Bloch vector, disclination and exotic quantum holonomy

Abstract

A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.

Path topology dependence of adiabatic time evolution

“…The eigenvalues of U are, as shown in Ref. [17], z ± = exp {−i(φ ± ∆)/2}, where ∆ = 2 arccos cos φ 2 cos B 2 .…”

“…Second, we will show that the discrepancy between two final stationary states corresponding to two different adiabatic path is characterized by a permutation matrix, which is governed by a homotopy equivalence. Our idea is an application of the topological formulation for the exotic quantum holonomy [15,16], which concerns the nontrivial change in eigenspaces induced by adiabatic cycles [17].…”

Adiabatic cycles in quantum systems with contact interactions

“…In general, an "exotic" quantum holonomy [2,3], in which one eigenstate turns into another eigenstate belonging to the same Hamiltonian, can be observed after the cyclic variation of the system parameters. In this paper, we examine an adiabatic cycle that excites a system consisting of Bose particles confined in a one-dimensional box.…”

Complete population inversion of Bose particles by an adiabatic cycle

“…There has been studies of the excitation of quantum systems by adiabatic cycles, which is referred to as the exotic quantum holonomy [11][12][13]. We also mention, in studies of atomic and molecular systems under the oscillating field, that an adiabatic cycle involving a level crossing may excite a quantum system [14,15].…”