Geometry of adiabatic Hamiltonians for two-level quantum systems

TL;DR

This paper introduces a geometric framework using plane curves to analyze the adiabatic dynamics of two-level quantum systems, linking curvature to nonadiabatic coupling for better physical understanding.

Contribution

It formulates the adiabatic quantum dynamics problem in terms of differential geometry, providing a novel geometric interpretation of nonadiabatic effects.

Findings

Curvature of the plane curve corresponds to nonadiabatic coupling.

Geometric quantities have clear physical interpretations.

Framework simplifies understanding of adiabatic quantum behavior.

Abstract

We present the formulation of the problem of the coherent dynamics of quantum mechanical two-level systems in the adiabatic region in terms of the differential geometry of plane curves. We show that there is a natural plane curve corresponding to the Hamiltonian of the system for which the geometrical quantities have a simple physical interpretation. In particular, the curvature of the curve has the role of the nonadiabatic coupling.