Topological concepts for robust directed amplification

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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.