Projective Hilbert space of non-Hermitian system and its application

by Dr 立斌 傅 (中国工程物理研究院)

Monday 28 Sep 2020, 17:00 → 19:00 Asia/Shanghai

腾讯会议id：839107238 (北京工业大学)

摘要：Projective Hilbert space lies in the heart of studying of geometric properties in quantum mechanics. A state on Projective Hilbert space represents a ray of Hilbert space, an equivalence class, of which the quantum states only differ from phases. Then, the evolution of quantum system can be fully described by the motion on Projective space and the phase shift on the ray can be acquired through a path integral on Projective space. For a system with a non-Hermitian Hamiltonian, the time evolution is not unitary. Hence the above equivalence property does not hold generally. However, we can still formally set up the Projective Hilbert space for a non-Hermitian system. Here, the states on the same ray not only differ by phases, but also by normalization factors. At this time, the changing in normalization factor and the phase shift can be still obtained by integral on the Projective space. Based on the Projective space, the dynamics of 2 × 2 matrix non-Hermitian systems is investigated in detail. It shows that there are four kinds of dynamical regions for such systems. The different regions are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken PT non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the system with exceptional point (EP). Furthermore, we investigate the connections between properties of Projective space and topological features of a PT-symmetric Su-Schrieffer-Heeger model. 简介：傅立斌，中国工程物理研究院研究生院研究员，中国工程物理研究院杰出专家，享受国务院特殊津贴，"新世纪百千万人才工程"国家级人选，主持“国家杰出青年基金”。1994年兰州大学物理系理论物理专业本科毕业，师从段一士先生，1999年获得博士学位。1999年至2001年在北京应用物理与计算数学研究所做博士后。2001年入北京应用物理与计算数学研究所工作，受聘副研究员。2005年破格晋升为研究员。2017年入职研究生院。期间，先后访问香港浸会大学非线性中心、新加坡国立大学工程计算系和量子工程中心、澳大利亚国立大学非线性物理系、香港中文大学物理系和理论物理所合作研究。2003年至2004年，曾作为洪堡学者在德国马克思－普郎克物理复杂系统研究所工作一年。在强场物理和量子物理前沿等诸多领域取得丰硕的研究成果。自1998年以来，共发表SCI论文159篇，其中Phys. Rev. Lett. 12篇，Phys. Rev. A/B/C/D/E 80余篇。从事的强激光与原子相互作用的研究，与所在单位承担的国家重大工程密切相关，由于这方面的贡献，他获得第十届于敏数理科学奖（2014年）和院科技创新一等奖（2016年）。