Exotic quantum holonomy in Hamiltonian systems

Abstract

We study the evolution of quantum eigenstates in the presence of level crossing under adiabatic cyclic change of environmental parameters. We find that exotic holonomies, indicated by exchange of the eigenstates after a single cyclic evolution, can arise from non-Abelian gauge potentials among non-degenerate levels. We illustrate our arguments with solvable two and three level models.Comment: LaTeX Elsevier double column format, 6 pages, 10 figures, See http://www.kochi-tech.ac.jp/~cheon/ for higher quality figure

Quantum anholonomies in time-dependent Aharonov-Bohm rings

“…We leave this as an open question. The conventional examples of eigenvalue and the eigenspace anholonomies requires a rank-1 perturbation and its cousins [1,2,5,20,21]. In contrast to this, the eigenspace anholonomy in the present example emerges from the quasi-periodic boundary condition.…”

Adiabatic excitation of a confined particle in one dimension with a variable infinitely sharp wall

“…The same concept has been utilized in Refs. [10,11,21] to realize the adiabatic excitations along cycles. Although the level crossings are generally fragile against perturbations according to Wigner and von Neumann's theorem [22], this theorem is inapplicable to our case since the system is completely divided into two parts.…”

“…the effect of tunneling, the spectral degeneracy may be lifted. There we need to resort to the diabatic evolution to go across the avoided crossing in order to approximately realize the adiabatic excitation [21].…”

Exotic quantum holonomy and higher-order exceptional points in quantum kicked tops

“…The first is equivalent to a nonresonant condition, ω / ∈ 2πq p q and p are integer and 0 < |p| < d . (8) The first condition together with the assumption J,M|v = 0 for all M = −J, . .…”

“…Namely, under the presence of such an exotic quantum holonomy, the initial and final states of an adiabatic cycle belong to different eigenspaces. The exotic quantum holonomy has been studied both in one-body [5][6][7][8][9][10] and in many-body systems [11]. Applications of the exotic quantum holonomy to quantum state manipulation and adiabatic quantum computation [12] were also proposed [5,6,13].…”