Topological concepts for robust directed amplification

项目描述：量子系统可以显示与拓扑属性相关的强大功能。这些获得精确的值，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对于电子和超导系统是很好理解的，但是对于光子系统和一般的玻色子系统，这些效应的范围要丰富得多。在这些系统中，粒子可以被创造和湮灭，这导致了损失、增益和非线性。近年来，人们已经看到了将这些玻色子系统与其电子对应物相适应的活动激增，主要是通过消除上述差异。然而，很快人们就意识到拓扑物理学已经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子凝聚的实验演示。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，以及如何以电子背景下实现的方式对系统进行分类。这个项目解决这个问题，一般来说，以及实际上通过检查特定的模型系统的实验兴趣，并询问如何增加其鲁棒性可能的应用。挑战是要充分捕捉非厄米，非互惠，非线性方面，这些系统的非线性。特别是，该项目将研究这种相互作用如何在非厄米光谱简并附近展开，称为例外点。首先，学生将开发一个全面的框架来评估在这些点附近运行的系统的光谱强度，同时考虑到所涉及的状态的完整双正交结构。这些见解将被应用到非互易系统，支持一种特殊类型的束缚态，其双正交态位于系统的相对两侧，并介导定向放大。最终的目标是发展这些定向放大器的自洽描述，考虑到量子噪声和非线性。该项目开发分析和数值建模技能。

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.