Exact quantization of nonreciprocal quasi-lumped electrical networks

TL;DR

The paper develops an exact, geometry-based framework for canonically quantizing nonlinear, nonreciprocal quasi-lumped electrical circuits, extending previous work to include transmission lines and nonreciprocal devices. Central to the approach is the Faddeev–Jackiw reduction of Kirchhoff constraints, yielding a reduced state manifold that mixes flux and charge fields and provides a UV cutoff, thereby avoiding renormalization. The authors demonstrate the method on TLs coupled to one- and multiport networks, NR blackboxes, and dissipative environments within the Caldeira–Leggett picture, deriving explicit canonical Lagrangians and Hamiltonians and showing how to diagonalize and quantize the resulting systems. A key outcome is that the resulting models are finite and well-behaved at high frequencies, with solid ground for applications in circuit QED and waveguide QED platforms. The work thus furnishes a robust toolkit for designing and analyzing nonreciprocal superconducting networks with precise input–output theory.

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.

Figures (20)

• Figure 1: Outside of the cloud, the set of dynamical lumped elements that are characterized by energy functions (energetic constitutive equations) such as (nonlinear) inductors ($\mathcal{L}$) and capacitors ($\mathcal{C}$), voltage sources ($\mathcal{V}$), and current sources ($\mathcal{I}$). Within the cloud, connections among these elements are implemented by constraints such as Kirchhoff's laws (KCL and KVL), and two more general ones, ideal transformers ($\mathcal{T}_F$), recently known as energy-participation ratios Nigg:2012Minev:2021bCiani:2023, and gyrators ($\mathcal{G}$), both of which can be used to express more general constraints between fluxes and charges effectively breaking time-reversal symmetry, see ParraRodriguez:2024a. As to gyrators, we depict the canonical two-port nonreciprocal element to represent its class.
• Figure 2: Lumped element circuit example: an LC oscillator is capacitively coupled to both a Josephson junction and a voltage source. Given that in this circuit there are only Kirchhoff's constraints, node fluxes and loop charges are a good basis for all the branch fluxes and charges.
• Figure 3: (a) A 2 active-node lumped-element $LC$ network is (b) deformed into two multiport capacitive and inductive linear elements connected in parallel, and characterized by $2\times 2$ (full-rank) $\mathop{\mathrm{\mathsf{C}}}\nolimits$ and $\mathop{\mathrm{\mathsf{L}}}\nolimits$ matrices, respectively. (c) Following standard circuit theory, the non-diagonal (possibly, even singular) capacitance matrix can be further decomposed in two capacitors connected to an ideal transfomer Newcomb:1966.
• Figure 4: The nonreciprocal circuit comprising capacitors, inductors, pure Josephson elements, and a gyrator introduced in Ref. Rymarz:2021 to stabilize GKP states.
• Figure 5: A microscopically-consistent topological ansatz for the branch manifold of lumped elements was put forward in Ref. ParraRodriguez:2024a. Here,