On "Consistent Quantization of Nearly Singular Superconducting Circuits"

TL;DR

This paper critically examines Rymarz and DiVincenzo’s 2023 claims that Kirchhoff-based variable elimination must be discarded for quantum circuit quantization. It emphasizes that RD’s Theorems assume extended (unbounded) flux variables and do not straightforwardly apply to compact (S^1) flux scenarios, where intrinsic scales and global topology alter the reduction. Through a classical slow-manifold analysis and Born-Oppenheimer considerations, the authors show that for extended φ_c the standard reduction can be consistent with quantization in the β<1 regime, while for compact φ_c the RD framework is inapplicable and its conclusions do not follow. They also point to substantial experimental support for conventional lumped-element models and nonsingular β<1 circuits, arguing that RD’s sweeping generalizations are not warranted. The work concludes that the extended-vs-compact debate remains nuanced and that careful, case-by-case analysis is essential for reliable quantum circuit descriptions.

Abstract

The analysis conducted by Rymarz and DiVincenzo (Phys. Rev. X 13, 021017 (2023)) regarding quantization of superconducting circuits is insufficient to justify their general conclusions, most importantly the need to discard Kirchhoff's laws to effect variable reductions. Amongst a variety of reasons, one source of several disagreements with experimental and theoretical results is the long-standing dispute between extended vs compact variables in the presence of Josephson junctions.

Figures (5)

• Figure 1: (a) In electrical circuit theory, a generic two-terminal (1-port) lumped element is described by two variables: a branch charge and a branch flux, and a constitutive (possibly nonlinear and nonlocal in time) relation between them Chua:1980. (b) When working with many-element circuits, there is a strong general interest in obtaining approximating low-energy descriptions with (c) fewer elements. For instance, in superconducting circuits, low-energy models with few nonlinear lumped elements have been used Manucharyan:2009Yan:2020Ye:2021 when capacitive parasitic effects have been neglected altogether Miano:2023.
• Figure 2: (a) The main results of RD (Theorems 1-4) pertain to the limit $C'\rightarrow 0$ for different classes of nonlinear inductors, all of them described by nonlinear constitutive relations between their branch charge $q_{\text{nl}}$ and flux $\phi_{\text{nl}}$. (b) According to the classification in RD, the pure Josephson element (with its parallel parasitic capacitance $C_J$), described by the nonlinear relation $\dot{q}_J=E_J\left(\frac{2\pi}{\Phi_Q}\right)\sin\left(\frac{2\pi \phi_J}{\Phi_Q}\right)$, is a type 1a inductor. Consequently, Theorem 1 should be applicable. RD explicitly shows that the circuit in (c), i.e., (b) without the Josephson parasitic capacitance, is singular in two ways. First, in having a defective kinetic term in its flux presentation. Second, in that obstructions to variable reduction exist for some parameter regimes of $L$ and $E_J$. They attribute this failure to the Dirac-Bergmann algorithm.
• Figure 3: RD performs a quantum adiabatic approximation (QAA) to the circuit in the top of the figure by taking the limit $C' \rightarrow 0$ while assuming that the phase space before quantization is $T^*\mathcal{X} = \mathbb{R}^4$, which includes $\phi_c \in \mathbb{R}$. RD asserts that the application of their Theorems 1 and 2 (Theorem 1 includes the unbiased Josephson junction, while Theorem 2 involves non-symmetric bounded potentials, such as SQUIDs and SNAILs) provides a general effective low-energy Hamiltonian dynamics approximated by the orange circuit for inductors of type 1, independent of the other parameters ($C$ and $L$). They also assert that such a limit is in strong contradiction to the specific case of a Josephson junction when the capacitance has been removed before the quantization ($C' = 0$), resulting in the effective circuit model in green for $\beta \equiv L E_J (2\pi/\Phi_Q)^2 \leq 1$. We show in the main text that for $\beta < 1$: (i) there is no such contradiction between the two models under their assumption, and (ii) a simple classical adiabatic approximation (CAA, black-to-green arrow) also renders the same model.
• Figure 4: (a) A conceptual sketch layout of a Josephson junction (small gap) shunted by a large capacitance (big gap) can have increasingly precise lumped element models accounting for various capacitive and inductive effects. In a first-order model, the response of the shunting linear environment is often approximated (e.g., Koch:2007 and subsequent literature) by only the pole at $s=-i\infty$ of the admittance function $\tilde{Y}(s)$. (b) In a second-order approximation, the inductive response of the superconducting loop can be added, reaching the motivating circuit used in RD. Note that here, and in the circuits of Figs. 3 and 4 of RD, the capacitive effects in parallel with the inductance $L$ are classically neglected before any quantum analysis is performed Rymarz:2018. (c) More systematically, the linear response of a one-port lossless environment can be modeled to the desired level of precision with a series sum of poles at infinity ($C_J$), zero ($L_0$), and finite frequencies (here, a first pole at $\omega=1/\sqrt{LC}$) Newcomb:1966Nigg:2012Solgun:2014Solgun:2015. Note that, from the purely mathematical blackbox perspective, not all the nodes of a lumped-element circuit must be connected through capacitors, nor do all the connecting wires have an associated inductance.
• Figure 5: Summary of the results from RD and this work. RD claims the failure of the classical circuit reduction methods (DB, FJ, etc.) due to the impossibility of obtaining standard Hamiltonian dynamics for an ill-posed classical circuit. E.g., for the circuit with a Josephson junction without its parasitic capacitance ($C'=0$), when $\beta>1$. Puzzled by this fact, they explore an alternative route, via a quantum adiabatic approximation. In particular, applying Theorem 1 to the case at hand, (JJ as a nonlinear inductor), they obtain low-energy open circuit dynamics (with an associated continuous spectrum), depicted in orange in the figure, which work, as we have proven, only under the assumption that $\phi_c\in \text{R}$. In addition, we have proven how their obtained spectrum matches that of the quantization of a model obtained under a classical adiabatic approximation for $\beta<1$ (left green circuit). In addition, we have shown how those spectra are fundamentally different from those obtained (via adiabatic analysis) under the assumption of $\phi_c\in S^1$, both classically and (naïvely) quantum mechanically (lower right-hand green and red circuits).