{"id":"6f6f1e33-7354-439c-abc0-d8e5479e588c","arxiv_id":"2412.18456","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In a single-level quantum dot, the Mpemba effect occurs when an initially farther-from-equilibrium state has a smaller charge-mode amplitude; attractive interaction strongly enhances the effect, and the internal energy reveals it only when interaction is finite.","lead":"A theoretical study shows that in a single-level quantum dot coupled to a heat bath, attractive electron-electron interaction makes the quantum Mpemba effect, where a hotter state relaxes faster, much easier to observe. It gives explicit criteria in terms of charge and parity decay modes and proposes gate and temperature quench protocols for experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's claim is analytic within a clearly stated model. All steps from the master equation to Eq. (23) check out: the rate hierarchy is correct, the relative-entropy expansion has the stated leading term, and the energy result (Eq. 21) vanishes at U = 0 as required. The reader flagged the Markovian sequential-tunneling master equation as the weakest assumption. I scrutinized the attractive-interaction regime where this concern is most acute. At the particle-hole symmetric point with negative U, single-electron states are energetically costly, but the transition rates out of those states remain of order Gamma, so the parity relaxation stays fast and the charge mode is the slow one. Higher-order cotunneling processes are suppressed by Gamma/|U| relative to Gamma and hence cannot overturn the hierarchy under the stated weak-coupling condition. Thus the central claim stands and the reader's ACCEPT verdict is unchanged.","tokens_in":26818,"tokens_out":30644,"duration_ms":281590,"concrete_test":"Recompute the eigenvalues of Eq. (7) for the parameters of Fig. 2(f) (beta U = -10, beta epsilon = 4.4) and verify gamma_c/gamma_p approx 0.008; then include the leading second-order cotunneling correction to the charge rate for hbar Gamma / k_B T = 0.01 and confirm that the correction is below 10% of gamma_c and that the Mpemba crossing time t_M from Eq. (18) changes by less than a factor of 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation is internally consistent: conditions (23) follow from the long-time limit of Eq. (18), the accessible-state triangle in Appendix A supports the state-preparation analysis, and the eigenvalue hierarchy gamma_c <= gamma_p with gamma_c/gamma_p << 1 for attractive U follows from Eqs. (10) and (14). The weakest point is the stated weak-coupling, sequential-tunneling master equation (Sec. II B). I examined whether second-order cotunneling in the attractive regime, where |0> and |2> are nearly degenerate, could break the hierarchy; it cannot at the plotted parameters, because the singly occupied intermediate state decays rapidly (Fermi factors near unity), keeping gamma_p approx 2Gamma, and any cotunneling correction to gamma_c is of order Gamma^2/|U|, parametrically smaller than Gamma for hbar Gamma << k_B T. The quantitative prediction at beta U = -10 is therefore robust, and no load-bearing concern was identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript analyzes the Mpemba effect in a single-level quantum dot with local electron-electron interaction U, coupled to a thermal reservoir. Using a Markovian master equation in the weak-coupling limit and the fermionic duality symmetry, the authors decompose the relaxation into charge, spin, and parity modes. They derive sufficient conditions for the Mpemba effect based on the relative entropy (Eq. (23): the initially hotter state has a smaller charge-mode amplitude) and based on the internal dot energy (Eq. (25), which requires finite U). They show analytically and numerically that a strongly attractive interaction creates a large separation between the charge and parity decay rates, making the Mpemba effect particularly pronounced, and they propose experimentally relevant state-preparation protocols via gate-voltage switches, temperature quenches, interaction quenches, and two-reservoir initial states.","tokens_in":26934,"tokens_out":17628,"duration_ms":152563,"significance":"This is a solid theoretical contribution with a parameter-free, analytically derived criterion for the Mpemba effect in a realistic mesoscopic system. The use of fermionic duality to identify charge and parity modes gives clear physical insight into the relaxation mechanism, and the prediction that attractive-U quantum dots exhibit a pronounced Mpemba effect is falsifiable with existing experimental platforms (e.g., Refs. [31-34]). The proposed state-preparation protocols are concrete and experimentally relevant, and the appendices provide useful proofs of the physical-state triangle and the long-time behavior of the relative entropy. If the results hold, this paper establishes quantum dots as a promising platform for observing and controlling the Mpemba effect.","major_comments":[],"minor_comments":[{"comment":"The statement \"For 0 ≲ ϵ ≲ |U|, γc ≪ γp becomes almost zero\" is too broad. For βU = -10, the ratio γc/γp is small only in a window around the particle-hole symmetry point ϵ ≈ -U/2 (e.g., βϵ ≈ 4.4 gives γc/γp ≈ 0.008), whereas at the edges ϵ = 0 and ϵ = |U| the ratio is approximately 0.25. Please correct this range to avoid overstating the region where the rate hierarchy holds.","section":"Sec. II C, around Eq. (10) and Fig. 1(d)"},{"comment":"The Taylor expansion of log[Pn(t)] around Pn^eq converges only when |Pn(t) - Pn^eq| < Pn^eq for all n. Since the paper uses Eq. (18) to extract the long-time limit, please state explicitly that Eq. (18) is an asymptotic expansion valid for sufficiently long times (or under the stated convergence condition), and that the Mpemba conditions (23) are derived from the leading long-time term. This does not affect the validity of the conclusions, but it clarifies the mathematical status of the expansion.","section":"Appendix B, Eqs. (B1)-(B3) and Eq. (18)"},{"comment":"It would be helpful to add a brief sentence clarifying why the sequential-tunneling master equation remains valid for strongly attractive U near the particle-hole symmetry point, where |0⟩ and |2⟩ are nearly degenerate. The authors could note that the relevant higher-order (cotunneling) corrections are suppressed by a factor of order Γ/|U| in the weak-coupling regime, so they do not alter the hierarchy γc ≪ γp.","section":"Sec. II B, around Eq. (7)"},{"comment":"Please ensure that Eq. (19a) is typeset unambiguously; the coefficient should read A_k = (-1)^k / [k(k-1)].","section":"Eq. (19a)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is well within the scope of the journal and the central derivation is sound. The minor comments above are local and easily addressable in revision; none of them affects the main conclusions. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid theory paper, and the new result is real. The authors show that in a single-level quantum dot, the sign of U matters: with attractive interaction, the charge decay rate becomes much smaller than the parity rate, so the Mpemba effect shows up early and is experimentally visible. They also give an energy-based diagnostic that only works for finite U, and they propose concrete preparation protocols (gate switch, temperature quench, combinations). The fermionic duality framework is their own prior work, but the systematic U-dependence and the criteria in Eqs. (23) and (25) are new.\n\nThe derivations check out. I followed the mode decomposition, the long-time expansion of the relative entropy, and the triangle of physical states in Appendix A. The sufficient conditions are exactly what they claim: if the hotter state has a smaller charge-mode amplitude, the relative-entropy curves cross. The attractive regime gamma_c << gamma_p is correctly identified, and the plots support the claim. The reliance on their own fermionic duality results is fine; those are established results, not a circular move.\n\nThe soft spot is the weak-coupling sequential-tunneling master equation. For strong attractive U, one might worry about cotunneling corrections near the degenerate |0> and |2> states. The stress-test estimate is that such corrections are of order Gamma^2/|U|, parametrically small, so the predicted hierarchy survives at the plotted parameters. This is a standard model choice, stated clearly, and not a load-bearing flaw. Minor gaps: the log expansion in Appendix B lacks a convergence discussion, and the energy observable can be non-monotonic, which the authors themselves acknowledge. Neither changes the conclusions. The paper is also honest about what a gate switch cannot prepare (no c = 0 with p != 0).\n\nI would send this to a referee--someone competent in master equations should check the algebra, but I see no red flags. It is a paper for quantum thermodynamics and mesoscopic transport people, and for experimentalists looking for a concrete recipe. I would cite it.\n\nRecommendation: accept; the paper is careful, honest, and the central claim about attractive interaction is well supported.","headline":"Solid theory paper: the interaction-sign dependence of the quantum Mpemba effect in a quantum dot is new, the attractive-U regime is the right place to look, and the paper deserves peer review.","tokens_in":27485,"tokens_out":3853,"would_cite":true,"duration_ms":35119,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.63.Kv","73.23.-b"],"model":"deepseek-v4-flash","headline":"A single-level quantum dot shows the Mpemba effect whenever the initially hotter state carries less charge-mode amplitude, and attractive electron-electron interaction makes the effect large.","keywords":["electron-electron interaction","quantum Mpemba effect","single-level quantum dot","fermionic duality","relaxation decay modes","nonequilibrium free energy","attractive interaction","energy relaxation"],"falsifier":"A time-resolved experiment on a single-level dot with attractive interaction ($\\beta U \\lesssim -5$) in the Coulomb-blockaded region could settle the claim: prepare two states by a temperature quench such that $D(\\varrho'_{\\rm in})>D(\\varrho_{\\rm in})$ and $|c'_{\\rm in}|<|c_{\\rm in}|$, then record the relative entropy or dot energy versus time. If no crossing appears before both curves decay at the slow charge rate, or if the measured ratio $\\gamma_c/\\gamma_p$ is not much smaller than 1, the central claim fails.","tokens_in":26606,"feed_emoji":"🧊","tokens_out":9408,"duration_ms":75661,"temperature":0.7,"pith_summary":"The paper asks when a single-level quantum dot, tunnel-coupled to a thermal bath, shows the Mpemba effect: an initial state that starts farther from equilibrium relaxes back to equilibrium faster than one that starts closer. Using the relative entropy (equivalently the nonequilibrium free energy) and the internal dot energy as distance measures, it identifies the condition as $D(\\varrho'_{\\rm in})>D(\\varrho_{\\rm in})$ combined with $|c'_{\\rm in}|<|c_{\\rm in}|$, where $c$ is the amplitude of the slow charge decay mode. The sign and magnitude of the electron-electron interaction $U$ then decide how visible the effect is: strongly attractive interaction makes the charge mode nearly frozen ($\\gamma_c \\ll \\gamma_p$), so the initially hotter state overtakes the colder one early and clearly. The paper also shows that the internal energy reveals the effect only when $U$ is finite. If the analysis is right, quantum dots with phonon-mediated attractive interaction are a practical platform for observing the quantum Mpemba effect.","feed_headline":"Attractive quantum dots make the Mpemba effect strong and fast","feed_subtitle":"The paper derives a simple condition on decay-mode amplitudes and predicts that negative-U dots show a clear, early crossing.","key_machinery":"The machinery is the fermionic-duality eigenmode decomposition of the quantum-dot master equation. Fermionic duality is a dissipative symmetry of the evolution kernel under energy inversion, $\\epsilon \\to -\\epsilon$, $U\\to -U$, and it turns the non-Hermitian relaxation matrix into three physically labeled modes: charge, spin, and parity, with rates $\\gamma_c$, $\\gamma_s$, and $\\gamma_p=2\\Gamma$. The charge and parity modes carry the argument: the charge amplitude $c=(c'|\\rho(0))$ and parity amplitude $p=(p'|\\rho(0))$ are the overlaps that decide whether a Mpemba crossing occurs, and the rate ratio $\\gamma_c/\\gamma_p$ sets how early the crossing appears. The same decomposition, together with a Taylor expansion of the logarithm in the relative entropy, produces the two-exponential form of the energy decay and the asymptotic relative-entropy difference that yield conditions (23) and (25).","core_discovery":"On its own terms, the paper's discovery is a mode-level criterion for anomalous relaxation in an interacting quantum dot. The relaxation of the dot populations is decomposed into a charge mode and a parity mode with rates $\\gamma_c$ and $\\gamma_p=2\\Gamma$, and the Mpemba effect occurs for two initial states when the one that is thermodynamically farther from equilibrium carries the smaller charge-mode amplitude: $D(\\varrho'_{\\rm in})>D(\\varrho_{\\rm in})$ and $|c'_{\\rm in}|<|c_{\\rm in}|$. When the charge amplitude of the hotter state is exactly zero, the relaxation is exponentially faster and the strong Mpemba effect appears. The interaction determines how strong the effect is because $\\gamma_c/\\gamma_p$ depends on $U$ and the level position; for strong attractive interaction inside the Coulomb-blockaded region this ratio is very small, so the parity mode dominates and the crossing in the relative entropy happens early. The energy-based version is not equivalent: for finite $U$, the energy decays as a sum of two exponentials with rates $\\gamma_c$ and $\\gamma_p$, and a crossing occurs under a different condition on $c'_{\\rm in}$ and $c_{\\rm in}$; at $U=0$ the parity contribution vanishes and the energy cannot show the Mpemba effect.","pith_inferences":["If the rate hierarchy holds, the same mode decomposition should predict where the Mpemba effect appears in other observables, such as heat currents or full counting statistics, even in devices where the steady state is a nonequilibrium one.","The criterion is tied to the relative-entropy distance; readers should not expect the same 'smaller charge amplitude' rule for other measures, since trace distance or entanglement asymmetry will weight the parity and charge modes differently.","A direct experimental test could use a phonon-mediated negative-$U$ dot, prepare two states by temperature quench, and check that the crossing time tracks $1/(\\gamma_p-\\gamma_c)$; a measured ratio near 1 would argue against the Markovian approximation.","Because the paper's preparation protocols realize initial states that are thermal at a different temperature or gate voltage, the effect could be probed without full state tomography, using only time-resolved energy or charge detection."],"forward_implications":["For any interaction strength, a pair of initial states satisfying $D(\\varrho'_{\\rm in})>D(\\varrho_{\\rm in})$ and $|c'_{\\rm in}|<|c_{\\rm in}|$ will show a Mpemba crossing in the relative entropy, becoming strong when $c'_{\\rm in}=0$.","A quantum dot with strong attractive interaction ($-\\beta U \\gg 1$) in the Coulomb-blockaded region has $\\gamma_c \\ll \\gamma_p$, so the crossing occurs early and the effect is pronounced; repulsive interaction gives a late, weak crossing.","The internal dot energy can show the Mpemba effect only when $U\\neq 0$; at $U=0$ the energy relaxes as a single exponential with rate $\\gamma_c$ and no crossing is possible.","Initial states suitable for observing the effect can be prepared experimentally by a rapid gate-voltage switch or by a temperature quench, and combining both quenches gives access to states with exponential speedup.","The energy decay is not necessarily monotonic: when the charge and parity contributions have opposite signs, the dot energy can overshoot equilibrium before relaxing."],"supporting_citations":[{"why":"Establishes the Markovian decay-mode explanation of the Mpemba effect that the paper generalizes to the quantum dot.","marker":"[9]"},{"why":"Prior prediction of the Mpemba effect in a single-level quantum dot with reservoirs, the setting this work extends to arbitrary interaction strength.","marker":"[12]"},{"why":"Introduces the nonequilibrium free-energy (relative-entropy) distance measure used to define and test the Mpemba effect here.","marker":"[14]"},{"why":"Experiments realizing effective attractive electron-electron interaction in quantum dots, making the predicted strong effect testable.","marker":"[31-34]"},{"why":"Provides the phonon-mediated negative-U mechanism that justifies treating attractive interaction as physically relevant.","marker":"[35]"},{"why":"Derives the fermionic duality and the charge, spin, and parity eigenmode decomposition of the quantum-dot master equation used throughout.","marker":"[36]"},{"why":"Supplies the rate-asymmetry analysis and the thermoelectric energy coefficient entering the energy-decay expression (21).","marker":"[38]"}],"fun_headline_variants":["Attractive electron interactions strengthen quantum Mpemba effect","Negative U dots speed up anomalous cooling","Mpemba effect in quantum dots depends on interaction sign","Strong attraction makes quantum Mpemba effect prominent","Interaction-controlled Mpemba effect in quantum dots"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the standard weak-coupling rate equation with a constant, spin-degenerate tunnel rate describes the dot faithfully for every interaction strength considered, including strongly attractive $U$; if higher-order tunneling, non-Markovian memory, or level broadening changes the relaxation at large $|\\beta U|$, the predicted hierarchy $\\gamma_c\\ll\\gamma_p$ and the Mpemba crossing could shift or disappear.","fun_headline_variants_meta":{"raw":{"variants":["Attractive electron interactions strengthen quantum Mpemba effect","Negative U dots speed up anomalous cooling","Mpemba effect in quantum dots depends on interaction sign","Strong attraction makes quantum Mpemba effect prominent","Interaction-controlled Mpemba effect in quantum dots"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1499,"prompt_tokens":1024,"completion_tokens":475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":640,"tokens_out":475,"duration_ms":4426,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:42:50.866078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A time-resolved experiment on a single-level dot with attractive interaction ($\\beta U \\lesssim -5$) in the Coulomb-blockaded region could settle the claim: prepare two states by a temperature quench such that $D(\\varrho'_{\\rm in})>D(\\varrho_{\\rm in})$ and $|c'_{\\rm in}|<|c_{\\rm in}|$, then record the relative entropy or dot energy versus time. If no crossing appears before both curves decay at the slow charge rate, or if the measured ratio $\\gamma_c/\\gamma_p$ is not much smaller than 1, the central claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Markovian decay-mode explanation of the Mpemba effect that the paper generalizes to the quantum dot."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior prediction of the Mpemba effect in a single-level quantum dot with reservoirs, the setting this work extends to arbitrary interaction strength."},{"cited_title":"Lasanta, F","cited_arxiv_id":null,"evidence_quote":"Introduces the nonequilibrium free-energy (relative-entropy) distance measure used to define and test the Mpemba effect here."},{"cited_title":"V olk, C","cited_arxiv_id":null,"evidence_quote":"Provides the phonon-mediated negative-U mechanism that justifies treating attractive interaction as physically relevant."},{"cited_title":"Hartman, C","cited_arxiv_id":null,"evidence_quote":"Derives the fermionic duality and the charge, spin, and parity eigenmode decomposition of the quantum-dot master equation used throughout."}],"review_version":1}