High efficiency superconducting diode effect in a gate-tunable double-loop SQUID

Jukka I. V\"ayrynen; Kevin Barrow; Michael J. Manfra; Teng Zhang; Tyler Lindemann; Wyatt Gibbons

arxiv: 2512.14909 · v1 · submitted 2025-12-16 · ❄️ cond-mat.supr-con

High efficiency superconducting diode effect in a gate-tunable double-loop SQUID

Wyatt Gibbons , Teng Zhang , Kevin Barrow , Tyler Lindemann , Jukka I. V\"ayrynen , Michael J. Manfra This is my paper

Pith reviewed 2026-05-16 21:21 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords superconducting diodeSQUIDJosephson junctionsgate-tunablecurrent-phase relationshipdiode efficiencyflux dependence

0 comments

The pith

A double-loop SQUID with two gate-tunable Josephson junctions per branch reaches superconducting diode efficiency above 50 percent.

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

The paper shows that a double-loop SQUID can produce a strong diode effect when each of its two interference branches contains two gate-tunable Josephson junctions placed in series. This arrangement lets experimenters adjust the amplitude and harmonic content of three separate current-phase relationships independently. Optimized gate settings then drive the maximum diode efficiency past 50 percent. The flux dependence of the effect matches a basic model of SQUID operation without additional corrections. The result improves diode performance by giving direct control over the interference conditions that generate the asymmetry.

Core claim

Placing two gate-tunable Josephson junctions in series in each branch of a double-loop SQUID allows independent control of the amplitude and harmonic content of three interfering current-phase relationships; optimized tuning of the individual Josephson energies produces diode efficiency exceeding 50 percent, with flux-dependent oscillations in quantitative agreement with a simple model of SQUID operation.

What carries the argument

The double-loop SQUID whose three current-phase relationships are formed by two gate-tunable Josephson junctions in series per branch, enabling separate adjustment of their amplitudes and harmonics to strengthen nonreciprocal interference.

If this is right

Diode efficiency above 50 percent becomes achievable in superconducting circuits by gate adjustment alone.
Flux-dependent behavior can be predicted quantitatively from the three-branch interference model.
Independent control of multiple current-phase relationships extends the design space for nonreciprocal superconducting devices.
Gate tuning of Josephson energies provides a practical route to optimize diode performance without altering device geometry.

Where Pith is reading between the lines

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

The same gate-tunable architecture could be used to test whether higher-order harmonics limit efficiency in other SQUID geometries.
Integration with semiconductor gates may allow on-chip switching between diode and reciprocal modes in hybrid circuits.
If crosstalk remains negligible at higher frequencies, the device could serve as a low-dissipation rectifier in superconducting electronics.

Load-bearing premise

The three current-phase relationships stay independent and can be tuned without significant crosstalk or extra harmonics that would lower the observed diode efficiency.

What would settle it

A measurement in which gate voltages are set to the reported optimum values yet the diode efficiency remains below 50 percent, or the flux oscillations deviate from the simple model predictions, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.14909 by Jukka I. V\"ayrynen, Kevin Barrow, Michael J. Manfra, Teng Zhang, Tyler Lindemann, Wyatt Gibbons.

**Figure 2.** Figure 2: FIG. 2. (a) Differential resistance ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (b) for an Rthresh of 10 Ω. The minima of I + c and |I − c | in each period of these SQUID oscillations are clearly shifted from each other, leading to a strong observed diode effect at specific B⊥ values. Through flux tuning, we demonstrate the ability to periodically tune η to any value between −54% and 47% in this voltage configuration, see [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Differential resistance ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Demonstration of single-loop and double-loop SQUID [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: (b). TABLE III. Gate voltages applied to each JJ to generate data shown in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: displays highly asymmetric SQUID oscillations measured in a second voltage configuration. To generate this figure, negative gate voltages are applied to V2 and V4 so that EJ1 ≫ EJ2, and EJ3 ≫ EJ4, and EJ5 ∼ EJ6. The difference between this configuration and the one we tune to in Section III B is that here, we keep V5 and [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

In superconducting quantum interference devices (SQUIDs), the superconducting diode effect may be generated by interference of multiple harmonic components in the current-phase relationships (CPRs) of different branches forming SQUID loops. Through the inclusion of two gate-tunable Josephson junctions in series in each interference branch of a double-loop SQUID, we demonstrate independent control over both the harmonic content and the amplitude of three interfering CPRs, facilitating significant improvement in the maximum diode efficiency. Through optimized gate-controlled tuning of individual Josephson energies, diode efficiency exceeding 50% is demonstrated. Flux-dependent oscillations show quantitative agreement with a simple model of SQUID operation.

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

A double-loop SQUID with two gate-tunable junctions per branch reaches over 50% diode efficiency through independent CPR tuning, though the paper leaves the no-crosstalk assumption unverified.

read the letter

The main result here is a measured diode efficiency above 50% in a double-loop SQUID where each interference branch contains two series Josephson junctions that can be gate-tuned separately. This geometry gives direct control over both the amplitude and the harmonic content of three CPRs at once, which lets them optimize the interference condition more finely than single-loop or fixed-junction versions have managed so far. The flux-dependent oscillations line up quantitatively with a simple model, which is the strongest part of the evidence that the tuning is actually doing what they intend. That agreement is not just qualitative; it tracks the expected period and shape, so the basic picture of how the loops combine holds up under the reported conditions. The work is a straightforward extension of existing SQUID diode ideas, but the extra control knobs are a practical improvement for anyone trying to push efficiency in this platform. The soft spot is the independence of the three CPRs. The model and the efficiency claim both assume each gate voltage affects only its target junction with no measurable crosstalk to the others. In a real multi-gate device on one chip, electrostatic or inductive coupling between gates is common and can shift neighboring critical currents by several percent. The abstract and the reported data do not include a calibration that would bound this effect, such as fixing one junction and sweeping a non-adjacent gate while watching the response. Without that check, the peak efficiency number could be slightly optimistic or require extra compensation steps that are not described. It is not a load-bearing flaw, but it is the kind of detail that needs tightening before the result can be taken as fully settled. This paper is useful for groups working on superconducting rectifiers, SQUID-based logic, or quantum circuit elements that need tunable non-reciprocal behavior. A reader who already knows the prior SQUID diode literature will see the incremental advance clearly and can judge whether the extra fabrication complexity is worth it for their application. I would send it to peer review. The central measurement is concrete, the model comparison adds credibility, and the remaining questions are addressable with standard additional data rather than a fundamental rethink.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a double-loop SQUID incorporating two gate-tunable Josephson junctions in series within each interference branch. Through gate-controlled tuning of individual Josephson energies, the authors demonstrate independent control over the amplitudes and harmonic contents of three current-phase relationships, achieving a superconducting diode efficiency exceeding 50%. Flux-dependent oscillations are reported to show quantitative agreement with a simple model of SQUID operation.

Significance. If the central experimental claims hold, this work advances superconducting diode devices by showing that gate tuning in a multi-junction geometry can yield efficiencies above 50% with a straightforward model description. The approach offers a scalable route to high-performance rectifiers for superconducting electronics and quantum circuits, building on prior SQUID-based diode demonstrations through explicit multi-parameter control.

major comments (2)

[Device characterization and gate tuning] The claim of independent control over three CPRs (abstract and results) rests on the assumption of negligible gate crosstalk, yet no dedicated calibration is reported, such as monitoring one junction's Ic while sweeping a non-local gate. This is load-bearing because even small crosstalk would detune the interference condition and cap the efficiency below the stated >50% value.
[Flux-dependent measurements] The quantitative agreement between flux oscillations and the simple model (results section) is presented without error bars on the data, details on parameter extraction, or goodness-of-fit metrics, making it difficult to evaluate whether the model truly validates the independent-tuning interpretation or merely reproduces qualitative trends.

minor comments (2)

[Figures] Figure captions should explicitly state the gate-voltage ranges used for each efficiency data point to allow readers to assess the tuning procedure.
[Device schematic] Notation for the three CPRs (e.g., labeling of junctions J1, J2, J3) is introduced without a dedicated schematic panel, which would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major point below and have revised the manuscript to incorporate additional data and clarifications that strengthen the claims.

read point-by-point responses

Referee: The claim of independent control over three CPRs (abstract and results) rests on the assumption of negligible gate crosstalk, yet no dedicated calibration is reported, such as monitoring one junction's Ic while sweeping a non-local gate. This is load-bearing because even small crosstalk would detune the interference condition and cap the efficiency below the stated >50% value.

Authors: We agree that explicit calibration for gate crosstalk is essential to support the independent-control claim. Although omitted from the original submission, we have performed the suggested measurements by monitoring the critical current of one junction while sweeping the non-local gate voltage. These data show crosstalk below 3% across the full operating range, consistent with the device layout. We will add this calibration as a new supplementary figure with accompanying text in the methods and results sections. revision: yes
Referee: The quantitative agreement between flux oscillations and the simple model (results section) is presented without error bars on the data, details on parameter extraction, or goodness-of-fit metrics, making it difficult to evaluate whether the model truly validates the independent-tuning interpretation or merely reproduces qualitative trends.

Authors: We accept that the model comparison requires more quantitative detail. In the revision we will add error bars to all flux-dependent data, describe the parameter-extraction procedure (fitting Josephson energies and harmonic amplitudes from gate sweeps), and report goodness-of-fit metrics including reduced chi-squared values. With these additions the agreement remains quantitative within experimental uncertainty and reinforces the independent-tuning interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; experimental result is self-contained

full rationale

The paper's core claim is an experimental demonstration of >50% diode efficiency achieved by gate tuning of Josephson energies in a double-loop SQUID, with flux oscillations showing agreement to a simple model. This is a measured quantity, not a derivation that reduces to its own inputs by construction. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The model serves only as post-hoc validation, leaving the result independent of any circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Josephson junction physics and the assumption that gate voltages independently set the Josephson energies without introducing unaccounted phase shifts.

free parameters (1)

Individual Josephson energies
Gate voltages are adjusted to set the amplitude and harmonic content of each CPR; these values are optimized experimentally rather than derived from first principles.

axioms (1)

domain assumption Interference among multiple harmonic components of CPRs in different SQUID branches produces the diode effect
Invoked in the opening sentence as the mechanism enabling the observed asymmetry.

pith-pipeline@v0.9.0 · 5419 in / 1136 out tokens · 49660 ms · 2026-05-16T21:21:09.975841+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

I2JJ(φ,EJ1,EJ2)=eσ/2ℏ τ_eff sin(φ)√(1−τ_eff sin²(φ/2)) with τ_eff=4ρ/(1+ρ)², ρ=EJ1/EJ2
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

Monte-Carlo optimization over EJi yields η>50% when one branch has τ_eff=1 and the others τ_eff≈0.5-0.6

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

44 extracted references · 44 canonical work pages

[1]

Banszerus, W

L. Banszerus, W. Marshall, C. W. Andersson, T. Linde- mann, M. J. Manfra, C. M. Marcus, and S. Vaitiek˙ enas, Voltage-controlled synthesis of higher harmonics in hy- brid josephson junction circuits, Phys. Rev. Lett. 133, 186303 (2024)

work page 2024
[2]

Banszerus, C

L. Banszerus, C. W. Andersson, W. Marshall, T. Linde- mann, M. J. Manfra, C. M. Marcus, and S. Vaitiek˙ enas, Hybrid josephson rhombus: A superconducting element with tailored current-phase relation, Phys. Rev. X 15, 011021 (2025)

work page 2025
[3]

Coraiola, A

M. Coraiola, A. E. Svetogorov, D. Z. Haxell, D. Sabo- nis, M. Hinderling, S. C. t. Kate, E. Cheah, F. Krizek, R. Schott, W. Wegscheider, J. C. Cuevas, W. Belzig, and F. Nichele, Flux-Tunable Josephson Diode Eﬀect in a Hy- brid Four-Terminal Josephson Junction, ACS Nano 18, 9221 (2024), arXiv:2312.04415 [cond-mat]

work page arXiv 2024
[4]

Ciaccia, R

C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Linde- mann, M. J. Manfra, C. Schrade, and C. Sch¨ onenberger, Gate-tunable josephson diode in proximitized inas su- percurrent interferometers, Phys. Rev. Res. 5, 033131 (2023)

work page 2023
[5]

Greco, Q

A. Greco, Q. Pichard, E. Strambini, and F. Giazotto, Double loop dc-SQUID as a tunable Josephson diode, Applied Physics Letters 125, 072601 (2024)

work page 2024
[6]

Leblanc, C

A. Leblanc, C. Tangchingchai, Z. S. Momtaz, E. Kiyooka, J.-M. Hartmann, G. T. Fernandez-Bada, Z. Scher¨ ubl, B. Brun, V. Schmitt, S. Zihlmann, et al. , From nonre- ciprocal to charge-4e supercurrent in ge-based josephson devices with tunable harmonic content, Physical Review Research 6, 10.1103/physrevresearch.6.033281 (2024)

work page doi:10.1103/physrevresearch.6.033281 2024
[7]

Baumgartner, L

C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Supercurrent rectiﬁcation and magne- tochiral eﬀects in symmetric Josephson junctions, Nature Nanotechnology 17, 39 (2022)

work page 2022
[8]

F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Obser- vation of superconducting diode eﬀect, Nature 584, 373 (2020)

work page 2020
[9]

Daido, Y

A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic supercon- ducting diode eﬀect, Phys. Rev. Lett. 128, 037001 (2022)

work page 2022
[10]

Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ili´ c, O. Glezakou- Elbert, et al. , Ubiquitous superconducting diode eﬀect in superconductor thin ﬁlms, Phys. Rev. Lett. 131, 027001 (2023)

work page 2023
[11]

Ili´ c and F

S. Ili´ c and F. S. Bergeret, Theory of the supercurrent diode eﬀect in rashba superconductors with arbitrary dis- order, Phys. Rev. Lett. 128, 177001 (2022)

work page 2022
[12]

J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. Li, Zero-ﬁeld superconducting diode eﬀect in small-twist-angle trilayer graphene, Na- ture Physics 18, 1221 (2022)

work page 2022
[13]

H. D. Scammell, J. I. A. Li, and M. S. Scheurer, Theory of zero-ﬁeld superconducting diode eﬀect in twisted trilayer graphene, 2D Materials 9, 025027 (2022)

work page 2022
[14]

H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, The ﬁeld-free Josephson diode in a van der Waals heterostructure, Nature 604, 653 (2022)

work page 2022
[15]

Zhang, Y

Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, General the- ory of josephson diodes, Phys. Rev. X 12, 041013 (2022)

work page 2022
[16]

C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Asymmetric josephson eﬀect in inversion symmetry breaking topological materials, Phys. Rev. B 98, 075430 (2018)

work page 2018
[17]

Kononov, G

A. Kononov, G. Abulizi, K. Qu, J. Yan, D. Mandrus, K. Watanabe, T. Taniguchi, and C. Sch¨ onenberger, One- Dimensional Edge Transport in Few-Layer WTe2, Nano Letters 20, 4228 (2020), publisher: American Chemical Society

work page 2020
[18]

B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, N. Yuan, N. B. M. Schr¨ oter, L. Fu, and S. S. P. Parkin, Josephson diode eﬀect from Cooper pair momentum in a topological semimetal, Nature Physics 18, 1228 (2022)

work page 2022
[19]

Davydova, S

M. Davydova, S. Prembabu, and L. Fu, Universal joseph- son diode eﬀect, Science Advances 8, eabo0309 (2022), https://www.science.org/doi/pdf/10.1126/sciadv.abo0309

work page doi:10.1126/sciadv.abo0309 2022
[20]

N. F. Q. Yuan and L. Fu, Supercurrent diode eﬀect and ﬁnite-momentum superconductors, Proceedings of the National Academy of Sciences 119, e2119548119 (2022), https://www.pnas.org/doi/pdf/10.1073/pnas.2119548119

work page doi:10.1073/pnas.2119548119 2022
[21]

Matsuo, T

S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lin- demann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Josephson diode eﬀect derived from short- range coherent coupling, Nature Physics 19, 1636 (2023)

work page 2023
[22]

Matsuo, T

S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lindemann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Phase engineering of anomalous josephson eﬀect derived from andreev 11 molecules, Science Advances 9, eadj3698 (2023), https://www.science.org/doi/pdf/10.1126/sciadv.adj3698

work page doi:10.1126/sciadv.adj3698 2023
[23]

Costa, C

A. Costa, C. Baumgartner, S. Reinhardt, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Man- fra, J. Fabian, D. Kochan, N. Paradiso, and C. Strunk, Sign reversal of the Josephson inductance magnetochi- ral anisotropy and 0– π-like transitions in supercurrent diodes, Nature Nanotechnology 18, 1266 (2023)

work page 2023
[24]

Reinhardt, T

S. Reinhardt, T. Ascherl, A. Costa, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, C. Strunk, and N. Paradiso, Link between supercurrent diode and anomalous Josephson eﬀect re- vealed by gate-controlled interferometry, Nature Com- munications 15, 4413 (2024)

work page 2024
[25]

Costa, O

A. Costa, O. Kanehira, H. Matsueda, and J. Fabian, Un- conventional josephson supercurrent diode eﬀect induced by chiral spin-orbit coupling, Phys. Rev. B 111, L140506 (2025)

work page 2025
[26]

Shaﬀer and A

D. Shaﬀer and A. Levchenko, Theories of supercon- ducting diode eﬀects (2025), arXiv:2510.25864 [cond- mat.supr-con]

work page arXiv 2025
[27]

R. S. Souto, M. Leijnse, and C. Schrade, Josephson diode eﬀect in supercurrent interferometers, Phys. Rev. Lett. 129, 267702 (2022)

work page 2022
[28]

Josephson, Possible new eﬀects in superconductive tunnelling, Physics Letters 1, 251 (1962)

B. Josephson, Possible new eﬀects in superconductive tunnelling, Physics Letters 1, 251 (1962)

work page 1962
[29]

A. A. Golubov, M. Y. Kupriyanov, and E. Il’ichev, The current-phase relation in josephson junctions, Rev. Mod. Phys. 76, 411 (2004)

work page 2004
[30]

K. K. Likharev, Superconducting weak links, Rev. Mod. Phys. 51, 101 (1979)

work page 1979
[31]

M. L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, D. Esteve, and C. Urbina, Measurement of the current- phase relation of superconducting atomic contacts, Phys. Rev. Lett. 99, 127005 (2007)

work page 2007
[32]

Ciaccia, R

C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Linde- mann, M. J. Manfra, C. Schrade, and C. Sch¨ onenberger, Charge-4e supercurrent in a two-dimensional InAs-Al superconductor-semiconductor heterostructure, Commu- nications Physics 7, 41 (2024)

work page 2024
[33]

Nichele, E

F. Nichele, E. Portol´ es, A. Fornieri, A. M. Whiticar, A. C. C. Drachmann, S. Gronin, T. Wang, G. C. Gardner, C. Thomas, A. T. Hatke, M. J. Manfra, and C. M. Mar- cus, Relating andreev bound states and supercurrents in hybrid josephson junctions, Phys. Rev. Lett. 124, 226801 (2020)

work page 2020
[34]

C. W. J. Beenakker, Universal limit of critical-current ﬂuctuations in mesoscopic josephson junctions, Phys. Rev. Lett. 67, 3836 (1991)

work page 1991
[35]

A. M. Bozkurt, J. Brookman, V. Fatemi, and A. R. Akhmerov, Double-Fourier engineering of Josephson energy-phase relationships applied to diodes, SciPost Phys. 15, 204 (2023)

work page 2023
[36]

Y. S. Barash, Proximity-reduced range of internal phase diﬀerences in double josephson junctions with closely spaced interfaces, Phys. Rev. B 97, 224509 (2018)

work page 2018
[37]

Tinkham and V

M. Tinkham and V. Emery, Introduction to Supercon- ductivity, Physics Today 49, 198 (1996)

work page 1996
[38]

M¨ uller and E

K.-H. M¨ uller and E. E. Mitchell, Theoretical model for parallel squid arrays with ﬂuxoid focusing, Phys. Rev. B 103, 054509 (2021)

work page 2021
[39]

Ambegaokar and B

V. Ambegaokar and B. I. Halperin, Voltage due to ther- mal noise in the dc josephson eﬀect, Phys. Rev. Lett. 22, 1364 (1969)

work page 1969
[40]

Kringhøj, B

A. Kringhøj, B. van Heck, T. W. Larsen, O. Erlandsson, D. Sabonis, P. Krogstrup, L. Casparis, K. D. Petersson, and C. M. Marcus, Suppressed charge dispersion via res- onant tunneling in a single-channel transmon, Phys. Rev. Lett. 124, 246803 (2020)

work page 2020
[41]

Bargerbos, W

A. Bargerbos, W. Uilhoorn, C.-K. Yang, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, B. van Heck, and A. Kou, Observation of vanishing charge dispersion of a nearly open superconducting island, Phys. Rev. Lett. 124, 246802 (2020)

work page 2020
[42]

Willsch, D

D. Willsch, D. Rieger, P. Winkel, M. Willsch, C. Dickel, J. Krause, Y. Ando, R. Lescanne, Z. Leghtas, N. T. Bronn, P. Deb, et al. , Observation of Josephson harmon- ics in tunnel junctions, Nature Physics 20, 815 (2024)

work page 2024
[43]

Schrade, C

C. Schrade, C. M. Marcus, and A. Gyenis, Protected hy- brid superconducting qubit in an array of gate-tunable josephson interferometers, PRX Quantum 3, 030303 (2022)

work page 2022
[44]

W. C. Smith, A. Kou, X. Xiao, U. Vool, and M. H. De- voret, Superconducting circuit protected by two-Cooper- pair tunneling, npj Quantum Information 6, 8 (2020)

work page 2020

[1] [1]

Banszerus, W

L. Banszerus, W. Marshall, C. W. Andersson, T. Linde- mann, M. J. Manfra, C. M. Marcus, and S. Vaitiek˙ enas, Voltage-controlled synthesis of higher harmonics in hy- brid josephson junction circuits, Phys. Rev. Lett. 133, 186303 (2024)

work page 2024

[2] [2]

Banszerus, C

L. Banszerus, C. W. Andersson, W. Marshall, T. Linde- mann, M. J. Manfra, C. M. Marcus, and S. Vaitiek˙ enas, Hybrid josephson rhombus: A superconducting element with tailored current-phase relation, Phys. Rev. X 15, 011021 (2025)

work page 2025

[3] [3]

Coraiola, A

M. Coraiola, A. E. Svetogorov, D. Z. Haxell, D. Sabo- nis, M. Hinderling, S. C. t. Kate, E. Cheah, F. Krizek, R. Schott, W. Wegscheider, J. C. Cuevas, W. Belzig, and F. Nichele, Flux-Tunable Josephson Diode Eﬀect in a Hy- brid Four-Terminal Josephson Junction, ACS Nano 18, 9221 (2024), arXiv:2312.04415 [cond-mat]

work page arXiv 2024

[4] [4]

Ciaccia, R

C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Linde- mann, M. J. Manfra, C. Schrade, and C. Sch¨ onenberger, Gate-tunable josephson diode in proximitized inas su- percurrent interferometers, Phys. Rev. Res. 5, 033131 (2023)

work page 2023

[5] [5]

Greco, Q

A. Greco, Q. Pichard, E. Strambini, and F. Giazotto, Double loop dc-SQUID as a tunable Josephson diode, Applied Physics Letters 125, 072601 (2024)

work page 2024

[6] [6]

Leblanc, C

A. Leblanc, C. Tangchingchai, Z. S. Momtaz, E. Kiyooka, J.-M. Hartmann, G. T. Fernandez-Bada, Z. Scher¨ ubl, B. Brun, V. Schmitt, S. Zihlmann, et al. , From nonre- ciprocal to charge-4e supercurrent in ge-based josephson devices with tunable harmonic content, Physical Review Research 6, 10.1103/physrevresearch.6.033281 (2024)

work page doi:10.1103/physrevresearch.6.033281 2024

[7] [7]

Baumgartner, L

C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Supercurrent rectiﬁcation and magne- tochiral eﬀects in symmetric Josephson junctions, Nature Nanotechnology 17, 39 (2022)

work page 2022

[8] [8]

F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Obser- vation of superconducting diode eﬀect, Nature 584, 373 (2020)

work page 2020

[9] [9]

Daido, Y

A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic supercon- ducting diode eﬀect, Phys. Rev. Lett. 128, 037001 (2022)

work page 2022

[10] [10]

Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ili´ c, O. Glezakou- Elbert, et al. , Ubiquitous superconducting diode eﬀect in superconductor thin ﬁlms, Phys. Rev. Lett. 131, 027001 (2023)

work page 2023

[11] [11]

Ili´ c and F

S. Ili´ c and F. S. Bergeret, Theory of the supercurrent diode eﬀect in rashba superconductors with arbitrary dis- order, Phys. Rev. Lett. 128, 177001 (2022)

work page 2022

[12] [12]

J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. Li, Zero-ﬁeld superconducting diode eﬀect in small-twist-angle trilayer graphene, Na- ture Physics 18, 1221 (2022)

work page 2022

[13] [13]

H. D. Scammell, J. I. A. Li, and M. S. Scheurer, Theory of zero-ﬁeld superconducting diode eﬀect in twisted trilayer graphene, 2D Materials 9, 025027 (2022)

work page 2022

[14] [14]

H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, The ﬁeld-free Josephson diode in a van der Waals heterostructure, Nature 604, 653 (2022)

work page 2022

[15] [15]

Zhang, Y

Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, General the- ory of josephson diodes, Phys. Rev. X 12, 041013 (2022)

work page 2022

[16] [16]

C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Asymmetric josephson eﬀect in inversion symmetry breaking topological materials, Phys. Rev. B 98, 075430 (2018)

work page 2018

[17] [17]

Kononov, G

A. Kononov, G. Abulizi, K. Qu, J. Yan, D. Mandrus, K. Watanabe, T. Taniguchi, and C. Sch¨ onenberger, One- Dimensional Edge Transport in Few-Layer WTe2, Nano Letters 20, 4228 (2020), publisher: American Chemical Society

work page 2020

[18] [18]

B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, N. Yuan, N. B. M. Schr¨ oter, L. Fu, and S. S. P. Parkin, Josephson diode eﬀect from Cooper pair momentum in a topological semimetal, Nature Physics 18, 1228 (2022)

work page 2022

[19] [19]

Davydova, S

M. Davydova, S. Prembabu, and L. Fu, Universal joseph- son diode eﬀect, Science Advances 8, eabo0309 (2022), https://www.science.org/doi/pdf/10.1126/sciadv.abo0309

work page doi:10.1126/sciadv.abo0309 2022

[20] [20]

N. F. Q. Yuan and L. Fu, Supercurrent diode eﬀect and ﬁnite-momentum superconductors, Proceedings of the National Academy of Sciences 119, e2119548119 (2022), https://www.pnas.org/doi/pdf/10.1073/pnas.2119548119

work page doi:10.1073/pnas.2119548119 2022

[21] [21]

Matsuo, T

S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lin- demann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Josephson diode eﬀect derived from short- range coherent coupling, Nature Physics 19, 1636 (2023)

work page 2023

[22] [22]

Matsuo, T

S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lindemann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Phase engineering of anomalous josephson eﬀect derived from andreev 11 molecules, Science Advances 9, eadj3698 (2023), https://www.science.org/doi/pdf/10.1126/sciadv.adj3698

work page doi:10.1126/sciadv.adj3698 2023

[23] [23]

Costa, C

A. Costa, C. Baumgartner, S. Reinhardt, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Man- fra, J. Fabian, D. Kochan, N. Paradiso, and C. Strunk, Sign reversal of the Josephson inductance magnetochi- ral anisotropy and 0– π-like transitions in supercurrent diodes, Nature Nanotechnology 18, 1266 (2023)

work page 2023

[24] [24]

Reinhardt, T

S. Reinhardt, T. Ascherl, A. Costa, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, C. Strunk, and N. Paradiso, Link between supercurrent diode and anomalous Josephson eﬀect re- vealed by gate-controlled interferometry, Nature Com- munications 15, 4413 (2024)

work page 2024

[25] [25]

Costa, O

A. Costa, O. Kanehira, H. Matsueda, and J. Fabian, Un- conventional josephson supercurrent diode eﬀect induced by chiral spin-orbit coupling, Phys. Rev. B 111, L140506 (2025)

work page 2025

[26] [26]

Shaﬀer and A

D. Shaﬀer and A. Levchenko, Theories of supercon- ducting diode eﬀects (2025), arXiv:2510.25864 [cond- mat.supr-con]

work page arXiv 2025

[27] [27]

R. S. Souto, M. Leijnse, and C. Schrade, Josephson diode eﬀect in supercurrent interferometers, Phys. Rev. Lett. 129, 267702 (2022)

work page 2022

[28] [28]

Josephson, Possible new eﬀects in superconductive tunnelling, Physics Letters 1, 251 (1962)

B. Josephson, Possible new eﬀects in superconductive tunnelling, Physics Letters 1, 251 (1962)

work page 1962

[29] [29]

A. A. Golubov, M. Y. Kupriyanov, and E. Il’ichev, The current-phase relation in josephson junctions, Rev. Mod. Phys. 76, 411 (2004)

work page 2004

[30] [30]

K. K. Likharev, Superconducting weak links, Rev. Mod. Phys. 51, 101 (1979)

work page 1979

[31] [31]

M. L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, D. Esteve, and C. Urbina, Measurement of the current- phase relation of superconducting atomic contacts, Phys. Rev. Lett. 99, 127005 (2007)

work page 2007

[32] [32]

Ciaccia, R

C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Linde- mann, M. J. Manfra, C. Schrade, and C. Sch¨ onenberger, Charge-4e supercurrent in a two-dimensional InAs-Al superconductor-semiconductor heterostructure, Commu- nications Physics 7, 41 (2024)

work page 2024

[33] [33]

Nichele, E

F. Nichele, E. Portol´ es, A. Fornieri, A. M. Whiticar, A. C. C. Drachmann, S. Gronin, T. Wang, G. C. Gardner, C. Thomas, A. T. Hatke, M. J. Manfra, and C. M. Mar- cus, Relating andreev bound states and supercurrents in hybrid josephson junctions, Phys. Rev. Lett. 124, 226801 (2020)

work page 2020

[34] [34]

C. W. J. Beenakker, Universal limit of critical-current ﬂuctuations in mesoscopic josephson junctions, Phys. Rev. Lett. 67, 3836 (1991)

work page 1991

[35] [35]

A. M. Bozkurt, J. Brookman, V. Fatemi, and A. R. Akhmerov, Double-Fourier engineering of Josephson energy-phase relationships applied to diodes, SciPost Phys. 15, 204 (2023)

work page 2023

[36] [36]

Y. S. Barash, Proximity-reduced range of internal phase diﬀerences in double josephson junctions with closely spaced interfaces, Phys. Rev. B 97, 224509 (2018)

work page 2018

[37] [37]

Tinkham and V

M. Tinkham and V. Emery, Introduction to Supercon- ductivity, Physics Today 49, 198 (1996)

work page 1996

[38] [38]

M¨ uller and E

K.-H. M¨ uller and E. E. Mitchell, Theoretical model for parallel squid arrays with ﬂuxoid focusing, Phys. Rev. B 103, 054509 (2021)

work page 2021

[39] [39]

Ambegaokar and B

V. Ambegaokar and B. I. Halperin, Voltage due to ther- mal noise in the dc josephson eﬀect, Phys. Rev. Lett. 22, 1364 (1969)

work page 1969

[40] [40]

Kringhøj, B

A. Kringhøj, B. van Heck, T. W. Larsen, O. Erlandsson, D. Sabonis, P. Krogstrup, L. Casparis, K. D. Petersson, and C. M. Marcus, Suppressed charge dispersion via res- onant tunneling in a single-channel transmon, Phys. Rev. Lett. 124, 246803 (2020)

work page 2020

[41] [41]

Bargerbos, W

A. Bargerbos, W. Uilhoorn, C.-K. Yang, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, B. van Heck, and A. Kou, Observation of vanishing charge dispersion of a nearly open superconducting island, Phys. Rev. Lett. 124, 246802 (2020)

work page 2020

[42] [42]

Willsch, D

D. Willsch, D. Rieger, P. Winkel, M. Willsch, C. Dickel, J. Krause, Y. Ando, R. Lescanne, Z. Leghtas, N. T. Bronn, P. Deb, et al. , Observation of Josephson harmon- ics in tunnel junctions, Nature Physics 20, 815 (2024)

work page 2024

[43] [43]

Schrade, C

C. Schrade, C. M. Marcus, and A. Gyenis, Protected hy- brid superconducting qubit in an array of gate-tunable josephson interferometers, PRX Quantum 3, 030303 (2022)

work page 2022

[44] [44]

W. C. Smith, A. Kou, X. Xiao, U. Vool, and M. H. De- voret, Superconducting circuit protected by two-Cooper- pair tunneling, npj Quantum Information 6, 8 (2020)

work page 2020