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  • Open Access

Global Becomes Local: Efficient Many-Body Dynamics for Global Master Equations

Alexander Schnell*

  • *Contact author: schnell@tu-berlin.de
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Phys. Rev. Lett. 134, 250401 – Published 23 June, 2025

DOI: https://doi.org/10.1103/1q5b-p2bf

Abstract

This Letter makes progress on the issue of global vs local master equations. Global master equations like the Redfield master equation (following from standard Born and Markov approximation) require a full diagonalization of the system Hamiltonian. This is especially challenging for interacting quantum many-body systems. We discuss a short-bath-correlation-time expansion in reciprocal (energy) space, leading to a series expansion of the jump operator, which avoids a diagonalization of the Hamiltonian. For a bath that is coupled locally to one site, this typically leads to an expansion of the global Redfield jump operator in terms of local operators. We additionally map the local Redfield master equation to a novel local Lindblad form, giving an equation which has the same conceptual advantages of traditional local Lindblad approaches while being applicable in a much broader class of systems. Our ideas give rise to a nonheuristic foundation of local master equations, which can be combined with established many-body methods.

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References (95)

  1. R. Labouvie, B. Santra, S. Heun, and H. Ott, Bistability in a driven-dissipative superfluid, Phys. Rev. Lett. 116, 235302 (2016).
  2. F. Letscher, O. Thomas, T. Niederprüm, M. Fleischhauer, and H. Ott, Bistability versus metastability in driven dissipative Rydberg gases, Phys. Rev. X 7, 021020 (2017).
  3. M. Hohmann, F. Kindermann, T. Lausch, D. Mayer, F. Schmidt, E. Lutz, and A. Widera, Individual tracer atoms in an ultracold dilute gas, Phys. Rev. Lett. 118, 263401 (2017).
  4. D. Husmann, M. Lebrat, S. Häusler, J.-P. Brantut, L. Corman, and T. Esslinger, Breakdown of the Wiedemann–Franz law in a unitary Fermi gas, Proc. Natl. Acad. Sci. U.S.A. 115, 8563 (2018).
  5. L. W. Clark, N. Schine, C. Baum, N. Jia, and J. Simon, Observation of Laughlin states made of light, Nature (London) 582, 41 (2020).
  6. Q. Bouton, J. Nettersheim, D. Adam, F. Schmidt, D. Mayer, T. Lausch, E. Tiemann, and A. Widera, Single-atom quantum probes for ultracold gases boosted by nonequilibrium spin dynamics, Phys. Rev. X 10, 011018 (2020).
  7. F. E. Öztürk, T. Lappe, G. Hellmann, J. Schmitt, J. Klaers, F. Vewinger, J. Kroha, and M. Weitz, Observation of a non-Hermitian phase transition in an optical quantum gas, Science 372, 88 (2021).
  8. R. Nyman, H. Dhar, J. Rodrigues, and F. Mintert, Phase transitions of light in a dye-filled microcavity: Observations and simulations, J. Phys. Conf. Ser. 1919, 012006 (2021).
  9. Q. Fontaine, D. Squizzato, F. Baboux, I. Amelio, A. Lemaître, M. Morassi, I. Sagnes, L. Le Gratiet, A. Harouri, M. Wouters, I. Carusotto, A. Amo, M. Richard, A. Minguzzi, L. Canet, S. Ravets, and J. Bloch, Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate, Nature (London) 608, 687 (2022).
  10. J. Bloch, I. Carusotto, and M. Wouters, Non-equilibrium Bose–Einstein condensation in photonic systems, Nat. Rev. Phys. 4, 470 (2022).
  11. N. Dogra, M. Landini, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger, Dissipation-induced structural instability and chiral dynamics in a quantum gas, Science 366, 1496 (2019).
  12. A. Vrajitoarea, R. Belyansky, R. Lundgren, S. Whitsitt, A. V. Gorshkov, and A. A. Houck, Ultrastrong light-matter interaction in a photonic crystal, arXiv:2209.14972.
  13. G. Roberts, A. Vrajitoarea, B. Saxberg, M. G. Panetta, J. Simon, and D. I. Schuster, Manybody interferometry of quantum fluids, arXiv:2309.05727.
  14. I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, C. Ciuti, and A. Imamoǧlu, Fermionized photons in an array of driven dissipative nonlinear cavities, Phys. Rev. Lett. 103, 033601 (2009).
  15. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7, 971 (2011).
  16. T. Prosen, Open xxz spin chain: Nonequilibrium steady state and a strict bound on ballistic transport, Phys. Rev. Lett. 106, 217206 (2011).
  17. T. Prosen, Exact nonequilibrium steady state of a strongly driven open xxz chain, Phys. Rev. Lett. 107, 137201 (2011).
  18. D. Karevski, V. Popkov, and G. M. Schütz, Exact matrix product solution for the boundary-driven Lindblad xxz chain, Phys. Rev. Lett. 110, 047201 (2013).
  19. D. Vorberg, W. Wustmann, R. Ketzmerick, and A. Eckardt, Generalized Bose-Einstein condensation into multiple states in driven-dissipative systems, Phys. Rev. Lett. 111, 240405 (2013).
  20. E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Robustness of many-body localization in the presence of dissipation, Phys. Rev. Lett. 116, 237203 (2016).
  21. M. H. Fischer, M. Maksymenko, and E. Altman, Dynamics of a many-body-localized system coupled to a bath, Phys. Rev. Lett. 116, 160401 (2016).
  22. L.-N. Wu, A. Schnell, G. De Tomasi, M. Heyl, and A. Eckardt, Describing many-body localized systems in thermal environments, New J. Phys. 21, 063026 (2019).
  23. M. Soriente, T. Donner, R. Chitra, and O. Zilberberg, Dissipation-induced anomalous multicritical phenomena, Phys. Rev. Lett. 120, 183603 (2018).
  24. H. J. Hesten, R. A. Nyman, and F. Mintert, Decondensation in nonequilibrium photonic condensates: When less is more, Phys. Rev. Lett. 120, 040601 (2018).
  25. L.-N. Wu and A. Eckardt, Bath-induced decay of stark many-body localization, Phys. Rev. Lett. 123, 030602 (2019).
  26. X. Xu, J. Thingna, C. Guo, and D. Poletti, Many-body open quantum systems beyond Lindblad master equations, Phys. Rev. A 99, 012106 (2019).
  27. H. Fröml, C. Muckel, C. Kollath, A. Chiocchetta, and S. Diehl, Ultracold quantum wires with localized losses: Many-body quantum Zeno effect, Phys. Rev. B 101, 144301 (2020).
  28. M. Gluza, J. Sabino, N. H. Ng, G. Vitagliano, M. Pezzutto, Y. Omar, I. Mazets, M. Huber, J. Schmiedmayer, and J. Eisert, Quantum field thermal machines, PRX Quantum 2, 030310 (2021).
  29. V. N. Gladilin and M. Wouters, Vortex-pair annihilation in arrays of photon cavities, Phys. Rev. A 105, 013527 (2022).
  30. K. Stitely, F. Finger, R. Rosa-Medina, F. Ferri, T. Donner, T. Esslinger, S. Parkins, and B. Krauskopf, Quantum fluctuation dynamics of dispersive superradiant pulses in a hybrid light-matter system, Phys. Rev. Lett. 131, 143604 (2023).
  31. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2007).
  32. U. Weiss, Quantum Dissipative Systems, 4th ed. (World Scientific, Singapore, 2012), https://www.worldscientific.com/doi/pdf/10.1142/8334.
  33. H. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer Science & Business Media, New York, 1999), Vol. 1, 10.1007/978-3-662-03875-8.
  34. Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, Phys. Soc. Jpn. 58, 101 (1989).
  35. B. B. Laird, J. Budimir, and J. L. Skinner, Quantum-mechanical derivation of the Bloch equations: Beyond the weak-coupling limit, J. Chem. Phys. 94, 4391 (1991).
  36. N. Makri and D. E. Makarov, Tensor propagator for iterative quantum time evolution of reduced density matrices. I. theory, J. Chem. Phys. 102, 4600 (1995).
  37. S. Jang, J. Cao, and R. J. Silbey, Fourth-order quantum master equation and its Markovian bath limit, J. Chem. Phys. 116, 2705 (2002).
  38. J. Iles-Smith, N. Lambert, and A. Nazir, Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems, Phys. Rev. A 90, 032114 (2014).
  39. J. Cerrillo and J. Cao, Non-Markovian dynamical maps: Numerical processing of open quantum trajectories, Phys. Rev. Lett. 112, 110401 (2014).
  40. A. Imamoḡlu, Stochastic wave-function approach to non-Markovian systems, Phys. Rev. A 50, 3650 (1994).
  41. B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).
  42. L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway, Sudden death and sudden birth of entanglement in common structured reservoirs, Phys. Rev. A 79, 042302 (2009).
  43. J. Thingna, H. Zhou, and J.-S. Wang, Improved Dyson series expansion for steady-state quantum transport beyond the weak coupling limit: Divergences and resolution, J. Chem. Phys. 141, 194101 (2014).
  44. A. Nazir and G. Schaller, The reaction coordinate mapping in quantum thermodynamics, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018), pp. 551–577.
  45. S. Alipour, A. T. Rezakhani, A. P. Babu, K. Mølmer, M. Möttönen, and T. Ala-Nissila, Correlation-picture approach to open-quantum-system dynamics, Phys. Rev. X 10, 041024 (2020).
  46. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  47. N. Anto-Sztrikacs and D. Segal, Strong coupling effects in quantum thermal transport with the reaction coordinate method, New J. Phys. 23, 063036 (2021).
  48. A. Trushechkin, Quantum master equations and steady states for the ultrastrong-coupling limit and the strong-decoherence limit, Phys. Rev. A 106, 042209 (2022).
  49. 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).
  50. T. Becker, L.-N. Wu, and A. Eckardt, Lindbladian approximation beyond ultraweak coupling, Phys. Rev. E 104, 014110 (2021).
  51. G. M. Timofeev and A. S. Trushechkin, Hamiltonian of mean force in the weak-coupling and high-temperature approximations and refined quantum master equations, Int. J. Mod. Phys. A 37, 2243021 (2022).
  52. F. Cerisola, M. Berritta, S. Scali, S. A. R. Horsley, J. D. Cresser, and J. Anders, Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling, arXiv:2204.10874.
  53. T. Becker, A. Schnell, and J. Thingna, Canonically consistent quantum master equation, Phys. Rev. Lett. 129, 200403 (2022).
  54. M. Winczewski and R. Alicki, Renormalization in the theory of open quantum systems via the self-consistency condition, arXiv:2112.11962.
  55. M. Łobejko, M. Winczewski, G. Suárez, R. Alicki, and M. Horodecki, Towards reconciliation of completely positive open system dynamics with the equilibration postulate, Phys. Rev. E 110, 014144 (2024).
  56. A. M. Lacerda, A. Purkayastha, M. Kewming, G. T. Landi, and J. Goold, Quantum thermodynamics with fast driving and strong coupling via the mesoscopic leads approach, Phys. Rev. B 107, 195117 (2023).
  57. K. Shiraishi, M. Nakagawa, T. Mori, and M. Ueda, Quantum master equation for many-body systems: Derivation based on the Lieb-Robinson bound, Phys. Rev. B 111, 184311 (2025).
  58. R. Hartmann and W. T. Strunz, Accuracy assessment of perturbative master equations: Embracing nonpositivity, Phys. Rev. A 101, 012103 (2020).
  59. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths, Phys. Rev. A 105, 032208 (2022).
  60. A. G. Redfield, The theory of relaxation processes, Adv. Magn. Reson. 1, 1 (1965).
  61. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Searching for Lindbladians obeying local conservation laws and showing thermalization, Phys. Rev. A 107, 062216 (2023).
  62. T. Mori and S. Miyashita, Dynamics of the density matrix in contact with a thermal bath and the quantum master equation, J. Phys. Soc. Jpn. 77, 124005 (2008).
  63. C. H. Fleming and N. I. Cummings, Accuracy of perturbative master equations, Phys. Rev. E 83, 031117 (2011).
  64. J. Thingna, J.-S. Wang, and P. Hänggi, Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order, J. Chem. Phys. 136, 194110 (2012).
  65. J. Thingna, J.-S. Wang, and P. Hänggi, Reduced density matrix for nonequilibrium steady states: A modified Redfield solution approach, Phys. Rev. E 88, 052127 (2013).
  66. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  67. J. D. Cresser and J. Anders, Weak and ultrastrong coupling limits of the quantum mean force Gibbs state, Phys. Rev. Lett. 127, 250601 (2021).
  68. 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).

  69. The master equation of Ref. [53] is not of Lindblad form, so complete positivity is not guaranteed. Nevertheless, positivity is typically preserved on second order.

  70. G. T. Landi, D. Poletti, and G. Schaller, Nonequilibrium boundary-driven quantum systems: Models, methods, and properties, Rev. Mod. Phys. 94, 045006 (2022).
  71. F. Garcia-Gaitan and B. K. Nikolić, Fate of entanglement in magnetism under lindbladian or non-Markovian dynamics and conditions for their transition to Landau-Lifshitz-Gilbert classical dynamics, Phys. Rev. B 109, L180408 (2024).
  72. H. Wichterich, M. J. Henrich, H.-P. Breuer, J. Gemmer, and M. Michel, Modeling heat transport through completely positive maps, Phys. Rev. E 76, 031115 (2007).
  73. G. T. Landi, E. Novais, M. J. de Oliveira, and D. Karevski, Flux rectification in the quantum xxz chain, Phys. Rev. E 90, 042142 (2014).
  74. L. Schuab, E. Pereira, and G. T. Landi, Energy rectification in quantum graded spin chains: Analysis of the XXZ model, Phys. Rev. E 94, 042122 (2016).
  75. M. Žnidarič, J. J. Mendoza-Arenas, S. R. Clark, and J. Goold, Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system, Ann. Phys. (N.Y.) 529, 1600298 (2017).
  76. K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Verdozzi, P. Samuelsson, and P. P. Potts, Entanglement and thermokinetic uncertainty relations in coherent mesoscopic transport, Phys. Rev. Res. 5, 023155 (2023).
  77. A. Schnell, D. Vorberg, R. Ketzmerick, and A. Eckardt, High-temperature nonequilibrium bose condensation induced by a hot needle, Phys. Rev. Lett. 119, 140602 (2017).
  78. See Supplemental Material at http://link.aps.org/supplemental/10.1103/1q5b-p2bf for details on (A) nonproduct bath interaction Hamiltonians, (B) a convergence criterion for the series expansion, (C) an application to Bose-Hubbard chain, (D) the bath-correlation time, (E) rewriting the Redfield equation in pseudo-Lindblad form, (F),(G) error of different truncation orders, (H) the local Lindblad equation, and (I) the Drude bath, as well as (J) nonequilibrium setups and (K) the trajectory algorithm.
  79. J. Thingna, Steady-state transport properties of anharmonic systems, Ph.D. thesis, National University of Singapore, 2013.
  80. A. Schnell, Dissipative state engineering in quantum many-body systems, Ph.D. thesis, Technische Universität Dresden, 2019.
  81. E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  82. M. Kliesch, C. Gogolin, and J. Eisert, Lieb-Robinson bounds and the simulation of time-evolution of local observables in lattice systems, in Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View, edited by V. Bach and L. Delle Site (Springer International Publishing, Cham, 2014), pp. 301–318.
  83. T. Becker, C. Netzer, and A. Eckardt, Quantum trajectories for time-local non-Lindblad master equations, Phys. Rev. Lett. 131, 160401 (2023).
  84. K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, J. Opt. Soc. Am. B 10, 524 (1993).
  85. C. W. Gardiner, A. S. Parkins, and P. Zoller, Wave-function quantum stochastic differential equations and quantum-jump simulation methods, Phys. Rev. A 46, 4363 (1992).
  86. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  87. H.-P. Breuer, B. Kappler, and F. Petruccione, Stochastic wave-function method for non-Markovian quantum master equations, Phys. Rev. A 59, 1633 (1999).
  88. I. Kondov, U. Kleinekathöfer, and M. Schreiber, Stochastic unraveling of Redfield master equations and its application to electron transfer problems, J. Chem. Phys. 119, 6635 (2003).
  89. H.-P. Breuer, Genuine quantum trajectories for non-Markovian processes, Phys. Rev. A 70, 012106 (2004).
  90. B. Donvil and P. Muratore-Ginanneschi, Quantum trajectory framework for general time-local master equations, Nat. Commun. 13, 4140 (2022).
  91. M. Žnidarič, T. c. v. Prosen, G. Benenti, G. Casati, and D. Rossini, Thermalization and ergodicity in one-dimensional many-body open quantum systems, Phys. Rev. E 81, 051135 (2010).
  92. C. Zanoci, Y. Yoo, and B. Swingle, Thermalization at low temperatures via weakly damped multisite baths, Phys. Rev. B 108, 035156 (2023).
  93. M. Cattaneo, G. De Chiara, S. Maniscalco, R. Zambrini, and G. L. Giorgi, Collision models can efficiently simulate any multipartite Markovian quantum dynamics, Phys. Rev. Lett. 126, 130403 (2021).
  94. F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Quantum collision models: Open system dynamics from repeated interactions, Phys. Rep. 954, 1 (2022), quantum collision models: Open system dynamics from repeated interactions.

  95. Our local Lindblad equation only slightly violates the correct behavior at the boundaries [78].

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Phys. Rev. Lett. 134, 250401– Published 23 June, 2025
  • Received 25 October 2023
  • Revised 6 September 2024
  • Accepted 2 June 2025
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