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Stochastic Representations of Open Systems

Student thesis: Doctoral ThesisDoctor of Philosophy

This thesis outlines the development and implementation of an exact tech-nique for the analysis of a particular class of open quantum systems. Start-ing from a generalised Caldeira-Leggett 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 appli-cable even in the case of external driving and strong environment cou-pling. They also permit a more general class of initial states, where the combined system and environment are in full thermal equilibrium. Col-lectively 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 two-time 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 non-trivially correlated and when averaged over realisations, give the exact reduced density matrix of the system.
A first application of the ESLE to a spin-boson 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 two-level sys-tem being driven from equilibrium is also discussed.
An equivalent classical analysis is performed using Koopman-von Neu-mann (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 self-adjoint ex-tensions as a measure of local entropy conservation is developed.
Original languageEnglish
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Award date2019

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