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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 $\left(\delta n(x, t) \equiv n_{\uparrow}(x, t)-n_{\downarrow}(x, t)\right)$ 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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