Semiclassical methods for investigating open quantum systems and decoherence

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

Abstract

In this thesis we use the phase space formulation of quantum mechanics to investigate open quantum systems via the Lindblad equation in the semiclassical limit with a focus on analysing the spread of decoherence. In the case of linear Lindblad operators and quadratic Hamiltonians where the phase space Lindblad equation becomes exact, we find that decoherence is intimately related to H\"{o}rmander's condition which describes when a partial differential equation is hypoelliptic. In particular, it motivates a natural orthogonal decomposition of phase space which encodes the timescales describing the onset of decoherence as well as the subspaces which are protected. Explicit results for the evolution of a Gaussian coherent state and its Hilbert-Schmidt norm evolving under this exact Lindblad equation are given. We use this to investigate a class of simple examples of networks of harmonic oscillators and show that by changing the macro structure of the network, the spread of decoherence throughout the system can change dramatically. Finally, by interpreting the Lindblad equation as a Schr\"{o}dinger equation on a doubled phase space, we use recent results in the theory of non-Hermitian Schr\"{o}dinger equations to determine the evolution of a Gaussian state under a Lindblad equation with general Hamiltonian and Lindblad operators.
Date of Award23 Jan 2020
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorRoman C V Schubert (Supervisor)

Keywords

  • semiclassical
  • quantum mechanics
  • Lindblad
  • decoherence
  • phase-space quantum mechanics
  • Hörmander
  • gaussian states
  • coherent states

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