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This paper claims that the global fermion parity of a delocalized superconducting state can be read out locally from the phase of a nonlocal Josephson response: in a carbon nanotube hosting two coupled quantum-dot Josephson junctions, the s

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In a carbon nanotube Andreev molecule, the phase of the nonlocal Josephson current flips by π when the global fermion parity of the molecular ground state changes.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection A solid experimental demonstration of the nonlocal Josephson effect in a nanotube Andreev molecule, with robust π-phase shifts; the parity interpretation is plausible but not yet secured. the 2 major comments →

arxiv 2607.15786 v1 pith:CW2EOTPE submitted 2026-07-17 cond-mat.mes-hall cond-mat.supr-conquant-ph

Fermion parity of an Andreev molecule probed by nonlocal Josephson effect

S. Annabi , H. Riechert , K. Watanabe , T. Taniguchi , J. Griesmar , E. Arrighi , L. Bretheau , J.-D. Pillet This is my paper
classification cond-mat.mes-hall cond-mat.supr-conquant-ph
keywords Andreev moleculefermion paritynonlocal Josephson effectcarbon nanotubequantum dotπ-phase shiftsuperconducting circuit
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.

The reading

The paper tries to establish that the global fermion parity of a delocalized superconducting state—an Andreev molecule—can be detected locally. It reports that the nonlocal Josephson effect, where the supercurrent through one junction is modulated by the phase across another, shows characteristic π-phase shifts when the molecule's ground-state parity changes. The contrast of this modulation is largest at avoided crossings, connecting the nonlocal response directly to wavefunction delocalization. A minimal double-quantum-dot model reproduces the π shifts and ties each one to a change in total ground-state charge. If correct, this makes global parity an experimentally accessible transport signature in hybrid superconducting circuits.

Core claim

The central claim is that the nonlocal Josephson effect in an Andreev molecule encodes two defining properties of the delocalized ground state: the spatial delocalization, revealed by the strength of the nonlocal response, and the global fermion parity, revealed by the phase of the periodic response. The experiment shows periodic modulation of the left junction's critical current (device B) or zero-bias conductance (device T) with the right-junction phase; the modulation phase flips by π at boundaries where the ground-state charge changes by one electron. This behavior is reproduced by a minimal model of two coupled quantum dots with induced pairing, where each π-phase shift coincides with a

What carries the argument

The key object is the Andreev molecule: two quantum-dot Josephson junctions coupled through a common superconducting electrode, described by a minimal double-quantum-dot Hamiltonian with on-dot Coulomb repulsion U, induced pairing Γ, asymmetry δΓ, and inter-dot tunneling t. The nonlocal response is the critical current of the left junction, Ic(δR) = max_δL I_L(δL, δR), where I_L = (2e/ℏ)∂E0/∂δL is obtained from the exact ground-state energy of the full many-body Hilbert space. The phase of Ic(δR) is tied to the total occupation ⟨n_L+n_R⟩; a one-electron change in that occupation flips the sign of the supercurrent modulation, producing the observed π-phase shift.

Load-bearing premise

The central claim rests on the assumption that each observed π-phase shift corresponds to a genuine change in the global fermion parity of the Andreev-molecule ground state, as inferred from a deliberately minimal model that omits inter-dot Coulomb interactions and nonlocal pairing terms.

What would settle it

A measurement that simultaneously tracks the ground-state charge (for instance via a nearby charge sensor) and the phase of the nonlocal response: if the response flips by π without the total dot charge changing by one electron, or if adding inter-dot Coulomb repulsion and nonlocal pairing to the model removes the phase flip without removing a parity crossing, the parity interpretation is falsified.

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

If this is right

  • The phase of the nonlocal Josephson response can serve as a local, transport-based readout of the global fermion parity of a delocalized Andreev molecule.
  • Strong nonlocal Josephson coupling arises from hybridization: the modulation contrast is maximal at avoided crossings and can approach unity, nearly suppressing one junction's supercurrent by changing the phase across the other.
  • Parity changes in the molecule produce sharp π-phase shifts in the nonlocal response, analogous to 0–π transitions but appearing in a nonlocal signal.
  • The effect is robust to the detection scheme: it appears both in switching-current measurements and in zero-bias conductance.
  • Extending the architecture to larger superconducting networks could enable controlled parity of delocalized fermionic states relevant for topological phases and quantum information processing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inference: If the parity–phase mapping holds, a circuit-QED version of this device could perform high-frequency, quantum non-demolition readout of global parity by measuring the phase response dispersively, not just by switching-current measurements.
  • Inference: Because the nonlocal contrast tracks wavefunction delocalization, the same technique could serve as a spatial coherence probe in multi-junction networks, mapping hybridization beyond the simplest dimer case.
  • Inference: The minimal model deliberately omits inter-dot Coulomb interaction (ULR) and nonlocal pairing (ΓLR); including those terms may shift the quantitative location or sharpness of the π flips, so a direct quantitative fit to the measured phase boundaries would test whether parity alone explains the data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The manuscript reports the observation of a nonlocal Josephson effect in carbon-nanotube-based Andreev molecules formed by two coupled quantum-dot Josephson junctions. Device B shows periodic modulation of the left-junction switching current with the flux-controlled phase difference across the right junction, with contrast up to near unity. In device T, local gates reveal a double-dot stability diagram; the nonlocal contrast is largest at avoided crossings, interpreted as delocalization of Andreev molecular states. In a strong-pairing regime, the nonlocal response exhibits π-phase shifts across boundaries inferred as global parity transitions, reproduced in both devices and qualitatively captured by a minimal double-dot model that correlates phase shifts with changes in the ground-state total charge parity. The authors conclude that the nonlocal Josephson effect can locally probe global fermion parity.

Significance. If the central claim holds, the paper would establish the nonlocal Josephson effect as a local probe of global fermion parity in delocalized Andreev states, with potential relevance for parity readout in hybrid superconducting circuits. The experimental strengths are substantial: two devices, two detection schemes (switching current and zero-bias conductance), validation by switching-probability measurements, a quantitative flux-period calibration to Φ0, and a clear qualitative separation between weak- and strong-pairing stability diagrams. The main weakness is that the parity interpretation is not independently verified and rests on a simplified model in which the crossed-Andreev pairing Γ_LR is set to zero without an estimate or robustness check.

major comments (2)
  1. [Methods C, Eq. (4)] The model sets U_LR=0 and, more importantly, Γ_LR=0, with the justification that neither is required to reproduce the central observations. This is the key weak point for the parity claim. The experiment does not measure parity independently; it infers parity from the gate positions of the observed π-phase shifts. Γ_LR is not a perturbative detail: it is a crossed-Andreev term d†_{L↑}d†_{R↓} that directly couples the two dots and contributes its own δ_R-dependent nonlocal Josephson current. Such a term can produce a gate-dependent sign change of the nonlocal coupling, which would appear as a π-phase shift with no change of total fermion parity. Since no estimate of Γ_LR from the device geometry is given, and no calculation with finite Γ_LR is shown to preserve the phase-to-parity correspondence, the central attribution is an assumption rather than a demonstrated property. An independent
  2. [Sec. IV, Fig. 4f] The parameters used in Fig. 4f (Γ=0.5, δΓ=0.25, t=2, U=5) are hand-picked, with no fit or uncertainty budget, and the manuscript explicitly states that the model is not meant to be quantitative. The statement that 'each phase shift indeed coincides with a change of the total charge by one electron' is therefore a property of this particular parameter point, not of the measured device. To make the parity interpretation load-bearing, the authors should demonstrate that the coincidence between phase shifts and charge-parity changes persists over a broad parameter range, including finite Γ_LR and U_LR, or provide an independent parity-sensitive measurement. Without this, Figs. 4b–e establish reproducible phase jumps, but not that these are parity jumps.
minor comments (3)
  1. [Supplementary IV and VI] Several passages contain garbled font-encoding artifacts (e.g., '/uni00000014/uni00000011/...' and '/uni00000010/...') where equations or axis labels should appear. These must be repaired before publication.
  2. [Fig. 4c] The phase assignment (max vs min at zero flux) is described qualitatively. A fitted phase map or overlaid parity boundaries on the conductance map would make the correlation between phase shifts and inferred parity transitions easier to assess quantitatively.
  3. [Sec. III, C_G] The conductance contrast C_G is said to be 'directly analogous' to Eq. (1), but the exact normalization used for conductance is not given. Please state explicitly whether the same min/max formula is applied to the zero-bias conductance traces.

Circularity Check

0 steps flagged

No significant circularity: the experimental phase-shift data are external, the model parameters are not fitted to those data, and the parity mapping is derived from an independent ground-state charge calculation.

full rationale

The central claim is that π-phase shifts in the nonlocal Josephson response track global fermion parity. The experimental evidence (periodic Ic,L(δR) modulations, contrast maxima at avoided crossings, abrupt π shifts across gate-tuned boundaries) is directly measured, not generated by the model. In Methods C the model is explicitly non-quantitative: 'The purpose of the model is not to provide a quantitative description of the device...' and parameters are 'chosen to capture the qualitative regimes observed experimentally rather than to provide a quantitative fit' (Supplementary VI). The model's phase-to-parity connection is not fitted: the critical current Ic(δR)=max_δL (2e/ℏ ∂E0/∂δL) and the ground-state charge N=Σ⟨Ψ0|nασ|Ψ0⟩ are both computed from the same diagonalized Hamiltonian, and the coincidence of their changes is a derived consistency statement, not an input constraint. No fitted parameter is renamed as a prediction. The paper omits U_LR and Γ_LR with stated justification; that is a modeling approximation relevant to correctness, not circularity. Self-citations are used for device fabrication (Ref. 45) and prior Andreev-molecule platform work (Refs. 39,41,45) but are not the load-bearing justification for the parity interpretation; no uniqueness theorem or ansatz is smuggled in via self-citation. The derivation is therefore self-contained with respect to circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim rests on standard proximity-effect modeling and on the qualitative model of two coupled dots with induced pairing. The model introduces four hand-chosen energy parameters (Γ, δΓ, t, U) that set the regime but are not fitted to the data. No new particles, forces, or mediators are introduced. The main burden is the assumption that the minimal model, with U_LR = Γ_LR = 0, captures the parity dependence of the nonlocal response.

free parameters (4)
  • Γ_L = Γ_R (induced pairing amplitude) = 0.5 (Fig 4f); 1 (weak) / 2 (strong) in Supp Fig S6
    Chosen by hand to reproduce the weak- and strong-induced-pairing regimes of the stability diagrams; the paper explicitly states the model is not a quantitative fit.
  • δΓ_L = δΓ_R (left-right asymmetry of pairing) = 0.25 (Fig 4f); 0.5 / 1 in Supp Fig S6
    Hand-chosen asymmetry parameter in the pairing term Ĥ_S; controls the phase-dependence of the induced pairing.
  • t (interdot tunnel coupling) = 2 (Fig 4f); 3 (weak) / 1 (strong) in Supp Fig S6
    Chosen to produce avoided crossings of qualitatively the right size in the stability diagrams.
  • U_L = U_R (dot charging energy) = 5 (Fig 4f); 10 (weak) / 5 (strong) in Supp Fig S6
    Coulomb scale chosen to qualitatively match the charge stability patterns and the parity-island structure.
axioms (5)
  • domain assumption Proximity-induced local pairing on each dot is described by the BCS-like term Ĥ_S = Σ_α (Γ_α cos(δ_α/2) + i δΓ_α sin(δ_α/2)) d†_α↑ d†_α↓ + h.c.
    Standard mean-field description of superconducting proximity effect on a quantum dot, invoked in Methods C. All parity conclusions depend on this form of induced pairing.
  • domain assumption The right junction is phase-biased by a flux loop with negligible loop inductance (L_g ≈ 120 pH << L_J ≈ 30 nH).
    Methods B. This justifies that δ_R is directly controlled by the applied magnetic flux; if the assumption fails, the flux-to-phase relation would be distorted.
  • domain assumption Zero-bias conductance of device T is a faithful proxy for the Josephson critical current in the strong phase-diffusion regime.
    Supplementary II. All device T measurements of the nonlocal response and its π-phase shifts use this proxy; if the conductance were dominated by normal-state effects, the parity conclusions would be compromised.
  • ad hoc to paper Interdot Coulomb interaction (U_LR) and nonlocal pairing (Γ_LR) are negligible for the central observations; the paper sets U_LR = 0 and Γ_LR = 0.
    Methods C. The paper states these omissions 'are not required to reproduce the central observations', but this claim is not tested against models that include them; the parity assignment could change if these terms matter.
  • domain assumption Carbon nanotube transport is ballistic and effectively one-dimensional, providing clean single-mode weak links.
    Supplementary I (Fabry-Pérot interference). This justifies the double-quantum-dot description with well-defined Andreev levels and no significant multimode broadening.

reviewed 2026-08-01 · how reviews work

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Cite this review

Pith. "Pith review of Fermion parity of an Andreev molecule probed by nonlocal Josephson effect." pith.science (2026). https://pith.science/paper/CW2EOTPE

@misc{pith2026260715786,
  author       = {Pith},
  title        = {Pith review of: Fermion parity of an Andreev molecule probed by nonlocal Josephson effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CW2EOTPE}},
  note         = {Machine review of arXiv:2607.15786}
}
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abstract

Fermion parity is a fundamental property of superconducting many-body states. Here, we show that the global fermion parity of a delocalized superconducting state can be detected locally by exploiting the nonlocal Josephson effect. Using a carbon nanotube-based Andreev molecule formed by two coupled quantum-dot Josephson junctions, we observe a pronounced nonlocal Josephson response and demonstrate the formation of delocalized Andreev molecular states extending across both junctions. We further show that changes in the molecular ground-state parity manifest as characteristic $\pi$-phase shifts in the nonlocal response. Supported by a minimal theoretical model, these results identify global fermion parity as an experimentally accessible degree of freedom in hybrid superconducting circuits that can be readily revealed through the nonlocal Josephson effect.

Figures

Figures reproduced from arXiv: 2607.15786 by E. Arrighi, H. Riechert, J.-D. Pillet, J. Griesmar, K. Watanabe, L. Bretheau, S. Annabi, T. Taniguchi.

Figure 1
Figure 1. Figure 1: Fermion parity detection in a carbon nan￾otube Andreev molecule. (a) Schematic of an Andreev molecule formed by two quantum-dot Josephson junctions (blue and red) coupled through a common superconduct￾ing electrode (grey). The hybridized molecular state |Ψhyb⟩ (purple) extends across both junctions and mediates a non￾local Josephson effect (NLJE), whereby the supercurrent IL through the left junction depen… view at source ↗
Figure 2
Figure 2. Figure 2: Nonlocal Josephson effect. (a)-(b) Current￾voltage characteristics IL(V ) measured at different back-gate voltages Vg. The current is swept from negative to positive values, and the switching current Isw,L is defined as the maxi￾mum current sustained by the supercurrent branch. (c) Isw,L as a function of the flux current IδR controlling the phase difference δR across the right junction, shown at three diff… view at source ↗
Figure 3
Figure 3. Figure 3: Hybridization of Andreev molecular states. (a) Zero-bias conductance GL of the left junction in device T averaged over the flux current IδR as a function of the local gate voltages Vg,L/R. Resonance lines correspond to the align￾ment of discrete quantum-dot levels with the chemical po￾tential of the superconducting electrodes, providing access to the electronic spectrum of the double quantum dot. Conduc￾ta… view at source ↗
Figure 4
Figure 4. Figure 4: Probing the global fermion parity through the nonlocal Josephson effect. (a) Zero-bias conductance GL of device T averaged over IδR , as a function of the local gate voltages Vg,L/R. The highlighted region exhibits superconducting avoided crossings connecting charge states differing by two electrons, specified by (nL, nR). (b) Flux-dependent nonlocal response measured at two operating points of opposite pa… view at source ↗

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