AI 中文总结
本文将静态系统的Bloch带量子几何框架扩展至Floquet系统，定义含时Berry曲率与量子度量，推导光学求和规则，提出非绝热量子化电荷泵浦机制，经数值模拟验证，为周期驱动量子系统研究提供统一框架。
AI 中文摘要
以Berry曲率和量子度量为特征的Bloch带量子几何，是静态系统中广泛线性与非线性响应的基础。本文将该框架扩展至周期驱动（Floquet）系统，直接在Floquet-Bloch基中定义含时Berry曲率与量子度量。我们推导了光学求和规则，将这些几何量的傅里叶分量与光导率关联，并证明在理想Floquet带占据下，谐波频率处的一阶直流霍尔响应与纵向响应完全消失。我们进一步引入了含时间与动量导数的混合Berry曲率，该曲率会产生非绝热的量子化电荷泵浦机制，此机制自然发生于每个驱动周期内，无需绝热演化。此外，我们将时域量子度量识别为Floquet带能量涨落的度量，并将其混合分量解释为量化极化-能量关联。全面的对称性分析揭示了时间反演、子晶格（手征）、粒子-空穴、反演、旋转及反射对称性如何约束含时量子几何张量及其相关拓扑不变量。对Rudner-Lindner-Berg-Levin模型及全对称Floquet模型的数值模拟验证了这些分析预测。这些结果确立了含时量子几何张量作为描述周期驱动量子系统几何、拓扑与动力学性质的统一框架，对光谱学、量子输运及拓扑电荷泵浦实验具有直接意义。
英文摘要
The quantum geometry of Bloch bands, characterized by the Berry curvature and the quantum metric, underpins a wide range of linear and nonlinear responses in static systems. Here, we extend this framework to periodically driven (Floquet) systems by introducing a time-dependent Berry curvature and quantum metric defined directly in the Floquet-Bloch basis. We derive optical sum rules that relate the Fourier components of these geometric quantities to the optical conductivity and demonstrate that, under ideal Floquet-band occupations, the first-order DC Hall and longitudinal responses at harmonic frequencies vanish identically. We further introduce a mixed Berry curvature involving time and momentum derivatives, which gives rise to a non-adiabatic quantized charge-pumping mechanism that occurs naturally during each driving period without requiring adiabatic evolution. In addition, we identify the time-domain quantum metric as a measure of the energy fluctuations of a Floquet band and interpret its mixed components as quantifying polarization-energy correlations. A comprehensive symmetry analysis reveals how time-reversal, sublattice (chiral), particle-hole, inversion, rotational, and reflection symmetries constrain the time-dependent quantum geometric tensor and its associated topological invariant. Numerical simulations of the Rudner-Lindner-Berg-Levin model and a fully symmetric Floquet model confirm the analytical predictions. These results establish the time-dependent quantum geometric tensor as a unified framework for describing the geometric, topological, and dynamical properties of periodically driven quantum systems, with direct implications for optical spectroscopy, quantum transport, and topological charge-pumping experiments.
