We derive semiclassical equations of motion for an accelerated wave packet in a two-band system. We show that these equations can be formulated in terms of the static band geometry described by the quantum metric. We consider the specific cases of the Rashba Hamiltonian with and without a Zeeman term. The semiclassical trajectories are in full agreement with the ones found by solving the Schrödinger equation. This formalism successfully describes the adiabatic limit and the anomalous Hall effect traditionally attributed to Berry curvature. It also describes the opposite limit of coherent band superposition, giving rise to a spatially oscillating Zitterbewegung motion, and all intermediate cases. At $k=0$, such a wave packet exhibits a circular trajectory in real space, with its radius given by the square root of the quantum metric. This quantity appears as a universal length scale, providing a geometrical origin of the Compton wavelength. The quantum metric semiclassical approach could be extended to an arbitrary number of bands.

中文翻译：

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基于量子度量的双波段系统通用半经典方程

我们推导出了双波段系统中加速波包的半经典运动方程。我们表明，这些方程可以根据量子度量描述的静态带几何来表述。我们考虑有和没有塞曼项的拉什巴哈密顿量的具体情况。半经典轨迹与通过求解薛定谔方程得到的轨迹完全一致。这种形式主义成功地描述了传统上归因于贝里曲率的绝热极限和异常霍尔效应。它还描述了相干带叠加的相反限制，产生空间振荡的 Zitterbewegung 运动，以及所有中间情况。在$克=0$，这样的波包在真实空间中呈现出圆形轨迹，其半径由量子度量的平方根给出。该量表现为通用长度标度，提供康普顿波长的几何原点。量子度量半经典方法可以扩展到任意数量的波段。