Linear impurity modes in an electrical lattice: Theory and Experiment

TL;DR

This paper combines theoretical analysis and experimental validation to study localized and extended impurity modes in an electrical lattice, providing insights into their frequencies, spatial profiles, and responses to parameter changes.

Contribution

It offers a systematic analytical framework for impurity modes in electrical lattices and validates it through experiments, including quench dynamics.

Findings

Analytical expressions for impurity mode frequencies and profiles

Excellent agreement between theory and experiment

Understanding of quench experiment outcomes

Abstract

We examine theoretically and experimentally the localized %and extended electrical modes existing in a bi-inductive electrical lattice containing a bulk or a surface capacitive impurity. By means of the formalism of lattice Green's functions, we are able to obtain closed-form expressions for the frequencies of the impurity (bound-state) eigenmodes and for their associated spatial profiles. This affords us a systematic understanding of how these mode properties change as a function of the system parameters. We test these analytical results against experimental measurements, in both the bulk and surface cases, and find very good agreement. Lastly, we turn to a series of quench experiments, where either a parameter of the lattice or the lattice geometry itself is rapidly switched between two values or configurations. In all cases, we are able to naturally explain the results of such quench experiments from the larger analytical picture that emerges as a result of the detailed characterization of the impurity-mode solution branches.