Liouvillian analysis of dynamics at exceptional points incorporating quantum jumps

在这一年里，我在两条与特殊点动力学问题有关的轨道上取得了进展。 虽然最初的意图是专注于刘维级别的特殊点动力学，但我发现了一个新的发现，这在一定程度上改变了我的注意力。 我发现了拓扑绝缘体的一个简单模型的一个扩展，它产生了一个具有独特性质的特殊点，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缘态（或零能量模式），并且作为导电表面态，尽管系统的大部分表现为导体。 这些国家部分地受到保护，不受混乱的影响。 我发现，通过采取一个半无限扩展的苏-施里弗-海格尔（SSH）模型，它具有交替耦合沿着1-D晶格，我可以获得一个边缘状态的特征值正好为零，这样的保护，对无序是最大化的。 此外，通过在系统的端点处引入杂质，我可以证明出现了两个新的参数区域，它们在均匀晶格中没有对应关系。 此外，所有的本征值出现在这两个区域，这是从“平凡”的参数空间分开的例外点，具有拓扑properties.On一个单独的轨道上，我也取得了一些初步进展的问题上写的Lindblad方程的一个简单的系统，并将考虑如何扩展到包含量子跳跃在特殊点在未来的工作。

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.