AI 中文总结
本文推导了不同一维势场束缚态间矩阵元的经典极限公式，通过假设势场与算符的函数形式验证了其正确性，未来将用该公式提升光晶格钟性能。
AI 中文摘要
本文推导了不同一维势场束缚态间矩阵元的经典极限公式。我们暂不明确这些势场的物理解释，但通常设想它们是玻恩-奥本海默型势场，其中位置变量$x$对应系统中缓慢演化的自由度。例如，$x$可表示双原子分子的核间距，而这些势场是不同电子态的势能曲线。在这种情况下，矩阵元可以是常规的弗兰克-康登因子。为了验证所推导公式的正确性，我们假设势场和算符具有可给出矩阵元解析解的函数形式。当趋近经典极限时，计算得到的矩阵元呈现出向经典极限公式靠拢的明确趋势，这为该公式提供了有力的验证。在未来的工作中，我们预计将使用该公式来模拟一维光晶格中原子的非均匀激发谱，以期提升光晶格钟的性能。
英文摘要
A classical-limit formula is derived for matrix elements between bound states of distinct one-dimensional potentials. We leave open the physical interpretation of the potentials, but generally envision them to be Born-Oppenheimer-type potentials with the position variable $x$ identifying with a slowly evolving degree of freedom of the system. For instance, $x$ could represent the internuclear separation in a diatomic molecule, with the potentials being potential energy curves for different electronic states. In this scenario, the matrix elements could be, e.g., conventional Franck-Condon factors. To test the derived formula, we assume functional forms for the potentials and operator that afford analytical solutions for the matrix elements. As the classical limit is approached, the computed matrix elements exhibit a clear tendency towards the classical-limit formula, providing strong validation for the formula. In future work, we anticipate using the formula to model inhomogeneous excitation spectra of atoms in one-dimensional optical lattices, with an eye towards improved optical lattice clock performance.
Comments11 pages, 4 figures
Journal refAPS Open Sci. 1, 000077 (2026)
