Numerical verification of the exact WKB formula for the generalized Landau-Zener-Stueckelberg problem
Numerical verification of the exact WKB formula for the generalized Landau-Zener-Stueckelberg problem
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广义 Landau-Zener-Stueckelberg 问题精确 WKB 公式的数值验证
DOI：
10.1103/physreva.102.022213
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发表时间：
2020
影响因子：
2.9
通讯作者：
Shudo Akira
中科院分区：
文献类型：
--
作者：
Shimada Naoaki;Shudo Akira
高阶微分方程组的WKB分析直到最近所谓的精确WKB方法才得到发展，因此一般很难找到多态非绝热模型的WKB解，而对于广义Landau-Zener模型的研究则做了很大的努力。在这种情况下，Aoki-Kawai-Takei给出了含时多态非绝热系统的矩阵的封闭表达式，其中非绝热势能曲线是时间的多项式函数，并且它们都是相交的。他们的结果是完全明确的，而且该模型涵盖了比迄今所研究的多态非绝热模型更广泛的系统类别。在这里，我们数值地展示了他们的精确的WKB公式是如何工作的，只要非绝热势能曲线的交叉点被很好地分开。与广义Landau-Zener模型不同，所研究的模型包含了非绝热势能函数为非线性的情况。我们还讨论了虚拟转向点和新的Stokes曲线的情况，它们是Stokes几何中只出现在高阶微分方程中的新分量。特别是，我们用几个具体的例子研究了新的斯托克斯曲线的存在如何影响最终的跃迁概率。
The WKB analysis for higher order differential equations had not been developed until recent progress of the so-called exact WKB method, so it was difficult to find WKB solutions for multistate nonadiabatic models in general, whereas enormous efforts were made especially for the study of generalized Landau-Zener models. Under such circumstances Aoki-Kawai-Takei provided a closed expression for thematrix of a time-dependent multistate nonadiabatic system, in which diabatic potential curves are given as polynomial functions of time and they all intersect with each other. Their result is completely explicit, and the model covers much wider classes of systems than multistate nonadiabatic models studied so far. Here we numerically show how their exact WKB formula works as long as crossing points of diabatic potential curves are well separated. The model studied includes the cases where diabatic potential functions are nonlinear, not like generalized Landau-Zener models. We also discuss the situations where virtual turning points and new Stokes curves, new components in the Stokes geometry appearing only for higher order differential equations, come into play. In particular, we examine how the presence of new Stokes curves affects final transition probabilities using a couple of concrete examples.