Abstract
This thesis outlines the development and implementation of an exact technique for the analysis of a particular class of open quantum systems. Starting from a generalised CaldeiraLeggett model, a set of coupled stochastic di˙erential equations are derived as an evolution equation for the reduced density matrix of an arbitrary open system interacting (in a generalised manner) with a bath of harmonic oscillators. These equations are applicable even in the case of external driving and strong environment coupling. They also permit a more general class of initial states, where the combined system and environment are in full thermal equilibrium. Collectively these equations are known as the Extended Stochastic Liouville Equation (ESLE).The ESLE is derived by casting the system+environment density matrix as a path integral in both real and imaginary time. In this form, it is possible to obtain the reduced system density matrix using influence functional techniques. Applying the twotime Hubbard Stratonovich transformation to this path integral, one obtains the ESLE. This consists of two evolution equations, accounting for a propagation in imaginary time followed by real time. Both equations contain stochastic terms which are nontrivially correlated and when averaged over realisations, give the exact reduced density matrix of the system.
A first application of the ESLE to a spinboson model is also discussed. This is used as a proof of principle that the noises required by the ESLE can be generated numerically, and amenable to practical calculation. The impact of the ESLE’s generalisations in the description of a twolevel system being driven from equilibrium is also discussed.
An equivalent classical analysis is performed using Koopmanvon Neumann (KvN) mechanics (an operational Hilbert space formalism which puts the quantum and classical descriptions on the same footing). In this setting, the ESLE derivation reproduces the Langevin equation directly from classical mechanics. Finally, the KvN formalism is used to explore some adjacent topics. In particular, a theory of classical selfadjoint extensions as a measure of local entropy conservation is developed.
Date of Award  2019 

Original language  English 
Awarding Institution 

Supervisor  Lev Kantorovich (Supervisor) & Chris Lorenz (Supervisor) 