The influence of the elementary charge on the canonical quantization of LC-circuits

Abstract

In this paper one deals with the quantization of mesoscopic LC-circuits under the influence of an external time dependent voltage. The canonically conjugated variables, such as given by the electric charge and the magnetic flux, get established by resorting to the Hamiltonian equations of motion provided by both Faraday and Kirchhoff laws. This time the discretization of the electric charge is accounted for, so that magnetic flux operators one looks for should proceed in terms of discrete derivatives. However, the flux operators one deals with are not Hermitian, which means that subsequent symmetrizations are in order. The eigenvalues characterizing such operators can be readily established in terms of twisted boundary conditions. Besides the discrete Schrödinger equation with nearest-neighbor hopping, a nontrivial next nearest neighbor generalization has also been established. Such issues open the way to the derivation of persistent currents in terms of effective k-dependent Hamiltonians. Handling the time dependent voltage within the nearest neighbor description leads to the derivation of dynamic localization effects in L-ring configurations, such as discussed before by Dunlap and Kenkre. The onset of the magnetic flux quantum has also been discussed in some more detail.