Cheon's anholonomies in Floquet operators

TL;DR

This paper investigates Cheon's anholonomies in Floquet operators of periodically driven quantum systems, revealing new types of eigenvalue and eigenspace anholonomies, analyzing their stability, and proposing applications in quantum state manipulation.

Contribution

It introduces and analyzes Cheon's anholonomies in Floquet operators, demonstrating their stability and potential for quantum control, expanding understanding beyond geometric phase phenomena.

Findings

Eigenvalues can exhibit anholonomy independent of Floquet operator periodicity.

Eigenspaces also show anholonomy, affecting eigenvector directions.

Stability of anholonomies is confirmed through geometrical analysis.

Abstract

Anholonomies in the parametric dependences of the eigenvalues and the eigenvectors of Floquet operators that describe unit time evolutions of periodically driven systems, e.g., kicked rotors, are studied. First, an example of the anholonomies induced by a periodically pulsed rank-1 perturbation is given. As a function of the strength of the perturbation, the perturbed Floquet operator of the quantum map and its spectrum are shown to have a period. However, we show examples where each eigenvalue does not obey the periodicity of the perturbed Floquet operator and exhibits an anholonomy. Furthermore, this induces another anholonomy in the eigenspaces, i.e., the directions of the eigenvectors, of the Floquet operator. These two anholonomies are previously observed in a family of Hamiltonians [T. Cheon, Phys. Lett. A 248, 285 (1998)] and are different from the phase anholonomy known as geometric phases. Second, the stability of Cheon's anholonomies in periodically driven systems is established by a geometrical analysis of the family of Floquet operators. Accordingly, Cheon's anholonomies are expected to be abundant in systems whose time evolutions are described by Floquet operators. As an application, a design principle for quantum state manipulations along adiabatic passages is explained.