{"id":"5b6b60d4-5924-4076-94f3-0526fd04f267","arxiv_id":"2607.22235","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"HEOM simulations reproduce finite-bias and thermoelectric currents in Cooper pair splitters, going beyond Markovian and weak-coupling approximations.","lead":"Using a numerical method called HEOM, the authors compute electric currents in Cooper pair splitters at finite voltages and temperatures, going beyond simpler Markovian descriptions. The paper argues these calculations match recent experiments and capture a thermoelectric effect, offering a tool for designing and interpreting hybrid superconductor-quantum-dot devices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central 'quantitative agreement' claim rests on fitting the current maximum; no measured data are overlaid, so the abstract overstates the evidence.","rationale":"Reading the paper in good faith, the HEOM framework is credible, the analytic large-bias limit provides a useful benchmark, and the code is openly available. The central assertion is specifically that HEOM yield quantitative agreement with measured currents; this is what would have to be true for the paper's headline claim. The least secure condition is not the large-gap effective Hamiltonian (the experiments cited use Al gaps of order 200 µeV, much larger than the small bias and temperature scales, so Eq. (1) is likely adequate) nor the HEOM convergence (a technical matter that the reader has already flagged). The real soft spot is that the paper never places theory and experiment on the same axes: the only quantitative number comes from matching the maximum current by choosing Γ, which is a free scale. This does not distinguish HEOM from any other model with a tunable coupling, and it does not validate the full level-position dependence shown in the experimental stability diagrams. The thermoelectric comparison is similarly indirect, since it is benchmarked against a weak-coupling theory rather than the experimental data. These concerns reinforce the reader's CONDITIONAL verdict rather than overturning it: the claim is plausible but unevidenced in its current form. I therefore recommend UNCHANGED, with the explicit condition that the authors provide a direct overlay and residual analysis. My agreement is partial because the reader's identified weakest assumption (large-gap Hamiltonian plus convergence) is not the same as my primary concern, although the reader's rationale does mention the one-point Gamma fit and missing overlays.","tokens_in":922,"tokens_out":2136,"duration_ms":62173,"concrete_test":"Digitize (or obtain from the authors) the measured current stability diagrams from Ref. [22] (and, if available, Refs. [23,24]) for the same voltage and temperature configurations as in Fig. 2. Run HEOM with γ/Γ, κ/Γ, k_B T, and µ as in the paper, allowing only a global scale Γ to be fit, and overlay the full two-dimensional current maps. Compute the normalized residual map (e.g., (I_HEOM − I_exp)/max(I_exp)) and report the maximum and root-mean-square deviation. If the HEOM map cannot capture the measured current within a stated tolerance (say 10–20% of the peak) across the resonance region, the 'quantitative agreement' assertion in the abstract fails. This test is sufficient to settle whether the central claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and Sec. VI assert that 'HEOM yield quantitative agreement with the measured currents' for the finite-bias experiments in Refs. [22]–[24]. The support provided is a single-point calibration: the authors set the HEOM current maximum in Fig. 2(a) to the reported current maximum (0.4 nA) by choosing Γ ≈ 6 GHz (Sec. VI). No measured current map is overlaid, no line traces are compared, and no residual or goodness-of-fit statistic is reported. Because Γ is a free scale, matching one extremum is not a quantitative test; it merely converts units. Similarly, the thermoelectric discussion in Sec. VII compares HEOM with the weak-coupling results of Ref. [59], not with the experimental data of Ref. [16], so the claim to 'account for an experimentally observed thermoelectric effect' is not quantitatively demonstrated either. The central claim would be secure only if the HEOM results reproduce the measured level-position dependence of the currents, not just the peak value. That condition is currently unverified, and it is the least secure part of the paper's argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a hierarchical-equations-of-motion (HEOM) treatment of charge transport in a Cooper pair splitter with two quantum dots coupled to a superconductor and normal leads. It derives analytic large-bias Lindblad limits for elastic cotunneling and Cooper pair splitting, then uses HEOM to compute currents for finite voltage and temperature biases. The paper claims quantitative agreement with recent transport experiments and an account of an experimentally observed thermoelectric effect.","tokens_in":18675,"tokens_out":3996,"duration_ms":39240,"significance":"If the central claim is correct, the paper would establish HEOM as a practical non-Markovian, strong-coupling framework for superconducting hybrid devices. The analytic large-bias reduction provides a useful internal benchmark, and the open-source code is a valuable resource. However, the headline quantitative comparison rests on a one-point calibration, and the thermoelectric section does not compare directly with experimental data. The paper's contribution is therefore methodologically useful, but the evidence presented does not support the abstract's 'quantitative agreement' claim as currently stated.","major_comments":[{"comment":"The text states that the maximum calculated current I=0.4Γe is matched to the measured 0.4 nA by choosing Γ≈6 GHz. This is a single-point scale calibration, not a quantitative test. No measured current map or line trace is overlaid, and no residual or goodness-of-fit statistic is provided. Because Γ is a free parameter, matching one extremum only converts units. Supporting 'quantitative agreement' requires comparing the level-position dependence, resonance width, and off-resonant tails with the experimental data of Refs. [22–24].","section":"Sec. VI, Fig. 2"},{"comment":"The thermoelectric section compares HEOM results with the weak-coupling theory of Ref. [59] and reports qualitative agreement. It does not overlay or quantitatively compare with the experimental data of Ref. [16]. The conclusion that the calculations 'reproduce key features observed in recent experiments' and the abstract's claim to 'account for an experimentally observed thermoelectric effect' are therefore not supported by the presented evidence. A direct comparison of current maps, cuts, or stopping voltages with the experimental data, or a clear weakening of the claim, is required.","section":"Sec. VII, Figs. 7–9"},{"comment":"The HEOM results are used for quantitative statements, but the manuscript does not report the truncation order of the hierarchy, the number of Padé terms N in Eq. (35), or any convergence tests. Without this information the reader cannot judge whether the finite-bias currents are numerically converged. Please report the hierarchy depth, the number of exponentials used, and a convergence check (e.g., results versus increasing truncation order).","section":"Appendix A and Sec. V"},{"comment":"The effective Hamiltonian integrates out the superconductor under the assumption of a large superconducting gap. The quantitative comparison with experiments of Refs. [22–24] presumes that those devices are in this regime. Please state the relevant gap values and justify the approximation for those experiments, or discuss how omitted quasiparticle and higher-order Andreev processes could affect the currents. Otherwise the agreement, even if obtained, does not test the model.","section":"Sec. II, Eq. (1)"}],"minor_comments":[{"comment":"Typo: 'superconductcor' should be 'superconductor'. Also 'which each are tunnel coupled' should be 'which are each tunnel coupled'.","section":"Sec. II"},{"comment":"The caption refers to 'Fig. 8(a)' and 'Fig. 8(b)' when it should refer to Fig. 7(a) and Fig. 7(b).","section":"Fig. 8 caption"},{"comment":"The parameter sets differ between Figs. 1–2 (γ=0.4ℏΓ, κ=2ℏΓ) and Figs. 3–6 (γ=2ℏΓ, κ=3ℏΓ). The choice is not explained; please clarify why different parameters are used.","section":"Figs. 2–6"},{"comment":"The sign convention for particle currents in the leads is not consistent between Eq. (29) and the caption description for Fig. 1(b); please clarify the convention.","section":"Eq. (29) and Fig. 1(b)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is overstated relative to the evidence presented. The authors should either add direct experimental comparisons (or clearly weaken the claim in the abstract and conclusions) and provide HEOM convergence data. With those revisions, the methodological contribution could be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper before reading it. The HEOM machinery is applied carefully and the finite-bias current maps are genuinely new and useful. But the central claim in the abstract—that HEOM yield quantitative agreement with measured currents—is not supported by the evidence in the manuscript. The agreement rests on setting Γ so the HEOM peak matches one measured maximum current (0.4 nA), and no measured current map is overlaid, no line traces are compared, and no residuals are shown. That is calibration, not a test.\n\nWhat the paper actually does well: it provides a clean derivation of the fermionic influence functional, a transparent reduction to the Lindblad limit at large bias, and then systematically applies HEOM to a Cooper pair splitter with finite voltages, finite temperatures, finite Coulomb interactions, and structured spectral densities. The large-bias analytic formulas (Eqs. 29 and 31) serve as a useful internal benchmark, and the numerics appear to follow standard published machinery. The HEOM implementation is based on QuTiP and code/data are openly available, which is real evidence of reproducibility. The finite-bias maps themselves—showing how transport windows, temperature broadening, and spectral width reshape the ECT and CPS resonances—are a solid contribution, and the stopping voltage calculations go beyond the weak-coupling treatment of Ref. [59].\n\nThe soft spots are the ones the stress-test flags, and they are real. First, the abstract's \"quantitative agreement\" is an overstatement given the one-point fit. As a method paper, that is fixable: overlay the measured traces, report residuals, or soften the claim. Second, the paper does not report HEOM truncation parameters, Padé decomposition order, or convergence tests. That is an addressable reproducibility gap; a serious referee should ask for it. Third, the thermoelectric section compares HEOM results to weak-coupling theory, not to the experimental data of Ref. [16], so the claim to \"account for an experimentally observed thermoelectric effect\" is not quantitatively demonstrated either. The qualitative match may be fine, but the evidence is currently missing.\n\nNone of this is fatal to the paper's core contribution as a method demonstration. The derivation is coherent, the large-bias limit checks out, and the finite-bias results are the kind of thing people in the field will want to cite. But the abstract and conclusions overreach relative to the data actually presented. I'd tell a graduate student: read it for the HEOM formulation and the finite-bias maps; ignore the experimental-agreement claim until the authors back it up.\n\nRecommendation: this deserves a serious referee. The methods are solid, the topic is relevant, and the flaws are concrete and fixable. I would send it to peer review with the expectation that the authors either provide the missing overlay/residuals and convergence tests or substantially soften the language.","headline":"Competent HEOM application to Cooper pair splitters with useful new finite-bias results, but the advertised quantitative agreement with experiment is a one-point fit, not a demonstrated match.","tokens_in":19103,"tokens_out":2389,"would_cite":true,"duration_ms":26946,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Hierarchical equations of motion can capture finite-bias and non-Markovian transport in Cooper pair splitters, reproducing measured currents and a thermoelectric effect.","keywords":["Cooper pair splitter","hierarchical equations of motion","non-Markovian transport","finite bias","thermoelectric effect","quantum dot","superconducting hybrid device","elastic cotunneling"],"falsifier":"Repeat the same HEOM calculation with a substantially deeper hierarchy and a higher-resolution Padé fit of the lead correlation functions: if the computed current changes by more than the experimental error bars, the reported agreement is a truncation artifact. Alternatively, measure the same splitter with the dot levels tuned close to the superconducting gap, where quasiparticle tunneling excluded from the effective Hamiltonian should produce features the model cannot reproduce.","tokens_in":18264,"feed_emoji":"⚛️","tokens_out":6484,"duration_ms":55043,"temperature":0.7,"pith_summary":"This paper argues that hierarchical equations of motion (HEOM) can describe charge transport in Cooper pair splitters when the coupling to the leads is not weak and the bias or temperature differences are finite—conditions under which Markovian master equations are expected to fail. The authors compute the electric current through a double quantum dot coupled to a superconductor and two normal leads, resolving the resonances of elastic cotunneling and Cooper pair splitting as functions of the dot level positions. They show that in the large-bias limit the HEOM results reduce to known analytic Lindblad formulas, while at finite bias or temperature they reproduce the measured currents from recent experiments and capture a thermoelectric current generated by a temperature difference. If correct, HEOM becomes a reliable nonperturbative framework for transport in interacting superconducting hybrid devices beyond the wide-band, weak-coupling regime.","feed_headline":"HEOM model reproduces measured Cooper pair splitter currents","feed_subtitle":"Finite bias, temperature, and thermoelectric effects that Markovian master equations miss are captured quantitatively.","key_machinery":"The central object is the fermionic influence superoperator, which expresses the reduced density matrix of the two quantum dots as a time-ordered exponential of lead correlation functions. A Padé decomposition of the correlation functions into sums of exponentials converts the exact integro-differential equation into a closed hierarchy of auxiliary density operators (the HEOM), terminated in practice at a finite depth; the particle current into a lead is extracted from the first-tier auxiliary operators. The effective system Hamiltonian assumes a large superconducting gap and keeps only a cotunneling amplitude κ, local pair-transfer amplitudes γℓ, and a nonlocal splitting amplitude γ that cr","core_discovery":"The central claim is that HEOM yield quantitative agreement with the measured electric currents in Cooper pair splitters operated at finite voltage or temperature differences, a regime where the Markovian and weak-coupling descriptions are insufficient. The superconductor is integrated out into an effective double-dot Hamiltonian with local and nonlocal pair tunneling amplitudes, and the normal leads are treated through an exact fermionic influence functional whose memory is retained by a hierarchy of auxiliary density operators. With large bias voltages, the hierarchy collapses to analytic Lindblad results: a resonance along the diagonal ε_L=ε_R for elastic cotunneling and along the anti-di","pith_inferences":["The paper leaves implicit that the extracted coupling parameters should predict other observables in the same device, such as current noise, full counting statistics, or spin correlations; testing those predictions would strengthen the claimed quantitative agreement beyond the mean current.","Because the hierarchy retains memory, it is natural to apply the same machinery to time-dependent driving (gate pulses, pumping), where non-Markovian effects should be most visible; the authors list this direction but do not carry it out.","The effective-Hamiltonian starting point assumes the experimental devices sit deep in the large-gap regime; if a device is operated near the gap, quasiparticle tunneling omitted here should produce transport features the model cannot account for, marking the boundary of the method's validity."],"forward_implications":["In the large-bias limit, HEOM reproduces the analytic Lindblad expressions for the elastic-cotunneling and Cooper-pair-splitting currents, validating the numerical implementation.","At finite bias, the computed current vanishes when the dot levels leave the transport window set by the lead chemical potentials, matching the suppression seen in measurements but absent from large-bias formulas.","The comparison yields concrete device parameters (tunnel rate around 6 GHz, temperature around 50 mK for the reported current scale), which can be used for quantitative predictions.","The same calculation reproduces a thermally induced current and the stopping voltage that compensates it, giving a handle on particle–hole asymmetry in the device.","Finite Coulomb interactions and non-flat lead spectral densities can be included, so the framework extends beyond the Coulomb-blockade, wide-band approximations."],"fun_headline_variants":["HEOM matches measured splitter currents beyond Markovian limits","HEOM nails splitter currents at finite bias and temperature","HEOM reproduces thermoelectric splitter measurements","HEOM solves splitter quantum transport without Markovian assumptions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The quantitative agreement with experiment rests on the effective Hamiltonian that integrates out the superconductor under the assumption of a large superconducting gap, together with an assumed-converged truncation of the HEOM hierarchy; if the devices are not deep in the large-gap regime or the hierarchy is under-converged, the computed currents would differ.","fun_headline_variants_meta":{"raw":{"variants":["HEOM matches measured splitter currents beyond Markovian limits","HEOM nails splitter currents at finite bias and temperature","HEOM reproduces thermoelectric splitter measurements","HEOM solves splitter quantum transport without Markovian assumptions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":2835,"prompt_tokens":667,"completion_tokens":2168,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":2103}},"tokens_in":411,"tokens_out":2168,"duration_ms":14059,"temperature":1.0,"reasoning_tokens":2103,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T05:21:26.302193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the same HEOM calculation with a substantially deeper hierarchy and a higher-resolution Padé fit of the lead correlation functions: if the computed current changes by more than the experimental error bars, the reported agreement is a truncation artifact. Alternatively, measure the same splitter with the dot levels tuned close to the superconducting gap, where quasiparticle tunneling excluded from the effective Hamiltonian should produce features the model cannot reproduce.","supporting_citations":[],"review_version":1}