Bloch sphere representation for Rabi oscillation driven by Rashba field in the two-dimensional harmonic confinement

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) Schrödinger equation numerically and rewrite the resulting TD wavefunction onto the Bloch sphere (BS) using two BS parameters of the zenith $θ_B$ and azimuthal $φ_B$ angles, extracting the phase information φ_B as well as the mixing ratio θ_B between the two BS-pole states. We employed a two-state rotating wave (TSRW) approach and studied the fundamental features of θ_B and φ_B over time. The TSRW approach reveals a triangular wave formation in θ_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 φ_B. The slope of φ_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 π 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 θ_B^eff and φ_B^eff exhibit mixing and phase difference between two spin states α and β, 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.