Liouvillian analysis of dynamics at exceptional points incorporating quantum jumps

在这一年里，我在与例外点处的动力学问题有关的两个轨道上取得了进展。虽然最初的目的是关注刘维利安级别的特殊点数动态，但我发现了一个新发现，在某种程度上改变了我的注意力。我发现了一个拓扑绝缘体的简单模型的扩展，它产生了一个具有独特性质的异常点。通常，拓扑绝缘体的模型是有限的，并且表现出具有几乎为零的能量本征值的边态(或零能模)，并且充当导电表面态，尽管系统的大部分行为是导体。这些州在一定程度上受到保护，不会出现混乱。我发现，通过对SU-Schrieffer-Heeger(SSH)模型的半无限推广，我可以得到一个本征值正好为零的边态，从而最大化对无序的保护。此外，通过在系统的端点处引入杂质，我可以证明在均匀晶格中出现了两个没有对应关系的新的参数区域。此外，所有的本征值都出现在这两个区域的体隙内，这两个区域由一个具有拓扑性质的例外点从平凡的参数空间中分离出来。在另一个单独的轨道上，我也在写简单系统的Lindblad方程的问题上取得了一些初步进展，并将在未来的工作中考虑如何扩展这一点，以包含例外点处的量子跃迁。

In this year, I have made progress on two tracks related to the dynamical problem at the exceptional point. While the original intention was to focus on the exceptional point dynamics at the level of the Liouvillian, I have found a new discovery that redirects my focus somewhat. I have found an extension of a simple model for a topological insulator that gives rise to an exceptional point with unique properties.Usually the model for a topological insulator is finite and exhibits edge states (or zero-energy modes) that have nearly zero energy eigenvalue and act as conducting surface states despite that the bulk of the system behaves as a conductor. These states are partially protected against disorder. I have found that by taking a semi-infinite extension of the Su-Schrieffer-Heeger (SSH) model, which has alternating couplings along a 1-D lattice, I can obtain an edge state with eigenvalue exactly zero such that the protection against disorder is maximized. Further, by introducing an impurity at the endpoint of the system, I can show that two new parameter regimes appear that have no correspondence in the uniform lattice. Further, all of the eigenvalues appear inside the bulk gap in these two regions, which are separated from the 'trivial' parameter space by an exceptional point that has topological properties.On a separate track, I have also made some preliminary progress on the problem of writing the Lindblad equation for a simple system and will consider how to extend this to incorporate quantum jumps at exceptional point in future work.