Quantum-classical dynamics of Rashba spin-orbit coupling

Abstract

Mixed quantum-classical models are widely used to reduce the computational cost of fully quantum simulations. However, their general applicability across different classes of problems remains an open question. Here, we address this issue for systems featuring spin-orbit coupling. In particular, we study the interaction dynamics of quantum spin-1/2 and classical orbital momentum in one-dimensional models of Rashba nanowires. We tackle this problem by resorting to a new quantum-classical Hamiltonian model that, unlike conventional approaches, retains the Heisenberg principle and captures correlation effects beyond the common Ehrenfest approach. Based on Koopman wavefunctions in classical mechanics, the new model was recently implemented numerically via a particle scheme -- the koopmon method -- which is extended here to treat spin-orbit coupling. We apply the koopmon method to study the quantum-classical dynamics of nanowire models, with and without the presence of a harmonic potential and in both Rashba-dominated (strong coupling) and Zeeman-dominated (weak coupling) regimes. Considering realistic semiconductor parameters, the results are contrasted with both fully quantum and quantum-classical Ehrenfest dynamics. In the absence of external potential, the koopmon method qualitatively reproduces the features of the fully quantum evolution for all coupling regimes. While it exhibits a slight loss in spin accuracy compared to Ehrenfest simulations, the latter fail to capture the orbital dynamics. In the presence of a harmonic potential, the koopmon scheme reproduces the full quantum results with accuracy levels that are unachievable by the Ehrenfest model in both quantum and classical sectors. We conclude by presenting a test case that exhibits the formation of cat-like states.

Figures (18)

• Figure 1: Sketch of a quantum nanowire with its axis along the $\mathbf{e}_x$ direction. The Rashba magnetic field points along the $\mathbf{e}_y$ axis, while the external magnetic field ${\mathbf{B}}$ (Zeeman effect) is directed along the axis of the nanowire. In the case of a non-ballistic nanowire, a quantum dot is modeled by a harmonic potential with energy level spacing $\hbar\omega/2$.
• Figure 2: Energy surfaces in momentum space for the ballistic test cases. Left: Zeeman-dominated regime (InSb). Right: Rashba-dominated regime (InAs). Red: $\lambda_1$; black: $\lambda_2$.
• Figure 3: Time evolution in the classical sector for the ballistic test case in the Zeeman-dominated regime (InSb). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: $t=0,62.2,155.5$, and $217.7$ ps. Phase space: position $[q]=\mu\text{m}$ (horizontal) and momentum $[p]=\text{eV}/\text{c}$ (vertical). Particle plots with $N=500$ and $\alpha=0.5$ include the smoothed density $D({\mathbf{{\boldsymbol{z}}}},t)$.
• Figure 4: Time evolution of the Bloch vector components and purity (bottom right) for the ballistic, Zeeman-dominated case (InSb).
• Figure 5: Time evolution of the spin-momentum correlation components along $\widehat{\sigma}_x$ (top left), $\widehat{\sigma}_y$ (top right), and $\widehat{\sigma}_z$ (bottom) for the ballistic, Zeeman-dominated case (InSb).