Consistent Quantization of Nearly Singular Superconducting Circuits

arXiv (Cornell University)

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.

Authors 2

1. Martin Rymarz Aachen JARA-Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   JARA-Institute for Quantum Information , RWTH Aachen University , D-52056 Aachen , Germany

2. David Peter Divincenzo Aachen JARA-Institute for Quantum Information

   RWTH Aachen University · Forschungszentrum Jülich

   Affiliation as printed

   JARA-Institute for Quantum Information , RWTH Aachen University , D-52056 Aachen , Germany

   Peter Grünberg Institute , Theoretical Nanoelectronics , Forschungszentrum Jülich , D-52425 Jülich , Germany