Low-energy spectrum of double-junction superconducting circuits in the Born-Oppenheimer approximation

David Feldstein-Bofill; Karsten Flensberg; Ksenia Shagalov; Leo Uhre Jakobsen; Morten Kjaergaard; Svend Kr{\o}jer

REVIEW 2 major objections 3 minor 1 cited by

A Born-Oppenheimer reduction turns a double Josephson junction into an accurate single-mode qubit with a correction from its internal mode.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-14 20:07 UTC pith:XMGZ75XT

load-bearing objection Useful BO reduction of a shunted double-junction element with an internal-mode correction and honest caveats on PBCs/non-uniqueness; body text is unreadable so we cannot audit the math or numerics. the 2 major comments →

arxiv 2603.26374 v2 pith:XMGZ75XT submitted 2026-03-27 quant-ph

Low-energy spectrum of double-junction superconducting circuits in the Born-Oppenheimer approximation

Leo Uhre Jakobsen , Ksenia Shagalov , David Feldstein-Bofill , Morten Kjaergaard , Karsten Flensberg , Svend Kr{\o}jer This is my paper

classification quant-ph PACS 85.25.Cp74.50.+r03.67.Lx

keywords Josephson junctiondouble-junction circuitBorn-Oppenheimer approximationsuperconducting qubithigher harmonicscharge noiseeffective single-mode model

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Two Josephson junctions in series create a nonlinear circuit element whose energy-phase relation contains higher harmonics useful for multimode superconducting designs, but the series pair also hosts an internal mode fixed by the junctions' own capacitances. This paper treats that internal mode as fast relative to the slow qubit mode set by a large shunt capacitor, and applies a Born-Oppenheimer approximation to eliminate it. The result is an effective single-mode Hamiltonian for the qubit that includes an explicit correction generated by the internal degree of freedom. In parameter regimes typical of present experiments, numerical comparisons show that the reduced model reproduces the low-energy spectrum of the full circuit. The authors also show that removing the internal mode changes how periodic boundary conditions must be imposed and that the approximation is not unique, and they quantify the harmonic content of the double-junction element together with its residual charge-noise sensitivity. A reader who designs superconducting circuits cares because the reduction supplies a practical, accurate description that keeps the design advantages of higher harmonics while restoring a single-mode picture.

Core claim

When a double-junction element is shunted by a large capacitor, a Born-Oppenheimer treatment that regards the internal mode as fast yields an effective single-mode qubit Hamiltonian containing a correction term from that mode; numerical diagonalization confirms that this reduced model accurately captures the low-energy spectrum in experimentally relevant regimes.

What carries the argument

The Born-Oppenheimer approximation that freezes the fast internal mode (set by individual junction capacitances) and produces an effective single-mode potential for the slow qubit mode, including the correction generated by adiabatic elimination of the internal degree of freedom.

Load-bearing premise

The whole reduction rests on a clean separation of timescales: the internal mode must be fast enough compared with the qubit mode that the Born-Oppenheimer elimination remains valid.

What would settle it

Compute or measure the low-energy spectrum of a large-capacitor-shunted double-junction circuit in a regime where the internal-mode frequency is deliberately lowered toward the qubit frequency; if the effective single-mode model systematically fails to match the spectrum once the timescale separation is lost, the claim is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Useful BO reduction of a shunted double-junction element with an internal-mode correction and honest caveats on PBCs/non-uniqueness; body text is unreadable so we cannot audit the math or numerics.

read the letter

The one thing worth knowing is that this is a standard circuit-QED adiabatic-elimination paper aimed at a real design niche: series double junctions that give higher harmonics, but that also bring an internal mode set by the individual junction capacitances. They treat that mode as fast, derive an effective single-mode qubit Hamiltonian with an explicit correction from the internal degree of freedom, and claim numerical agreement for the low-energy spectrum when a large shunt capacitor enforces the timescale separation. They also flag that eliminating the internal mode messes with periodic boundary conditions and makes the BO procedure non-unique, then look at harmonic content and charge-noise sensitivity.

What is actually new is modest but concrete: the correction term itself, the PBC/non-uniqueness discussion, and the packaging of those caveats for people who want to use double-junction elements as design primitives. The workflow is clean on paper—two-mode Hamiltonian, timescale separation, reduced model, spectrum check in experimentally relevant regimes. No circular fitting is advertised; free parameters are the usual circuit ones (EJ, Cj, Cshunt, asymmetry). That is the right shape for this kind of claim.

The soft spots are real but proportional. The load-bearing premise is the clean separation between the internal mode and the qubit mode; if capacitances do not enforce it, the reduced model fails. The abstract states that condition and limits the accuracy claim to regimes where it holds, so this is not hidden. The bigger practical problem is that the supplied full text is garbled (encoding/OCR garbage plus a mismatched arXiv header), so we cannot check the derivation of the correction, the numerics, the figures, or the citation pattern. That is a verification barrier, not an internal contradiction in the argument as stated. Novelty is extension-level; BO elimination of fast modes is already common in multi-mode Josephson circuits.

This is for circuit designers and theorists who already work with higher-harmonic junctions or multimode qubits and need a usable effective Hamiltonian plus the boundary-condition warnings. It is not a broad conceptual advance. I would still send it to peer review: the abstract-level claim is plausible, the validation shape is correct, and the caveats look honest. A referee with the clean PDF can decide whether the numerics and the non-uniqueness discussion hold up. Engage if you care about double-junction design; otherwise file it as a solid specialized note once the body is readable.

Referee Report

2 major / 3 minor

Summary. The manuscript analyzes a double-junction superconducting circuit element shunted by a large capacitor, focusing on the internal mode set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats this internal mode as fast relative to the qubit mode, the authors derive an effective single-mode model of the qubit that includes a correction term arising from the internal degree of freedom. They report numerical agreement between this reduced model and the low-energy spectrum of the full circuit in experimentally relevant parameter regimes. The work also discusses how elimination of the internal mode affects periodic boundary conditions (leading to non-uniqueness of the Born-Oppenheimer procedure) and analyzes the harmonic content of the double-junction energy-phase relation together with its charge-noise sensitivity.

Significance. If the derivation and numerical validation hold, the paper supplies a practical reduced description for a circuit element that is already of design interest because of its higher-harmonic content. An effective single-mode Hamiltonian with an explicit correction from the internal mode would be useful for multimode circuit design and for assessing when the internal mode can be safely integrated out. The discussion of periodic-boundary-condition non-uniqueness is a conceptually interesting byproduct of the elimination procedure. The abstract frames the accuracy claim as limited to regimes with a clean timescale separation, which is the appropriate scope for a Born-Oppenheimer reduction. Because the supplied full-text stream is severely corrupted, independent verification of the correction term, the numerics, and the PBC analysis is not possible from the material provided; the significance assessment therefore rests on the abstract’s stated claims and the standard shape of the workflow.

major comments (2)

The full manuscript text supplied for review is unreadable (encoding/OCR corruption, repeated garbage blocks, and an arXiv header that does not match the stated paper). Consequently the central derivation of the effective single-mode model, the explicit form of the correction term, the numerical spectrum comparisons, and the periodic-boundary-condition/non-uniqueness discussion cannot be audited. A clean, complete manuscript is required before the load-bearing claims can be checked.
The validity of the Born-Oppenheimer reduction rests on a clean timescale separation between the internal double-junction mode and the qubit mode. The abstract correctly limits the accuracy claim to experimentally relevant regimes where this separation holds, but the manuscript must quantify the separation (e.g., ratios of plasma frequencies or capacitances) and show how the spectral error grows when the separation is reduced; without that, the domain of the effective model remains incompletely specified.

minor comments (3)

Once a clean manuscript is available, ensure that the effective Hamiltonian (including the correction term) is written explicitly and that the numerical comparisons are presented with clear parameter tables and error metrics.
Clarify notation for the two phase variables and the precise definition of the internal mode throughout the derivation so that the non-uniqueness discussion of the Born-Oppenheimer procedure is easy to follow.
The harmonic-content and charge-noise analysis should be cross-referenced to the effective model so that readers can see how the correction term modifies the usual single-junction Fourier content.

Circularity Check

0 steps flagged

No significant circularity: standard Born-Oppenheimer reduction of a two-mode circuit Hamiltonian, with independent numerical checks of the low-energy spectrum.

full rationale

The paper's claimed chain is: (i) write the two-mode double-junction + shunt Hamiltonian, (ii) impose a timescale separation so the internal junction mode is fast relative to the qubit mode, (iii) eliminate the fast mode via a Born-Oppenheimer procedure to obtain an effective single-mode qubit Hamiltonian that includes a correction from the internal mode, (iv) compare that reduced spectrum numerically to the full two-mode spectrum in experimentally relevant regimes, and (v) discuss side issues (periodic boundary conditions after elimination, non-uniqueness of the BO gauge choice, harmonic content, charge-noise sensitivity). None of these steps reduces the target spectrum to a fitted input by construction, nor does the abstract or readable framing rest on a load-bearing self-citation uniqueness theorem or a renamed empirical pattern. The numerical accuracy claim is an external check against the unreduced model, not a re-labeling of a fit. The load-bearing premise is a physical timescale separation (stated as a regime of validity), not a circular definition. Text corruption prevents equation-level audit of intermediate algebra, but nothing in the stated workflow matches the circularity patterns (self-definitional, fitted-as-prediction, self-citation uniqueness, ansatz smuggling, or renaming). Score 0 is therefore the honest finding.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

Central claim rests on standard circuit quantization of Josephson junctions, the Born-Oppenheimer/adiabatic elimination of a fast internal mode, and numerical diagonalization in chosen parameter regimes. No new particles or forces are invented. Free parameters are the usual circuit values (junction energies, capacitances, shunt C) chosen in 'experimentally relevant' windows rather than fitted to external data to force the claim. Domain assumptions include timescale separation and the validity of the lumped-element circuit model.

free parameters (1)

Experimentally relevant circuit parameters (E_J, C_j, C_shunt, asymmetry)
Numerical accuracy claims are made inside chosen parameter regimes typical of devices; specific values are not extractable from the corrupted text but act as the window where the effective model is asserted to work.

axioms (3)

domain assumption Lumped-element circuit quantization of SIS Josephson junctions with cosine energy-phase relation and capacitive charging terms is valid for the double-junction plus shunt circuit.
Standard superconducting-circuit premise underlying the full two-mode Hamiltonian before any reduction.
domain assumption The internal mode set by finite individual junction capacitances is sufficiently fast relative to the qubit mode that a Born-Oppenheimer elimination yields an accurate low-energy effective theory.
Load-bearing approximation stated in the abstract as the method of analysis.
standard math Standard mathematical machinery of adiabatic/Born-Oppenheimer reduction and numerical spectrum comparison is applicable to this circuit Hamiltonian.
Used to derive the effective single-mode model and validate it against the full spectrum.

pith-pipeline@v1.1.0-grok45 · 11360 in / 2736 out tokens · 27628 ms · 2026-07-14T20:07:37.280875+00:00 · methodology

0 comments

read the original abstract

The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. In experimentally relevant parameter regimes, we numerically find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.

discussion (0)

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