Enabling adiabatic passages between disjoint regions in parameter space through topological transitions

Abstract

We explore topological transitions in parameter space in order to enable adiabatic passages between regions adiabatically disconnected within a given parameter manifold. To this end, we study the Hamiltonian of two coupled qubits interacting with external magnetic fields, and make use of the analogy between the Berry curvature and magnetic fields in parameter space, with spectrum degeneracies associated to magnetic charges. Symmetry-breaking terms induce sharp topological transitions on these charge distributions, and we show how one can exploit this effect to bypass crossing degeneracies. We also investigate the curl of the Berry curvature, an interesting but as of yet not fully explored object, which together with its divergence uniquely defines this field. Finally, we suggest a simple method for measuring the Berry curvature, thereby showing how one can experimentally verify our results.

Topological charge distributions of an interacting two-spin system

“…A difference, however, is that the Berry curvature in our example exits the surface radially with respect to the origin (i.e., it is proportional to B/B), in contrast to the electric field which exits the conductor's surface in the surface normal direction [i.e., it is proportional to n(B)]. Another difference is that the curl of the Berry curvature is nonzero [45], however, the curl of the electric field induced by the charged conductor vanishes.…”

“…Again, we will follow the electrostatics analogy to determine the surface topological charge distribution on this ellipsoid, see also Ref. [45]. In electrostatics the surface charge density σ (r S ) of surface S and the electric field E(r) created by the surface charge density are related by the following formula:…”

“…The topological charge distribution along 1D or 2D degeneracy geometries in a 3D parameter space, which we studied throughout this paper, is a very natural and intuitive concept due to the strong analogy with electrostatics. So far, it has been rarely discussed and studied in the literature [45], but we expect an increasing interest in this concept for the following reasons.…”

Momentum space toroidal moment in a photonic metamaterial

“…The presence of MTM leads to various interesting phenomena including surface-dependent anomalous shifts, reconfigurable helical waveguide modes, and bulk transverse spin, etc. Moreover, the toroidal structure of Berry curvature shows non-vanishing curl 47 , corresponding to “electric” currents in the momentum space 48 . They behave as new sources for generating Berry curvature in parallel with the “magnetic” charges-Weyl points.…”

Topological charge distributions of an interacting two-spin system

“…Again, we will follow the electrostatics analogy to determine the surface topological charge distribution on this ellipsoid, see also Ref. [33]. In electrostatics the surface charge density σ(r S ) of surface S and the electric field E(r) created by the surface charge density are related by the following formula:…”