We studied the dynamical properties of Rabi oscillations driven by an alternating Rashba field applied to a two-dimensional (2D) harmonic confinement system. We solve the time-dependent (TD) Schr\"{o}dinger equation numerically and rewrite the resulting TD wavefunction onto the Bloch sphere (BS) using two BS parameters of the zenith ($\theta_B$) and azimuthal ($\phi_B$) angles, extracting the phase information $\phi_B$ as well as the mixing ratio $\theta_B$ between the two BS-pole states. We employed a two-state rotating wave (TSRW) approach and studied the fundamental features of $\theta_B$ and $\phi_B$ over time. The TSRW approach reveals a triangular wave formation in $\theta_B$. Moreover, at each apex of the triangular wave, the TD wavefunction passes through the BS pole, and the state is completely replaced by the opposite spin state. The TSRW approach also elucidates a linear change in $\phi_B$. The slope of $\phi_B$ vs. time is equal to the difference between the dynamical terms, leading to a confinement potential in the harmonic system. The TSRW approach further demonstrates a jump in the phase difference by $\pi$ when the wavefunction passes through the BS pole. The alternating Rashba field causes multiple successive Rabi transitions in the 2D harmonic system. We then introduce the effective BS (EBS) and transform these complicated transitions into an equivalent "single" Rabi one. Consequently, the EBS parameters $\theta_B^{\mathrm{eff}}$ and $\phi_B^{\mathrm{eff}}$ exhibit mixing and phase difference between two spin states $\alpha$ and $\beta$, leading to a deep understanding of the TD features of multi-Rabi oscillations. Furthermore, the combination of the BS representation with the TSRW approach successfully reveals the dynamical properties of the Rabi oscillation, even beyond the TSRW approximation.

翻译：我们研究了在二维谐振约束系统中施加交变Rashba场所驱动的拉比振荡的动力学特性。我们数值求解了含时薛定谔方程，并利用Bloch球的两个参数——天顶角($\theta_B$)和方位角($\phi_B$)，将得到的含时波函数重写至Bloch球上，从而提取相位信息$\phi_B$以及两个Bloch球极点态之间的混合比例$\theta_B$。我们采用双态旋转波近似方法，研究了$\theta_B$和$\phi_B$随时间变化的基本特征。TSRW方法揭示了$\theta_B$中三角波的形成。此外，在三角波的每个顶点处，含时波函数会穿过Bloch球极点，此时态完全被相反的自旋态所取代。TSRW方法还阐明了$\phi_B$的线性变化。$\phi_B$随时间变化的斜率等于动力学项之间的差值，这源于谐振系统中的约束势。TSRW方法进一步证明了当波函数穿过Bloch球极点时，相位差会发生$\pi$的跃变。交变Rashba场在二维谐振系统中引起了多次连续的拉比跃迁。随后，我们引入了有效Bloch球概念，并将这些复杂的跃迁转化为等效的“单次”拉比跃迁。因此，EBS参数$\theta_B^{\mathrm{eff}}$和$\phi_B^{\mathrm{eff}}$呈现了两种自旋态$\alpha$和$\beta$之间的混合与相位差，从而深化了对多拉比振荡含时特性的理解。此外，将Bloch球表示与TSRW方法相结合，成功地揭示了拉比振荡的动力学性质，其结果甚至超越了TSRW近似本身。