Geometric phase and its applications: topological phases, quantum walks and non-inertial quantum systems

TL;DR

This paper explores the geometric phase in quantum systems, proposing new geometric concepts, analyzing topological phases in quantum walks under environmental effects, and detecting non-inertial quantum phenomena.

Contribution

It introduces a new perspective on geodesics and null phase curves, and applies geometric phase analysis to topological quantum walks and non-inertial quantum systems.

Findings

Geodesics and null phase curves are geometrically decomposed on the Bloch sphere.

Topological phases in quantum walks persist under moderate environmental losses.

Geometric phase detects non-inertial effects on field correlators in rotating quantum systems.

Abstract

Geometric phase plays a fundamental role in quantum theory and accounts for wide phenomena ranging from the Aharanov-Bohm effect, the integer and fractional quantum hall effects, and topological phases of matter, including topological insulators, to name a few. In this thesis, we have proposed a fresh perspective of geodesics and null phase curves, which are key ingredients in understanding the geometric phase. We have also looked at a number of applications of geometric phases in topological phases, quantum walks, and non-inertial quantum systems. The shortest curve between any two points on a given surface is a (minimal) geodesic. They are also the curves along which a system does not acquire any geometric phase. In the same context, we can generalize geodesics to define a larger class of curves, known as null phase curves (NPCs), along which also the acquired geometric phase is zero; however, they need not be the shortest curves between the two points. We have proposed a geometrical decomposition of geodesics and null phase curves on the Bloch sphere, which is crucial in improving our understanding of the geometry of the state space and the intrinsic symmetries of geodesics and NPCs. We have also investigated the persistence of topological phases in quantum walks in the presence of an external (lossy) environment. We show that the topological order in one and two-dimensional quantum walks persist against moderate losses. Further, we use the geometric phase to detect the non-inertial modifications to the field correlators perceived by a circularly rotating two-level atom placed inside a cavity.