Mpemba effect and super-accelerated thermalization in the damped quantum harmonic oscillator
Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy & IFISC (UIB-CSIC), Instituto de Fisica Interdisciplinar y Sistemas Complejos - Palma de Mallorca, Spain
| Published: | 2025-03-26, volume 9, page 1677 |
| Editor: | Francesco Ciccarello |
| Eprint: | arXiv:2411.09589v2 |
| Doi: | https://doi.org/10.22331/q-2025-03-26-1677 |
| Citation: | Quantum 9, 1677 (2025). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
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.
Featured image: Cartoon of the Mpemba effect. A system far from equilibrium can relax to equilibrium more quickly than those closer to it.
Popular summary
► BibTeX data
► References
[1] E. B. Mpemba and D. G. Osborne. "Cool?". Phys. Educ. 4, 172 (1969).
https://doi.org/10.1088/0031-9120/4/3/312
[2] G. S. Kell. "The Freezing of Hot and Cold Water". Am. J. Phys. 37, 564 (1969).
https://doi.org/10.1119/1.1975687
[3] M. Jeng. "The Mpemba effect: when can hot water freeze faster than cold?". Am. J. Phys. 74, 514 (2006).
https://doi.org/10.1119/1.2186331
[4] 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).
https://doi.org/10.1007/s11814-016-0029-2
[5] 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).
https://doi.org/10.1103/PhysRevLett.119.148001
[6] 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).
https://doi.org/10.1073/pnas.1701264114
[7] I. Klich, O. Raz, O. Hirschberg, and M. Vucelja. "Mpemba index and anomalous relaxation". Phys. Rev. X 9, 021060 (2019).
https://doi.org/10.1103/PhysRevX.9.021060
[8] A. Lapolla and A. Godec."Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches". Phys. Rev. Lett. 125, 110602 (2020).
https://doi.org/10.1103/PhysRevLett.125.110602
[9] A. Kumar and J. Bechhoefer." Exponentially faster cooling in a colloidal system". Nature 584, 64 (2020).
https://doi.org/10.1038/s41586-020-2560-x
[10] J. Bechhoefer, A. Kumar, and R. Chétrite."A fresh understanding of the Mpemba effect". Nat. Rev. Phys. 3, 534 (2021).
https://doi.org/10.1038/s42254-021-00349-8
[11] G. Teza, R. Yaacoby, and O. Raz. "Relaxation Shortcuts through Boundary Coupling". Phys. Rev. Lett. 131, 017101 (2023).
https://doi.org/10.1103/PhysRevLett.131.017101
[12] 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).
https://doi.org/10.1063/5.0155855
[13] T. Van Vu and H. Hayakawa. "Thermomajorization Mpemba Effect". Phys. Rev. Lett. 134, 107101 (2025).
https://doi.org/10.1103/PhysRevLett.134.107101
[14] A. Nava and M. Fabrizio."Lindblad dissipative dynamics in the presence of phase coexistence". Phys. Rev. B 100, 125102 (2019).
https://doi.org/10.1103/PhysRevB.100.125102
[15] S.K. Manikandan."Equidistant quenches in few-level quantum systems". Phys. Rev. Res. 3, 043108 (2021).
https://doi.org/10.1103/PhysRevResearch.3.043108
[16] 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).
https://doi.org/10.1103/PhysRevLett.127.060401
[17] 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).
https://doi.org/10.1103/PhysRevA.106.012207
[18] 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).
https://doi.org/10.1103/PhysRevResearch.5.043036
[19] F. Ivander, N. Anto-Sztrikacs, and D. Segal. "Hyperacceleration of quantum thermalization dynamics by bypassing long-lived coherences: An analytical treatment". Phys. Rev. E 108, 014130 (2023).
https://doi.org/10.1103/PhysRevE.108.014130
[20] A.K. Chatterjee, S. Takada, and H. Hayakawa, "Quantum Mpemba Effect in a Quantum Dot with Reservoirs". Phys. Rev. Lett. 131, 080402 (2023).
https://doi.org/10.1103/PhysRevLett.131.080402
[21] F. Ares, S. Murciano, and P. Calabrese. "Entanglement asymmetry as a probe of symmetry breaking". Nat. Commun. 14, 2036 (2023).
https://doi.org/10.1038/s41467-023-37747-8
[22] 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).
https://doi.org/10.1103/PhysRevLett.133.010402
[23] C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini. "Microscopic origin of the quantum Mpemba effect in integrable systems". Phys. Rev. Lett. 133, 010401 (2024).
https://doi.org/10.1103/PhysRevLett.133.010401
[24] A. K. Chatterjee, S. Takada, and H. Hayakawa. "Multiple quantum Mpemba effect: Exceptional points and oscillations". Phys. Rev. A 110, 022213 (2024).
https://doi.org/10.1103/PhysRevA.110.022213
[25] 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).
https://doi.org/10.1038/s41467-024-54303-0
[26] S. Yamashika, F. Ares, and P. Calabrese."Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems". Phys. Rev. B 110, 085126 (2024).
https://doi.org/10.1103/PhysRevB.110.085126
[27] 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).
https://doi.org/10.1103/PhysRevLett.133.010403
[28] 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).
https://doi.org/10.48550/arXiv.2405.08913
[29] M. Moroder, O. Culhane, K. Zawadzki, and J. Goold. "Thermodynamics of the Quantum Mpemba Effect". Phys. Rev. Lett. 133, 140404 (2024).
https://doi.org/10.1103/PhysRevLett.133.140404
[30] D.J. Strachan, A. Purkayastha, and S.R. Clark. "Non-Markovian Quantum Mpemba effect". arXiv preprint quant-ph/2402.05756 (2024).
https://doi.org/10.48550/arXiv.2402.05756
arXiv:quant-ph/2402.05756
[31] 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).
https://doi.org/10.1103/PhysRevLett.133.140405
[32] S. Longhi. "Photonic Mpemba effect". Opt. Lett. 49, 5188 (2024).
https://doi.org/10.1364/OL.532503
[33] X. Wang and J. Wang. "Mpemba effects in nonequilibrium open quantum systems". Phys. Rev. Res. 6, 033330 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033330
[34] A. Nava and R. Egger. "Mpemba Effects in Open Nonequilibrium Quantum Systems". Phys. Rev. Lett. 133, 136302 (2024).
https://doi.org/10.1103/PhysRevLett.133.136302
[35] S. Longhi. "Bosonic Mpemba effect with non-classical states of light". APL Quantum 1, 046110 (2024).
https://doi.org/10.1063/5.0234457
[36] 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).
https://doi.org/10.48550/arXiv.2408.07750
arXiv:quant-ph/2408.07750
[37] 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).
https://doi.org/10.48550/arXiv.2410.14299
arXiv:2410.14299
[38] U. Warring. "Exploring Quantum Mpemba Effects". Physics 17, 105 (2024).
https://doi.org/10.1103/Physics.17.105
[39] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Oxford (2007).
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001
[40] W. Vogel and D.-G. Welsch. Quantum Optics. Wiley-VCH, Berlin (2006).
https://doi.org/10.1002/3527608524
[41] C. W. Gardiner and M. J. Collett. "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation". Phys. Rev. A 31, 3761 (1985).
https://doi.org/10.1103/PhysRevA.31.3761
[42] D. F. Walls and G. J. Milburn. "Effect of dissipation on quantum coherence". Phys. Rev. A 31, 2403 (1985).
https://doi.org/10.1103/PhysRevA.31.2403
[43] T. A. B. Kennedy and D. F. Walls. "Squeezed quantum fluctuations and macroscopic quantum coherence". Phys. Rev. A 37, 152 (1988).
https://doi.org/10.1103/PhysRevA.37.152
[44] W.H. Zurek. "Decoherence and the Transition from Quantum to Classical". Phys. Today 44, 36 (1991).
https://doi.org/10.1063/1.881293
[45] 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).
https://doi.org/10.1103/PhysRevLett.77.4887
[46] H. Saito and H. Hyuga. "Relaxation of Schroedinger Cat States and Displaced Thermal States in a Density Operator Representation". J. Phys. Soc. Jpn. 65, 1648 (1996).
https://doi.org/10.1143/JPSJ.65.1648
[47] 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).
https://doi.org/10.1088/1464-4266/2/3/309
[48] A. Isar and W. Scheid. "Quantum decoherence and classical correlations of the harmonic oscillator in the Lindblad theory". Physica A 373, 298 (2007).
https://doi.org/10.1016/j.physa.2006.04.065
[49] M.O. Scully and W.E. Lamb, Jr. "Quantum Theory of an Optical Maser. I. General Theory". Phys. Rev. 159, 208 (1967).
https://doi.org/10.1103/PhysRev.159.208
[50] A. Schell and Ri. Barakat. "Approach to equilibrium of single mode radiation in a cavity". J. Phys. A 6, 826 (1973).
https://doi.org/10.1088/0305-4470/6/6/011
[51] H. F. Arnoldus. "Density matrix for photons in a cavity". J. Opt. Soc. Am. B 13, 1099 (1996).
https://doi.org/10.1364/JOSAB.13.001099
[52] S. Longhi. "Laser Mpemba effect". Opt. Lett. 50, 2069 (2025).
https://doi.org/10.1364/OL.550728
[53] A. Isar, A. Sandulescu, and W. Scheid. "Density matrix for the damped harmonic oscillator within the Lindblad theory". J. Math. Phys. 34, 3887(1993).
https://doi.org/10.1063/1.530013
[54] R. Endo, K. Fujii, and T. Suzuki. "General Solution of the Quantum Damped Harmonic Oscillator". Int. J. Geom. Meth. Mod. Phys. 5, 653 (2008).
https://doi.org/10.1142/S0219887808003065
[55] 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).
https://doi.org/10.1016/j.aop.2020.168321
[56] P. N. Shivakumar, J.J. Williams, and N. Rudraiah. "Eigenvalues for Infinite Matrices". Linear Algebra Appl. 96, 55 (1987).
https://doi.org/10.1016/0024-3795(87)90335-1
[57] W. Ledermann and G. E. H. Reuter. "Spectral Theory for the Differential Equations of Simple Birth and Death Processes". Phil. Trans. R. Soc. Lond. A 246, 321 (1954).
https://doi.org/10.1098/rsta.1954.0001
[58] M.E.H Ismail, J. Letessier, and G. Valent. "Linear birth and death models and associated Laguerre and Meixner polynomials". J. Approx. Theory 55, 337 (1988).
https://doi.org/10.1016/0021-9045(88)90100-1
Cited by
[1] Stefano Longhi, "Dynamical Phase Transitions in Open Quantum Walks", Advanced Quantum Technologies 8 11, e00539 (2025).
[2] Stefano Longhi, "Pontus–Mpemba effect in cavity quantum electrodynamics", Physica A: Statistical Mechanics and its Applications 695, 131640 (2026).
[3] Ze-Zhou Zhang, Hong-Gang Luo, and Wei Wu, "Quantum Mpemba Effect Induced by Non-Markovian Exceptional Points", Physical Review Letters 136 21, 210402 (2026).
[4] Stefano Longhi, "Quantum Mpemba effect from initial system–reservoir entanglement", APL Quantum 2 2, 026133 (2025).
[5] Zijun Wei, Mingdi Xu, Xiang-Ping Jiang, Haiping Hu, and Lei Pan, "Quantum Mpemba effect in dissipative spin chains at criticality", Science China Physics, Mechanics & Astronomy 69 4, 240315 (2026).
[6] Sayan Mondal and Ujjwal Sen, "Mpemba effect in self-contained quantum refrigerators: Accelerated cooling", Physical Review A 113 2, 022211 (2026).
[7] Israel Klich and Marija Vucelja, "Mpemba effect on near-complete graphs with independence Metropolis–Hastings dynamics", Journal of Physics A: Mathematical and Theoretical 59 25, 255002 (2026).
[8] Ivan Medina, Oisín Culhane, Felix C. Binder, Gabriel T. Landi, and John Goold, "Anomalous Discharging of Quantum Batteries: The Ergotropic Mpemba Effect", Physical Review Letters 134 22, 220402 (2025).
[9] Sibo Guo, Shuai Yin, Shi-Xin Zhang, and Zi-Xiang Li, "Skin effect induced anomalous dynamics from charge-fluctuating initial states", Physical Review B 112 15, 155419 (2025).
[10] Stefano Longhi, "Giant Mpemba Effect via Weak Interactions in Open Quantum Systems", Entropy 28 4, 427 (2026).
[11] Théo Lejeune, Miha Papič, John Goold, Felix C Binder, François Damanet, and Mattia Moroder, "Accelerating qubit reset through the Mpemba effect", Journal of Physics A: Mathematical and Theoretical 59 26, 265303 (2026).
[12] Stefano Longhi, "Quantum Mpemba Effect from Non-Normal Dynamics", Entropy 27 6, 581 (2025).
[13] Stefano Longhi, "Quantum Pontus–Mpemba effect enabled by the Liouvillian skin effect", Journal of Physics A: Mathematical and Theoretical 59 6, 065304 (2026).
[14] Pitambar Bagui, Arijit Chatterjee, and Bijay Kumar Agarwalla, "Accelerated relaxation and Mpemba-like effect for operators in open quantum systems", Physical Review A 114 1, L010601 (2026).
[15] Filiberto Ares, Pasquale Calabrese, and Sara Murciano, "The quantum Mpemba effects", Nature Reviews Physics 7 8, 451 (2025).
[16] Gianluca Teza, John Bechhoefer, Antonio Lasanta, Oren Raz, and Marija Vucelja, "Speedups in nonequilibrium thermal relaxation: Mpemba and related effects", Physics Reports 1164, 1 (2026).
[17] Shion Yamashika and Filiberto Ares, "Quantum Mpemba Effect in Long-Range Spin Systems", Physical Review Letters 136 9, 090402 (2026).
[18] Asad Ali, Hamid Arian Zad, Muhammad Irtiza Hussain, Saif Al‐Kuwari, Hashir Kuniyil, Muhammad Talha Rahim, Michal Jaščur, and Saeed Haddadi, "Quantum Mpemba Effect in a Four‐Site Bose–Hubbard Model", Fortschritte der Physik 74 3, e70089 (2026).
[19] Mingdi Xu, Zijun Wei, Xiang-Ping Jiang, and Lei Pan, "Expedited thermalization dynamics in incommensurate systems", Physical Review A 112 4, 042210 (2025).
[20] Doruk Can Alyürük, Mahir H. Yeșiller, Vlatko Vedral, and Onur Pusuluk, "Thermodynamic limits of the Mpemba effect: A unified resource theory analysis of correlation-enabled mechanisms", arXiv:2502.00123, (2025).
[21] Stefano Longhi, "Laser Mpemba effect", Optics Letters 50 6, 2069 (2025).
[22] Mingdi Xu, Kaixiang Lu, Xiang-Ping Jiang, Haiping Hu, and Lei Pan, "Strong Quantum Mpemba Effect from Exact Slow-Mode Selection in Constrained Rydberg Chains", arXiv:2607.17975, (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 04:31:27) and SAO/NASA ADS (last updated successfully 2026-08-06 20:08:02). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch ADS cited-by data during last attempt 2026-08-08 04:31:27: Cannot retrieve data from ADS due to rate limitations.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.