Canonical Quantization of Superconducting Circuits

TL;DR

This paper develops mathematically rigorous Hamiltonian models for superconducting circuits, including nonreciprocal elements, enabling precise quantization and analysis of complex quantum superconducting networks.

Contribution

It introduces a systematic method to quantize superconducting circuits with arbitrary elements, including nonreciprocal devices, and clarifies previous misunderstandings in the literature.

Findings

Models are finite with no divergence issues.

Extended standard theory to include nonreciprocal elements.

Demonstrated quantization of frequency-dependent gyrators and circulators.

Abstract

In the quest to produce quantum technology, superconducting networks, working at temperatures just above absolute zero, have arisen as one of the most promising physical implementations. The precise analysis and synthesis of such circuits have required merging the fields of physics, engineering, and mathematics. In this dissertation, we develop mathematically consistent and precise Hamiltonian models to describe ideal superconducting networks made of an arbitrary number of lumped elements, such as capacitors, inductors, Josephson and phase-slip junctions, gyrators, etc., and distributed ones like transmission lines. We give formal proofs for the decoupling at high and low frequencies of lumped degrees of freedom from infinite-dimensional systems in different coupling configurations in models based on the effective Kirchhoff's laws. We extend the standard theory to quantize circuits that include ideal nonreciprocal elements all the way to their Hamiltonian descriptions in a systematic way. Finally, we pave the way on how to quantize general frequency-dependent gyrators and circulators coupled to both transmission lines and other lumped-element networks. We have explicitly shown, that these models, albeit ideal, are finite and present no divergence issues. We explain and dispel misunderstandings from the previous literature. Furthermore, we have demonstrated the usefulness of a redundant basis for performing separation of variables of the transmission line (1D) fields in the presence of point-like (lumped-element) couplings by time-reversal symmetry-breaking terms, i.e. nonreciprocal elements.