Schurly you're joking - quantum-classical correspondence in phase space for non-Hermitian systems
File(s)
Author(s)
Hall, Joseph
Type
Thesis
Abstract
In closed quantum systems described by Hermitian Hamiltonians the Husimi distributions of stationary states are closely related to classical phase space structures. To each state there is assigned a unique area of the Planck cell partitioned phase space. The localisation of the stationary states upon specific regions of phase space can be related to the classical dynamics and energies. In systems described by non-normal operators, and in particular non-Hermitian Hamiltonians, the general non-orthogonality of the stationary states stands in the way of the application of many standard ideas of quantum-classical correspondence. Many of these can be recovered if the Schur vectors are used instead of the eigenvectors. For specific classes of non-Hermitian systems, with simple loss profiles, this correspondence between the Schur vectors and features of the classical phase space are well investigated. For general non-Hermitian systems with more complex gain-loss profiles, however, there is a lack of methods and understanding. In this thesis we will address this problem by associating to each Schur vector an area of phase space described by the classical analogue of the quantum norm, which we describe as classical norm maps. An algorithm will be introduced that constructs from the norm maps a classical density that shows remarkably close correspondence to the Husimi distributions of the Schur vectors. We will demonstrate this correspondence in both regular and chaotic non-Hermitian systems.
Version
Open Access
Date Issued
2022-04
Date Awarded
2022-10
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Graefe, Eva-Maria
Sponsor
European Research Council
Grant Number
P66366_MATH
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)