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

TL;DR

This work examines Rabi oscillations driven by an alternating Rashba field in a two-dimensional harmonic confinement and maps the time-dependent wavefunction onto a Bloch sphere using the angles $\theta_B$ and $\phi_B$, enabling simultaneous access to state mixing and phase evolution. A two-state rotating-wave (TSRW) framework reveals a triangular evolution of $\theta_B$ and a linear progression of $\phi_B$, with $\pi$ phase jumps occurring when the wavefunction crosses Bloch-sphere poles. The presence of multiple, sequential transitions due to the harmonic level spacing is addressed first through a TSRW-based analysis and then via an Effective Bloch Sphere (EBS) that contracts the dynamics to an effective two-state problem between $|\alpha\rangle$ and $|\beta\rangle$; this captures the dominant TD features of multi-level Rabi oscillations and clarifies how higher-state involvement modulates the observed phase and mixing. Overall, the study provides a detailed, platform-specific picture of phase dynamics and mixing in Rashba-driven qubit-like systems and offers a practical framework (BS+TSRW and EBS) for interpreting complex TD spin-orbit driven transitions.

Abstract

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 $θ_B^{\mathrm{eff}}$ and $φ_B^{\mathrm{eff}}$ 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.

Figures (10)

• Figure 1: Potential profile of the hybrid confinement against the harmonic strength $\omega_0$ reproduced from our previous work jjap1jjap2jjap3inui (Copyright 2014, 2016, 2017, The Japan Society of Applied Physics, also 2018, IOP Publishing Ltd); a harmonic potential is surrounded by a cylindrical hard-wall. Solid lines are the energy eigenvalues calculated numerically by the finite difference method and dotted lines indicate those of the ideal harmonic confinement. We also show the resulting electron density profiles of the cylindrical hard-wall ($\omega_0=0$) and the harmonic potential ($\omega_0=15$) in the figure.
• Figure 2: Rashba SOI energetics of the ground (a) and 1st excited state (b) reproduced from our previous work jjap1jjap2jjap3inui (Copyright 2014, 2016, 2017, The Japan Society of Applied Physics, 2018, IOP Publishing Ltd). We show the energy difference from the unperturbed ground $(n, l)=(0, 0)$ and 1st excited $(0, |1|)$ state over varied $\theta$. The energy difference is normalized by $m^\ast\Xi_0^2$. We also show the spin densities $\rho^\alpha$ (left) and $\rho^\beta$ (right) for the Rashba SOI perturbed state at $\theta =0$. Hereafter, we expediently use the unperturbed notation $(n, l, \sigma)$ for the classification of the perturbed eigenstates. In the present InSb QD, the confinement strength of $1\hbar\omega$ a.u. corresponds to 11.9 meV.
• Figure 3: Snapshots of TD spin density profiles of $\alpha$ and $\beta$. An electron is initially confined by the harmonic potential $\omega_0=15$ having the spin $\alpha$, as described in text.
• Figure 4: State probability against time (a) and possible transitions driven by the alternating Rashba field in harmonic confinement. (b) Note that during $\Delta \tau$ as shown in figure (a), the component $\alpha$ is caused mainly by the state $(10\alpha)$ rather than by the state $(00\alpha)$.
• Figure 5: Change in the state probabilities, BS parameters $\theta_{\mathrm B}$ and $\phi_{\mathrm B}$ against time. Those for the TSRW approach are shown in figure (a) whereas those for the primary transition between $(00\alpha)$ and $(01\beta)$ are shown in figure (b). We further show each trajectory of the TD wavefunction during the single cyclic period (yellow region) in (c). Note that the trajectory for the TSRW approach passes exactly through the poles whereas that for the extract model only approaches the pole.