋-偶极以及自旋-呼吸四种模式对应观测量的含时演化行为
Figure 2. Time-dependent observables of the dipole, breathing, spin-dipole, and spin-breathing modes in the stripe phase.
图 3 偶极, 呼吸, 自旋-偶极及自旋-呼吸四种集体激发模式频率$ \omega_{d, b, sd, sb} $与拉比频率Ω的关系
Figure 3. Collective excitation frequencies $ \omega_{d, b, sd, sb} $ of the dipole mode, breathing mode, spin-dipole mode, and spin-breathing mode, as functions of the Rabi frequency Ω.
图 4 偶极模式(第一行)和呼吸模式(第二行)中的典型密度动力学特征: (a), (c) $ \Omega=1.0 E_r $处的条纹相; (b), (d) $ \Omega=3.0 E_r $处的平面波相. 其中, 三种颜色线分别表示$ 0, T/4, T/2 $三个时间点的密度分布
Figure 4. Typical density evolution of dipole-mode (top panel) and breathing-mode (bottom panel) oscillations: (a), (c) stripe phase at $ \Omega=1.0 E_r $; (b), (d) plane-wave phase at $ \Omega=3.0 E_r $. Here, three colorful lines indicate the density distributions at $ 0 $, $ T/4 $, and $ T/2 $ of one full period T.
图 5 $ \Omega=1.0 E_r $处条纹相中自旋-偶极振荡动力学. 不同颜色曲线分别表示不同自旋分量的密度分布
Figure 5. Oscillating behavior of the spin-dipole mode in the stripe phase at $ \Omega=1.0 E_r $. Here, the colorful lines indicate the spin-up and spin-down density distributions.
图 6 $ \Omega=1.0 E_r $处条纹相中自旋-呼吸振荡动力学. 不同颜色曲线分别表示不同自旋分量的密度分布
Figure 6. Oscillating behavior of the spin-breathing mode in the ST phase at $ \Omega=1.0 E_r $. Here, the colorful lines indicate the spin-up and spin-down density distributions.
图 7 自旋-偶极模式(第一行)和自旋-呼吸模式(第二行)中自旋密度分布$ \delta n(x, t)\equiv n_\uparrow(x, t)-n_\downarrow(x, t) $的t-x平面等高图: (a), (c) $ \Omega=1.0 E_r $处的条纹相; (b), (d) $ \Omega=5.0 E_r $处的零动量相. 其中, (a)—(d)中色条表示范围分别为: [–15, 15], [–0.1, 0.1], [–10, 15], [–0.14, 0]
Figure 7. Contour plots of time-dependent spin density $ \delta n(x, t)\equiv n_\uparrow(x, t)-n_\downarrow(x, t) $ of spin-dipole mode (top panel) and spin-breathing mode (bottom panel) in the t-x plane: (a), (c) stripe phase at $ \Omega=1.0 E_r $; (b), (d) zero-momentum phase at $ \Omega=5.0 E_r $. Color scales: (a): –15 to 15; (b): –0.1 to 0.1; (c): –10 to 15; (d): –0.14 to 0.
表 1 四种集体激发模式(偶极, 呼吸, 自旋-偶极, 自旋-呼吸)所对应的激发方式以及观测量. 其中, $x_0$和α均为小量以使得系统被微弱地扰动
Table 1. Exciting approaches and observables for four collective excitation modes, including dipole mode, breathing mode, spin-dipole mode, and spin-breathing mode. Here, $x_0$ and α are both small enough to perturb the systems slightly.
集体激发模式 $ V_1(x) $ 物理观测量 偶极模式 $ \dfrac{1}{2} m\omega_x^{2} (x-x_0)^2 $ $ \left\langle x\right\rangle $ 呼吸模式 $ \dfrac{1}{2} m(1+\alpha )^2\omega_x^{2} x^2 $ $ \left\langle x^2\right\rangle $ 自旋-偶极模式 $ \dfrac{1}{2} m\omega_x^{2} (x-\sigma_{z}x_0)^2 $ $ \left\langle\sigma_{z}x \right\rangle $ 自旋-呼吸模式 $ \dfrac{1}{2} m(1+\sigma_{z} \alpha )^2\omega_x^{2} x^2 $ $ \left\langle\sigma_{z}x^2\right\rangle $ 
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