Consistent Quantization of Nearly Singular Superconducting Circuits

TL;DR

This paper investigates the quantization of nearly singular superconducting circuits, revealing limitations of traditional methods and proposing a perturbative approach that captures universal low-energy dynamics.

Contribution

It demonstrates the failure of Dirac-Bergmann theory for these circuits and introduces a Born-Oppenheimer perturbation method with rigorous validation for accurate quantization.

Findings

Dirac-Bergmann theory fails for nearly singular circuits.

Born-Oppenheimer analysis provides a correct perturbative quantization.

Low-energy dynamics converge to universal fixed points as parasitic capacitances vanish.

Abstract

The theory of circuit quantum electrodynamics has successfully analyzed superconducting circuits on the basis of the classical Lagrangian, and the corresponding quantized Hamiltonian, describing these circuits. In many simplified versions of these networks, the modeling involves a Lagrangian that is singular, describing an inherently constrained system. In this work, we demonstrate the failure of the Dirac-Bergmann theory for the quantization of realistic, nearly singular superconducting circuits, both reciprocal and nonreciprocal. The correct treatment of nearly singular systems involves a perturbative Born-Oppenheimer analysis. We rigorously prove the validity of the corresponding perturbation theory using Kato-Rellich theory. We find that the singular limit of this regularized analysis is, in many cases, completely unlike the singular theory. Dirac-Bergmann, which uses the Kirchhoff's (and Tellegen's) laws to deal with constraints, predicts dynamics that depend on the detailed parameters of nonlinear circuit elements, e.g., Josephson inductances. By contrast, the limiting behavior of the low-energy dynamics obtained from the regularized Born-Oppenheimer approach exhibits a fixed point structure, flowing to one of a few universal fixed points as parasitic capacitance values go to zero.