Geomeric phases in Quantum Mechanics

TL;DR

This paper reviews geometric phases in quantum mechanics, explaining key phenomena through examples and applying concepts like connections and curvatures to phenomena such as the Aharonov-Bohm effect and holonomic quantum computation, including detailed gate implementation.

Contribution

It provides a comprehensive analysis of geometric phases, introduces two approaches to connections in Schrödinger equations, and demonstrates implementation of holonomic quantum computation with detailed calculations.

Findings

Connections in Schrödinger equations can be treated by two approaches

Holonomic quantum computation can be implemented using dark states

Anyons relate to wave function symmetries in quantum Hall effects

Abstract

Various phenomena related to geometric phases in quantum mechanics are reviewed and explained by analyzing some examples.The concepts of 'parallelism' ,'connections' and 'curvatures' are applied to Aharonov-Bohm (AB) effect, to U(1)phase rotation, to SU(2) phase rotation and to holonomic quantum computation (HQC). The connections in Schrodinger equations are treated by two alternative approaches. Implementation of HQC is demonstrated by the use of 'dark states' including detailed calculations with the connections, for implementing the quantum gates. 'Anyons' are related to the symmetries of the wave functions,in a two-dimensional space, and the use of this concept is demonstrated by analyzing an example taken from the field of Quantum Hall effects.