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Mpemba effect and super-accelerated thermalization in the damped quantum harmonic oscillator

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Matching the first r moments of the thermal state makes a damped quantum oscillator relax r+1 times faster.

desk verdict Solid population moment-matching result, but the general coherence theorem in Eq. (35) is misstated for s=0 and needs a fix before it can be used as stated. read the letter →

arxiv 2411.09589 v2 pith:VFASUHNR submitted 2024-11-14 quant-ph

Stefano Longhi This is my paper · ORCID
classification quant-ph PACS 03.65.Yz05.30.-d42.50.-p
keywords MpembaeffectquantumthermalizationdampedharmonicoscillatormethodofmomentsLindbladmasterequationsuper-acceleratedrelaxationpopulationcoherencesbosonicmode
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the simplest open quantum system, a harmonic oscillator damped by a thermal bath, and asks when it thermalizes faster than expected. Its central claim is that if the initial population distribution has its first r moments exactly equal to those of the equilibrium thermal distribution, the oscillator relaxes to equilibrium at rate $2\gamma(r+1)$, one step faster for every matched moment. Because a distribution can sit far from equilibrium and still match a few moments, this produces a genuine quantum Mpemba effect inside a paradigmatic model: farther-from-equilibrium states can overtake nearer ones. The statement is proven exactly by the method of moments and the Lindblad spectrum, and it extends to a broad class of initial states with coherences under explicit conditions.

What carries the argument

The carrier of the argument is the moment hierarchy of the Pauli master equation for the populations: the $l$-th moment $Q_l(t)$ evolves only through lower moments, so if the first $r$ moments start at their equilibrium values they remain locked there while the $(r+1)$-th moment relaxes at rate $2\gamma(r+1)$. Spectrally, this is backed by the exactly solvable Lindbladian spectrum ($\lambda_\alpha=-2\gamma\alpha$ in the population sector and $\lambda^{(s)}_\alpha=-2\gamma(\alpha+|s|/2)+i\omega_0 s$ for coherences), whose left eigenoperators are polynomials in the number operator — polynomial eigenfunctions are exactly why moment matching cancels the slow modes.

What would settle it

Numerically integrate the Pauli master equation starting from a two-Fock mixture $p|0\rangle\langle 0|+(1-p)|n_1\rangle\langle n_1|$ with $p=n_{\rm th}/n_1$ so the mean is exactly $n_{\rm th}$; if the Kullback-Leibler distance to equilibrium decays with exponent $4\gamma t$ rather than $8\gamma t$, the vanishing of the slow-mode amplitude predicted by the theory is disproved.

Watch

Extended reading notes

Core claim

For the damped quantum harmonic oscillator in a Markovian thermal bath at temperature $T$, the population spectrum of the Lindbladian is $\lambda_\alpha=-2\gamma\alpha$ and the coherence spectrum is $\lambda^{(s)}_\alpha=-2\gamma(\alpha+|s|/2)+i\omega_0 s$. The paper proves that any initial population distribution $P_n(0)$ whose first $r$ moments match those of the thermal distribution $P^{(S)}_n$ has zero amplitude in the $r$ slowest population modes, so it relaxes with leading rate $|\lambda_{r+1}|=2\gamma(r+1)$. With coherences present, the same super-accelerated relaxation survives provided $\rho_{n,m}(0)=0$ for $|m-n|\le 2r+1$; a general moment theorem on the transformed coherences guarantees relaxation no slower than $\gamma h$ for any positive integer $h$. This is the quantum Mpemba effect in a foundational model.

Load-bearing premise

The result assumes the oscillator is weakly coupled to a memoryless thermal bath so the Lindblad equation is valid, and that the initial population distribution has finite moments.

Editorial extensions

If this is right

  • A thermal initial state at any temperature $T'$ close to $T$ always excites the slowest population mode, so it relaxes at rate $2\gamma$; any state with the first $r$ moments matched relaxes at $2\gamma(r+1)$, making the Mpemba crossing generic.
  • Coherence does not destroy the speedup: if the low-lying coherences up to $|m-n|\le 2r+1$ are absent initially, the full density matrix still relaxes at a rate no slower than $2\gamma(r+1)$.
  • At zero bath temperature, population-only acceleration disappears because all thermal moments vanish, so the zero-temperature bosonic Mpemba effect must be carried by coherences rather than by population moments.
  • Moment-matched initial states provide a practical fast-reset mechanism for quantum devices: a state engineered to match a few thermal moments of the surrounding bath will reset faster than a nearby thermal state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same triangular moment hierarchy should appear in any Markovian thermalization map whose slow eigenmodes are polynomials of a conserved observable; the acceleration mechanism is therefore more general than the harmonic-oscillator model itself.
  • Editorial inference: an experiment could verify the mechanism by monitoring photon-number moments rather than the full distribution — matched low-order moments should be frozen in time while the first unmatched moment decays at the predicted rate.
  • Editorial inference: because the argument requires finite moments, distributions with power-law tails cannot be accelerated; a truncated experimental approximation of such a distribution should show the acceleration degrade as the cutoff is lowered.
  • Editorial inference: the crossing time in the paper's $n_{\rm th}=2$ example, near $\gamma t \simeq 0.28$, offers a sharp quantitative benchmark for cavity or trapped-ion implementations of the effect.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper analyzes thermalization of a damped quantum harmonic oscillator coupled to a thermal bath, described by the standard Markovian Lindblad master equation. Using generating-function methods for the spectral problem and a moment hierarchy, it shows that an initial diagonal state whose first r population moments equal those of the thermal equilibrium distribution does not excite the r slowest population eigenmodes and therefore relaxes at the accelerated rate 2γ(r+1). The same moment matching is used to construct explicit Mpemba-effect examples, comparing thermal states at nearby temperatures with specially prepared Fock-state mixtures. The paper then extends the discussion to initial states with coherences and states a general theorem, Eq. (35), intended to give a complete criterion for super-accelerated thermalization of the full density matrix. Numerical diagonalization of the Pauli master equation supports the population-subspace results.

Significance. The population-subspace result is a clean, parameter-free analytical criterion: matching the first r moments to the thermal distribution removes the r slowest population eigenmodes, giving a relaxation rate growing linearly in r. This is derived self-consistently and is confirmed numerically, and the constructed Mpemba examples are explicit and convincing. The paper therefore makes a useful contribution to the quantum Mpemba literature. However, the advertised general criterion in Sec. 4 is not correct as stated for the population block, and since this criterion is highlighted in the abstract and conclusions as one of the central results, the error is load-bearing and must be repaired.

major comments (3)
  1. [Sec. 4, Eq. (35); Appendix B.2, Eq. (82)] The general theorem is stated incorrectly for the s=0 block. Eq. (35) requires, for s=0 and l=1,2,... with 2l<h, that sum_n n^l rho_{n,n}(0)=0. This imposes vanishing initial population moments, not equality to the thermal moments. The correct population condition, proven in Sec. 3.3 and Appendix A.2, is sum_n n^l rho_{n,n}(0)=sum_n n^l P_n^{(S)}. Consequently the theorem as written is false. A concrete counterexample is n_th=2, h=3, and the initial vacuum state |0><0|: all conditions in Eq. (35) are satisfied because all coherences vanish and the only population condition is the vanishing first moment, but Eq. (26) gives C1=1, so the slowest decaying population mode has rate 2 gamma, not the claimed rate >=3 gamma. The theorem should either be restricted to s>=1, with the population block treated separately by the moment-matching condition, or Eq. (35) should be replaced by the condition that the moments of rho_{n,n}(0)-P_n^{(S)} vanish.
  2. [Appendix B.2, final paragraph] The statement that the coherence theorem 'also holds for s=0 and l=1,2,3,..., i.e. for populations, reproducing the result given in Appendix A.2' is directly contradicted by Appendix A.2 itself. In Appendix A.2 the accelerated-relaxation condition is Eq. (63), Q_l(0)=Q_l^{(S)} for l=1,...,r, not Q_l(0)=0. The vanishing-moment condition used for coherences cannot reproduce the population result, because the thermal distribution has nonzero population moments. This sentence must be corrected together with Eq. (35).
  3. [Abstract and Sec. 4] Because Eq. (35) is presented as the paper's general criterion for super-accelerated thermalization, the error in the s=0 case affects the central claim of the paper as summarized in the abstract, not merely a peripheral lemma. The population result remains valid, but the generality claimed for the coherence theorem should be restated carefully, for instance by formulating separate conditions for the population block and for the coherence blocks s>=1.
minor comments (3)
  1. [Sec. 3.3, Eq. (28)] The notation P_n^{(S)}(0) on the right-hand side is misleading because the stationary distribution does not depend on time; it should read P_n^{(S)}.
  2. [Sec. 3.3, footnote 2] The footnote says 'the condition given by Eq. (25) is never satisfied' when discussing distributions without finite moments; the intended reference appears to be Eq. (27), the condition C1=0, rather than Eq. (25).
  3. [Sec. 3.4] The numerical truncation at Nmax=1800 is adequate for the reported nth values, but the paper does not state how the truncation error was checked for the slowest decay exponents; a brief convergence statement would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central moment-matching acceleration theorem is derived self-containedly from the Lindblad generator; self-citations are only contextual.

full rationale

The central claims are derived analytically from Eq. (1), with the population and coherence spectra computed by explicit generating-function methods in Sec. 3.2 and Appendices A.1 and B.1; the eigenvalues lambda_alpha = -2 gamma alpha and lambda^(s)_alpha = -2 gamma(alpha + |s|/2) + i omega_0 s are obtained by solving the spectral problem, not assumed. The moment-matching condition is derived from the same Pauli master equation in Sec. 3.3 and Appendix A.2 (Eq. 61), and the vanishing of the low-order spectral amplitudes C_1, ..., C_r follows from the triangular property of the moment matrix (Eq. 68), not from the desired conclusion. No fitted parameter is renamed as a prediction, and no target result is fed back as an input. The self-citations [32,35,52] are contextual references to prior Mpemba and laser work and are not load-bearing. A separate correctness concern, not a circularity, is that the Sec. 4 general theorem as stated in Eq. (35) imposes vanishing population moments for s = 0 rather than matching the thermal moments; for example, the vacuum state with n_th = 2 and h = 3 satisfies Eq. (35) but still excites the 2 gamma population mode, contradicting the claimed gamma h bound. This is an internal mathematical inconsistency in the theorem statement, not a derivation that assumes its conclusion, so it does not raise the circularity score. Overall, the paper's main derivation is self-contained and exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters: γ and nth are model inputs, not fitted. Axioms are the Markovian master equation, the Hilbert-space choice for the spectral problem, and the finite-moment condition on initial states. No new entities are postulated.

assumptions (3)
  • domain assumption The damped quantum harmonic oscillator is governed by the Markovian Lindblad master equation (Eq. 1) under the Born-Markov and weak-coupling approximations.
    All results are derived from this master equation; if non-Markovian or strong-coupling effects are present, the predicted spectrum and relaxation rates may not apply.
  • domain assumption The spectral problem is posed on the Hilbert space of sequences with ∑|ψ_n| exp(σ+n)<∞, σ=0+, which makes the spectrum purely point (Appendix A.1).
    The eigenvalue set λα = -2γα depends on this particular Hilbert-space choice; other spaces could introduce a continuous spectrum.
  • domain assumption The initial population distribution has finite moments (Sec. 3.3).
    The paper explicitly assumes Pn(0) decays fast enough that all moments Il are finite; for heavy-tailed distributions without finite moments, accelerated relaxation is absent.

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Cite this review

Pith. "Pith review of Mpemba effect and super-accelerated thermalization in the damped quantum harmonic oscillator." pith.science (2026). https://pith.science/paper/VFASUHNR

@misc{pith2026241109589,
  author       = {Pith},
  title        = {Pith review of: Mpemba effect and super-accelerated thermalization in the damped quantum harmonic oscillator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFASUHNR}},
  note         = {Machine review of arXiv:2411.09589}
}
abstract

The behavior of systems far from equilibrium is often complex and unpredictable, challenging and sometimes overturning the physical intuition derived from equilibrium scenarios. One striking example of this is the Mpemba effect, which implies that non-equilibrium states can sometimes relax more rapidly when they are further from equilibrium. Despite a rich historical background, the precise conditions and mechanisms behind this phenomenon remain unclear. Recently, there has been growing interest in investigating accelerated relaxation and Mpemba-like effects within quantum systems. In this work, we explore a quantum manifestation of the Mpemba effect in a simple and paradigmatic model of open quantum systems: the damped quantum harmonic oscillator, which describes the relaxation of a bosonic mode in contact with a thermal bath at finite temperature $T$. By means of an exact analytical analysis of the relaxation dynamics based on the method of moments in both population and coherence subspaces, we demonstrate that any initial distribution of populations with the first $r$ moments exactly matching those of the equilibrium distribution shows a super-accelerated relaxation to equilibrium at a rate linearly increasing with $r$, leading to a pronounced Mpemba effect. In particular, one can find a broad class of far-from-equilibrium distributions that relax to equilibrium faster than any other initial thermal state with a temperature $T'$ arbitrarily close to $T$. The super-accelerated relaxation effect is shown to persist even for a broad class of initial states with non-vanishing coherences, and a general criterion for the observation of super-accelerated thermalization is presented.

Figures

Figures reproduced from arXiv: 2411.09589 by the authors.

Figure 1
Figure 1. (Color online) (a) Spectrum λ of the Lindbladian L = LP LLC of the damped quantum harmonic oscillator. The blue circles are the eigenvalues of the population sub-block LP , whereas the red squares are the eigenvalues of the coherence sub-block LC = L s L (s) C , with s = ±1, ±2, ±3, .... The λ = 0 eigenvalue corresponds to the equilibrium thermal distribution S. (b) Schematic of the absorption and emission of quanta… view at source ↗
Figure 2
Figure 2. (Color online) Mpemba effect in the relaxation dynamics of a damped quantum harmonic [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. (Color online) Thermalization dynamics (behavior of the distance [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (Color online) Mpemba effect in the relaxation dynamics of a damped quantum harmonic [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: (Color online) Contour paths in complex z plane entering in the calculation of the eigenfunction amplitude ψn via the inversion relation Eq.(48). The contour C is a circle of radius R, with z2 = 1 < R < z1 = 1 + 1/nth. The contour C can be deformed in the contour C ′ =…

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

58 extracted references · 54 canonical work pages · cited by 2 Pith papers

  1. [1]

    E. B. Mpemba and D. G. Osborne. "Cool?". Phys. Educ.4, 172 (1969). Accepted in Quantum 2025-03-22, click title to verify. Published under CC-BY 4.0. 20

    work page 1969

  2. [2]

    The Freezing of Hot and Cold Water

    G. S. Kell. "The Freezing of Hot and Cold Water". Am. J. Phys.37, 564 (1969)

    work page 1969

  3. [3]

    The Mpemba effect: when can hot water freeze faster than cold?

    M. Jeng. "The Mpemba effect: when can hot water freeze faster than cold?". Am. J. Phys. 74, 514 (2006)

    work page 2006

  4. [4]

    Experimental verifications of Mpemba- like behaviors of clathrate hydrates

    Y.-H. Ahn, H. Kang, D.-Y. Koh, and H. Lee. "Experimental verifications of Mpemba- like behaviors of clathrate hydrates". Korean J. Chem. Eng.33, 1903 (2016)

    work page 2016

  5. [5]

    When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids

    A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos. "When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids". Phys. Rev. Lett.119, 148001 (2017)

    work page 2017

  6. [6]

    Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse

    Z. Lu and O. Raz. "Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse". Proc. Natl. Acad. Sci. U.S.A.114, 5083 (2017)

    work page 2017

  7. [7]

    Mpemba index and anomalous relaxation

    I. Klich, O. Raz, O. Hirschberg, and M. Vucelja. "Mpemba index and anomalous relaxation". Phys. Rev. X9, 021060 (2019)

    work page 2019

  8. [8]

    Faster Uphill Relaxation in Thermodynamically Equidis- tant Temperature Quenches

    A. Lapolla and A. Godec."Faster Uphill Relaxation in Thermodynamically Equidis- tant Temperature Quenches". Phys. Rev. Lett.125, 110602 (2020)

    work page 2020

Show all 58 references
  1. [9]

    Exponentially faster cooling in a colloidal system

    A. Kumar and J. Bechhoefer." Exponentially faster cooling in a colloidal system". Nature 584, 64 (2020)

    2020

  2. [10]

    A fresh understanding of the Mpemba effect

    J. Bechhoefer, A. Kumar, and R. Chétrite."A fresh understanding of the Mpemba effect". Nat. Rev. Phys.3, 534 (2021)

    2021

  3. [11]

    Relaxation Shortcuts through Boundary Cou- pling

    G. Teza, R. Yaacoby, and O. Raz. "Relaxation Shortcuts through Boundary Cou- pling". Phys. Rev. Lett.131, 017101 (2023)

    2023

  4. [12]

    Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results

    A. Biswas, R. Rajesh, and A. Pal."Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results". J. Chem. Phys.159, 044120 (2023)

    2023

  5. [13]

    Thermomajorization Mpemba Effect

    T. Van Vu and H. Hayakawa. "Thermomajorization Mpemba Effect". Phys. Rev. Lett. 134, 107101 (2025)

    2025

  6. [14]

    Lindblad dissipative dynamics in the presence of phase coexistence

    A. Nava and M. Fabrizio."Lindblad dissipative dynamics in the presence of phase coexistence". Phys. Rev. B100, 125102 (2019)

    2019

  7. [15]

    Equidistant quenches in few-level quantum systems

    S.K. Manikandan."Equidistant quenches in few-level quantum systems". Phys. Rev. Res. 3, 043108 (2021)

    2021

  8. [16]

    Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect

    F. Carollo, A. Lasanta and I. Lesanovsky. "Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect". Phys. Rev. Lett.127, 060401 (2021)

    2021

  9. [17]

    Accelerating the approach of dissipative quantum spin systems towards stationarity through global spin rotations

    S. Kochsiek, F. Carollo, and I. Lesanovsky."Accelerating the approach of dissipative quantum spin systems towards stationarity through global spin rotations". Phys. Rev. A 106, 012207 (2022)

    2022

  10. [18]

    Accelerating relaxation through Liouvillian exceptional point

    Y.-L. Zhou, X.-D. Yu, C.-W. Wu, X.-Q. Li, J. Zhang, W. Li, and P.X. Chen."Accelerating relaxation through Liouvillian exceptional point". Phys. Rev. Res. 5, 043036 (2023)

    2023

  11. [19]

    Hyperacceleration of quantum thermal- ization dynamics by bypassing long-lived coherences: An analytical treatment

    F. Ivander, N. Anto-Sztrikacs, and D. Segal. "Hyperacceleration of quantum thermal- ization dynamics by bypassing long-lived coherences: An analytical treatment". Phys. Rev. E 108, 014130 (2023)

    2023

  12. [20]

    Quantum Mpemba Effect in a Quan- tum Dot with Reservoirs

    A.K. Chatterjee, S. Takada, and H. Hayakawa, "Quantum Mpemba Effect in a Quan- tum Dot with Reservoirs". Phys. Rev. Lett.131, 080402 (2023)

    2023

  13. [21]

    Entanglement asymmetry as a probe of symmetry breaking

    F. Ares, S. Murciano, and P. Calabrese. "Entanglement asymmetry as a probe of symmetry breaking". Nat. Commun.14, 2036 (2023). Accepted in Quantum 2025-03-22, click title to verify. Published under CC-BY 4.0. 21

    2023

  14. [22]

    Observing the Quantum Mpemba Effect in Quantum Simulations

    L. Kh Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C.F. Roos, and M.K. Joshi. "Observing the Quantum Mpemba Effect in Quantum Simulations". Phys. Rev. Lett.133, 010402 (2024)

    2024

  15. [23]

    Micro- scopic origin of the quantum Mpemba effect in integrable systems

    C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini. "Micro- scopic origin of the quantum Mpemba effect in integrable systems". Phys. Rev. Lett. 133, 010401 (2024)

    2024

  16. [24]

    Multiple quantum Mpemba effect: Exceptional points and oscillations

    A. K. Chatterjee, S. Takada, and H. Hayakawa. "Multiple quantum Mpemba effect: Exceptional points and oscillations". Phys. Rev. A110, 022213 (2024)

    2024

  17. [25]

    Observation of quantum strong Mpemba effect

    J. Zhang, G. Xia, C.-W. Wu, T. Chen, Q. Zhang, Y. Xie, W.-B. Su, W. Wu, C.-W. Qiu, P. xing Chen, W. Li, H. Jing, and Y.-L. Zhou. "Observation of quantum strong Mpemba effect" Nature Commun.16, 301 (2025)

    2025

  18. [26]

    Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems

    S. Yamashika, F. Ares, and P. Calabrese."Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems". Phys. Rev. B110, 085126 (2024)

    2024

  19. [27]

    Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit

    S.A. Shapira, Y. Shapira, J. Markov, G. Teza, N. Akerman, O. Raz, and R. Ozeri. "Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit". Phys. Rev. Lett. 133, 010403 (2024)

    2024

  20. [28]

    The quantum Mpemba effect in free-fermionic mixed states

    F. Ares, V. Vitale, and S. Murciano. "The quantum Mpemba effect in free-fermionic mixed states". arXiv preprint cond-mat.stat-mech/2405.08913 (2024)

    2024 arXiv

  21. [29]

    Thermodynamics of the Quan- tum Mpemba Effect

    M. Moroder, O. Culhane, K. Zawadzki, and J. Goold. "Thermodynamics of the Quan- tum Mpemba Effect". Phys. Rev. Lett.133, 140404 (2024)

    2024

  22. [30]

    Non-Markovian Quantum Mpemba effect

    D.J. Strachan, A. Purkayastha, and S.R. Clark. "Non-Markovian Quantum Mpemba effect". arXiv preprint quant-ph/2402.05756 (2024)

    2024

  23. [31]

    "Symmetry Restoration and Quantum Mpemba Effect in Symmetric Random Circuits

    S. Liu, H.-K. Zhang, S. Yin, and S.-X. Zhang. ""Symmetry Restoration and Quantum Mpemba Effect in Symmetric Random Circuits". Phys. Rev. Lett.133, 140405 (2024)

    2024

  24. [32]

    Photonic Mpemba effect

    S. Longhi. "Photonic Mpemba effect". Opt. Lett.49, 5188 (2024)

    2024

  25. [33]

    Mpemba effects in nonequilibrium open quantum systems

    X. Wang and J. Wang. "Mpemba effects in nonequilibrium open quantum systems". Phys. Rev. Res.6, 033330 (2024)

    2024

  26. [34]

    Mpemba Effects in Open Nonequilibrium Quantum Systems

    A. Nava and R. Egger. "Mpemba Effects in Open Nonequilibrium Quantum Systems". Phys. Rev. Lett.133, 136302 (2024)

    2024

  27. [35]

    Bosonic Mpemba effect with non-classical states of light

    S. Longhi. "Bosonic Mpemba effect with non-classical states of light". APL Quantum 1, 046110 (2024)

    2024

  28. [36]

    Quantum Mpemba effects in many-body localization systems

    S. Liu, H.-K. Zhang, S. Yin, S.-X. Zhang, and H. Yao. "Quantum Mpemba effects in many-body localization systems". arXiv preprint quant-ph/2408.07750 (2024)

    2024

  29. [37]

    Quenching from superfluid to free bosons in two dimensions: entanglement, symmetries, and quantum Mpemba effect

    S. Yamashika, P. Calabrese, and F. Ares. "Quenching from superfluid to free bosons in two dimensions: entanglement, symmetries, and quantum Mpemba effect". arXiv preprint cond-mat.stat-mec/arXiv:2410.14299 (2024)

    2024 arXiv

  30. [38]

    Exploring Quantum Mpemba Effects

    U. Warring. "Exploring Quantum Mpemba Effects". Physics17, 105 (2024)

    2024

  31. [39]

    Breuer and F

    H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems . Oxford University Press, Oxford (2007)

    2007

  32. [40]

    Vogel and D.-G

    W. Vogel and D.-G. Welsch.Quantum Optics. Wiley-VCH, Berlin (2006)

    2006

  33. [41]

    Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation

    C. W. Gardiner and M. J. Collett. "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation". Phys. Rev. A31, 3761 (1985). Accepted in Quantum 2025-03-22, click title to verify. Published under CC-BY 4.0. 22

    1985

  34. [42]

    Effect of dissipation on quantum coherence

    D. F. Walls and G. J. Milburn. "Effect of dissipation on quantum coherence". Phys. Rev. A 31, 2403 (1985)

    1985

  35. [43]

    Squeezed quantum fluctuations and macroscopic quantum coherence

    T. A. B. Kennedy and D. F. Walls. "Squeezed quantum fluctuations and macroscopic quantum coherence". Phys. Rev. A37, 152 (1988)

    1988

  36. [44]

    Decoherence and the Transition from Quantum to Classical

    W.H. Zurek. "Decoherence and the Transition from Quantum to Classical". Phys. Today 44, 36 (1991)

    1991

  37. [45]

    Observing the Progressive Decoherence of the ’Meter’ in a Quantum Measurement

    M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche. "Observing the Progressive Decoherence of the ’Meter’ in a Quantum Measurement". Phys. Rev. Lett.77, 4887 (1996)

    1996

  38. [46]

    RelaxationofSchroedingerCatStatesandDisplacedThermal States in a Density Operator Representation

    H.SaitoandH.Hyuga."RelaxationofSchroedingerCatStatesandDisplacedThermal States in a Density Operator Representation". J. Phys. Soc. Jpn.65, 1648 (1996)

    1996

  39. [47]

    Decoherence and thermalization dynamics of a quantum oscillator

    V.V. Dodonov, S.S. Mizrahi, A.L. de Souza Silva and S.S. Mizrahi. "Decoherence and thermalization dynamics of a quantum oscillator". J. Opt. B: Quantum Semiclass. Opt. 2, 271 (2000)

    2000

  40. [48]

    Quantum decoherence and classical correlations of the har- monic oscillator in the Lindblad theory

    A. Isar and W. Scheid. "Quantum decoherence and classical correlations of the har- monic oscillator in the Lindblad theory". Physica A373, 298 (2007)

    2007

  41. [49]

    Quantum Theory of an Optical Maser. I. General Theory

    M.O. Scully and W.E. Lamb, Jr. "Quantum Theory of an Optical Maser. I. General Theory". Phys. Rev.159, 208 (1967)

    1967

  42. [50]

    Approach to equilibrium of single mode radiation in a cavity

    A. Schell and Ri. Barakat. "Approach to equilibrium of single mode radiation in a cavity". J. Phys. A6, 826 (1973)

    1973

  43. [51]

    Density matrix for photons in a cavity

    H. F. Arnoldus. "Density matrix for photons in a cavity". J. Opt. Soc. Am. B13, 1099 (1996)

    1996

  44. [52]

    Laser Mpemba effect

    S. Longhi. "Laser Mpemba effect". Opt. Lett.50, 2069 (2025)

    2025

  45. [53]

    Density matrix for the damped harmonic oscillator within the Lindblad theory

    A. Isar, A. Sandulescu, and W. Scheid. "Density matrix for the damped harmonic oscillator within the Lindblad theory". J. Math. Phys.34, 3887(1993)

    1993

  46. [54]

    General Solution of the Quantum Damped Har- monic Oscillator

    R. Endo, K. Fujii, and T. Suzuki. "General Solution of the Quantum Damped Har- monic Oscillator". Int. J. Geom. Meth. Mod. Phys.5, 653 (2008)

    2008

  47. [55]

    Time evolution of the quantized field coupled to a thermal bath: A phase space approach

    E.P. Mattos and A. Vidiella-Barranco. "Time evolution of the quantized field coupled to a thermal bath: A phase space approach". Ann. Phys. (N.Y.)442, 168321 (2020)

    2020

  48. [56]

    EigenvaluesforInfiniteMatrices

    P.N.Shivakumar, J.J.Williams, andN.Rudraiah."EigenvaluesforInfiniteMatrices". Linear Algebra Appl.96, 55 (1987)

    1987

  49. [57]

    Spectral Theory for the Differential Equations of Simple Birth and Death Processes

    W. Ledermann and G. E. H. Reuter. "Spectral Theory for the Differential Equations of Simple Birth and Death Processes". Phil. Trans. R. Soc. Lond. A246, 321 (1954)

    1954

  50. [58]

    Linear birth and death models and asso- ciated Laguerre and Meixner polynomials

    M.E.H Ismail, J. Letessier, and G. Valent. "Linear birth and death models and asso- ciated Laguerre and Meixner polynomials". J. Approx. Theory55, 337 (1988). Accepted in Quantum 2025-03-22, click title to verify. Published under CC-BY 4.0. 23

    1988

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