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Exceptional points and PT symmetric systems in adiabatic quantum theory

Pap, Eric (2018) Exceptional points and PT symmetric systems in adiabatic quantum theory. Master's Thesis / Essay, Physics.

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We investigate the theory of exceptional points (EPs); degeneracies of non-hermitian systems famous for the swaps of eigenstates that arise when encircling the EP according to the adiabatic approximation. We review various characterizations of EPs that are used in the literature. We observe that they are not equivalent and consequently a unanimous definition is lacking. The theory of PT symmetric systems is studied as an aid in the study of EPs. In particular, we deal explicitly with a three-channel waveguide system originating from PT theory and supporting various EPs. We show how one can properly compose the eigenvalue permutations arising from EPs and how this generates a non-abelian group which we call the Λ-group. We finish with the needed geometry as induced by the adiabatic approximation. The adiabatic change as measured in experiment can be modelled naturally on a principal C× o Sn-bundle consisting of the eigenframes of the operator family. To the best of our knowledge this bundle is not found in the literature to date. The adiabatic approximation induces a canonical connection on this bundle. The special shape of the holonomy group induces three groups, one of which is the Λ-group, that fit together in a short exact sequence (SES). This SES can be used to describe and distinguish degeneracy structures. In particular, the Λ-group allows one to phrase a new definition of exceptional point that specializes existing definitions. Explicitly, one can define an EP by demanding that swaps of eigenvalues occur arbitrarily close to the EP. This definition is widely applicable, yet guarantees that the eigenvalue and eigenstates sheets are connected in a non-trivial way.

Item Type: Thesis (Master's Thesis / Essay)
Supervisor name: Boer, D. and Waalkens, H.
Degree programme: Physics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 27 Aug 2018
Last Modified: 15 Nov 2018 13:54

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