Trapped modes in finite quantum waveguides

TL;DR

This paper develops a variational approach to identify trapped eigenmodes in finite quantum waveguides, showing how branch length influences trapping and extending the understanding beyond infinite waveguide models.

Contribution

It introduces a general variational formalism for detecting trapped modes in finite waveguides with cylindrical branches, including a criterion for minimal branch length for trapping.

Findings

Trapped modes can be characterized by exponential decay inside finite branches.

Branch length determines whether eigenmodes are trapped or not.

The theory of trapping in infinite waveguides needs extension for microscopic quantum devices.

Abstract

The Laplace operator in infinite quantum waveguides (e.g., a bent strip or a twisted tube) often has a point-like eigenvalue below the essential spectrum that corresponds to a trapped eigenmode of finite L2 norm. We revisit this statement for resonators with long but finite branches that we call "finite waveguides". Although now there is no essential spectrum and all eigenfunctions have finite L2 norm, the trapping can be understood as an exponential decay of the eigenfunction inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped states. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two stripes, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for being applied to microscopic quantum devices.