In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

中文翻译：

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布洛赫方程传播子的解析解和表达式

在本文中，我们提出了布洛赫方程的解析解。在求解从 Bloch 方程导出的特征值的长期方程后，在任意初始条件下获得磁化矢量随时间演化的解析解。随后，得到了布洛赫方程和光学布洛赫方程的传播子的显式解析表达式。与现有分析研究的结果相比，目前的结果更加简洁和严谨，可以预测传播者在参数空间不同区域的行为。传播子的解析解可以直接用于复合激光脉冲光谱。