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Quantization shifts Mpemba effect to ultra-cold temperatures

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load-bearing objection Quantum Mpemba in the bistable potential: clean mechanism, schematic figures, load-bearing two-level approximation the 4 major comments →

arxiv 2607.06071 v1 pith:AZ4SEJOI submitted 2026-07-07 quant-ph

Quantization of the classical Mpemba effect

Jannis Melles , Hartmut L\"owen , Benno Liebchen , Michael te Vrugt , Giovanna Morigi , Artur Widera , Alexander P. Antonov This is my paper
classification quant-ph
keywords Mpemba effectquantum relaxationbistable potentialanomalous relaxationLindblad dynamicsultracold atomsmetastabilitynon-equilibrium thermodynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Mpemba effect—where a hotter system cools faster than a warm one—has a well-understood classical mechanism in asymmetric double-well potentials: relaxation is bottlenecked by thermally activated barrier crossing, and anomalous relaxation arises from non-monotonic temperature dependence of the population imbalance between the two wells. This paper asks what happens when the same double-well system is quantized. The authors find that quantization replaces the classical barrier-crossing bottleneck with a fundamentally different one: relaxation is governed by transitions between discrete energy eigenstates, and the slowest mode is set by weakly coupled low-lying states whose energy splittings lie far below the barrier height. This shifts the Mpemba effect to temperatures orders of magnitude below the classical regime. The quantum system also exhibits an inverse Mpemba effect (anomalous heating) and a double-inverse Mpemba effect with two disconnected initial-temperature windows, neither of which has a classical counterpart in the same potential. The mechanism is traced to the non-monotonic temperature dependence of the population of the weakly coupled first excited state, which controls the slowest relaxation channel.

Core claim

When the paradigmatic classical Mpemba benchmark—an asymmetric bistable potential—is quantized, the relaxation bottleneck changes from thermal barrier crossing (governed by barrier height V_0) to tunneling between discrete eigenstates (governed by energy splittings Δϵ ≪ V_0). This relocation of the bottleneck shifts anomalous relaxation to ultra-cold temperatures inaccessible classically and generates new Mpemba phenomena: an inverse Mpemba effect under heating and a double-inverse Mpemba effect with two distinct initial-temperature windows. The quantum Mpemba effect requires the existence of a weakly coupled eigenstate whose population P_1(T) is non-monotonic in temperature; when all eigen-

What carries the argument

The central object is the population of the first excited state P_1(T) = e^{-βϵ_1} / Σ_n e^{-βϵ_n}, which is non-monotonic in temperature (vanishing at both T→0 and T→∞) and controls the slowest Liouvillian relaxation mode. The prefactor |P_1(T_init) - P_1(T_fin)| in the exponential relaxation law determines whether anomalous relaxation occurs, analogous to the classical domain-occupation imbalance P_L(T) but operating on the quantum energy-splitting scale Δϵ rather than the barrier height V_0.

Load-bearing premise

The quantum Mpemba effect requires the existence of at least one eigenstate that is weakly coupled to the rest of the system, making it metastable and governing the slowest relaxation mode. If no such weakly coupled state exists—as can happen for certain potential shapes—the quantum Mpemba effect is suppressed even though the classical effect may persist.

What would settle it

If experiments in ultracold-atom double-well systems at temperatures below the barrier height fail to show anomalous relaxation tied to the population of a specific weakly coupled eigenstate, or if the relaxation time does not exhibit the predicted non-monotonic dependence on initial temperature, the central claim would be challenged.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Ultracold atoms in tailored optical double-well potentials could serve as experimental platforms for observing quantum Mpemba effects at nanokelvin temperatures, using existing trapping and thermalization techniques.
  • The mechanism suggests a route to rapid quantum state preparation: by choosing initial temperatures where the population of the bottleneck eigenstate matches the target, one can bypass the slowest relaxation channel entirely.
  • The double-inverse Mpemba effect under heating protocols indicates that quantum systems can exhibit multiple anomalous-relaxation windows, potentially enabling fine-grained temperature control strategies.
  • The requirement of a weakly coupled eigenstate identifies a design principle for engineering or suppressing Mpemba behavior in quantum systems via spectral engineering of the potential landscape.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the weakly coupled eigenstate is the load-bearing structure, then systems with controllable tunneling (e.g., optical lattices with tunable barrier heights) could switch Mpemba behavior on and off by tuning the coupling strength of specific eigenstates.
  • The shift from barrier-height to energy-splitting scales suggests that the quantum Mpemba effect could persist in systems where classical barrier crossing is exponentially suppressed, making it relevant for low-temperature quantum annealing or adiabatic quantum computing where thermal activation is frozen out.
  • The appearance of a double-inverse effect driven by two extrema in |P_1(T_init) - P_1(T_fin)| raises the question of whether higher-dimensional potentials with multiple weakly coupled states could produce triple or higher-order Mpemba effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This letter investigates how quantization modifies the Mpemba effect in the paradigmatic asymmetric bistable potential. The authors show that, whereas the classical Mpemba effect is governed by thermal barrier crossing at temperatures comparable to the barrier height $V_0$, the quantum analog arises from non-monotonicity of the Boltzmann population $P_1(T)$ of a weakly-coupled metastable eigenstate, shifting the effect to ultra-cold temperatures set by the level splitting $k_B T ~ Δε ≪ V_0$. The paper further predicts an inverse and a double-inverse quantum Mpemba effect under heating protocols, which are absent in the classical counterpart. The analytical framework reduces both the classical (Eq. 2, Eq. 15) and quantum (Eq. 23) Mpemba effects to a non-monotonicity condition on a population probability, and the quantum derivation proceeds via a two-level truncation of the Liouvillian spectral decomposition.

Significance. The paper addresses a well-motivated gap: prior quantum Mpemba studies have focused on spin models and open quantum systems, while the classical bistable-potential benchmark—where the mechanism is most transparent—has not been quantized. The parameter-free derivation of the quantum Mpemba condition from the Boltzmann distribution over discrete eigenstates (Eq. 3) is a clean analytical result. The prediction of a double-inverse quantum Mpemba effect is novel and falsifiable. The connection to ultracold-atom experiments in optical double-well potentials is concrete and well-supported by realistic parameter estimates. These strengths are partly offset by the schematic nature of the figures and the reliance on approximations whose quantitative validation is deferred to the Supplemental Material.

major comments (4)
  1. End Matter, Eqs. (18)–(20): The reduction from the full spectral sum $c_1 = Σ_n α_n P_n(T_{init})$ (Eq. 18) to the two-level form $c_1 = (α_1 - α_0)[P_1(T_{init}) - P_1(T_{fin})]$ (Eq. 20) is the load-bearing step for the central analytical result (Eq. 23) and especially for the double-inverse Mpemba claim. The paper states that the dominant contribution arises from $α_1$ 'in agreement with numerical results' (deferred to SM) but does not quantify $|α_n|/|α_1|$ for $n ≥ 2$ in the main text. This matters acutely for the double-inverse claim: the number of extrema of $|P_1(T_{init}) - P_1(T_{fin})|$ (Fig. 4) determines whether a 'double' effect exists, but the full $c_1(T) = Σ_n α_n P_n(T)$ could have different extremal structure if multiple $P_n(T)$ contribute with comparable weights, since each $P_n(T)$ is individually non-monotonic. The main text should either (i) provide a quantitative
  2. bound on $|α_n|/|α_1|$ for $n ≥ 2$ demonstrating that the truncation is controlled, or (ii) show numerically that the full $c_1(T)$ and the truncated expression have the same number of extrema for the parameter regime of Fig. 3(b). Without this, the double-inverse Mpemba effect—a headline claim of the paper—is not rigorously established by the analytical argument presented.
  3. End Matter, Eqs. (16) and (23): The single-mode approximation for the Liouvillian dynamics requires a spectral gap $|Re(Λ_1)| ≪ |Re(Λ_2)|$ so that the slowest mode dominates at long times. This gap condition is not demonstrated in the main text. For the double-inverse effect, which involves two distinct initial-temperature windows where relaxation is anomalously fast, it is important to verify that the single-mode regime is reached within experimentally relevant timescales for both windows. The SM should be referenced explicitly for this verification, or a brief statement of the gap ratio should be included.
  4. Fig. 3 and Fig. 4: These figures are described as schematic/illustrative, but they are the primary evidence for the inverse and double-inverse Mpemba effects. Fig. 3(b) in particular shows the relaxation time with two extrema and distinct temperature windows, but no numerical data or parameter values are given. Since the double-inverse effect is a central novel claim, at least one panel with quantitative numerical results—showing the actual relaxation time $t_{relax}(T_{init})$ from the full Lindblad dynamics (not just the schematic of Fig. 3 or the population imbalance of Fig. 4)—should be included in the main text or explicitly cross-referenced to a specific SM figure.
minor comments (5)
  1. The notation $P_n(T)$ in Eq. (3) and the End Matter uses $P_1$ for both the population of state $|1⟩$ and the left-domain occupation probability $P_L$ in the classical case. While context disambiguates, a brief clarifying remark would help.
  2. Eq. (3): The expression $P_1(T) = e^{-βΔε} / (1 + Σ_{n≥1} e^{-β(ε_n - ε_0)})$ has a potential notational issue—the sum in the denominator starts at $n ≥ 1$ but the numerator already accounts for $n=1$. This should be $Σ_{n≥2}$ in the denominator, or the expression should be written as $e^{-β ε_1} / Σ_n e^{-β ε_n}$ as stated in the first equality.
  3. The potential $V(x)$ and its parameters (barrier height $V_0$, well-depth asymmetry $ΔV$, level splitting $Δε$) are introduced qualitatively but never specified explicitly. A figure or table with the concrete parameter values used for each panel of Fig. 3 would make the results reproducible.
  4. The reference to 'quantum active matter' [58–64] in the conclusion feels tangential to the main results. Consider trimming or motivating the connection more explicitly.
  5. Ref. [31] (Hayakawa and Takada, 'Quantum tunneling Mpemba effect') appears closely related to the present work and is cited only once in passing. The relationship between that work and the present results should be clarified to establish novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive reading of our manuscript. The referee correctly identifies the key analytical steps on which our central claims rest, and we agree that the main text should provide stronger quantitative support for these steps. Below we respond to each major comment in turn.

read point-by-point responses

  1. Referee: End Matter, Eqs. (18)–(20): The reduction from the full spectral sum to the two-level form is the load-bearing step for the central analytical result (Eq. 23) and especially for the double-inverse Mpemba claim. The paper does not quantify |α_n|/|α_1| for n ≥ 2 in the main text. The full c_1(T) could have different extremal structure if multiple P_n(T) contribute with comparable weights.

    Authors: The referee is correct that the truncation from Eq. (18) to Eq. (20) is the critical step, and we agree that the main text should justify it quantitatively rather than deferring entirely to the SM. We will address this in revision as follows. First, we will add to the main text a quantitative statement of the ratio |α_n|/|α_1| for n ≥ 2, computed from the Lindblad eigenvectors for the parameter regime of Fig. 3(b), demonstrating that the truncation is controlled (the metastable state |1⟩ is weakly coupled to the rest of the spectrum by construction, so α_1 dominates by orders of magnitude). Second, we will include a numerical comparison between the full c_1(T) = Σ_n α_n P_n(T) and the truncated expression (α_1 − α_0)[P_1(T_init) − P_1(T_fin)] over the relevant temperature range, showing that both exhibit the same number of extrema. This directly addresses the concern that higher-order terms could alter the extremal structure underlying the double-inverse claim. We note that the physical mechanism—non-monotonicity of a single weakly-coupled metastable state's Boltzmann population—makes it structurally unlikely that subleading terms would introduce additional extrema, but we agree this should be shown, not asserted. revision: yes

  2. Referee: End Matter, Eqs. (16) and (23): The single-mode approximation requires a spectral gap |Re(Λ_1)| ≪ |Re(Λ_2)|. This gap condition is not demonstrated in the main text. For the double-inverse effect, it is important to verify that the single-mode regime is reached within experimentally relevant timescales for both temperature windows.

    Authors: This is a fair point. The single-mode approximation is standard in the metastability literature (Refs. [46, 47] in our manuscript), but we agree that the gap ratio should be stated explicitly for the specific parameters used in Fig. 3(b). In the revised manuscript, we will include the numerical value of |Re(Λ_1)|/|Re(Λ_2)| for the relevant parameter regime and confirm that it is sufficiently small for both initial-temperature windows associated with the double-inverse effect. We will also add an explicit cross-reference to the SM figure where the full time-dependent relaxation is shown, so the reader can verify that the single-mode regime is reached on experimentally accessible timescales. If the SM does not currently contain this verification in sufficient detail, we will expand it accordingly. revision: yes

  3. Referee: Fig. 3 and Fig. 4: These figures are described as schematic/illustrative, but they are the primary evidence for the inverse and double-inverse Mpemba effects. At least one panel with quantitative numerical results—showing the actual relaxation time t_relax(T_init) from the full Lindblad dynamics—should be included in the main text or explicitly cross-referenced to a specific SM figure.

    Authors: We agree. The schematic figures serve to communicate the qualitative structure of the effect, but the double-inverse Mpemba effect is a central novel claim and should be backed by quantitative numerical evidence in the main text. In the revised version, we will add a panel showing t_relax(T_init) computed from the full Lindblad dynamics (not the truncated two-level model) for the parameter regime of Fig. 3(b), with explicit parameter values stated. This will directly demonstrate the two extrema and the two distinct temperature windows. We will also ensure that the SM figure containing the full numerical verification is explicitly referenced at the relevant location in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity. The derivation is parameter-free and self-contained; self-citations are contextual rather than load-bearing.

full rationale

The paper's central result—that quantization shifts the Mpemba effect to ultra-cold temperatures and produces inverse/double-inverse variants—follows from the non-monotonicity of P₁(T) = e^{-βΔε}/(1 + Σ_{n≥1} e^{-β(ε_n−ε₀)}) (Eq. 3), which is a direct consequence of the Boltzmann distribution over discrete eigenstates. No parameter is fitted to Mpemba data and then presented as a prediction. The classical comparison uses the standard Arrhenius/Boltzmann result (Eq. 2). The analytical derivation (Eqs. 16→23) reduces the relaxation dynamics to |P₁(T_init)−P₁(T_fin)| via a two-level approximation (Eq. 19) and a single-mode approximation (Eq. 23), but these are stated as approximations validated by numerical results (deferred to SM), not as definitions that circularly produce the target result. The weakly-coupled eigenstate assumption is openly acknowledged as load-bearing (Fig. 3c shows the effect is suppressed without it), but this is a physical assumption about the system's spectral structure, not a circular definition. Self-citations (Refs. [6], [62], [69]) appear in the context of quantum active matter motivation and dissipative modeling, not as load-bearing premises for the Mpemba derivation itself. Refs. [46, 47] (Macieszczak et al.) are external citations for metastability theory. The derivation chain is: Boltzmann distribution → non-monotonic P₁(T) → non-monotonic prefactor in relaxation law → Mpemba effect. Each step is a genuine physical deduction, not a renaming or a fit repackaged as prediction. The two-level truncation (Eq. 18→19) and the claim that α₁ dominates are approximations whose validity is a correctness concern (noted by the skeptic), not a circularity concern. No step reduces to its own inputs by construction. Score 1 reflects minor self-citations that are contextual rather than load-bearing on the central claim.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper does not invent new physical entities. It applies standard quantum mechanical constructs (eigenstates, Lindblad dynamics, double-well potentials) to a known classical problem. The free parameters are structural choices for the potential and dissipation model, not fitted constants. The axioms are domain assumptions about the system's spectral properties, with the weakly-coupled eigenstate being the most critical.

free parameters (3)
  • Potential shape parameters (asymmetric double-well) = Not specified in main text; deferred to SM
    The specific potential V(x) parameters (barrier height V0, well depths, asymmetry) are chosen to exhibit the desired phenomena (e.g., ground-state inversion, weakly-coupled eigenstate). These are structural choices that enable the effect.
  • Transition rates γ(ω) = Not specified; dipole coupling assumed
    The Lindblad transition rates γ(ω) for the quantum dissipative dynamics are not derived from first principles but are assumed to exist and lead to thermalization. Their specific form affects the Liouvillian spectrum.
  • Tolerance distance ε0 = Arbitrary small
    The relaxation time (Eq. 12) depends on the choice of ε0, the tolerance threshold for defining 'relaxed.' This is a standard but ultimately arbitrary parameter in Mpemba effect definitions.
axioms (4)
  • domain assumption The quantum system exhibits metastability due to a weakly-coupled eigenstate (e.g., |1⟩).
    Stated in the main text: 'For the quantum Mpemba effect to emerge, multiple slow modes need to exist in the system for the metastability of one of the states... Without loss of generality, we assume it to be |1⟩.' This is a load-bearing assumption for the single-mode approximation.
  • domain assumption The two-level approximation (involving only |0⟩ and |1⟩) is valid for the long-time dynamics.
    Eq. 19 and Eq. 23 rely on reducing the system to two levels. The paper states 'Within the two-level approximation involving only |0⟩ and |1⟩, this expression reduces to...' without proving the approximation is quantitatively justified in the main text.
  • standard math The dissipative dynamics relaxes to a thermal stationary state.
    The Lindblad dynamics (Eq. 6) is constructed to have the thermal state as a stationary state. This is a standard assumption for studying thermal relaxation.
  • domain assumption The emergence of Mpemba effects is governed by spectral properties of the Liouvillian rather than microscopic details.
    The paper states: 'For the considered dissipative dynamics... the emergence of the Mpemba effects is governed by the spectral properties of the Liouvillian and the presence of metastable states rather than by microscopic details of the Liouvillian.' This justifies using a generic thermalizing Lindbladian.

pith-pipeline@v1.1.0-glm · 15310 in / 2913 out tokens · 561454 ms · 2026-07-08T17:23:31.647223+00:00 · methodology

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read the original abstract

The Mpemba effect refers to the counterintuitive phenomenon that an initially hot system can freeze faster than an initially warm one. Recent years have brought major advances in both classical and quantum realizations, with the asymmetric bistable potential emerging as the paradigmatic classical benchmark because its mechanism is especially transparent and controllable. Yet for precisely this benchmark problem, the impact of quantization remains unexplored. Here we show that quantization shifts Mpemba behavior to qualitatively new regimes, moving it to ultra-cold temperatures that are orders of magnitude lower than those relevant for classical thermal barrier crossing. In addition, quantization produces inverse and double inverse Mpemba effects that are absent in the corresponding classical dynamics. Our results establish quantization as a robust route to quantum-enabled Mpemba effects inaccessible in classical regimes.

Figures

Figures reproduced from arXiv: 2607.06071 by Alexander P. Antonov, Artur Widera, Benno Liebchen, Giovanna Morigi, Hartmut L\"owen, Jannis Melles, Michael te Vrugt.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the cooling process in a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Key ingredient for the Mpemba effect: non-monotonic [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) A Mpemba effect (reduction of the relaxation time upon increasing the initial temperature) for the cooling process [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The population imbalance [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

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Forward citations

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