Phase space approximations for non-hermitian quantum systems
File(s)
Author(s)
Rehman, Wasim
Type
Thesis
Abstract
In this thesis, we study phase-space approximations to quantum dynamics for both Hermitian and non-Hermitian quantum systems, motivated by the need to efficiently describe quantum effects in models that are analytically intractable but admit classical analogues offering insight into their complex behaviour. Such systems arise in diverse areas, include but are not limited to quantum optics, condensed matter, open quantum systems, and quantum technologies.
We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and $PT$-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques.
To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach.
We then apply this framework to both the $\mathfrak{su}_2$ algebra and non-Lie algebraic bosonic conversion systems. For $\mathfrak{su}_2$, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of $PT$-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds.
Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.
We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and $PT$-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques.
To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach.
We then apply this framework to both the $\mathfrak{su}_2$ algebra and non-Lie algebraic bosonic conversion systems. For $\mathfrak{su}_2$, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of $PT$-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds.
Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.
Version
Open Access
Date Issued
2025-06-16
Date Awarded
01/12/2025
License URL
Advisor
Graefe, Eva-Maria
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)