Topological concepts for robust directed amplification

项目描述：量子系统可以显示与拓扑特性相关的鲁棒特征。这些获得的精确值只能在状态改变其拓扑性质的相变中改变。虽然这些效应的范围在电子和超导系统中得到了很好的理解，但在光子系统和一般玻色子系统中得到了更广泛的解释。在这些系统中，粒子可以被创造和湮灭，从而导致损失、增益和非线性。近年来，人们对这些玻色子系统进行了大量的研究，主要是通过消除上述差异来使其适应电子系统。然而，人们很快意识到，拓扑物理学超越了这些简单的类比，导致了激光、微波谐振器阵列和极化凝聚体的实验演示。缺少的是对这些扩展的实际范围的详细理解——如何系统地定义拓扑不变量，并以电子环境中实现的方式对系统进行分类。这个项目从总体上解决了这个问题，同时也通过检查实验感兴趣的特定模型系统来实际解决这个问题，并探讨如何为可能的应用增加它们的鲁棒性。挑战在于充分捕捉这些系统的非厄米、非互反和非线性特征。特别是，该项目将研究这种相互作用如何在非厄米谱简并附近展开，称为异常点。首先，学生将开发一个综合框架来评估在这些点附近运行的系统的光谱强度，考虑到所涉及状态的完整双正交结构。这些见解将被应用于非互易系统，它支持一种特殊类型的束缚态，其双正交态位于系统的两侧，并介导定向放大。最终目标是在考虑量子噪声和非线性的情况下，发展这些定向放大器的自洽描述。该项目发展分析和数值模拟技能。

Description of Project: Quantum systems can display robust features related to topological properties. These attain precise values that can only change in phase transitions where the states change their topological properties. While the scope of these effects is well understood for electronic and superconducting systems, a much richer range is accounted for photonic and in general bosonic systems. In these systems particles can be created and annihilated, which results in loss, gain, and nonlinearity. Recent years have seen a surge of activity to tailor these bosonic systems to their electronic counterparts, mostly by eliminating the mentioned differences. However, it was soon realised that topological physics extends beyond these mere analogies, leading to experimental demonstrations for laser, microwave resonator arrays, and polaritonic condensates.What is missing is a detailed understanding of the actual scope of these extensions - how to systematically define the topological invariants, and classify systems in the manner achieved in the electronic context. This project tackles this question both generally, as well as practically by examining specific model systems of experimental interest, and inquiring how to increase their robustness for possible applications. The challenge is to fully capture the non-Hermitian, non-reciprocal, and non-linear aspects that characterise these systems. In particular, the project will examine how this interplay unfolds near non-Hermitian spectral degeneracies, known as exceptional points. First, the student will develop a comprehensive framework to evaluate the spectral strength of systems operating in the vicinity of these points, taking the full biorthogonal structure of the involved states into account. These insights will then be applied to nonreciprocal systems, which support a special type of bound state whose biorthogonal states are located at opposite sides of a system, and mediates directed amplification. The ultimate goal is to develop the selfconsistent description of these directed amplifiers, taking quantum noise and nonlinearities into account. This project develops both analytical and numerical modelling skills.