Overlapping resonances in open quantum systems

Haifeng Song

Published May 1, 2014• 0 citations

Abstract

In this thesis, we study the dynamics of open quantum systems in the overlapping resonances regime. An open quantum system consists of a small, finite-dimensional system S and a reservoir R, which interacts with the system S via some operator. The reservoir is, for example, modeled by a spatially infinitely extended free Bose gas in thermal equilibrium at positive temperature. One of the main tasks in the theory of open quantum system is to examine quantum phenomena observed in the small system, such as return to equilibrium and decoherence, brought about by the interaction with the environment. To deal with such a problem, one tries to study the dynamics of the total system S + R, which is generated by the total Hamiltonian. By tracing out the degrees of freedom of the reservoir, we obtain the dynamics of the small system. Over the last decade and a half, a perturbation theory based on quantum resonance methods has been developed in the isolated resonances regime. The perturbation theory developed so far permits a mathematically rigorous treatment of the dynamics for fixed, small system-reservoir coupling parameter λ. However, in complex quantum systems, e.g. when the dimension of small system is large, the problem belongs to the overlapping resonances regime. The theory mentioned above is not applicable in this regime. In this thesis, we adapt the perturbation theory for the treatment of such a regime. We first obtain a representation formula of the reduced dynamics involving resonances of an associated operator. By analyzing the behaviour of resonances, we derive the reduced density matrix of the small system. Furthermore, we consider the spin-boson model and analyze the transition from the isolated resonances regime to the overlapping one by using the Feshbach map. We also find a critical value, related to the parameters, which separates the two regimes, marking a sharp transition in the behaviour of the resonance energies.

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