{"id":"f8efa188-91da-4615-b928-34d9511de538","arxiv_id":"2412.13259","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Squeezing-charged single-mode quantum batteries discharge their extractable work faster than displacement-charged batteries with identical initial charge, an effect the authors derive exactly.","lead":"A quantum battery stored in a single light mode charged by squeezing loses its extractable energy faster than one charged by displacement, even when both start with the same charge. The authors derive an exact formula that turns the stored charge into a phase-space relative entropy, making this 'ergotropic Mpemba effect' fully analytical.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The operational 'displacement is better' claim assumes instantaneous work extraction; finite-duration extraction could, in principle, reverse the ordering.","rationale":"I checked the central derivation and found Eqs. (5)–(16) internally consistent. The ergotropy formula follows from the single-mode Gaussian identity f(βπ)=√|Θ|, and the equal-initial-charge ordering E_s(t) ≤ E_d(t) follows directly from f(βt) ≥ Δβ in Eq. (13), so the core physics is well supported. The reader's flagged textual issues—Eq. (15) as printed and the |v|² normalization in Appendix C—are real defects but do not affect the main result. The one genuinely load-bearing element is the explicit fast-extraction assumption: it connects the instantaneous ergotropy curve to the operational claim that displacement charging is a better resource. Because that assumption is stated clearly and is standard in the quantum battery literature, it does not undermine the mathematical result, but it does delimit the practical scope of the conclusion. The proposed finite-extraction simulation would settle whether the ordering survives outside the ideal limit. There is no fitting, no circularity, and no internally inconsistent step affecting the central claim.","tokens_in":14679,"tokens_out":21558,"duration_ms":216191,"concrete_test":"Numerically solve the master equation (8) with an added time-dependent control term H_c(t) that implements the optimal extraction unitary at time t0 over a finite duration τ, using H_c(t)=λ(t)(i(µ* a − µ a†)) for the displaced battery and H_c(t)=λ(t)(i(e^{iθ} a†^2 − e^{-iθ} a^2)/2) for the squeezed battery, with λ(t) chosen so that the unitary is completed in time τ. Compute the actually extracted work W_ext(t0, τ) for τ ranging from 0 up to 1/γ. If W_ext^disp > W_ext^sq persists for all τ ≤ 1/γ, the assumption is not load-bearing; if the ordering reverses for some τ, the operational claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central operational claim—that charging by displacement is a better resource than charging by squeezing—depends on interpreting the instantaneous ergotropy E(t) as the charge that can actually be discharged at time t. This reading requires the assumption stated after Eq. (10): extraction unitaries must act on a timescale much shorter than the dissipative dynamics. If extraction takes a finite time τ comparable to 1/γ, the battery keeps dissipating while work is being extracted, so the achievable extracted work is no longer the instantaneous ergotropy E(t) but the solution of a finite-time control problem. The optimal extraction unitary for a squeezed thermal state is an anti-squeezing quadratic operation, whereas for a displaced thermal state it is a displacement; these have different control Hamiltonians and may respond differently to the same dissipative environment during a finite-duration protocol. Thus the ordering E_s(t) < E_d(t) proven analytically for instantaneous extraction is not automatically an ordering of realistically extractable work. This concern is operational rather than mathematical: Eqs. (5)–(16) are internally consistent, and the single-mode Gaussian restriction is explicit, but the resource-theoretic conclusion is only secured in the ideal instantaneous-extraction limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a single bosonic-mode Gaussian quantum battery weakly coupled to a thermal bath. Its central results are: (i) an exact rewriting of ergotropy as ω f(βπ) K[W||Wπ] for Gaussian states; (ii) an analytic solution of the dissipative dynamics for displaced and squeezed thermal states, expressed through Eqs. (11)-(14); (iii) the identification of an ergotropic Mpemba effect in which a squeezed-charged battery with higher initial ergotropy discharges faster than a displaced-charged one; and (iv) a proof, via Eq. (16), that if the two batteries start with the same ergotropy, the squeezed battery loses its charge faster at all later times. The appendices contain complete derivations of the Gaussian relative entropy, the passive-state relation f(βπ)=sqrt(|Θ|), and the Lyapunov solution.","tokens_in":14768,"tokens_out":10732,"duration_ms":103140,"significance":"The work is a valuable analytic contribution to the quantum-battery and Mpemba-effect literature. The derivation is parameter-free: no constants are fitted, and the crossing time and the equal-ergotropy comparison are derived consequences of the Lindblad dynamics rather than inputs. The phase-space formulation connects battery physics to experimentally accessible Wigner distributions and provides a rare exactly solvable Mpemba setting. Within its stated scope, the mathematics is internally consistent and the appendices are self-contained. The main caveat is scope: single-mode Gaussian states and instantaneous work extraction; the resource-theoretic conclusion requires an explicit operational qualification.","major_comments":[{"comment":"The abstract and the concluding sentence state that charging by displacement is \"a better resource\" than charging by squeezing. This operational claim is only established for the instantaneous ergotropy E(t), as the assumption stated after Eq. (10) makes explicit. If work extraction takes a finite time comparable to 1/γ, the state continues to dissipate during the extraction protocol, and the achievable extracted work is no longer the instantaneous ergotropy but the solution of a finite-time control problem; the ordering of E_s(t) and E_d(t) need not carry over to that setting. Since this is load-bearing for the central resource-theoretic conclusion, please either restrict all operational statements to the instantaneous-extraction limit or provide a quantitative discussion, and ideally numerical evidence, of the finite-duration extraction case.","section":"After Eq. (10) and in the Conclusion"}],"minor_comments":[{"comment":"The phrase \"the higher the temperature βπ is\" is inconsistent with βπ being an inverse temperature; it should read \"the higher the temperature (i.e., the larger \\bar n_π and the smaller βπ)\".","section":"Discussion of Fig. 3"},{"comment":"The crossing-time formula is presented without derivation; please include a short derivation in an appendix or state explicitly that it follows from solving E_s(τc)=E_d(τc) using Eqs. (E6) and (E16).","section":"Eq. (15)"},{"comment":"The equal-initial-ergotropy claim would be much clearer with the displayed inequality E_s(t)=E_d(t)+ω(Δβ−f(βt))≤E_d(t), which follows from f(βt)≥Δβ; the current verbal argument hides the essential step.","section":"Paragraph after Eq. (16)"},{"comment":"There are minor typographical issues: \"Reyni-2\" should be \"Rényi-2\", the heading of Appendix C has a stray space in \"V ector\", and reference [43] has \"Inroductory\" instead of \"Introductory\".","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the mathematical core is sound; I do not see circularity or parameter fitting. The only substantive concern is the operational interpretation of the main resource claim under finite-duration work extraction, which the authors should qualify or address explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a solid, exactly solvable example of a Mpemba-like effect in quantum battery discharging. The central identity, Eq. (5), connecting ergotropy to relative Wigner entropy for single-mode Gaussian states is new as far as I know, and the authors use it to get closed-form discharge curves for displaced and squeezed thermal states. I re-derived the key relations and they are consistent; no free parameters, no fitting, and the figures are direct evaluations of the formulas. That is real, careful work.\n\nThe main message—that displacement-charged batteries retain ergotropy longer than squeezing-charged ones, even when starting from equal stored charge—follows cleanly from the math. The equal-initial-charge result (Eq. 16, Fig. 4) is a nice touch.\n\nSoft spots, in order of importance:\n\n1. The operational claim is that displacement is the better resource. That is proven for instantaneous ergotropy extraction. The paper explicitly assumes the extraction unitary acts on a timescale much shorter than dissipation. If that timescale separation fails, the achievable work is a finite-time control problem and the ordering could in principle change. The stress-test note lands here. It does not undermine the analytic results, but it does mean the resource statement is conditional. I would have liked the authors to phrase it as \"under the standard fast-extraction assumption.\"\n\n2. The single-mode Gaussian restriction is explicit and fine for a Letter, but the resource ordering is established only within this class. A scope limitation, not an error.\n\n3. Presentation defects: Eq. (15) as printed cannot be evaluated as written; the intended formula matches Fig. 3. Also, the |v|² normalization in Eqs. (C4)-(C5) is inconsistent with their definition of v. Both are typos, since all final results are correct, but they should be fixed.\n\nNo issues with the citation pattern; the existing bosonic Mpemba refs are cited and the difference is properly framed as ergotropy versus energy-based measures.\n\nBottom line: this deserves a serious referee. I would accept it for peer review and ask for minor revisions—fix the typos and soften the resource claim to match the instantaneous-extraction assumption. The paper is a useful addition for the quantum battery and Mpemba communities.","headline":"Solid analytic result: ergotropy as Wigner relative entropy gives exact discharge curves and a clean Mpemba example, but the resource claim rests on the explicitly stated instantaneous-extraction assumption.","tokens_in":15418,"tokens_out":2196,"would_cite":true,"duration_ms":22191,"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":"The ergotropy of a single-mode Gaussian quantum battery equals a phase-space relative entropy, and squeezed states discharge faster than displaced ones even at equal initial charge.","keywords":["quantum batteries","ergotropy","Mpemba effect","Gaussian states","Wigner relative entropy","open quantum systems","work extraction","bosonic mode"],"falsifier":"Prepare a single bosonic mode in a squeezed thermal state and a displaced thermal state with equal initial ergotropy, couple each weakly to the same thermal bath, and measure the maximum work extractable over time; the claim predicts the squeezed state's ergotropy falls below the displaced state's at the analytic crossing time $\\tau_c$ of Eq. (15).","tokens_in":14377,"feed_emoji":"🔋","tokens_out":11647,"duration_ms":85632,"temperature":0.7,"pith_summary":"This paper establishes that for a single bosonic mode charged by Gaussian operations and weakly coupled to a thermal bath, the extractable energy (ergotropy) equals a phase-space relative entropy, which makes the discharge curves analytically solvable. In this setting, a squeezed thermal state loses its stored charge faster than a displaced thermal state, even when both start with the same ergotropy. This is an ergotropic Mpemba effect: a more highly charged battery can discharge faster than a less charged one. The authors conclude that charging with displacement is a better resource than charging with squeezing, because the ergotropy dissipates more slowly.","feed_headline":"Squeezed quantum batteries lose charge faster than displaced ones","feed_subtitle":"Even at equal initial charge, squeezing discharges faster than displacement, a quantum Mpemba effect.","key_machinery":"The central object is the Wigner relative entropy, $K[W_1||W_2] = \\int d^2\\alpha\\, W_1 \\ln(W_1/W_2)$, together with the identity $E(W) = \\omega f(\\beta_\\pi) K[W||W_\\pi]$ that ties ergotropy to a phase-space divergence. The companion constraint $f(\\beta_\\pi) = \\sqrt{|\\Theta|}$, which follows from the unitary connection between the state and its passive version, reduces the problem to the covariance matrix and mean vector, splitting the ergotropy into a displacement part $E_d = \\omega|\\mu|^2$ and a squeezing part $E_s = \\omega f(\\beta_\\pi)[\\cosh(2r)-1]$. The analytic solution of the Lyapunov equation for the covariance matrix then yields explicit time-dependent $f(\\beta_t)$ and $r_t$, exposing the non-monotonic passive-state energy of the squeezed state that drives the effect.","core_discovery":"For a single Gaussian bosonic mode with Hamiltonian $\\hat{H} = \\omega(\\hat{a}^\\dagger\\hat{a} + 1/2)$ weakly coupled to a thermal bath, the paper shows that the ergotropy at any time is $E(W) = \\omega f(\\beta_\\pi) K[W||W_\\pi]$, where $W_\\pi$ is the thermal passive state unitarily connected to $W$ and $K$ is the Wigner relative entropy. Because the passive state of the squeezed state carries a positive energy contribution from the squeezing parameter, its ergotropy relaxes faster than that of a displaced state, and the two curves can cross even when the initial ergotropies are identical. The central claim is that displacement charging is therefore a better resource than squeezing charging, since it makes the extractable charge dissipate more slowly.","pith_inferences":["The same relative-entropy identity likely extends to multimode Gaussian states, where intermodal correlations could also raise the passive-state energy and produce analogous ergotropic crossings.","If the extraction unitaries cannot be performed much faster than the dissipative dynamics, the true achievable discharge profile will lag the computed ergotropy, and the Mpemba crossing may be delayed or hidden.","The analytic crossing time could be used to design charging operations that deliberately maximize charge retention by keeping the passive-state energy monotonic."],"forward_implications":["The ergotropy of a Gaussian state is exactly a Wigner relative entropy, so discharge curves for Gaussianity-preserving dissipative dynamics can be computed in closed form.","A squeezed thermal state can discharge faster than a displaced thermal state even when the initial ergotropies are equal, because squeezing adds a positive, non-monotonic contribution to the passive-state energy.","The ergotropic Mpemba crossing time is given by a closed-form expression (Eq. 15) that depends only on the charging parameters and the bath temperature.","Charging with a displacement operation is a better resource than charging with squeezing, since the ergotropy dissipates more slowly in the first case.","The phase-space formulation makes the effect directly testable in quantum optical and mesoscopic platforms where Wigner functions are measured."],"supporting_citations":[{"why":"Defines ergotropy as the maximum work extractable via cyclic unitaries; the central quantity the paper analyzes.","marker":"[9]"},{"why":"Introduces the relative Wigner entropy used to recast ergotropy as a phase-space relative entropy.","marker":"[37]"},{"why":"Establishes that the Lindblad dynamics preserves Gaussianity, justifying the analytic treatment of the Wigner function.","marker":"[41]"},{"why":"Supplies the Lyapunov equation for the covariance matrix under the dissipative dynamics.","marker":"[42]"},{"why":"Provides the transformation properties of displacement and squeezing operators used to split the ergotropy into displacement and squeezing contributions.","marker":"[43]"}],"fun_headline_variants":["Squeezed quantum batteries discharge faster than displaced","Quantum Mpemba effect: squeezed batteries drain quicker","Even at equal charge, squeezed quantum batteries lose power faster","Ergotropic Mpemba: squeezing speeds up quantum battery discharge","Squeezed quantum batteries hit Mpemba-like faster drain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume that the unitary operations used to extract the stored work can be performed almost instantly compared with the dissipative timescale; if extraction is slow, the computed ergotropy curves are not the physically achievable discharge profiles.","fun_headline_variants_meta":{"raw":{"variants":["Squeezed quantum batteries discharge faster than displaced","Quantum Mpemba effect: squeezed batteries drain quicker","Even at equal charge, squeezed quantum batteries lose power faster","Ergotropic Mpemba: squeezing speeds up quantum battery discharge","Squeezed quantum batteries hit Mpemba-like faster drain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000437,"raw_usage":{"total_tokens":2205,"prompt_tokens":910,"completion_tokens":1295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1214}},"tokens_in":526,"tokens_out":1295,"duration_ms":10274,"temperature":1.0,"reasoning_tokens":1214,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:21:10.002637+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Prepare a single bosonic mode in a squeezed thermal state and a displaced thermal state with equal initial ergotropy, couple each weakly to the same thermal bath, and measure the maximum work extractable over time; the claim predicts the squeezed state's ergotropy falls below the displaced state's at the analytic crossing time $\\tau_c$ of Eq. (15).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines ergotropy as the maximum work extractable via cyclic unitaries; the central quantity the paper analyzes."},{"cited_title":"Adesso, D","cited_arxiv_id":null,"evidence_quote":"Introduces the relative Wigner entropy used to recast ergotropy as a phase-space relative entropy."},{"cited_title":"Linowski, A","cited_arxiv_id":null,"evidence_quote":"Establishes that the Lindblad dynamics preserves Gaussianity, justifying the analytic treatment of the Wigner function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov equation for the covariance matrix under the dissipative dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the transformation properties of displacement and squeezing operators used to split the ergotropy into displacement and squeezing contributions."}],"review_version":1}