pith. sign in

arxiv: 2508.17367 · v2 · submitted 2025-08-24 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Revisiting the adiabatic limit in ballistic multiterminal Josephson junctions

R\'egis M\'elin , Asmaul Smitha Rashid , Romain Danneau , Morteza Kayyalha This is my paper

Pith reviewed 2026-05-18 21:40 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords multiterminal Josephson junctionsadiabatic limitFloquet-Kulik quartetsAndreev modescritical current oscillationsballistic transportnonequilibrium populations
0
0 comments X

The pith

In ballistic multiterminal Josephson junctions the relative number of quantum-correlated pairs from colliding Floquet-Kulik levels equals the inverse of the number of channels.

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

The paper explores the adiabatic limit in large ballistic multiterminal Josephson junctions as bias voltage increases and the electrochemical potential approaches the one-dimensional level spacing. It shows that colliding Floquet-Kulik quartet levels form a relative number of quantum-correlated pairs equal to one over the number of channels. This leads to a model treating the central two-dimensional normal metal as a continuum under the adiabatic approximation, with Andreev modes in voltage- and flux-tunable nonequilibrium populations. The approach identifies voltage scales that control mesoscopic oscillations in the critical current, relevant to experiments in quartets, topology, and Floquet physics.

Core claim

The relative number of quantum-correlated pairs formed by colliding Floquet-Kulik quartet levels is equal to the inverse of the number of channels. This observation motivates a model for the intermediate regime in which the ballistic central two-dimensional normal metal is treated as a continuum under the adiabatic approximation, while Andreev modes propagate in a background of voltage- and flux-tunable nonequilibrium electronic populations. The model predicts characteristic voltage scales that govern the mesoscopic oscillations of the critical current.

What carries the argument

Adiabatic continuum treatment of the two-dimensional normal metal with Andreev modes propagating in voltage- and flux-tunable nonequilibrium electronic populations.

If this is right

  • The model predicts characteristic voltage scales that govern mesoscopic oscillations of the critical current.
  • These scales lie at the intersection of quartet, topology, and Floquet interpretations for multiterminal Josephson junction experiments.
  • The continuum treatment addresses the intermediate regime where electrochemical potential becomes comparable to one-dimensional level spacing.

Where Pith is reading between the lines

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

  • Varying channel number experimentally could directly test the inverse scaling of correlated pairs.
  • The nonequilibrium populations may influence additional transport properties such as conductance.
  • This approach could connect discrete-level descriptions in small devices to continuum models in larger ones.

Load-bearing premise

The ballistic central two-dimensional normal metal can be treated as a continuum under the adiabatic approximation while Andreev modes propagate in a background of voltage- and flux-tunable nonequilibrium electronic populations.

What would settle it

Measuring whether the relative number of quantum-correlated pairs equals the inverse of the number of channels in devices with controlled channel counts would confirm or refute the central claim.

Figures

Figures reproduced from arXiv: 2508.17367 by Asmaul Smitha Rashid, Morteza Kayyalha, R\'egis M\'elin, Romain Danneau.

Figure 2
Figure 2. Figure 2: FIG. 2. The four-terminal JJ: The device (a) and the circuit contain [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The two-terminal JJ: The horizontal Andreev tubes that carry [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the cross-over diagrams in the plane of the separation R between the contacts and the bias voltage V. Devices of small linear dimension L have discrete spectrum and, close to the y-axis, they are candidates for implementing the physics of the Floquet theory. Such small quantum dots sustaining a few ABS within the superconducting gap have not yet been experimentally realized. One of the challenges is … view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The second model for the interference [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The first model for the interference [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The interference [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: summarizes the sensitivity of the quartet critical cur￾rent Iq,c(δ µ′ N , Φ Φ0 ) on the value δ µ′ N of the electrochemical potential, in zero field Φ Φ0 = 0 (magenta) and at half-flux quan￾tum Φ Φ0 = 1 2 (green). The different panels of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The three-terminal quartets. Two pairs are emitted from [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The four-terminal split-quartets. Two pairs are emitted from [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

Motivated by recent experiments on multiterminal Josephson junctions (MJJs) that probe different ranges of the size and bias voltage parameters, we explore the regime of increasing bias voltage in large-scale devices, where the electrochemical potential becomes comparable to the 1D energy level spacing. We find that the relative number of quantum-correlated pairs formed by colliding Floquet--Kulik quartet levels is equal to the inverse of the number of channels. This observation motivates a model for the intermediate regime in which the ballistic central two-dimensional normal metal is treated as a continuum under the adiabatic approximation, while Andreev modes propagate in a background of voltage- and flux-tunable nonequilibrium electronic populations. The model predicts characteristic voltage scales that govern the mesoscopic oscillations of the critical current, and these scales are at the crossroads of interpreting experiments in all sectors of the MJJs: quartets, topology, and Floquet theory. Our model is specifically inspired by the recent Harvard and Penn State group experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript explores the regime of increasing bias voltage in large-scale ballistic multiterminal Josephson junctions where the electrochemical potential becomes comparable to the 1D energy level spacing. It reports that the relative number of quantum-correlated pairs formed by colliding Floquet-Kulik quartet levels equals the inverse of the number of channels. This observation is used to motivate a model in which the central ballistic 2D normal metal is treated as a continuum under the adiabatic approximation, while Andreev modes propagate against a background of voltage- and flux-tunable nonequilibrium electronic populations. The model predicts characteristic voltage scales that govern mesoscopic oscillations of the critical current and are relevant to interpreting experiments in the quartets, topology, and Floquet sectors.

Significance. If the pair-counting result holds and the continuum model is justified, the work would supply a practical bridge between microscopic quantum correlations and mesoscopic observables, offering voltage-scale predictions that could help unify interpretations across different experimental regimes of multiterminal Josephson junctions.

major comments (1)
  1. [paragraph on model motivation] Paragraph on model motivation: The claim that the relative number of quantum-correlated pairs formed by colliding Floquet-Kulik quartet levels equals 1/N_channels is invoked to justify treating the ballistic central 2D normal metal as a continuum under the adiabatic approximation. However, the explored regime is explicitly defined by an electrochemical potential comparable to the 1D level spacing, which is the regime in which discrete transverse modes and finite-size quantization are expected to produce corrections to any continuum integral. The adiabatic condition itself risks violation when nonequilibrium distributions vary on the scale of the level spacing rather than varying slowly, directly undermining the load-bearing approximation used to derive the characteristic voltage scales.
minor comments (2)
  1. [abstract] The abstract states that the model is 'specifically inspired by the recent Harvard and Penn State group experiments' but does not identify which specific data sets or voltage ranges are being addressed; adding a short sentence with the relevant experimental references would improve clarity.
  2. [abstract] Notation for the number of channels (N_channels) and the inverse relation is introduced without an explicit definition or reference to a prior equation; a brief parenthetical definition on first use would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the regime of validity of the continuum adiabatic approximation. We address the concern directly below and propose revisions to clarify the connection between our discrete numerical results and the continuum model.

read point-by-point responses

  1. Referee: Paragraph on model motivation: The claim that the relative number of quantum-correlated pairs formed by colliding Floquet-Kulik quartet levels equals 1/N_channels is invoked to justify treating the ballistic central 2D normal metal as a continuum under the adiabatic approximation. However, the explored regime is explicitly defined by an electrochemical potential comparable to the 1D level spacing, which is the regime in which discrete transverse modes and finite-size quantization are expected to produce corrections to any continuum integral. The adiabatic condition itself risks violation when nonequilibrium distributions vary on the scale of the level spacing rather than varying slowly, directly undermining the load-bearing approximation used to derive the characteristic voltage scales.

    Authors: We appreciate the referee's identification of this potential tension. Our central numerical finding—that the fraction of quantum-correlated pairs is precisely 1/N_channels—is obtained from fully discrete multichannel calculations in the regime where the electrochemical potential is comparable to the level spacing. This exact scaling with channel number indicates that the correlated component remains a parametrically small fraction of the total spectrum even as discrete effects are retained; the remaining majority of modes can therefore be integrated over in a continuum description without altering the leading voltage scales. Regarding the adiabatic condition, the nonequilibrium populations are set by the applied bias voltages, whose energy variation occurs on the scale of eV. In the large-device limit relevant to the experiments we discuss, this scale is slow compared with the local transverse quantization energy once the 2D continuum limit is taken, preserving the adiabatic separation. We will add a dedicated paragraph in the revised manuscript that (i) explicitly states the regime of validity, (ii) shows how the 1/N result bridges the discrete numerics to the continuum model, and (iii) discusses the expected size of finite-N corrections to the predicted voltage scales. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from independent finding to motivated model

full rationale

The paper states it finds that the relative number of quantum-correlated pairs formed by colliding Floquet-Kulik quartet levels equals the inverse of the number of channels; this result is then used to motivate treating the ballistic 2D normal metal as a continuum under the adiabatic approximation with nonequilibrium populations. Characteristic voltage scales are subsequently predicted from the resulting model. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the initial counting result is presented as an output of analysis rather than an input assumption, and the continuum treatment is an explicit modeling choice justified by that result rather than presupposed. Questions of whether the continuum remains valid when electrochemical potential approaches 1D level spacing concern model applicability, not circularity in the logical chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the adiabatic approximation for the 2D normal metal and the assumption of voltage-tunable nonequilibrium populations; no free parameters or new entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption The central 2D normal metal region can be treated as a continuum under the adiabatic approximation in the intermediate bias regime.
    Invoked to justify replacing discrete level structure with a smooth background for Andreev mode propagation.
  • domain assumption Andreev modes propagate in a background of voltage- and flux-tunable nonequilibrium electronic populations.
    Required for the model to predict characteristic voltage scales governing critical current oscillations.

pith-pipeline@v0.9.0 · 5717 in / 1502 out tokens · 36617 ms · 2026-05-18T21:40:13.701901+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    7 and the Appendix B

    Three-terminal quartets In this subsubsection, we present the three-terminal quartet current-phase relations, see Ref. 7 and the Appendix B. In per- turbation theory in the tunneling amplitudes, the current-phase relations of the three-terminal quartets originate from two Cooper pairs emanating from the condensate of the grounded superconducting lead ST ....

    work page

  2. [2]

    22 and the Ap- pendix B

    Four-terminal split-quartets In this subsubsection, we provide the expression of the split-quartet current-phase relations, see Ref. 22 and the Ap- pendix B. Those complementary four-terminal split-quartets exchange partners among the single Cooper pairs coming from each of the grounded superconducting leads SR or SB. The resulting outgoing pairs are tran...

    work page

  3. [3]

    36 and 63

    Phase-Andreev reflection in an Andreev interferometer In this subsubsection, we summarize recent results on a ballistic Andreev interferometer, see Refs. 36 and 63. We note that a general analogy holds with the two-fluid pic- ture of superconductivity, where the central- N quasiparti- cle electrochemical potential δ µN eV, Φ Φ0 parameterizes the dissipati...

    work page

  4. [4]

    The latter are produced in the central- N simply by voltage-biasing in the presence of nonsymmetrical contact transparencies

    Even-in-phase current-phase relation In this subsubsection, we calculate the quartet current- phase relations in the presence of a coupling to the back- ground nonequilibrium electronic populations. The latter are produced in the central- N simply by voltage-biasing in the presence of nonsymmetrical contact transparencies. We implement the simplifying ass...

    work page

  5. [5]

    Numerical results In this subsubsection, we present numerical calculations that illustrate the above theory. We first assume that the crit- ical current results from the following interference between both types of the even-in-phase quartets emitted by the con- tacts SR or by SB at the extremities of the grounded supercon- ducting loop: Iq,c (Φ, δ µN) = M...

    work page

  6. [6]

    (24) The critical currents are then parameterized by δ µ′ N, and they also depend on the strength s0 of the electroflux effect 70,71

    We use the notations δ µN eV, Φ Φ0 = 0 = δ µ′ N − s0 (23) δ µN eV, Φ Φ0 = 1 2 = δ µ′ N + s0. (24) The critical currents are then parameterized by δ µ′ N, and they also depend on the strength s0 of the electroflux effect 70,71. The shift δ µ′ N ∝ V in the electrochemical potential is propor- tional to the bias voltageV in the presence of nonsymmetrical con...

    work page

  7. [7]

    8a for Φ Φ0 = 0 and for Φ Φ0 = 1 2

    This is why the same colorcode is used on Fig. 8a for Φ Φ0 = 0 and for Φ Φ0 = 1 2. This panel confirms the oscillations in the quartet critical cur- rent as δ µ′ N is increased, or, equivalently, as the bias volt- age V is increased. Fig. 8c now shows the full Eq. (22) with Iq,3T,c = 1 and ic,R,L = 0.5, still with Isq,c = 0. This panel c reveals that Iq,c...

    work page

  8. [8]

    Nambu-trace

    and the inversion (i.e. the critical cur- rent is smaller at Φ Φ0 = 0 than at Φ Φ0 = 1 2). VI. CONCLUSIONS In summary, we have presented a phenomenological ap- proach to understanding the emergence of characteristic low- bias voltages in intermediate-scale MJJs. The mesoscopic en- ergy associated with the contact separation was taken to be comparable to t...

    work page

  9. [9]

    (A3) for ˆΣa,α ˆG+,− α,a into both terms ˆΣa,α ˆG+,− α,a S and ˆΣa,α ˆG+,− α,a N having the bare Keldysh Green’s function in the superconducting or normal leads, respectively

    Odd-in-phase/even-in-voltage three-terminal quartets We previously decomposed Eq. (A3) for ˆΣa,α ˆG+,− α,a into both terms ˆΣa,α ˆG+,− α,a S and ˆΣa,α ˆG+,− α,a N having the bare Keldysh Green’s function in the superconducting or normal leads, respectively. In this subsection, we now evaluate ˆΣa,α ˆG+,− α,a S in perturbation in the hopping amplitudes, an...

    work page

  10. [10]

    We specifically cal- culate the corresponding quartet current as a function of both the quartet phase and the bias voltage

    Even-in-phase/odd-in-voltage three-terminal quartets In this subsection, we consider the long-junction limit67–69, which hosts a discrete spectrum of ABS, at all energy scales with respect to the superconducting gap. We specifically cal- culate the corresponding quartet current as a function of both the quartet phase and the bias voltage. We focus on smal...

    work page

  11. [11]

    2, sustaining the process of the split-quartets

    Odd-in-phase/even-in-voltage four-terminal split-quartets In this subsection, we go back to the short-junction limit and consider the four-terminal device in Fig. 2, sustaining the process of the split-quartets. Namely, the two incoming pairs 12 S a ϕa ϕbS b ’γ S ϕ S ϕ c c dd b β c c’ a e (a) e h hh α h γ d d’ δ ’δ S a ϕa ϕbS b ’γ S ϕ S ϕ c c dd b β c c’ ...

    work page

  12. [12]

    1,1” and “2,2

    Even-in-phase/odd-in-voltage four-terminal split-quartets We first make the Nambu structure explicit in the first term of ˆΣa,α ˆGα,a (A),1,1 N , see Eq. (B27): ˆΣ1,1 a,α ˆg1,1 α,γ ˆΣ1,1 γ,c ˆg1,1 c,c′ ˆΣ1,1 c′,γ ′ ˆg1,1 γ ′,δ ′ ˆΣ1,1 δ ′,d′ ˆg1,2 d′,d ˆΣ2,2 d,δ ˆg2,2 δ ,β ˆΣ2,2 β ,b ˆg2,1 b,b = Σ2 aΣ2 bΣ2 cΣ2 c′Σ2 d′g2,1 a,ag2,1 b,bg1,1 c,c′g1,2 c′,c × g...

    work page arXiv 1994