{"id":"6c3ee51b-1f09-43db-8ec6-7e0a2448ddad","arxiv_id":"2501.11609","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A Green's function approach predicts both types of quantum Mpemba effect in phase-quenched Josephson dots, including spin-orbit and Zeeman-coupled junctions.","lead":"Scientists derived the transition rates for electrons in tiny superconducting wires with a Green's function method, and used them to predict a quantum version of the Mpemba effect, where a more disturbed system relaxes faster than a less disturbed one. The work suggests phase-quenched Josephson dots, readable by microwave spectroscopy, as a practical platform to test this counterintuitive quantum cooling shortcut.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Phase-quench protocol ignores the change of the Andreev eigenbasis; the reported quantum Mpemba effect may be an artifact of initializing the post-quench master equation with the pre-quench populations.","rationale":"The reader's weakest assumption (ad hoc Tqp, continuum equilibrium) is legitimate and indeed load-bearing, because the QME appears only for Tqp > Tb. However, I find a more fundamental problem that the reader did not flag: the quench protocol does not account for the change of the Andreev eigenbasis. A rapid phase quench is an instantaneous change of the single-particle Hamiltonian h(ω) (Eq. (2.31)); the ABS creation operators and many-body states are defined with respect to ϕ0. The pre-quench stationary state, when expressed in the post-quench basis, has coherences and modified populations. The paper simply reuses the old populations as initial conditions for the new master equation, which is only valid if the ABS wavefunctions are independent of ϕ0 or the quench is adiabatic—both contradict the stated 'rapid quench'. This is an internal inconsistency, not a matter of parameter choice, and it directly affects the central QME predictions. The green's function derivation of rates may be correct, but the application to QME is not established. Therefore, the paper's main claim is unverified. The concrete numerical test described above would settle whether the QME survives a correct treatment of the quench; without it, the claim should not be accepted as presented. The reader's CONDITIONAL verdict is reasonable but the condition should include the basis-change issue; I set UNVERDICTED to indicate that the central claim currently lacks support from the presented analysis.","tokens_in":30863,"tokens_out":10129,"duration_ms":122569,"concrete_test":"For the parameters of Fig. 5(e) (T=0.99, Γ=6.5, Tb=0.2, Tqp=0.5), numerically solve Eq. (2.34) for the ABS spinors η(ϕ0) at pre- and post-quench phases. Construct the overlap matrix S_{νν'} = η†_ν(ϕ_i) η_{ν'}(ϕ_eq) and the corresponding many-body unitary U (including vacuum overlaps) that maps the pre-quench ABS basis to the post-quench basis. Set the initial density matrix at t=0+ to U ρ_stat(ϕ_i) U† (restricted to the ABS sector), then evolve with the Pauli master equation (3.15). Recompute the τ and DM(P(0)) maps in Fig. 5(e). If the Mpemba arcs survive with the same topology, the concern is resolved; if they shift or vanish, the QME prediction is not robust.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The most load-bearing concern is the treatment of the phase quench itself. In Sec. IIIA2, the protocol initializes the post-quench Pauli master equation with the pre-quench stationary populations P_{r,stat}^{(c,f)} (see text after Eq. (3.15)). However, the ABS eigenspinors ην(ϕ0) in Eq. (2.34) and the quasiparticle operators a_λ in Eq. (2.33) depend on ϕ0. A sudden phase change therefore rotates the Andreev basis: at t=0+ the pre-quench density matrix is not diagonal in the new basis, and the diagonal elements (populations of the new ABS levels) are not equal to the old occupations. The paper never computes the overlap matrix between old and new ABS spinors or the many-body projection. In the short-junction limit Γ≫Δ, the ABS spinor rotates substantially with ϕ0 (the dispersion Eq. (3.4) spans from Δ to Δ√(1−T)≈0.1Δ for T=0.99), so the overlap can deviate significantly from unity. Consequently, the calculated relaxation times and the 'Mpemba arcs' in Figs. 5, 6, and 8 may be artifacts of an unstated assumption that ABS occupations are conserved during the quench. This is an internal inconsistency in the stated physical scenario, independent of the ad hoc Tqp issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a Green's-function-based equation-of-motion approach for multi-level quantum dots coupled to superconducting leads and a bosonic environment, and uses it to derive Markovian transition rates for Andreev bound states interacting with phase fluctuations. It then applies the resulting Lindblad/Pauli dynamics to study quantum Mpemba effects after a sudden phase quench, claiming both type-I and type-II QMEs in short junctions and type-II-dominated QMEs in intermediate-length spin-orbit-coupled junctions. The central technical results are the rate formulas Eqs. (2.56), (2.77), and (2.79), and the central predictions are encoded in the phase-quench protocol of Sec. IIIA2 and the numerical maps of Figs. 5, 6, and 8.","tokens_in":31172,"tokens_out":8348,"duration_ms":94078,"significance":"If the results hold, the paper provides a useful simplification: transition rates that previously required explicit BdG wavefunction matching are obtained from dot-level matrices, and the explicit comparison with Refs. [9,10] lends credibility to the GF route. The application to QMEs is concrete and experimentally falsifiable in principle, since the trace distance in Eq. (3.22) is tied to microwave-spectroscopy measurements of ABS populations, and the paper is unusually explicit about its assumptions (Tqp, spectral-density forms, parameter choices). The numerical results are supported by a data-availability statement. However, the credibility of the QME predictions hinges on two load-bearing steps that are not fully justified: the mapping of pre-quench to post-quench populations under the phase quench, and the truncation of principal-value contributions in the ABS-continuum rate derivation.","major_comments":[{"comment":"The quench protocol initializes the post-quench Pauli master equation with the pre-quench stationary populations P^{(c,f)}_{r,stat}. This equates occupations in the old and new Andreev bases, but the ABS eigenspinors η_ν(ϕ0) in Eq. (2.34) depend on ϕ0, so a sudden change from ϕ0^(i) to ϕ0^(eq) rotates the quasiparticle basis. A density matrix that was diagonal in the old basis acquires coherences in the new basis, and the new occupations are not equal to the old ones: they are given by a projection involving overlaps such as |⟨η_n(ϕ_eq)|η_ν(ϕ_i)⟩|² in the occupation sector. This overlap is never computed or estimated. For the high-transparency case T=0.99 used in Figs. 5 and 6, the short-junction dispersion (3.4) spans from Δ down to about 0.1Δ, so the spinor rotation with ϕ0 is substantial and the approximation P(t=0)=P_{r,stat}(ϕ_i) is not obviously negligible. Since the Mpemba arcs and the type-I/type-II classification in Figs. 5, 6, and 8 are defined by relaxation times obtained from these initial conditions, the central QME claim is affected. Please compute the projection of the pre-quench stationary state onto the post-quench ABS basis, or provide a controlled argument for why the overlap correction can be dropped.","section":"Sec. IIIA2, after Eq. (3.15)"},{"comment":"The replacement G^R_{nk}(E) = -iπ δ_{nk} ρ_n(E) discards the principal-value integrals in Eq. (2.69) under the assumption that ρ_m(z) is smooth. Smoothness alone does not make a principal-value integral vanish: the integrand also contains the overlap factors ξ†_n(E)ξ_m(z), and the principal value generally gives a nonzero off-shell contribution whose magnitude depends on the model parameters. This step enters directly into the continuum rates Γ^out_λ and Γ^in_λ in Eqs. (2.77) and (2.79), and hence into every QME prediction in Sec. III. The agreement with the BdG-based rates of Refs. [9,10] is reassuring, but the derivation given here does not establish the truncation. The authors should either prove the cancellation for this model, or quantify the omitted principal-value terms by evaluating Eq. (2.69) numerically for the multi-level examples of Sec. IIIB.","section":"Sec. IIC3, Eq. (2.74)"},{"comment":"The continuum of BCS quasiparticles is modeled by a global thermal Fermi function at an effective quasiparticle temperature Tqp, and the QME regimes shown in Figs. 3 and 4 appear only for Tqp > Tb (compare panels (a) with (b,c)). The parameter Tqp is introduced with a qualitative physical discussion (local cooling of the dot region, phonon-induced lead equilibration), but it is not derived from a microscopic calculation or tied to an experimentally controllable quantity. As a result, the central prediction is conditional on an unquantified free parameter. The authors should specify how Tqp is set in a concrete experimental protocol, or map the QME region in the (Tqp,Tb) plane and state how robust the type-I/type-II classification is to plausible Tqp variations.","section":"Sec. IIB2 and Sec. IIIA1"}],"minor_comments":[{"comment":"The notation Q_T is overloaded: Eq. (2.65) states Q_T(E) = T Q_T(E) with Q_T defined in Eq. (2.51), but Eq. (2.71) is only correct for the redefined object. Please use a separate symbol, e.g., R_T(E) = T Q_T(E), and make the factor of T explicit in Eq. (2.65).","section":"Eqs. (2.51), (2.65), and (2.71)"},{"comment":"The population curves are distinguished only by dashed/dot-dashed/dotted styles; in printed or grayscale versions these may be difficult to tell apart, especially where the curves cross. Adding markers or labels inside the panels would improve readability.","section":"Figs. 3 and 4"},{"comment":"The sentence immediately after Eq. (3.22) notes that DM(P(t)) differs from DM(Pr(t)) because the second component of Pr is counted twice. It would be helpful to write the explicit relation to avoid confusion about the factor 1/2 in the definition.","section":"Sec. IIIA2, Eq. (3.22)"},{"comment":"The statement that the existence of at least one Mpemba arc is a necessary and sufficient condition for the QME is stronger than what is shown; the numerical evidence establishes sufficiency-type behavior for the studied parameters, but a proof or a more cautious wording would be appropriate.","section":"Sec. IIIA3"},{"comment":"The truncation of the infinite Hamiltonian matrix to eigenstates below an energy cutoff comparable to Δ is mentioned, but the convergence of the hybridization matrices (2.25) with respect to this cutoff is not discussed. A brief convergence test for the L=1.7ξ0 case would strengthen the numerical results in Sec. IIIB.","section":"Appendix, Eq. (A6)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe real content here is the Green's function derivation of transition rates for Andreev states coupled to a bosonic environment. That part is clean and useful: it reproduces known BdG results in the atomic limit, extends them to finite Δ, avoids the need to compute full BdG wavefunctions, and the numerical implementation is backed by uploaded data on Zenodo. Credit where due: the rate formulas in Sec. IIC are a solid technical contribution.\n\nThe advertised headline—quantum Mpemba effects in phase-quenched Josephson dots—is where I have a substantial concern. The quench protocol initializes the post-quench Pauli master equation with the stationary populations of the pre-quench phase. But the Andreev spinors themselves depend on the phase. A sudden change of φ0 rotates the single-particle basis; at t=0+ the old occupations are not the populations of the new ABS levels. The paper never computes the overlap between old and new spinors. For a short junction with transparency 0.99 the spinor rotation is large, so the error is not obviously small. This makes the \"Mpemba arcs\" and the type-I/II classification in Figs. 5, 6, and 8 conditional on an unstated approximation. If the authors want to claim a real physical prediction, they need to either justify the approximation (e.g., by showing the overlap is near unity in the relevant parameter range) or solve the quench dynamics within the GF framework.\n\nThat is the main soft spot. Secondary issues: the effective quasiparticle temperature Tqp enters the rates and the QME appears only for Tqp > Tb; this is a reasonable model but not a derivation. The ABS-to-continuum rate derivation discards principal-value integrals in Eq. (2.74) on a smoothness assumption; plausible but worth checking. And the claim that a \"Mpemba arc\" is necessary and sufficient for QME is stated without proof; it's probably true for the two-exponential relaxation here, but it should be a one-line argument.\n\nOverall: worth a serious referee, but the referee should push on the quench initialization before accepting the QME predictions. The rate formalism alone justifies reading the paper.","headline":"Solid GF-based rate derivation for Andreev levels, but the quantum Mpemba predictions rest on an unjustified sudden-quench initialization that ignores the rotation of the Andreev spinors.","tokens_in":31688,"tokens_out":6470,"would_cite":true,"duration_ms":69180,"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":"A Green's-function derivation shows quantum Mpemba effects in phase-quenched Josephson dots.","keywords":["Green's function","Josephson junction","Andreev bound states","quantum Mpemba effect","Lindblad master equation","spin-orbit coupling","Zeeman field","phase quench"],"falsifier":"Measure the relaxation time $\\tau$ as a function of pre-quench phase $\\phi_0^{(i)}$ for fixed post-quench phase $\\phi_0^{(eq)}$ in a short, high-transparency junction with $T\\approx 0.99$ and $T_{qp}>T_b$; the central claim predicts non-monotonic $\\tau$ with local minima ('Mpemba arcs') and both QME types, so a flat, monotonic landscape in this regime would falsify it. Observing a QME at $T_{qp}=T_b$, where the paper predicts none, would also falsify it.","tokens_in":30646,"feed_emoji":"🧊","tokens_out":5859,"duration_ms":59155,"temperature":0.7,"pith_summary":"This paper develops a Green's-function method for computing quasiparticle transition rates in multi-level quantum dots coupled to superconducting leads and a bosonic environment, avoiding explicit Bogoliubov–de Gennes eigenstate calculations. It applies the method to a Josephson dot whose average phase difference is suddenly quenched, and argues that the resulting open-system relaxation can exhibit quantum Mpemba effects, where a copy prepared farther from the final phase difference reaches the steady state faster. For a short single-channel junction, the paper finds parameter windows for both type-I and type-II quantum Mpemba effects; for an intermediate-length junction with spin-orbit coupling and a Zeeman field, it predicts that type-II effects dominate. The predictions matter because they identify a concrete, tunable superconducting platform where Mpemba accelerations could be observed through microwave spectroscopy of Andreev-level populations.","feed_headline":"Quantum Mpemba effect predicted in phase-quenched Josephson dots","feed_subtitle":"A Green's-function analysis shows both Mpemba types in short junctions and type-II dominance with spin-orbit coupling.","key_machinery":"The central object is the retarded boundary Green's function of the BCS leads, $g^R(\\omega) = -\\pi\\nu_F (\\omega\\tau_0 + \\Delta\\tau_x)/\\zeta(\\omega)$, which is used to integrate out the leads and obtain a time-nonlocal effective Schrödinger equation for the $4\\ell$-component dot wavefunction. Transition rates follow from first-order perturbation theory in the phase fluctuation, yielding $\\Gamma_{\\lambda\\to n} = 2\\pi |I_{n\\lambda}|^2 J(\\omega) n_B(\\omega)$ at $\\omega = E_n - E_\\lambda$, with current matrix elements $I_{n\\lambda} = \\eta_n^\\dagger[\\tau_z I(E_\\lambda) - I(E_n)\\tau_z]\\eta_\\lambda$. Rates to the continuum are obtained by resolving the spectral function into eigenstates $\\xi_n(\\omega)$, and the resulting expressions are consistent with earlier Bogoliubov–de Gennes calculations while avoiding explicit wavefunction matching in the leads. This machinery converts the nonequilibrium dynamics of Andreev levels into a Pauli master equation whose rates depend only on matrices in dot-level space.","core_discovery":"The paper's central claim is that the low-energy Andreev sector of a Josephson dot obeys a Markovian Lindblad equation whose rates can be obtained from the poles and spectral eigenvectors of the retarded Green's function of the dot-lead system, and that these rates produce quantum Mpemba effects after a rapid phase quench. In the short-junction limit with high transparency, the steady-state Andreev populations become non-monotonic functions of the phase bias once the quasiparticle temperature exceeds the bath temperature, and this non-monotonicity is what lets a 'far' pre-quench phase relax faster than a 'close' one. The paper distinguishes type-I QME, where the far copy is closer to equilibrium for all times, from type-II QME, where the far copy crosses the close copy at a finite time, and shows both can be realized. In longer junctions with spin-orbit and Zeeman terms, the broken phase-reflection symmetry removes the mirror symmetry of the relaxation-time map, making type-II QME the dominant outcome. The authors identify 'Mpemba arcs' in the pre- versus post-quench phase plane, curves where stationary populations coincide, as necessary and sufficient for the effect.","pith_inferences":["Editorial inference: the same Green's-function rate formulas apply to a dot with Coulomb interactions, and the paper states the formalism allows this but does not pursue it; an interacting Josephson dot is therefore a natural next place to search for QMEs.","Editorial inference: because the relaxation-time map loses mirror symmetry when spin-orbit and Zeeman terms break $\\phi_0\\to 2\\pi-\\phi_0$, comparing quenches on the two sides of the phase interval would give a sharp test of the predicted type-II dominance.","Editorial inference: the 'avoided QME' crossings visible in the authors' time traces suggest that experiments using only crossing times as a QME diagnostic could misclassify an avoided QME; a population-resolved measurement of the full distance function would be needed to confirm the type."],"forward_implications":["For a short, high-transparency junction, both type-I and type-II quantum Mpemba effects can be produced by a rapid phase quench, with relaxation times set by the slowest eigenvalue of the Pauli master equation.","The existence of at least one 'Mpemba arc' in the pre- versus post-quench phase plane is necessary and sufficient for a quantum Mpemba effect in this system.","For an intermediate-length junction with spin-orbit coupling and a Zeeman field, the type-II effect dominates, because the broken $\\phi_0\\to 2\\pi-\\phi_0$ symmetry makes Mpemba arcs invisible in the initial distance but visible in relaxation time.","The effect is robust to the shape of the environment spectral density: similar QME regimes appear for both Lorentzian microwave-resonator and Ohmic environments.","The steady-state current-phase relation develops pronounced minima near the phase values where Andreev population components have extrema, providing an experimentally accessible indicator of the $T_{qp}>T_b$ regime."],"supporting_citations":[{"why":"Supplies the Bogoliubov–de Gennes based transition-rate framework for Andreev and continuum states that the present rates reproduce.","marker":"[5]"},{"why":"Provides BdG-derived rates and dynamical parity selection in superconducting weak links, used as a benchmark for the Green's-function rates.","marker":"[9]"},{"why":"Previous many-body dynamics of spin-orbit coupled Andreev states in a Zeeman field, whose Lindblad structure this paper extends.","marker":"[10]"},{"why":"Supplies the bosonic bath correlation function and spectral-density conventions used to model the electromagnetic environment.","marker":"[13]"},{"why":"Defines the two-type quantum Mpemba protocol and trace-distance criterion adopted throughout Section III.","marker":"[38]"},{"why":"Provides the short-junction Andreev dispersion relation used in the analytical $\\Gamma\\gg\\Delta$ limit.","marker":"[54]"}],"fun_headline_variants":["Mpemba arcs traced in phase-quenched Josephson dots","Green's function predicts both Mpemba types in dots","Phase quench yields Mpemba effect in Josephson dots","Spin-orbit coupling favors type-II Mpemba in long dots","Both Mpemba effects possible in short Josephson dots"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The continuum of BCS quasiparticles stays in a thermal Fermi distribution at an effective temperature $T_{qp}$, chosen independently of the bosonic bath and with $T_{qp}\\ge T_b$; without that assumption the predicted Mpemba windows do not emerge.","fun_headline_variants_meta":{"raw":{"variants":["Mpemba arcs traced in phase-quenched Josephson dots","Green's function predicts both Mpemba types in dots","Phase quench yields Mpemba effect in Josephson dots","Spin-orbit coupling favors type-II Mpemba in long dots","Both Mpemba effects possible in short Josephson dots"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2967,"prompt_tokens":949,"completion_tokens":2018,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":1932}},"tokens_in":565,"tokens_out":2018,"duration_ms":15475,"temperature":1.0,"reasoning_tokens":1932,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:03:45.282634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the relaxation time $\\tau$ as a function of pre-quench phase $\\phi_0^{(i)}$ for fixed post-quench phase $\\phi_0^{(eq)}$ in a short, high-transparency junction with $T\\approx 0.99$ and $T_{qp}>T_b$; the central claim predicts non-monotonic $\\tau$ with local minima ('Mpemba arcs') and both QME types, so a flat, monotonic landscape in this regime would falsify it. Observing a QME at $T_{qp}=T_b$, where the paper predicts none, would also falsify it.","supporting_citations":[{"cited_title":"However, they cannot be described by eigenstates ofh(ω) ̸= h†(ω) for |ω| > ∆","cited_arxiv_id":null,"evidence_quote":"Supplies the Bogoliubov–de Gennes based transition-rate framework for Andreev and continuum states that the present rates reproduce."},{"cited_title":"Mpemba arcs","cited_arxiv_id":null,"evidence_quote":"Provides BdG-derived rates and dynamical parity selection in superconducting weak links, used as a benchmark for the Green's-function rates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous many-body dynamics of spin-orbit coupled Andreev states in a Zeeman field, whose Lindblad structure this paper extends."},{"cited_title":"Giuliano and I","cited_arxiv_id":null,"evidence_quote":"Supplies the bosonic bath correlation function and spectral-density conventions used to model the electromagnetic environment."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the short-junction Andreev dispersion relation used in the analytical $\\Gamma\\gg\\Delta$ limit."}],"review_version":1}