Exact quantization of nonreciprocal quasi-lumped electrical networks

TL;DR

This paper introduces an exact, geometrically consistent method for quantizing nonlinear, nonreciprocal electrical networks, resolving singularities and extending canonical quantization to complex circuit configurations.

Contribution

It provides a systematic approach for Hamiltonian descriptions of nonreciprocal circuits, including solutions to singular cases and extensions to dissipative environments, advancing quantum circuit analysis.

Findings

Unified framework for quantizing nonreciprocal networks

Resolution of singularities in Hamiltonian formulations

Extension to dissipative and frequency-dependent environments

Abstract

Following a consistent geometrical description previously introduced in Parra-Rodriguez et al. (2024), we present an exact method for obtaining canonically quantizable Hamiltonian descriptions of nonlinear, nonreciprocal quasi-lumped electrical networks. We identify and classify singularities arising in the quest for Hamiltonian descriptions of general quasi-lumped element networks via the Faddeev-Jackiw technique. We offer systematic solutions to cases previously considered singular--a major challenge in the context of canonical circuit quantization. The solution relies on the correct identification of the reduced classical circuit-state manifold, i.e., a mix of flux and charge fields and functions. Starting from the geometrical description of the transmission line, we provide a complete program including lines coupled to one-port lumped-element networks, as well as multiple lines connected to multiport nonreciprocal lumped-element networks, with intrinsic ultraviolet cutoff. On the way we naturally extend the canonical quantization of transmission lines coupled through frequency-dependent, nonreciprocal linear systems, such as practical circulators. Additionally, we demonstrate how our method seamlessly facilitates the characterization of general nonreciprocal, dissipative linear environments. This is achieved by extending the Caldeira-Leggett formalism, using continuous limits of series of immittance matrices. We provide a tool in the analysis and design of electrical circuits and of special interest in the context of canonical quantization of superconducting networks. For instance, this work will provide a solid ground for a precise non-divergent input-output theory in the presence of nonreciprocal devices, e.g., within (chiral) waveguide QED platforms.