量子弹球和量子图高精度普适涨落规律测量与研究

During the funding period high-precision experiments will be performed with flat superconducting microwave resonators and with microwave networks in order to simulate quantum billiards and quantum graphs. The experiments are primarily motivated by open problems from quantum chaos, semiclassical theory and condensed matter physics. We will construct microwave billiards, i.e., flat resonators, simulating singular quantum billiards, or quantum billiards with mixed boundary conditions and microwave graphs, using material which is superconducting at liquid helium temperature. Microwave power will be coupled into and out of such systems and the moduli and phases of the associated complex scattering-matrix elements are measured. The key aspects of research are (i) the experimental determination of complete sequences of several thousand energy levels and of wave functions of quantum systems with regular, mixed or chaotic dynamics exhibiting certain characteristic features and the study of their universal properties, (ii) high-precision measurements of the complex scattering-matrix elements of quantum chaotic scattering systems and the study of the fluctuation properties and correlation functions, in order to explore their universal properties. For this we use the scattering formalism of compound-nucleus reaction theory. Quantum billiards and quantum graphs serve as most appropriate systems for the study of universal features of quantum systems. Therefore, due to the analogy between such systems and their microwave realizations, the results of this research have broad applications in the understanding of generic properties of energy spectra and scattering matrices of systems like nuclei, atoms or molecules and condensed matter structures (graphene, fullerene, ...).

本项目将在液氦温度下实现超导微波弹球和超导微波网络，用来模拟量子弹球和量子图，对能谱统计和散射矩阵的涨落性质进行高精度实验研究，检验相关半经典理论和随机矩阵理论的最新进展。这些规律具有普适性，具有相同经典动力学和对称性的微波（量子）系统的涨落性质相同。我们将实验构建超导的奇异微波弹球、具有混合边界的微波弹球和微波网络，并实现超导开放微波弹球的测量：（i）在超导条件下可以精确测得数千个本征频率（能级），考察可积、混合和混沌动力学的能谱特性及普适规律；（ii）通过测量输入和输出端口的微波幅值和相位，可以精确测得复数的微波散射矩阵元，研究不同情况下它们的涨落特性和关联函数。这些研究将有效的推进半经典理论的发展，增加人们对散射矩阵普遍规律的认识。由于模型和方法的通用性，这些实验和理论上的进展将能够广泛应用于原子核、复合核、原子、分子和凝聚态物质（石墨烯量子点、富勒烯等）的能谱及散射行为的预测。

在基金资助期间，我研究了量子混沌和量子混沌散射的以下几个方面：.我们利用微波网络进行了实验，研究了具有正交、幺正和辛普遍性类的封闭和开放（散射）量子图的性质。目前，我正在写两遍关于实验和理论的相关内容的文章。.我们研究了相对论中微子弹球的特征值和波函数的理论性质。这些性质包括弹球形状和边界条件对谱的影响，以及之前的理论是否能应用于超相对论和非相对论极限之间的过渡区域。一篇关于具有可积经典弹球形状的相对论性量子弹球谱性质的文章已提交给Journal of Physics A。.我们从理论和实验上研究了不同边界条件的非相对论量子弹球的特征值和波函数（部分通过插入“奇异散射体”实现）的性质，以了解它们对边界条件的依赖关系。目前，我们用含有外部磁化铁氧体的超导微波弹球进行实验，以诱导破坏时反演不变（仍在进行中）。.我们从理论上研究了具有三重旋转形状的离散蜂窝状弹球的谱性质和波函数。这些结果对微波光子晶体的实验结果具有解释意义。.我将半经典方法应用于具有可积和混沌经典弹球形状的相对论性中微子弹球中，并推导了谱密度的半经典近似表达式，以深入了解它们的半经典极限。这些半经典近似为理解相对论量子弹球的谱特性提供了一个工具，即用相应形状的经典台球的纯经典量来理解相对性量子混沌。