- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Nonequilibrium steady states in multibath quantum collision models

Ronan McElvogue1,2, Andrew K. Mitchell1,2, Gabriel T. Landi3,4, and Steve Campbell1,2

PDF

Phys. Rev. A 113, 022206 – Published 5 February, 2026

DOI: https://doi.org/10.1103/ypyy-ql4c

Abstract

Collision models provide a simple and versatile setting to capture the dynamics of open quantum systems. The standard approach to thermalization in this setting involves an environment of independent and identically prepared thermal qubits, interacting sequentially for a finite duration Δt with the system. We compare this to a two-bath scenario in which collisional qubits are prepared in either their ground or excited states and the environment temperature is encoded in system-environment couplings. The system reaches the same thermal steady state for both settings, although even in this limit they describe fundamentally different physical processes, with the two-bath setup yielding a nonequilibrium state with finite heat currents. Non-Markovian dynamics arises when intraenvironment interactions in either setting are introduced. Here the system in the single-bath setup again reaches a steady state at the canonical temperature of the bath, but the nonequilibrium steady state of the two-bath setup tends to a different temperature due to the generation of strong system-environment and intraenvironment correlations. The two-bath setting is particularly suited to studying quantum trajectories, which are well defined also for the non-Markovian case. We showcase this with a trajectory analysis of the heat currents within a two-point measurement scheme. Finally, we consider how our results are impacted when the system-environment interaction leads to strict homogenization. Our results provide insights into the dynamics and thermodynamics of thermalization towards nonequilibrium steady states and the role of non-Markovian interactions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. S. N. Filippov, S. Maniscalco, and G. García-Pérez, Scalability of quantum error mitigation techniques: From utility to advantage, arXiv:2403.13542.
  2. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  3. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  4. 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).
  5. D. Ferracin, A. Smirne, S. F. Huelga, M. B. Plenio, and D. Tamascelli, Spectral density modulation and universal Markovian closure of fermionic environments, J. Chem. Phys. 161, 174114 (2024).
  6. G. T. Landi, D. Poletti, and G. Schaller, Nonequilibrium boundary-driven quantum systems: Models, methods, and properties, Rev. Mod. Phys. 94, 045006 (2022).
  7. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  8. M. Brenes, J. J. Mendoza-Arenas, A. Purkayastha, M. T. Mitchison, S. R. Clark, and J. Goold, Tensor-network method to simulate strongly interacting quantum thermal machines, Phys. Rev. X 10, 031040 (2020).
  9. F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Quantum collision models: Open system dynamics from repeated interactions, Phys. Rep. 954, 1 (2022).
  10. S. Campbell and B. Vacchini, Collision models in open system dynamics: A versatile tool for deeper insights? Europhys. Lett. 133, 60001 (2021).
  11. S. Cusumano, Quantum collision models: A beginner guide, Entropy 24, 1258 (2022).
  12. J. Rau, Relaxation phenomena in spin and harmonic oscillator systems, Phys. Rev. 129, 1880 (1963).
  13. V. Scarani, M. Ziman, P. Štelmachovič, N. Gisin, and V. Bužek, Thermalizing quantum machines: Dissipation and entanglement, Phys. Rev. Lett. 88, 097905 (2002).
  14. M. Ziman, P. Štelmachovič, V. Bužek, M. Hillery, V. Scarani, and N. Gisin, Diluting quantum information: An analysis of information transfer in system-reservoir interactions, Phys. Rev. A 65, 042105 (2002).
  15. P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017).
  16. G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J. Roncaglia, and M. Antezza, Reconciliation of quantum local master equations with thermodynamics, New J. Phys. 20, 113024 (2018).
  17. S. Campbell, B. Çakmak, Ö. E. Müstecaplıoğlu, M. Paternostro, and B. Vacchini, Collisional unfolding of quantum Darwinism, Phys. Rev. A 99, 042103 (2019).
  18. E. Ryan, E. Carolan, S. Campbell, and M. Paternostro, Commutativity and the emergence of classical objectivity, J. Phys. Commun. 6, 095005 (2022).
  19. G. Karpat, I. Yalçınkaya, and B. Çakmak, Quantum synchronization in a collision model, Phys. Rev. A 100, 012133 (2019).
  20. P. Taranto, F. Bakhshinezhad, P. Schüttelkopf, F. Clivaz, and M. Huber, Exponential improvement for quantum cooling through finite-memory effects, Phys. Rev. Appl. 14, 054005 (2020).
  21. J. Langbehn, G. Mouloudakis, E. King, R. Menu, I. Gornyi, G. Morigi, Y. Gefen, and C. P. Koch, Universal cooling of quantum systems via randomized measurements, arXiv:2506.11964.
  22. G. Guarnieri, D. Morrone, B. Çakmak, F. Plastina, and S. Campbell, Non-equilibrium steady-states of memoryless quantum collision models, Phys. Lett. A 384, 126576 (2020).
  23. D. Heineken, K. Beyer, K. Luoma, and W. T. Strunz, Quantum-memory-enhanced dissipative entanglement creation in nonequilibrium steady states, Phys. Rev. A 104, 052426 (2021).
  24. F. Tian, J. Zou, L. Li, H. Li, and B. Shao, Effect of inter-system coupling on heat transport in a microscopic collision model, Entropy 23, 471 (2021).
  25. A. Prositto, M. Forbes, and D. Segal, Equilibrium and nonequilibrium steady states with the repeated interaction protocol: Relaxation dynamics and energetic cost, Quantum Sci. Technol. 10, 025061 (2025).
  26. S. Palafox, R. Román-Ancheyta, B. Çakmak, and Ö. E. Müstecaplıoğlu, Heat transport and rectification via quantum statistical and coherence asymmetries, Phys. Rev. E 106, 054114 (2022).
  27. M. Santiago-García, O. Pusuluk, Ö. E. Müstecaplıoğlu, B. Çakmak, and R. Román-Ancheyta, Quantum thermal machine as a rectifier, Quantum Sci. Technol. 10, 025018 (2025).
  28. K. Hammam, G. Manzano, and G. De Chiara, Quantum coherence enables hybrid multitask and multisource regimes in autonomous thermal machines, Phys. Rev. Res. 6, 013310 (2024).
  29. S. Cusumano and G. De Chiara, Structured quantum collision models: Generating coherence with thermal resources, New J. Phys. 26, 023001 (2024).
  30. K. Hammam, Y. Hassouni, R. Fazio, and G. Manzano, Optimizing autonomous thermal machines powered by energetic coherence, New J. Phys. 23, 043024 (2021).
  31. A. Corr, S. Cusumano, and G. De Chiara, Continuous variable structured collision models, Quantum Sci. Technol. 10, 045056 (2025).
  32. J. Ehrich, M. Esposito, F. Barra, and J. M. R. Parrondo, Micro-reversibility and thermalization with collisional baths, Physica A 552, 122108 (2020).
  33. S. L. Jacob, M. Esposito, J. M. R. Parrondo, and F. Barra, Thermalization induced by quantum scattering, PRX Quantum 2, 020312 (2021).
  34. J. Tabanera-Bravo, J. M. R. Parrondo, M. Esposito, and F. Barra, Thermalization and dephasing in collisional reservoirs, Phys. Rev. Lett. 130, 200402 (2023).
  35. S. Lorenzo, A. Farace, F. Ciccarello, G. M. Palma, and V. Giovannetti, Heat flux and quantum correlations in dissipative cascaded systems, Phys. Rev. A 91, 022121 (2015).
  36. S. Cusumano, A. Mari, and V. Giovannetti, Interferometric quantum cascade systems, Phys. Rev. A 95, 053838 (2017).
  37. S. Cusumano, A. Mari, and V. Giovannetti, Interferometric modulation of quantum cascade interactions, Phys. Rev. A 97, 053811 (2018).
  38. A. Pedram, B. Çakmak, and Ö. E. Müstecaplıoğlu, Environment-assisted modulation of heat flux in a bio-inspired system based on collision model, Entropy 24, 1162 (2022).
  39. R. Erbanni, X. Xu, T. F. Demarie, and D. Poletti, Simulating quantum transport via collisional models on a digital quantum computer, Phys. Rev. A 108, 032619 (2023).
  40. K. Garg, Z. Ahmed, S. Mitra, and S. Chakraborty, Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers, Phys. Rev. A 112, 022425 (2025).
  41. M. Ziman, P. Štelmachovič, and V. Bužek, Description of quantum dynamics of open systems based on collision-like models, Open Syst. Inf. Dyn. 12, 81 (2005).
  42. 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).
  43. G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quantum 5, 020201 (2024).
  44. J. P. Cherian, S. Chakraborty, and S. Ghosh, On thermalization of two-level quantum systems, Europhys. Lett. 126, 40003 (2019).
  45. S. Lorenzo, F. Ciccarello, and G. M. Palma, Composite quantum collision models, Phys. Rev. A 96, 032107 (2017).
  46. S. Cusumano, V. Cavina, M. Keck, A. De Pasquale, and V. Giovannetti, Entropy production and asymptotic factorization via thermalization: A collisional model approach, Phys. Rev. A 98, 032119 (2018).
  47. R. McCloskey and M. Paternostro, Non-Markovianity and system-environment correlations in a microscopic collision model, Phys. Rev. A 89, 052120 (2014).
  48. H. T. Şenyaşa, Ş. Kesgin, G. Karpat, and B. Çakmak, Entropy production in non-Markovian collision models: Information backflow vs. system-environment correlations, Entropy 24, 824 (2022).
  49. B. Çakmak, M. Pezzutto, M. Paternostro, and Ö. E. Müstecaplıoğlu, Non-Markovianity, coherence, and system-environment correlations in a long-range collision model, Phys. Rev. A 96, 022109 (2017).
  50. M. Pezzutto, M. Paternostro, and Y. Omar, Implications of non-Markovian quantum dynamics for the Landauer bound, New J. Phys. 18, 123018 (2016).
  51. Q. Zhang, Z.-X. Man, and Y.-J. Xia, Non-Markovianity in the presence of multiple thermal environments via collision model, Phys. Lett. A 383, 2456 (2019).
  52. G. Pleasance, Á. E. Neira, M. Merkli, and F. Petruccione, Non-Markovianity in collision models with initial intra-environment correlations, New J. Phys. 27, 114514 (2025).
  53. H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).
  54. S. Campbell, F. Ciccarello, G. M. Palma, and B. Vacchini, System-environment correlations and Markovian embedding of quantum non-Markovian dynamics, Phys. Rev. A 98, 012142 (2018).
  55. S. Kretschmer, K. Luoma, and W. T. Strunz, Collision model for non-Markovian quantum dynamics, Phys. Rev. A 94, 012106 (2016).
  56. C. Addis, B. Bylicka, D. Chruściński, and S. Maniscalco, Comparative study of non-Markovianity measures in exactly solvable one- and two-qubit models, Phys. Rev. A 90, 052103 (2014).
  57. H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems, Phys. Rev. Lett. 103, 210401 (2009).
  58. E.-M. Laine, J. Piilo, and H.-P. Breuer, Measure for the non-Markovianity of quantum processes, Phys. Rev. A 81, 062115 (2010).
  59. S. Wißmann, A. Karlsson, E.-M. Laine, J. Piilo, and H.-P. Breuer, Optimal state pairs for non-Markovian quantum dynamics, Phys. Rev. A 86, 062108 (2012).
  60. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  61. C. Sabín and G. García-Alcaine, A classification of entanglement in three-qubit systems, Eur. Phys. J. D 48, 435 (2008).
  62. T. A. Brun, A simple model of quantum trajectories, Am. J. Phys. 70, 719 (2002).
  63. S. J. Whalen, Collision model for non-Markovian quantum trajectories, Phys. Rev. A 100, 052113 (2019).
  64. D. Cilluffo, G. Buonaiuto, I. Lesanovsky, A. Carollo, S. Lorenzo, G. M. Palma, F. Ciccarello, and F. Carollo, Microscopic biasing of discrete-time quantum trajectories, Quantum Sci. Technol. 6, 045011 (2021).
  65. A. Prositto, C. Ramon-Escandell, and D. Segal, Collisional model with dissipative and dephasing baths: Nonadditive effects at strong coupling, New J. Phys. 28, 014502 (2025).
  66. D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. Kastner, Kondo effect in a single-electron transistor, Nature (London) 391, 156 (1998).
  67. J. Shim, H.-S. Sim, and S.-S. B. Lee, Numerical renormalization group method for entanglement negativity at finite temperature, Phys. Rev. B 97, 155123 (2018).
  68. P. Barthelemy and L. M. Vandersypen, Quantum dot systems: A versatile platform for quantum simulations, Ann. Phys. (Berlin) 525, 808 (2013).
  69. Z. Iftikhar, A. Anthore, A. Mitchell, F. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, and F. Pierre, Tunable quantum criticality and super-ballistic transport in a “charge” Kondo circuit, Science 360, 1315 (2018).
  70. W. Pouse, L. Peeters, C. L. Hsueh, U. Gennser, A. Cavanna, M. A. Kastner, A. K. Mitchell, and D. Goldhaber-Gordon, Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit, Nat. Phys. 19, 492 (2023).
  71. N. Petropoulos, X. Wu, A. Sokolov, P. Giounanlis, I. Bashir, A. K. Mitchell, M. Asker, D. Leipold, R. B. Staszewski, and E. Blokhina, Nanoscale single-electron box with a floating lead for quantum sensing: Modeling and device characterization, Appl. Phys. Lett. 124, 173503 (2024).
  72. T. Lacroix, D. Cilluffo, S. F. Huelga, and M. B. Plenio, Making quantum collision models exact, Commun. Phys. 8, 268 (2025).
  73. A. Harwood, M. Brunelli, and A. Serafini, Unified collision model of coherent and measurement-based quantum feedback, Phys. Rev. A 108, 042413 (2023).
  74. R. McElvogue, NM-CM-SS code, version 1.0.0, https://github.com/ronanmcelvogue/non-markovian-cm-ss (GitHub, San Francisco, 2026).
  75. G. Karpat and B. Çakmak, Transient dynamics and homogenization in incoherent collision models, Entropy 27, 206 (2025).

Outline

Information

Phys. Rev. A 113, 022206– Published 5 February, 2026
  • Received 24 July 2025
  • Accepted 15 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation