Cheon’s anholonomies in Floquet operators

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.

Adiabatic quantum computation along quasienergies

“…Accordingly the adiabatic passage along the ground quasienergy transports | − at s = 0 to | + at s = 2π. Note that the "first excited" quasienergy that connects (s, E) = (0, EP) and (2π, Emax) has crossings with other quasienergies (not depicted), which are s-independent and, whose eigenspaces are orthogonal to | ± [16].…”

“…This offers a systematic design principle for adiabatic passages, in particular, adiabatic quantum computation, along parametric changes of unitary operators [16]. We assume that the spectrum of an "unperturbed" HamiltonianĤ 0 is discrete, finite and nondegenerate.…”

“…Extensions of the quasienergy anholonomy for multilevels are straightforward [13,16]. For example, if we choose |v that has non-zero overlapping with all eigenvectors ofĤ 0 , E 0 (2πn) is the n-th excited energy of H 0 [24].…”

“…A family of DAQC that essentially relies on the adiabatic passage along a quasienergy is recently proposed by one of the authors [13]. Here the adiabatic passage is composed with a help of Cheon's eigenvalue anholonomy [13,14,15,16], which enables us to design adiabatic passages that visit all eigenstates of a given unperturbed Hamiltonian. This adiabatic scheme, which will be called anholonomic adi-abatic quantum computation (AAQC), composes an interesting and nontrivial family of DAQC.…”

Quantum anholonomies in time-dependent Aharonov-Bohm rings

“…This is the origin of the second factor in Eq. (20). Although this factor is interpreted as a part of dynamical phase factor in the Byers-Yang gauge, this is classified as a part of the geometric factor in the calculation through the periodic gauge.…”