Manipulation Of Quantum States In One-Dimensional Topological Superconductors

Majorana particles are unique quasiparticles that exhibit a non-abelian exchange statistic, and by exploiting that property one could braid them. In condensed matter systems Majorana quasiparticles occur as zero-energy edge states of topological superconductors, for example in the Kitaev chain. The common protocols proposed to braid these edge states are carried out in real space. In this thesis we investigate the possibility to braid Majorana quasiparticles within a one-dimensional superconductor via their parameter space. We utilise the geometrical phases that particles can obtain from adiabatic changes of the Hamiltonian along closed curves in the parameter space. In the first part, the mathematical background is established that is essential for understanding geometrical phases. In the second part of the study, several extensions of the Kitaev chain are proposed and examined for the presence of loops that give rise to nontrivial geometrical phases. We show how several...Show moreMajorana particles are unique quasiparticles that exhibit a non-abelian exchange statistic, and by exploiting that property one could braid them. In condensed matter systems Majorana quasiparticles occur as zero-energy edge states of topological superconductors, for example in the Kitaev chain. The common protocols proposed to braid these edge states are carried out in real space. In this thesis we investigate the possibility to braid Majorana quasiparticles within a one-dimensional superconductor via their parameter space. We utilise the geometrical phases that particles can obtain from adiabatic changes of the Hamiltonian along closed curves in the parameter space. In the first part, the mathematical background is established that is essential for understanding geometrical phases. In the second part of the study, several extensions of the Kitaev chain are proposed and examined for the presence of loops that give rise to nontrivial geometrical phases. We show how several topological phase transitions occur in those models. Finally, we present a noncontractible loop in the parameter space of a one-dimensional topological superconductor, albeit with a trivial corresponding geometrical phase.Show less

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Manipulation Of Quantum States In One-Dimensional Topological Superconductors