Quantum dynamics of curves on curved surfaces embedded in Euclidean space

Abstract

Fabrication of one-dimensional curved waveguide on two-dimensional curved manifold may give an impetus to the investigation on new devices. Understanding how the combined geometric quantities of the curve and surface affect the quantum dynamics calls for an effective theory. Here, we delve into the effective Hamiltonian governing a quantum particle confined to a curve that lies on a curved surface. By categorizing the constraints, we derive the effective Hamiltonians for two distinct scenarios: one where the curve is encapsulated in the surface, and another where the curve is a narrow ridge on the surface. The surface’s geometry substantially influences the dynamics of the curve, with the gauge potential in the former case being proportional to the geodesic torsion, and the emerging surface geodesic potential in the latter case being determined by both the geodesic curvature and geodesic torsion. We apply both the Hamiltonians to model quantum dynamics in circles on a sphere and helices on a cylinder, analyzing how the effective potential varies with different geometric parameters. This research serves as a foundational reference for the design of future waveguides, offering insights into how to manipulate quantum states within curved nanostructures.