Inconsistencies in the Quantization of Singular Superconducting Circuits

The theory of circuit quantum electrodynamics successfully describes superconducting circuits based on the Hamiltonian formalism of quantum mechanics. In the process, mathematical descriptions of an electrical network at hand might involve effective models, which, however, can easily lead to singular Lagrangians that describe constrained systems in which not all variables are independent.

Here, we apply the Dirac-Bergmann algorithm to derive the Hamiltonian of constrained electrical networks. We compare the results from the singular treatment with the low-energy dynamics obtained from a Born-Oppenheimer approach describing the electrical network with lifted singularities. We demonstrate inconsistencies between these two approaches, and we propose effective replacement rules for different classes of singular networks. Finally, we also revise frequently applied rules for the analysis of classical non-reciprocal electrical networks.