Geometric Phase in Eigenspace Evolution of Invariant and Adiabatic Action Operators

Abstract

The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with $N$-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this geometric phase captures the inherent geometric feature of the state evolution. Moreover, the geometric phase in the evolution of the eigenspace of an adiabatic action operator is also addressed, which is elaborated by a pullback U(N)-bundle. Several intriguing physical examples are illustrated.

arXiv commentAdded Refs. and corrected typos; 4 pages

References in corpus (5)

• Holonomic Quantum Computation
• Implementation of universal quantum gates based on nonadiabatic geometric phases
• Non-Abelian Berry connections for quantum computation
• Unconventional geometric quantum computation
• Adiabatic Geometric Phase for a General Quantum States

Cited by in corpus (5)

• Experimental implementation of high-fidelity unconventional geometric quantum gates using NMR interferometer
• Geometric phase for SU(N) through fiber bundles and unitary integration
• Implementation of quantum gates based on geometric phases accumulated in the eigenstates of periodic invariant operators
• Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies
• Geometric and holonomic quantum computation