Liouvillian analysis of dynamics at exceptional points incorporating quantum jumps

在这一年中，我在与动力问题相关的两个轨道上取得了进展。虽然最初的意图是关注Liouvillian水平的特殊点动态，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，并且表现出具有接近零能量特征值的边缘状态（或零能量模式），并且充当导电表面状态，尽管系统的大部分表现为导体。这些国家受到部分保护，免遭混乱。我发现，通过对沿1-D晶格交替耦合的Su-Schrieffer-Heeger （SSH）模型进行半无限扩展，我可以获得特征值完全为零的边缘状态，从而使对无序的保护最大化。此外，通过在系统的端点引入杂质，我可以证明在均匀晶格中出现了两个新的不对应的参数区。此外，所有特征值都出现在这两个区域的体积间隙内，这两个区域通过具有拓扑性质的异常点与“平凡”参数空间分开。在另一个单独的轨道上，我也在为一个简单系统编写Lindblad方程的问题上取得了一些初步的进展，并将考虑如何在未来的工作中扩展它以纳入特殊点的量子跳变。

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.