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  • Open Access

Open system dynamics in interacting quantum field theories

Brenden Bowen1,*, Nishant Agarwal1,†, and Archana Kamal1,2,‡

  • *Contact author: brenden_bowen@student.uml.edu
  • †Contact author: nishant_agarwal@uml.edu
  • ‡Contact author: archana.kamal@northwestern.edu
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Phys. Rev. Research 7, 043311 – Published 17 December, 2025

DOI: https://doi.org/10.1103/kjph-9z8l

Abstract

A quantum system that interacts with an environment generally undergoes nonunitary evolution described by a non-Markovian or Markovian master equation. In this paper, we construct the non-Markovian Redfield master equation for a quantum scalar field that interacts with a second field through a bilinear or nonlinear interaction on a Minkowski background. We use the resulting master equation to set up coupled differential equations that can be solved to obtain the equal-time two-point function of the system field. We show how the equations simplify under various approximations including the Markovian limit and argue that the Redfield equation-based solution provides a perturbative resummation to the standard second-order Dyson series result. For the bilinear interaction, we explicitly show that the Redfield solution is closer to the exact solution compared to the perturbation theory-based one. Further, the environment correlation function is oscillatory and nondecaying in this case, making the Markovian master equation a poor approximation. For the nonlinear interaction, on the other hand, the environment correlation function is sharply peaked and the Redfield solution matches that obtained using a Markovian master equation in the late-time limit.

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References (91)

  1. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n level systems, J. Math. Phys. 17, 821 (1976).
  2. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  3. A. G. Redfield, On the theory of relaxation processes, IBM J. Res. Dev. 1, 19 (1957).
  4. The Redfield equation is also referred to as the Born-Redfield equation [16].
  5. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91 (1974).
  6. E. B. Davies, Markovian master equations. II. Math. Ann. 219, 147 (1976).
  7. P. Gaspard and M. Nagaoka, Slippage of initial conditions for the redfield master equation, J. Chem. Phys. 111, 5668 (1999).
  8. R. Whitney, Staying positive: Going beyond lindblad with perturbative master equations, J. Phys. A: Math. Theor. 41, 175304 (2008).
  9. G. Schaller and T. Brandes, Preservation of positivity by dynamical coarse graining, Phys. Rev. A 78, 022106 (2008).
  10. C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar, Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations, Phys. Rev. A 88, 012103 (2013).
  11. D. Farina and V. Giovannetti, Open-quantum-system dynamics: Recovering positivity of the redfield equation via the partial secular approximation, Phys. Rev. A 100, 012107 (2019).
  12. E. Mozgunov and D. Lidar, Completely positive master equation for arbitrary driving and small level spacing, Quantum 4, 227 (2020).
  13. D. Davidović, Completely positive, simple, and possibly highly accurate approximation of the redfield equation, Quantum 4, 326 (2020).
  14. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  15. A. Trushechkin, Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation, Phys. Rev. A 103, 062226 (2021).
  16. A. Keefe, N. Agarwal, and A. Kamal, Quantifying spectral signatures of non-Markovianity beyond the Born-Redfield master equation, Quantum 9, 1863 (2025).
  17. A. Levy and R. Kosloff, The local approach to quantum transport may violate the second law of thermodynamics, Europhys. Lett. 107, 20004 (2014).
  18. F. Lombardo and F. D. Mazzitelli, Coarse graining and decoherence in quantum field theory, Phys. Rev. D 53, 2001 (1996).
  19. J. F. Koksma, T. Prokopec, and M. G. Schmidt, Decoherence in an interacting quantum field theory: The vacuum case, Phys. Rev. D 81, 065030 (2010).
  20. J. F. Koksma, T. Prokopec, and M. G. Schmidt, Decoherence in an interacting quantum field theory: Thermal case, Phys. Rev. D 83, 085011 (2011).
  21. C. Agón, V. Balasubramanian, S. Kasko, and A. Lawrence, Coarse grained quantum dynamics, Phys. Rev. D 98, 025019 (2018).
  22. D. Boyanovsky, Effective field theory out of equilibrium: Brownian quantum fields, New J. Phys. 17, 063017 (2015).
  23. C. Agón and A. Lawrence, Divergences in open quantum systems, J. High Energy Phys. 04 (2018) 008.
  24. D. Boyanovsky, Information loss in effective field theory: Entanglement and thermal entropies, Phys. Rev. D 97, 065008 (2018).
  25. C. Burrage, C. Käding, P. Millington, and J. Minář, Open quantum dynamics induced by light scalar fields, Phys. Rev. D 100, 076003 (2019).
  26. S. Banerjee, S. Choudhury, S. Chowdhury, J. Knaute, S. Panda, and K. Shirish, Thermalization in quenched open quantum cosmology, Nucl. Phys. B 996, 116368 (2023).
  27. C. Käding and M. Pitschmann, New method for directly computing reduced density matrices, Phys. Rev. D 107, 016005 (2023).
  28. C. Käding and M. Pitschmann, Density matrices in quantum field theory: Non-Markovianity, path integrals and master equations, arXiv:2503.08567.
  29. F. C. Lombardo and D. L. Nacir, Decoherence during inflation: The generation of classical inhomogeneities, Phys. Rev. D 72, 063506 (2005).
  30. D. Boyanovsky, Effective field theory during inflation: Reduced density matrix and its quantum master equation, Phys. Rev. D 92, 023527 (2015).
  31. C. P. Burgess, R. Holman, and G. Tasinato, Open EFTs, IR effects & late-time resummations: Systematic corrections in stochastic inflation, J. High Energy Phys. 01 (2016) 153.
  32. T. J. Hollowood and J. I. McDonald, Decoherence, discord and the quantum master equation for cosmological perturbations, Phys. Rev. D 95, 103521 (2017).
  33. S. Shandera, N. Agarwal, and A. Kamal, Open quantum cosmological system, Phys. Rev. D 98, 083535 (2018).
  34. D. Boyanovsky, Imprint of entanglement entropy in the power spectrum of inflationary fluctuations, Phys. Rev. D 98, 023515 (2018).
  35. S. Akhtar, S. Choudhury, S. Chowdhury, D. Goswami, S. Panda, and A. Swain, Open quantum entanglement: A study of two atomic system in static patch of de Sitter space, Eur. Phys. J. C 80, 748 (2020).
  36. J.-O. Gong and M.-S. Seo, Quantum non-linear evolution of inflationary tensor perturbations, J. High Energy Phys. 05 (2019) 021.
  37. S. Brahma, O. Alaryani, and R. Brandenberger, Entanglement entropy of cosmological perturbations, Phys. Rev. D 102, 043529 (2020).
  38. S. Banerjee, S. Choudhury, S. Chowdhury, R. N. Das, N. Gupta, S. Panda, and A. Swain, Indirect detection of cosmological constant from interacting open quantum system, Ann. Phys. 443, 168941 (2022).
  39. S. Brahma, A. Berera, and J. Calderón-Figueroa, Universal signature of quantum entanglement across cosmological distances, Classical Quantum Gravity 39, 245002 (2022).
  40. J.-T. Hsiang and B.-L. Hu, No intrinsic decoherence of inflationary cosmological perturbations, Universe 8, 27 (2022).
  41. T. Colas, J. Grain, and V. Vennin, Benchmarking the cosmological master equations, Eur. Phys. J. C 82, 1085 (2022).
  42. A. D. Hammou and N. Bartolo, Cosmic decoherence: Primordial power spectra and non-Gaussianities, J. Cosmol. Astropart. Phys. 04 (2023) 055.
  43. C. P. Burgess, R. Holman, G. Kaplanek, J. Martin, and V. Vennin, Minimal decoherence from inflation, J. Cosmol. Astropart. Phys. 07 (2023) 022.
  44. T. Colas, J. Grain, and V. Vennin, Quantum recoherence in the early universe, Europhys. Lett. 142, 69002 (2023).
  45. C. P. Burgess, T. Colas, R. Holman, G. Kaplanek, and V. Vennin, Cosmic purity lost: Perturbative and resummed late-time inflationary decoherence, J. Cosmol. Astropart. Phys. 08 (2024) 042.
  46. S. A. Salcedo, T. Colas, and E. Pajer, The open effective field theory of inflation, J. High Energy Phys. 10 (2024) 248.
  47. S. Brahma, J. Calderón-Figueroa, X. Luo, and D. Seery, The special case of slow-roll attractors in de Sitter: Non-Markovian noise and evolution of entanglement entropy, J. Cosmol. Astropart. Phys. 04 (2025) 050.
  48. F. Lopez and N. Bartolo, Quantum signatures and decoherence during inflation from deep subhorizon perturbations, arXiv:2503.23150.
  49. Y.-Z. Li, Stochastic inflation as an open quantum system, arXiv:2507.02070.
  50. S. Cao and D. Boyanovsky, Nonequilibrium dynamics of axionlike particles: The quantum master equation, Phys. Rev. D 107, 063518 (2023).
  51. C. Käding, M. Pitschmann, and C. Voith, Dilaton-induced open quantum dynamics, Eur. Phys. J. C 83, 767 (2023).
  52. A. Bassi, A. Großardt, and H. Ulbricht, Gravitational decoherence, Classical Quantum Gravity 34, 193002 (2017).
  53. M. Sharifian, M. Zarei, M. Abdi, N. Bartolo, and S. Matarrese, Open quantum system approach to the gravitational decoherence of spin-1/2 particles, Phys. Rev. D 109, 043510 (2024).
  54. M. Zarei, N. Bartolo, D. Bertacca, A. Ricciardone, and S. Matarrese, Non-Markovian open quantum system approach to the early universe: Damping of gravitational waves by matter, Phys. Rev. D 104, 083508 (2021).
  55. G. Kaplanek and C. P. Burgess, Hot accelerated qubits: Decoherence, thermalization, secular growth and reliable late-time predictions, J. High Energy Phys. 03 (2020) 008.
  56. G. Kaplanek and C. P. Burgess, Hot cosmic qubits: Late-time de sitter evolution and critical slowing down, J. High Energy Phys. 02 (2020) 053.
  57. G. Kaplanek and E. Tjoa, Effective master equations for two accelerated qubits, Phys. Rev. A 107, 012208 (2023).
  58. C. Jana, R. Loganayagam, and M. Rangamani, Open quantum systems and Schwinger-Keldysh holograms, J. High Energy Phys. 07 (2020) 242.
  59. R. Loganayagam, K. Ray, and A. Sivakumar, Fermionic open EFT from holography, arXiv:2011.07039.
  60. R. Loganayagam, M. Rangamani, and J. Virrueta, Holographic open quantum systems: Toy models and analytic properties of thermal correlators, J. High Energy Phys. 03 (2023) 153.
  61. P. Pelliconi and J. Sonner, The influence functional in open holography: Entanglement and Rényi entropies, J. High Energy Phys. 06 (2024) 185.
  62. M. Srednicki, Quantum Field Theory (Cambridge University Press, Cambridge, 2007).
  63. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, Berlin, Heidelberg, 1999).
  64. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  65. M. A. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, Berlin, 2007).
  66. E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I. O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, Heidelberg, 2003).
  67. We will see in Secs. 5 and 6 that for both interactions we consider, this environment correlator only depends on the magnitude k.
  68. Also see Ref. [89] for an effective time-local master equation construction and Ref. [90] for an effective field theory-inspired approach.
  69. Since we have used definite integrals in the integration by parts, there are terms that depend on the initial time t0. It is straightforward to show, however, that these terms cancel identically.
  70. Since we are working at O(λ2), only tree-level diagrams contribute to the environment correlation function. This will be convenient when we renormalize the λϕχ2 theory in Sec. 6.
  71. This qualitative equivalence between quantum master equations expressed in Schrödinger and interaction pictures does not hold when the free evolution of the system itself is nonunitary [91].
  72. J. Martin and V. Vennin, Non-Gaussianities from quantum decoherence during inflation, J. Cosmol. Astropart. Phys. 06 (2018) 037.
  73. We never have to explicitly choose a value for Mm since the explicit dependence on M and m appears as their ratio for this choice of parameters.
  74. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995).
  75. E. A. Calzetta and B.-L. B. Hu, Nonequilibrium Quantum Field Theory (Cambridge University Press, Cambridge, 2008).
  76. J. Baacke, K. Heitmann, and C. Patzold, On the choice of initial states in nonequilibrium dynamics, Phys. Rev. D 57, 6398 (1998).
  77. J. Baacke, D. Boyanovsky, and H. J. de Vega, Initial time singularities in nonequilibrium evolution of condensates and their resolution in the linearized approximation, Phys. Rev. D 63, 045023 (2001).
  78. H. Collins and R. Holman, Renormalization of initial conditions and the trans-Planckian problem of inflation, Phys. Rev. D 71, 085009 (2005).
  79. H. Collins, R. Holman, and T. Vardanyan, Renormalizing an initial state, J. High Energy Phys. 10 (2014) 124.
  80. S. Chaykov, N. Agarwal, S. Bahrami, and R. Holman, Loop corrections in Minkowski spacetime away from equilibrium. Part II. Finite-time results, J. High Energy Phys. 02 (2023) 094.
  81. We note that one could obtain the same final results using the dimensional regularization procedure of Ref. [80]. We have instead used the iε regularization scheme here as it is more instructive since the delta function piece of the environment correlation function emerges naturally.
  82. A. O. Caldeira and A. J. Leggett, Influence of dissipation on quantum tunneling in macroscopic systems, Phys. Rev. Lett. 46, 211 (1981).
  83. A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587 (1983).
  84. G. S. Agarwal, Quantum statistical theories of spontaneous emission and their relation to other approaches, Quantum Optics (Springer, Heidelberg, 1974), pp. 1–128.
  85. G. Timofeev and A. 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).
  86. K. Ganguly and B. K. Agarwalla, Study of non-equilibrium Green's functions beyond Born approximation in open quantum systems, arXiv:2309.03776.
  87. H. Hosseinabadi, D. E. Chang, and J. Marino, Far from equilibrium field theory for strongly coupled light and matter: Dynamics of frustrated multimode cavity QED, Phys. Rev. Res. 6, 043314 (2024).
  88. S. Khan, L. S. Rathore, and S. Jain, Steady-state correlation function beyond the standard weak-coupling limit and consistency with the Kubo-Martin-Schwinger relation, Phys. Rev. A 111, 032214 (2025).
  89. S. Prudhoe and S. Shandera, Classifying the non-time-local and entangling dynamics of an open qubit system, J. High Energy Phys. 02 (2023) 007.
  90. N. Agarwal and Y.-Z. Chu, Initial value formulation of a quantum damped harmonic oscillator, Phys. Rev. Res. 6, 023113 (2024).
  91. T. Thorbeck, Z. Xiao, A. Kamal, and L. C. G. Govia, Readout-induced suppression and enhancement of superconducting qubit lifetimes, Phys. Rev. Lett. 132, 090602 (2024).

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Phys. Rev. Research 7, 043311– Published 17 December, 2025
  • Received 8 August 2024
  • Accepted 14 October 2025
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