Electric and Heat Transport in One-Dimensional SNS Junctions with Energy-Dependent Transmission

arXiv:2610.11713 · cond-mat.supr-con, cond-mat.quant-gas · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Electric and Heat Transport in One-Dimensional SNS Junctions with Energy-Dependent Transmission".

Kai: The gist: This study investigates how different energy-dependent transmission profiles—energy-independent, quantum-dot-type,

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So, moving on, this paper is titled "Electric and Heat Transport in One-Dimensional SNS Junctions with Energy-Dependent Transmission." The authors are Kai Tsutsumi, Shun Uchino, and M. M¨uhlematter from Waseda University in Tokyo.

Mira: They’re looking at how the transmission profile of a one-dimensional superconductor-normal metal junction affects electric and heat currents when you introduce chemical potential and temperature differences between two superconducting reservoirs. It’s about comparing energy-independent, quantum-dot type, and quantum point contact profiles.

Lev: I wonder what the actual difficulty is for these researchers in setting up the theoretical framework to handle those energy dependencies consistently when they incorporate multiple Andreev reflections into the Bogoliubov–de Gennes equations.

Kai: That’s a good question, Lev. They tackle that by using a scattering approach based on those Bogoliubov–de Gennes equations specifically to incorporate multiple Andreev reflections while retaining the energy dependence of transmission through the normal region.

Mira: So, they aren't just ignoring the energy dependence; they are keeping it in because MARs couple scattering processes at different energies, meaning a quasiparticle samples its transmission over a sequence of bias-shifted energies five twenty-one <ref:2610.11713#pg2,samples its transmission over a sequence of bias-shifted energies 5, 21>. This is what drives their analysis of how that interplay affects the electric and heat currents.

Lev: From an error correction perspective, if you were trying to build an actual device on this model, how hard is it to keep track of all those coupled scattering processes described by the unitary scattering matrix S = " r (e) L (E) t (e) R (E) t (e) L (E) r (e) R (E)".

Kai: Well, they handle that complexity by introducing those coupling relations at energies shifted by integer multiples of V, which leads to coupled equations ten <ref:2610.11713#pg1>. They then solve for the reflection and transmission amplitudes using those source coefficients I(p)j and J(p)j.

Mira: And the resulting scattering relations involve terms like R(E) and T(E), with unitarity imposing conditions like R(E) + T(E) = one which are standard constraints in scattering theory three <ref:2610.11713#pg3,R(E) and T(E>. They also have to deal with relating electron and hole scattering amplitudes through a relation involving complex conjugation.

Lev: It sounds incredibly tedious for a simulation setup, Mira. Having all those coupled relations from the unitary conditions means you’ve got a lot of bookkeeping just to make the simulation run correctly before you even start analyzing the currents.

Kai: They did it numerically by solving that three-term recurrence relation, which they found they had to solve using a procedure outlined in Appendix B, which uses a tridiagonal matrix M(p)j(N).

Mira: And that numerical solution involves writing that recurrence relation in matrix form using a tridiagonal matrix M(p)j(N) and defining specific vectors and matrices for two cases of injection channels.

Lev: So, they’ve essentially created a very specific numerical pipeline just to get the results, which is typical when you're pushing the limits of what existing methods can do.

Kai: It shows how much effort goes into getting these results because they aren't just plugging in some simple parameters; they are building this entire framework from scratch for this level of detail.

Mira: And that whole setup is designed to give them the necessary tools to analyze how different transmission profiles influence the transport properties, which is what made the paper so important.

The paper's summary: Kai: So, looking at the summary of "Electric and Heat Transport in One-Dimensional SNS Junctions with Energy-Dependent Transmission," they’re essentially setting up a framework to compare how transmission profiles matter for electric and heat currents when you introduce chemical potential and temperature differences.

Mira: They summarize that the paper investigates the phase-independent dc components of these currents under chemical potential bias and temperature difference between two superconducting reservoirs, using a scattering approach based on Bogoliubov–de Gennes equations, incorporating multiple Andreev reflections while retaining energy dependence of transmission through the normal region.

Lev: The key summary here is that they are comparing three distinct profiles: energy-independent, quantum-dot type, and quantum point contact. They aren't just looking at one scenario; they’s a comparison across these different geometric setups.

Kai: Right. And the paper emphasizes that the comparison is done by calculating the phase-independent dc components of both electric and heat currents under chemical potential bias and temperature difference. It’s a joint analysis of electric and heat transport that needs to be done because they argue it’s necessary to see how these different transmission profiles interact with superconducting conversion processes.

Mira: They stress that the influence of an energy-dependent transmission involves its interplay with superconducting conversion processes, rather than just a change in the normal-state conductance. This is a deeper way of looking at what's happening inside the junction.

Lev: So, what does this mean for us practically? It means we have to look at both currents simultaneously because one doesn't tell the whole story without considering the other.

Kai: Precisely. And it tells us that if you only focus on electric current, you miss a lot of information about how the system is actually responding to those external conditions.

Mira: It’s about seeing that interplay between energy-dependent transmission and the superconducting conversion processes, which is what gives this analysis its unique insight into nonlinear transport.

The paper's improvements: Kai: Now, let’s talk about the improvements they suggest they made to their original work. They suggest focusing on how the comparison of profiles dictates whether you see different temperature responses for these currents.

Mira: They highlight that the key improvement is seeing how different transmission profiles lead to different temperature responses for the electric current, specifically showing that the energy-independent and quantum-dot-like profiles depend only weakly on Delta T when there's a small finite bias.

Lev: That’s a very specific result, but it means we have a clear way to predict when you’re going to see that weak dependence versus when you are going to see the pronounced nonlinear temperature response.

Kai: And they point out that the quantum-point contact profile produces a pronounced nonlinear increase in the electric current with Delta T. It’s that sharp distinction.

Mira: They also confirm that the heat current exhibits a nonlinear dependence on Delta T for all three profiles over every parameter range they investigated, which is important because it shows temperature effects are pervasive across all setups.

Lev: So, if you want to predict transport behavior accurately, you’d use this finding to know exactly which profile gives you the most robust prediction for thermal response versus the electric current.

Kai: That’s the practical implication: the transmission profile choice is crucial for predicting nonlinear responses in both currents. It directs how we set up our experimental measurements based on what we expect to see from these different theoretical setups.

Mira: They essentially give us a clear roadmap: choose your profile based on whether you are prioritizing a weak temperature dependence or a strong nonlinear one.

Conclusion: Kai: So, bringing it all together on "Electric and Heat Transport in One-Dimensional SNS Junctions with Energy-Dependent Transmission," the main point is that the role of the transmission profile is crucial in determining both electric and heat transport beyond linear response.

Mira: They conclude that while electric current is weakly temperature-dependent for some profiles, it exhibits a pronounced nonlinear temperature dependence for the QPC profile. But they also confirm that the heat current depends nonlinearly on Delta T for all three profiles over the parameter range investigated.

Lev: For me, this means we need to treat both currents together because one might look linear while the other isn't, and you can’t ignore those nonlinearities when characterizing transport in these systems.

Kai: It really is about understanding how that transmission profile shapes everything from bias dependence to thermal response. We’ve got a good foundation now for future work in this area.

Mira: This paper provides a microscopic reference for related transport studies in mesoscopic fermionic superfluid junctions, giving us a concrete way to describe these complex effects using energy-dependent transmission.

Lev: I think the comparison between the profiles and their different temperature dependencies is what makes this work so valuable for our field.

Kai: That’s it for this session on the paper. We’ve got some really interesting stuff to unpack there, but we’re ready to look at what’s next in the arXiv feed.

Department of Electronic and Physical Systems, Waseda University

cond-mat.supr-con, cond-mat.quant-gas

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 17 pages, 7 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist: This study investigates how different energy-dependent transmission profiles—energy-independent, quantum-dot-type, and quantum-point-contact—affect nonlinear electric and heat currents

Key concepts

Transmission Profiles
These describe how easily electrons can pass through a normal metal region within a superconducting junction. The study compares three types: energy-independent, Lorentzian quantum-dot-type, and smooth step-like quantum-point-contact (QPC) profiles, each characterized by different transmission probabilities.
Multiple Andreev Reflections (MARs)
These are nonlinear effects occurring at low bias where electrons tunnel across the superconducting gap multiple times. They are responsible for the nonlinear features observed in both electric and heat currents at small energy biases.
Temperature Difference Dependence
This examines how a temperature difference ($\Delta T$) between reservoirs affects transport. The results show that while the electric current is weakly dependent on $\Delta T$ for some profiles, the QPC profile shows a pronounced nonlinear increase, highlighting profile importance in thermoelectric response.

Summary

The gist: This study investigates how different energy-dependent transmission profiles—energy-independent, quantum-dot-type, and quantum-point-contact—affect nonlinear electric and heat currents in one-dimensional superconductor-normal-metal (SNS) junctions when subjected to chemical potential and temperature differences.

Model and Formulation

The investigation uses a scattering approach based on the Bogoliubov–de Gennes equations to incorporate multiple Andreev reflections while retaining the energy dependence of transmission through the normal region The model considers a one-dimensional SNS junction with reservoirs characterized by temperatures, chemical potentials, and phases The phase-independent dc components of the electric and heat currents are calculated subject to chemicalpotential bias and temperature difference.

Transmission Profiles Compared

Three representative transmission profiles are compared in the analysis:

  1. an energy-independent profile

  2. a Lorentzian quantumdot-type profile

  3. a smooth step-like quantum-pointcontact (QPC) profile

These profiles are characterized by transmission probabilities Tconst, TQdot, and TQPC respectively, which are defined in terms of parameters like T0, Γ, and ħomega The QPC profile increases monotonically through its threshold at E∗.

Bias Dependence

At low bias, both electric and heat currents exhibit nonlinear features associated with the superconducting gap and multiple Andreev reflections (MARs) At the upper end of the plotted bias range, the electric current becomes approximately linear for the energy-independent and QPC-like profiles over the range studied The quantum-dotlike profile shows a stronger dependence on the resonance width, where broad resonances yield a more nearly linear current-bias characteristic, whereas narrow resonances produce pronounced nonmonotonic features and a weak increase, or even a decrease, of the current at higher bias.

Temperature Difference Dependence

The temperature difference response varies significantly depending on the transmission profile For the energy-independent and quantum-dot-like profiles, the electric current depends only weakly on ∆T for small finite bias, whereas the quantum-pointcontact profile produces a pronounced nonlinear temperature response The heat current exhibits a nonlinear dependence on ∆T for all three profiles over the investigated parameter range

Conclusion

The results highlight that the role of the transmission profile is crucial in determining both electric and heat transport beyond linear response The comparison shows that while the electric current is weakly temperature-dependent for some profiles, it exhibits a pronounced nonlinear temperature dependence for the QPC profile Furthermore, the heat current depends nonlinearly on ∆T for all three profiles over the parameter range investigated. These findings underscore why both currents should be considered together when characterizing nonlinear transport through SNS junctions. The study provides a microscopic reference for related transport studies in mesoscopic fermionic superfluid junctions

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B. Electric current

The electric current J(DC)ele is derived by substituting the field-operator expansion into the definition of the electric current and retaining only the time-independent dc component The expression for J(DC)ele involves integrals over energy E, involving D(E - ζqeV), scattering amplitudes, and occupation factors from reservoir Sk.

Heat Current Derivation

The heat current Jheat(x, t) is given by an expression derived from the kinetic-energy-current contribution and the chemical-potentialweighted particle-current contribution This expression is obtained by making a replacement in the electric current expression, yielding Eq. (26).

Numerical Results Summary

The numerical results are presented in Figures 4 and 5, comparing the three transmission profiles at ∆T = 0 and showing the effect of a temperature difference for selected transmission parameters The figures illustrate the bias dependence of both J(DC)ele and J(DC)heat across different profiles.

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C. Temperature-difference dependence

The temperature-difference dependence is shown in Figures 6 and 7, where panels (a)-(c) show the response at V = 0.05∆ For the energy-independent and quantum-dot-like profiles, the electric current depends only weakly on ∆T In contrast, the QPC profile produces a pronounced nonlinear increase in the electric current with ∆T The heat current exhibits a nonlinear dependence on ∆T for all three transmission profiles over the parameter range investigated.

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D. Model and Scattering Formalism

The model describes a one-dimensional SNS junction with reservoirs characterized by temperatures Tj, chemical potentials µj, and constant superconducting phases θj The scattering is described by a unitary scattering matrix S where r(e)j(E) and t(e)j(E) are the reflection and transmission amplitudes for an electron incident from side j.

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B. Scattering Relations

The scattering relations couple amplitudes at energies shifted by integer multiples of V, leading to coupled equations (10) These relations involve source coefficients I(p)j and J(p)j, which are defined by comparison with Eq. (10).

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C. Recurrence relation for a specified injection channel

The three-term recurrence relation (16) is derived by eliminating scattering amplitudes other than β(p)j This recurrence relation is solved numerically using the procedure described in Appendix B.

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Appendix A: Derivation of the recurrence relation

The derivation involves several steps, including shifting sideband indices and substituting relations to eliminate scattering amplitudes, leading to Eq. (A17) The explicit expressions for the auxiliary coefficients R(p)j(n±2), S(p)j(n), and T(p)j(n±2) are provided in Appendix A 1.

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Appendix B: Numerical solution of the recurrence relation

The numerical solution involves writing Eq. (16) in matrix form using a tridiagonal matrix M(p)j(N) The vectors and matrices are defined explicitly for two cases of injection channels.

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Appendix C: Derivation of electric and heat currents

The derivation starts with the wave-function representation in the scattering-state representation in Ref. [21] The electric current is obtained by substituting the field-operator expansions into Eq. (22) and retaining only the time-independent dc component, yielding Eq. (23).

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Appendix C: Derivation of electric and heat currents

The heat current expression is obtained from Eq. (23) by making a replacement in the prefactors, yielding Eq. (26).

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V. Conclusion

The study concludes that the role of the transmission profile is crucial in determining both electric and heat transport beyond linear response Superconducting junctions can also exhibit phasedependent thermoelectric and thermal responses, including contributions to the linear-response coefficients L and K [6–8, 20, 34]

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ACKNOWLEDGMENTS

SU thanks D. Bertolusso, T. Bonaccorsi, T. Esslinger, T. Giamarchi, M. M¨uhlematter, M. Talebi, S. Wili, and Y. Yudkin for discussions SU is supported by JST PRESTO (JPMJPR2351) and JSPS KAKENHI (JP25K07191)

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Appendix A: Derivation of the recurrence relation

The derivation of the recurrence relation in Eq. (A8) involves shifting indices and rearranging terms to obtain Eq. (A9) The final form of the recurrence relation is obtained by restricting Eq. (A15) to the injection channel specified by j = k and p = q, yielding Eq. (A17).

--- Page 12 ---

Appendix A: Derivation of the recurrence relation

The explicit expressions for the auxiliary coefficients are derived in Appendix A 1 The final expression for beta(p)j(n ± 2) is obtained by substituting Eq. (A13) and (A14) into Eq. (A9), resulting in Eq. (A16).

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Appendix A: Derivation of the recurrence relation

The final expression for beta(p)j(n ± 2) is obtained by substituting Eqs. (A13) and (A14) into Eq. (A9), resulting in Eq. (A15).

--- Page 14 ---

Appendix A: Derivation of the recurrence relation

Restricting Eq.

Improvements for AI systems

  1. No longer relying on energy-independent transmission approximations for transport calculations; instead, implement a framework that explicitly incorporates the energy dependence of transmission through the normal region, allowing for a more accurate description of transport when scattering properties vary over the relevant energy window.

  2. Develop an AI system capable of distinguishing between different transport regimes based on transmission profiles; specifically, the system can predict whether a junction exhibits a pronounced nonlinear temperature response (as seen in the quantum-point-contact profile) versus a weak dependence (depends only weakly on the temperature difference for the energy-independent and centered quantum-dot-like profiles).

  3. Create a model for predicting heat current nonlinearity; this system will be able to determine if the heat current depends nonlinearly on the temperature difference for all three profiles over the parameter range investigated, providing a more nuanced understanding beyond simple linear terms proportional to K∆T.

  4. Enhance bias-dependent transport prediction by incorporating resonance width effects; the AI can predict how broad resonances yield a more nearly linear current-bias characteristic, whereas narrow resonances produce pronounced nonmonotonic features and a weak increase, or even a decrease, of the current at higher bias.

  5. Implement phase-dependent transport analysis; this allows the system to account for oscillatory contributions from the ac Josephson effect and determine if they are suppressed by averaging over times much longer than the Josephson period, providing quantitative comparisons consistent with experimental preparation.

Sources

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