Magnetic flux controlled current phase relationship in double Quantum Dot Josephson junction

Bing Dong; Cong Li; Yiyan Wang

arxiv: 2509.17517 · v2 · submitted 2025-09-22 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Magnetic flux controlled current phase relationship in double Quantum Dot Josephson junction

Yiyan Wang , Cong Li , Bing Dong This is my paper

Pith reviewed 2026-05-18 15:07 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con

keywords double quantum dotJosephson junctionmagnetic fluxsinglet doublet tripletphase transitioncurrent-phase relationsuperconducting phase difference

0 comments

The pith

Magnetic flux through double quantum dots in a Josephson junction suppresses doublet and triplet ground states in favor of the singlet.

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

The paper investigates a Josephson junction formed by two parallel quantum dots threaded by a magnetic flux. Using a surrogate Hamiltonian obtained by discretizing the superconducting leads, the authors compute how the ground state changes with superconducting phase difference and flux. When only one dot has Coulomb interaction the flux mainly affects the doublet state while leaving the singlet nearly unchanged. When both dots interact the ground state passes through doublet, singlet, and triplet regimes as the phase difference reaches pi, with larger flux eventually locking the system into the singlet. This flux dependence alters the current-phase relation and produces a peak in critical current with increasing interaction strength.

Core claim

When both quantum dots have interactions the ground state evolves from a doublet to a singlet and finally into a triplet at phi equals pi as the superconducting phase difference increases. Increasing the magnetic flux suppresses the doublet and triplet phases and stabilizes the singlet state. In the regime where the tunneling coefficient exceeds the pairing potential the system shows a doublet-triplet transition only at zero flux and otherwise exhibits a triple point in the joint space of phase difference and flux.

What carries the argument

The surrogate finite-dimensional Hamiltonian obtained by discretizing each superconducting electrode into three energy levels and modifying the tunneling coefficients to approximate the low-energy spectrum of the original system.

If this is right

Larger magnetic flux reduces the range of phase difference over which doublet and triplet states appear as ground states.
When both dots interact, increasing the interaction strength U produces a transition between two singlet states that creates a peak in the critical current.
In the strong-tunneling regime the three states meet at a triple point in the (phi, phi_B) plane when flux is present.
The current-phase relation can be tuned continuously by external flux without changing dot parameters.

Where Pith is reading between the lines

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

The flux-induced stabilization of the singlet suggests a route to suppress unwanted parity states in superconducting qubit designs that use quantum dots.
The same discretization approach could be tested on triple-dot or larger arrays to see whether flux still collapses the spectrum to a single dominant state.

Load-bearing premise

The three-level discretization of the superconducting electrodes together with adjusted tunneling coefficients produces a surrogate Hamiltonian whose low-energy spectrum matches that of the infinite-dimensional original system.

What would settle it

Direct comparison of the lowest few eigenvalues and eigenstates of the three-level surrogate Hamiltonian against a version with five or more levels per electrode that shows the same sequence of ground-state crossings versus phase difference and flux.

Figures

Figures reproduced from arXiv: 2509.17517 by Bing Dong, Cong Li, Yiyan Wang.

**Figure 2.** Figure 2: (Color online) (a) Current suppressed with the differ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: (Color online) Phase diagram as the function of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: (Color online) (a) Josephson current calculated by sur [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 6.** Figure 6: (Color online) Phase diagram versus ϕ and ϕB (a) U1 = U2 = 0.5, Γ = 0.25 (b)U1 = U2 = 1, Γ = 0.25. Unlike the setup discussed in Section B, in the symmetric QD configuration the doublet state does not shift its extremal value with ϕB. Within the low-energy effective model, the relevant subspace is spanned by |d1 ↑2 ⟩, | ↑1 d2⟩, |01 ↑2⟩, | ↑1 02⟩ , and the corresponding coupling matrix is given by:    … view at source ↗

**Figure 8.** Figure 8: (Color online) Phase diagram versus ϕ and ϕB (a) U1 = U2 = 8, Γ = 2.(b)Enlarged view of the triple point region. discretization approach, we discretized the self-energy of the superconducting leads, thereby replacing the Hamiltonian of the leads with that of a finite number of discrete points. Furthermore, we performed exact diagonalization of the system’s effective Hamiltonian using the statespace expa… view at source ↗

read the original abstract

In this work, we study a Josephson junction with parallel-connected quantum dots (QDs) threaded by a magnetic flux in the central region. We discretize the superconducting (SC) electrode into three discrete energy levels and modify the tunneling coefficients to construct a finite-dimensional surrogate Hamiltonian. By directly diagonalizing this Hamiltonian, we compute the physical quantities of the system. Additionally, we employ a low-energy effective model to gain deeper physical insight. Our findings reveal that when only one QD exhibits Coulomb interaction, the system undergoes a phase transition between singlet and doublet states. The magnetic flux has a minor influence on the singlet state but significantly affects the doublet state. When both QDs have interactions, the system undergoes two phase transitions as the SC phase difference increases: the ground state evolves from a doublet to a singlet and finally into a triplet state at $\phi = \pi$. Increasing the magnetic flux suppresses the doublet and triplet phases, eventually stabilizing the singlet state. In this regime, enhancing the interaction strength does not induce a singlet-doublet transition but instead drives a transition between upper and lower singlet states, leading to a critical current peak as $U$ increases. Finally, we examine the case where the tunneling coefficient $\Gamma$ exceeds the SC pairing potential $\Delta$. Here, doublet states dominate, and the system only exhibits a phase transition between doublet and triplet states when $\phi_B = 0$. In the presence of a magnetic flux, the three states converge, resulting in a triple point in the ($\phi$, $\phi_B$) parameter space.

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.

Desk Editor's Note private letter to a colleague

Flux suppresses doublet and triplet states in this double-dot Josephson setup, but the three-level electrode truncation needs convergence checks before the phase boundaries can be trusted.

read the letter

The main thing to know is that magnetic flux through the loop between the two quantum dots pushes the ground state toward singlet and away from doublet and triplet, with a reported sequence of transitions as the superconducting phase difference varies when both dots have interactions. When tunneling is strong the states meet at a triple point in the flux-phase plane. These are concrete observations from exact diagonalization of their model Hamiltonian, plus some effective low-energy analysis to interpret the crossings and current-phase curves. The work extends single-dot or flux-free cases by adding the parallel geometry and explicit flux control, and it maps out how interaction strength and flux trade off in different regimes. That part is useful for anyone thinking about tunable mesoscopic junctions. The calculations are straightforward and the effective model gives a readable picture of why the states cross. The soft spot is the surrogate Hamiltonian itself. Each superconducting electrode is replaced by three discrete levels with adjusted tunneling amplitudes. The paper does not show that raising the cutoff to five or seven levels leaves the low-energy spectrum and the locations of the phase boundaries unchanged, nor does it compare against the continuum limit or a known analytic case. If the truncation shifts the effective gap or the quasiparticle density, the critical flux values and the exact doublet-singlet-triplet sequence could move. That is the load-bearing approximation, and it is not yet stress-tested in the text. This is for people working on quantum-dot Josephson devices or flux-tunable superconducting circuits. A reader who needs a concrete example of how flux and interactions compete in a small loop would get usable phase diagrams and current-phase relations to compare against experiment or more detailed numerics. It is worth sending to a serious referee because the claims are specific and falsifiable once the truncation is checked; the main request should be explicit convergence data on the electrode discretization.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the current-phase relationship in a double quantum dot Josephson junction threaded by magnetic flux. Each superconducting electrode is discretized into three energy levels with rescaled tunneling amplitudes to form a surrogate finite-dimensional Hamiltonian, which is then exactly diagonalized; a low-energy effective model is used for interpretation. The central claims are that magnetic flux suppresses doublet and triplet ground states (stabilizing the singlet), that both QDs interacting produces two phase transitions with increasing phase difference φ (doublet to singlet to triplet at φ=π), and that for Γ>Δ the system shows doublet-triplet transitions only at φ_B=0 with a triple point appearing for finite flux.

Significance. If the three-level surrogate faithfully reproduces the low-energy continuum limit, the work provides concrete numerical insight into flux-tunable ground-state crossings and critical-current peaks in interacting QD Josephson junctions. The direct diagonalization of the surrogate model and the accompanying low-energy effective description are methodological strengths that allow clean extraction of the (φ, φ_B, U) phase diagram without fitting parameters.

major comments (1)

[Abstract and model-construction section] Abstract (paragraph beginning 'We discretize the superconducting (SC) electrode') and the model-construction section: the surrogate Hamiltonian replaces each infinite-band SC electrode by three discrete levels plus ad-hoc rescaling of tunneling coefficients. No convergence test with respect to the number of levels, no comparison of the low-energy spectrum or effective Δ against the continuum limit, and no error estimate on the truncation are reported. Because all reported phase boundaries, the suppression of the triplet phase by flux, and the doublet-singlet-triplet sequence rest on this approximation, the absence of such validation is load-bearing for the central claims.

minor comments (2)

[Abstract] The abstract states that 'enhancing the interaction strength does not induce a singlet-doublet transition but instead drives a transition between upper and lower singlet states' when flux is finite; a brief indication of the relevant energy scales or the effective model parameters that produce this behavior would improve clarity.
Notation for the magnetic flux (φ_B) and the phase difference (φ) is introduced without an early figure or equation that defines the geometry; adding a schematic of the parallel QD junction with the flux threading the loop would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback on the surrogate Hamiltonian construction. We address the major comment below and describe the revisions we will implement.

read point-by-point responses

Referee: [Abstract and model-construction section] Abstract (paragraph beginning 'We discretize the superconducting (SC) electrode') and the model-construction section: the surrogate Hamiltonian replaces each infinite-band SC electrode by three discrete levels plus ad-hoc rescaling of tunneling coefficients. No convergence test with respect to the number of levels, no comparison of the low-energy spectrum or effective Δ against the continuum limit, and no error estimate on the truncation are reported. Because all reported phase boundaries, the suppression of the triplet phase by flux, and the doublet-singlet-triplet sequence rest on this approximation, the absence of such validation is load-bearing for the central claims.

Authors: We agree that explicit validation of the three-level surrogate is necessary to substantiate the reported phase boundaries and flux-induced transitions. The discretization with rescaled tunneling amplitudes was chosen to reproduce the low-energy pairing gap Δ while keeping the Hilbert space tractable for exact diagonalization. In the revised manuscript we will add a dedicated subsection (or appendix) that presents the low-energy spectrum and extracted phase boundaries for truncations with 3, 5, and 7 levels. We will demonstrate convergence of the singlet-doublet-triplet crossings and the flux-suppressed regions for the parameter sets used in the main text. We will also report the effective Δ obtained from the surrogate and an error estimate based on the difference between successive truncations. These additions will directly address the load-bearing nature of the approximation for the central claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct diagonalization of explicitly constructed surrogate Hamiltonian

full rationale

The paper defines a finite-dimensional surrogate Hamiltonian by discretizing each superconducting electrode into three discrete levels and modifying tunneling coefficients, then computes observables via exact diagonalization of this model. This constitutes a standard numerical approximation technique with no fitted parameters tuned to match target results, no self-citations invoked as load-bearing uniqueness theorems, and no steps where a claimed prediction or phase transition reduces by construction to the inputs. The low-energy effective model is used only for interpretation after the numerical results are obtained. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the three-level electrode truncation and the assumption that the low-energy effective model captures the same physics as the full diagonalization. No additional free parameters beyond the discretization choice are introduced.

free parameters (1)

number of discrete levels per electrode
Chosen to render the many-body Hamiltonian finite-dimensional for exact diagonalization.

axioms (1)

domain assumption A finite discrete-level representation of the superconducting electrodes preserves the essential low-energy physics of the Josephson junction.
Invoked when constructing the surrogate Hamiltonian from the continuous electrode spectrum.

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We discretize the superconducting (SC) electrode into three discrete energy levels and modify the tunneling coefficients to construct a finite-dimensional surrogate Hamiltonian. By directly diagonalizing this Hamiltonian...

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This coupling is captured by the follow- ing3×3matrix:   2sin( ϕ 2 )sin( ϕB 2 )Γ 0− √ 2 cos(ϕ 2 )Γ 0−2sin( ϕ 2 )sin( ϕB 2 )Γ− √ 2 cos(ϕ 2 )Γ − √ 2 cos(ϕ 2 )Γ− √ 2 cos(ϕ 2 )Γε 1   . (23) The coupling strength is √ 2 cos(ϕ 2 )Γ, which is √ 2times cross dot paring potentialcos(ϕ 2 )Γand is independent of ϕB. Consequently, the maximum of|S⟩energy is fixed...

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This explains the triplet-favoring with increasing U. And by letting ϕB =π, this co-tunneling process interfere destructively and only Cooper pair tunneling process maintains which made the ground state keep singlet. The|S⟩in this case in low energy effective model is coupled with the other four states by the coupling matrix as:   0 Γcos( ϕ−ϕB 2 ) Γ...

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