Linear impurity modes in an electrical lattice: Theory and Experiment

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.