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Non-Markovian Quantum Mpemba Effect

David J. Strachan1,*, Archak Purkayastha2,†, and Stephen R. Clark1,‡

  • *Contact author: david.strachan@bristol.ac.uk
  • †Contact author: archak.p@phy.iith.ac.in
  • ‡Contact author: stephen.clark@bristol.ac.uk
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Phys. Rev. Lett. 134, 220403 – Published 5 June, 2025

DOI: https://doi.org/10.1103/PhysRevLett.134.220403

Abstract

Since its rediscovery in the twentieth century, the Mpemba effect, where a far-from-equilibrium state may relax faster than a state closer to equilibrium, has been extensively studied in classical systems and has recently received attention in quantum systems. Many theories explaining this counter-intuitive behavior in classical systems rely on memory effects. However, in quantum systems, the relation between the Mpemba effect and memory has remained unexplored. In this Letter, we consider general non-Markovian open quantum systems and reveal new classes of quantum Mpemba effects, with no analog in Markovian quantum dynamics. Generically, open quantum dynamics possess a finite memory time and a unique steady state. Because of non-Markovian dynamics, even if the system is initialized in the steady state it can take a long time to relax back. We find other initial states that reach the steady state much faster. Most notably, we demonstrate that there can be an initial state in which the system reaches the steady state within the finite memory time itself, giving the fastest possible relaxation to stationarity. We verify the effect for quantum dot systems coupled to electronic reservoirs in equilibrium and nonequilibrium setups at weak, intermediate and strong coupling. Our Letter provides new insights into the rich physics underlying accelerated relaxation in quantum systems.

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References (93)

  1. E. B. Mpemba and D. G. Osborne, Cool?, Phys. Educ. 4, 172 (1969).
  2. W. D. Ross, The Works of Aristotle, Translated into English under the Editorship of W.D. Ross (Oxford Clarendon Press, New York, 1931).
  3. R. Descartes, Discourse on Method (Harmondsworth, Penguin, Harmondsworth, 1950).
  4. J. Brownridge, When does hot water freeze faster then cold water? a search for the Mpemba effect, Am. J. Phys. 79, 78 (2011).
  5. M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvión, A. Gordillo-Guerrero, D. Iñiguez, A. Lasanta, A. Maiorano, E. Marinari et al., The Mpemba effect in spin glasses is a persistent memory effect, Proc. Natl. Acad. Sci. U.S.A. 116, 15350 (2019).
  6. A. Biswas, V. V. Prasad, O. Raz, and R. Rajesh, Mpemba effect in driven granular Maxwell gases, Phys. Rev. E 102, 012906 (2020).
  7. 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).
  8. P. Greaney, G. Lani, G. Cicero, and J. Grossman, Mpemba-like behavior in carbon nanotube resonators, Metall. Mater. Trans. A 42, 3907 (2011).
  9. A. Biswas, R. Rajesh, and A. K. Pal, Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results, J. Chem. Phys. 159, 044120 (2023).
  10. D. M. Busiello, D. Gupta, and A. Maritan, Inducing and optimizing Markovian Mpemba effect with stochastic reset, New J. Phys. 23, 103012 (2021).
  11. 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).
  12. S. K. Manikandan, Equidistant quenches in few-level quantum systems, Phys. Rev. Res. 3, 043108 (2021).
  13. A. Nava and M. Fabrizio, Lindblad dissipative dynamics in the presence of phase coexistence, Phys. Rev. B 100, 125102 (2019).
  14. A. K. Chatterjee, S. Takada, and H. Hayakawa, Quantum Mpemba effect in a quantum dot with reservoirs, Phys. Rev. Lett. 131, 080402 (2023).
  15. 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).
  16. 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).
  17. A. K. Chatterjee, S. Takada, and H. Hayakawa, Multiple quantum Mpemba effect: Exceptional points and oscillations, Phys. Rev. A 110, 022213 (2024).
  18. 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, Nat. Commun. 16 (2024).
  19. X. Wang and J. Wang, Mpemba effects in nonequilibrium open quantum systems, Phys. Rev. Res. 6, 033330 (2024).
  20. S. Aharony 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).
  21. L. K. 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).
  22. 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).
  23. R. Bao and Z. Hou, Accelerating relaxation in Markovian open quantum systems through quantum reset processes, arXiv:2212.11170.
  24. F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nat. Commun. 14, 2036 (2023).
  25. 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).
  26. F. Caceffo, S. Murciano, and V. Alba, Entangled multiplets, asymmetry, and quantum Mpemba effect in dissipative systems, J. Stat. Mech. (2024) 063103.
  27. C. Timm, Time-convolutionless master equation for quantum dots: Perturbative expansion to arbitrary order, Phys. Rev. B 83, 115416 (2011).
  28. D. Chruściński and A. Kossakowski, Non-Markovian quantum dynamics: Local versus nonlocal, Phys. Rev. Lett. 104, 070406 (2010).
  29. D. Chruściński, Dynamical maps beyond Markovian regime, Phys. Rep. 992, 1 (2022).

  30. The NMQME doesn’t rely on the convergence of L^(τ), it only requires the convergence of its fixed point ρ^TDFP(τ)→ρ^(∞) where L^(τ)[ρ^TDFP(τ)]=0.

  31. A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and J. Goold, Periodically refreshed baths to simulate open quantum many-body dynamics, Phys. Rev. B 104, 045417 (2021).
  32. V. Bruch, K. Nestmann, J. Schulenborg, and M. Wegewijs, Fermionic duality: General symmetry of open systems with strong dissipation and memory, SciPost Phys. 11, 053 (2021).
  33. U. Geigenmüller, U. Titulaer, and B. Felderhof, Systematic elimination of fast variables in linear systems, Physica A 119, 41 (1983).
  34. F. Haake and M. Lewenstein, Adiabatic drag and initial slip in random processes, Phys. Rev. A 28, 3606 (1983).
  35. F. Haake and R. Reibold, Strong damping and low-temperature anomalies for the harmonic oscillator, Phys. Rev. A 32, 2462 (1985).
  36. P. Gaspard and M. Nagaoka, Slippage of initial conditions for the redfield master equation, J. Chem. Phys. 111, 5668 (1999).
  37. K. Nestmann, V. Bruch, and M. R. Wegewijs, How quantum evolution with memory is generated in a time-local way, Phys. Rev. X 11, 021041 (2021).
  38. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  39. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2007).
  40. S. F. H. Angel Rivas, Open Quantum Systems (Springer, Berlin, Heidelberg, 2012).
  41. V. Reimer, M. R. Wegewijs, K. Nestmann, and M. Pletyukhov, Five approaches to exact open-system dynamics: Complete positivity, divisibility, and time-dependent observables, J. Chem. Phys. 151, 044101 (2019).
  42. Y. Li, B. Luo, J.-W. Zhang, and H. Guo, Time-convolutionless non-Markovian master equation in strong-coupling regime, arXiv:1202.3288.
  43. A. S. Hegde, K. P. Athulya, V. Pathak, J. Piilo, and A. Shaji, Open quantum dynamics with singularities: Master equations and degree of non-Markovianity, Phys. Rev. A 104, 062403 (2021).
  44. A. Purkayastha, A. Dhar, and M. Kulkarni, Out-of-equilibrium open quantum systems: A comparison of approximate quantum master equation approaches with exact results, Phys. Rev. A 93, 062114 (2016).
  45. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.134.220403 which includes Refs. [46–56] for additional details.
  46. L. Kohn and G. E. Santoro, Quench dynamics of the anderson impurity model at finite temperature using matrix product states: Entanglement and bath dynamics, J. Stat. Mech. (2022) 063102.
  47. A. Dorda, M. Nuss, W. von der Linden, and E. Arrigoni, Auxiliary master equation approach to nonequilibrium correlated impurities, Phys. Rev. B 89, 165105 (2014).
  48. S.-A. Cheong and C. L. Henley, Many-body density matrices for free fermions, Phys. Rev. B 69, 075111 (2004).
  49. A. W. Chin, Á. Rivas, S. F. Huelga, and M. B. Plenio, Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials, J. Math. Phys. (N.Y.) 51, 092109 (2010).
  50. G. G. Amosov and S. N. Filippov, Spectral properties of reduced fermionic density operators and parity superselection rule, Quantum Inf. Process. 16, 2 (2016).
  51. M. Baity-Jesi et al., The Mpemba effect in spin glasses is a persistent memory effect, Proc. Natl. Acad. Sci. U.S.A. 116, 15350 (2019).
  52. M. Sánchez-Barquilla and J. Feist, Accurate truncations of chain mapping models for open quantum systems, Nanomater. Nanotechnol. 11, 2104 (2021).
  53. M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 (1975).
  54. A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
  55. N. Anto-Sztrikacs, F. Ivander, and D. Segal, Quantum thermal transport beyond second order with the reaction coordinate mapping, J. Chem. Phys. 156, 214107 (2022).
  56. M. Esposito and P. Gaspard, Quantum master equation for a system influencing its environment, Phys. Rev. E 68, 066112 (2003).

  57. Note that the NMQME is not dependent on this choice, we have confirmed it for a flat and Lorentzian band also.

  58. A. W. Chin, S. F. Huelga, and M. B. Plenio, Chain representations of open quantum systems and their numerical simulation with time-adaptive density matrix renormalisation group methods, in Semiconductors and Semimetals (Elsevier, New York, 2011), Vol. 85 pp. 115–143.
  59. T. Mühlpfordt, F. Zahn, V. Hagenmeyer, and T. Faulwasser, Polychaos.jl—a Julia package for polynomial chaos in systems and control, IFAC-PapersOnLine 53, 7210 (2020), 21st IFAC World Congress.
  60. Y. Takahashi and H. Umezawa, Thermo field dynamics, Int. J. Mod. Phys. B 10, 1755 (1996).
  61. R. Borrelli and M. F. Gelin, Finite temperature quantum dynamics of complex systems: Integrating thermo-field theories and tensor-train methods, Wiley Interdiscip. Rev. 11, e1539 (2021).
  62. M. Yang and S. R. White, Time-dependent variational principle with ancillary Krylov subspace, Phys. Rev. B 102, 094315 (2020).
  63. M. Fishman, S. White, and E. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  64. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  65. D. Ryndyk, Theory of quantum transport at nanoscale, MRS Bull. 42, 75 (2015).
  66. A. Purkayastha, Classifying transport behavior via current fluctuations in open quantum systems, J. Stat. Mech. (2019) 043101.
  67. H. D. Cornean, A. Jensen, and V. Moldoveanu, A rigorous proof of the Landauer–Büttiker formula, J. Math. Phys. (N.Y.) 46 (2005).
  68. J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwenhoven, Single-shot read-out of an individual electron spin in a quantum dot, Nature (London) 430, 431 (2004).
  69. Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, Quantum optimal control in quantum technologies. strategic report on current status, visions and goals for research in Europe, Eur. Phys. J. Quantum Technol. 9, 19 (2022).
  70. C. P. Koch, Controlling open quantum systems: Tools, achievements, and limitations, J. Phys. Condens. Matter 28, 213001 (2016).
  71. G. Guarnieri, M. Kolář, and R. Filip, Steady-state coherences by composite system-bath interactions, Phys. Rev. Lett. 121, 070401 (2018).
  72. A. Purkayastha, G. Guarnieri, M. T. Mitchison, R. Filip, and J. Goold, Tunable phonon-induced steady-state coherence in a double-quantum-dot charge qubit, npj Quantum Inf. 6, 27 (2020).
  73. J. D. Cresser and J. Anders, Weak and ultrastrong coupling limits of the quantum mean force Gibbs state, Phys. Rev. Lett. 127, 250601 (2021).
  74. A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, Open quantum system dynamics and the mean force Gibbs state, AVS Quantum Sci. 4, 012301 (2022).
  75. A. Tavakoli, G. Haack, N. Brunner, and J. B. Brask, Autonomous multipartite entanglement engines, Phys. Rev. A 101, 012315 (2020).
  76. J. Bohr Brask, F. Clivaz, G. Haack, and A. Tavakoli, Operational nonclassicality in minimal autonomous thermal machines, Quantum 6, 672 (2022).
  77. R. Dann, A. Tobalina, and R. Kosloff, Shortcut to equilibration of an open quantum system, Phys. Rev. Lett. 122, 250402 (2019).
  78. M. Boubakour, S. Endo, T. Fogarty, and T. Busch, Dynamical invariant based shortcut to equilibration in open quantum systems, Quantum Sci. Technol. 10, 025036 (2025).
  79. A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013).
  80. H.-B. Liu, W. L. Yang, J.-H. An, and Z.-Y. Xu, Mechanism for quantum speedup in open quantum systems, Phys. Rev. A 93, 020105(R) (2016).
  81. X. Meng, C. Wu, and H. Guo, Minimal evolution time and quantum speed limit of non-Markovian open systems, Sci. Rep. 5, 16357 (2015).
  82. K. Funo, N. Shiraishi, and K. Saito, Speed limit for open quantum systems, New J. Phys. 21, 013006 (2019).
  83. E. O’Connor, G. Guarnieri, and S. Campbell, Action quantum speed limits, Phys. Rev. A 103, 022210 (2021).
  84. L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical speed limits on observables, Phys. Rev. X 12, 011038 (2022).
  85. K. Lan, S. Xie, and X. Cai, Geometric quantum speed limits for Markovian dynamics in open quantum systems, New J. Phys. 24, 055003 (2022).
  86. Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chem. Rev. 119, 10856 (2019).
  87. J. Cao, R. J. Cogdell, D. F. Coker, H.-G. Duan, J. Hauer, U. Kleinekathöfer, T. L. C. Jansen, T. Mančal, R. J. D. Miller, J. P. Ogilvie, V. I. Prokhorenko, T. Renger, H.-S. Tan, R. Tempelaar, M. Thorwart, E. Thyrhaug, S. Westenhoff, and D. Zigmantas, Quantum biology revisited, Sci. Adv. 6, eaaz4888 (2020).
  88. J. del Pino, F. A. Y. N. Schröder, A. W. Chin, J. Feist, and F. J. Garcia-Vidal, Tensor network simulation of non-Markovian dynamics in organic polaritons, Phys. Rev. Lett. 121, 227401 (2018).
  89. J. Prior, A. W. Chin, S. F. Huelga, and M. B. Plenio, Efficient simulation of strong system-environment interactions, Phys. Rev. Lett. 105, 050404 (2010).
  90. H. Vroylandt, On the derivation of the generalized Langevin equation and the fluctuation-dissipation theorem, Europhys. Lett. 140, 62003 (2022).
  91. H. Kirchberg and M. Thorwart, Time-resolved probing of the nonequilibrium structural solvation dynamics by the time-dependent Stokes shift, J. Phys. Chem. B 124, 5717 (2020).
  92. A. Nava and R. Egger, Mpemba effects in open nonequilibrium quantum systems, Phys. Rev. Lett. 133, 136302 (2024).
  93. 10.5523/bris.22vr3sy29xzrf2qd2nx8zd1egt

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Phys. Rev. Lett. 134, 220403– Published 5 June, 2025
  • Received 21 March 2024
  • Accepted 4 April 2025
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