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Non-Markovian dynamics in nonstationary Gaussian baths: A hierarchy of pure states approach

Vladislav Sukharnikov1,2,*, Stasis Chuchurka1, and Frank Schlawin1,2,3,†

  • *Contact author: vladislav.sukharnikov@desy.de
  • †Contact author: frank.schlawin@mpsd.mpg.de
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Phys. Rev. Research 8, 013123 – Published 3 February, 2026

DOI: https://doi.org/10.1103/yt37-s9hz

Abstract

Building on the standard hierarchy of pure states (HOPS) approach, we construct a generalized formulation suitable for open quantum systems interacting with nonstationary Gaussian baths, potentially extending its applicability to nonequilibrium baths. This is achieved by extending the conventional exponential decomposition of a bath-correlation function (BCF) for nonstationary cases. Using our formulation of HOPS, we derive the corresponding hierarchy of master equations and, when each term in the BCF expansion can be associated with an independent physical bath, we show how the formalism connects to the well-known pseudomode representation. We demonstrate the method's performance on two examples of nonstationary squeezed reservoirs generated via uniform squeezing and degenerate parametric amplification in a one-sided cavity. Benchmarking against the hierarchy of master equations shows that HOPS is more efficient under hierarchy truncation. The pseudomode representation is shown to be more efficient in the strongly non-Markovian regime. Our results highlight HOPS as a versatile and powerful tool for simulating open quantum systems in nonstationary baths, with potential applications ranging from squeezed light-matter interactions to driven-quantum materials and dissipative phase transitions.

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References (75)

  1. A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, and M. A. Sentef, Colloquium: Nonthermal pathways to ultrafast control in quantum materials, Rev. Mod. Phys. 93, 041002 (2021).
  2. M. Först, C. Manzoni, S. Kaiser, Y. Tomioka, Y. Tokura, R. Merlin, and A. Cavalleri, Nonlinear phononics as an ultrafast route to lattice control, Nat. Phys. 7, 854 (2011).
  3. R. Mankowsky, M. Först, and A. Cavalleri, Non-equilibrium control of complex solids by nonlinear phononics, Rep. Prog. Phys. 79, 064503 (2016).
  4. A. Subedi, Light-control of materials via nonlinear phononics, Compt. Rendus. Phys. 22, 161 (2021).
  5. A. S. Disa, T. F. Nova, and A. Cavalleri, Engineering crystal structures with light, Nat. Phys. 17, 1087 (2021).
  6. R. Mankowsky, A. Subedi, M. Först, S. O. Mariager, M. Chollet, H. T. Lemke, J. S. Robinson, J. M. Glownia, M. P. Minitti, A. Frano, M. Fechner, N. A. Spaldin, T. Loew, B. Keimer, A. Georges, and A. Cavalleri, Nonlinear lattice dynamics as a basis for enhanced superconductivity in YBa2Cu3O6.5, Nature (London) 516, 71 (2014).
  7. M. Budden, T. Gebert, M. Buzzi, G. Jotzu, E. Wang, T. Matsuyama, G. Meier, Y. Laplace, D. Pontiroli, M. Riccò, F. Schlawin, D. Jaksch, and A. Cavalleri, Evidence for metastable photo-induced superconductivity in K3C60, Nat. Phys. 17, 611 (2021).
  8. A. S. Disa, J. Curtis, M. Fechner, A. Liu, A. von Hoegen, M. Först, T. F. Nova, P. Narang, A. Maljuk, A. V. Boris, B. Keimer, and A. Cavalleri, Photo-induced high-temperature ferromagnetism in YTiO3, Nature (London) 617, 73 (2023).
  9. T. F. Nova, A. S. Disa, M. Fechner, and A. Cavalleri, Metastable ferroelectricity in optically strained SrTiO3, Science 364, 1075 (2019).
  10. M. Buzzi, et al., Photomolecular high-temperature superconductivity, Phys. Rev. X 10, 031028 (2020).
  11. J. Tindall, F. Schlawin, M. Buzzi, D. Nicoletti, J. R. Coulthard, H. Gao, A. Cavalleri, M. A. Sentef, and D. Jaksch, Dynamical order and superconductivity in a frustrated many-body system, Phys. Rev. Lett. 125, 137001 (2020).
  12. J. Tindall, F. Schlawin, M. A. Sentef, and D. Jaksch, Analytical solution for the steady states of the driven Hubbard model, Phys. Rev. B 103, 035146 (2021).
  13. J. Tindall, F. Schlawin, M. Sentef, and D. Jaksch, Lieb's theorem and maximum entropy condensates, Quantum 5, 610 (2021).
  14. M. Delor, I. V. Sazanovich, M. Towrie, and J. A. Weinstein, Probing and exploiting the interplay between nuclear and electronic motion in charge transfer processes, Acc. Chem. Res. 48, 1131 (2015).
  15. S. Valianti and S. S. Skourtis, Vibrational control of molecular electron transfer reactions, Mol. Phys. 117, 2618 (2019).
  16. M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered open systems and quantum simulations with atoms and ions, in Advances in Atomic, Molecular, and Optical Physics, edited by P. Berman, E. Arimondo and C. Lin (Academic Press, Cambridge, 2012), pp. 1–80.
  17. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016).
  18. B. Debecker, J. Martin, and F. Damanet, Controlling matter phases beyond Markov, Phys. Rev. Lett. 133, 140403 (2024).
  19. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
  20. S. Nakajima, On quantum theory of transport phenomena: Steady diffusion, Prog. Theor. Phys. 20, 948 (1958).
  21. R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33, 1338 (1960).
  22. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  23. D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Nonperturbative treatment of non-Markovian dynamics of open quantum systems, Phys. Rev. Lett. 120, 030402 (2018).
  24. L. Diósi and W. T. Strunz, The non-Markovian stochastic Schrödinger equation for open systems, Phys. Lett. A 235, 569 (1997).
  25. L. Diósi, N. Gisin, and W. T. Strunz, Non-Markovian quantum state diffusion, Phys. Rev. A 58, 1699 (1998).
  26. Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101 (1989).
  27. Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J. Phys. Soc. Jpn. 75, 082001 (2006).
  28. B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).
  29. P. Stenius and A. Imamoglu, Stochastic wavefunction methods beyond the Born-Markov and rotating-wave approximations, Quantum Semiclass. Opt. 8, 283 (1996).
  30. A. Imamoglu, Stochastic wave-function approach to non-Markovian systems, Phys. Rev. A 50, 3650 (1994).
  31. G. Pleasance, B. M. Garraway, and F. Petruccione, Generalized theory of pseudomodes for exact descriptions of non-Markovian quantum processes, Phys. Rev. Res. 2, 043058 (2020).
  32. F. Mascherpa, A. Smirne, A. D. Somoza, P. Fernández-Acebal, S. Donadi, D. Tamascelli, S. F. Huelga, and M. B. Plenio, Optimized auxiliary oscillators for the simulation of general open quantum systems, Phys. Rev. A 101, 052108 (2020).
  33. K. H. Hughes, C. D. Christ, and I. Burghardt, Effective-mode representation of non-Markovian dynamics: A hierarchical approximation of the spectral density. I. Application to single surface dynamics, J. Chem. Phys. 131, 024109 (2009).
  34. K. H. Hughes, C. D. Christ, and I. Burghardt, Effective-mode representation of non-Markovian dynamics: A hierarchical approximation of the spectral density. II. Application to environment-induced nonadiabatic dynamics, J. Chem. Phys. 131, 124108 (2009).
  35. G. Park, Z. Huang, Y. Zhu, C. Yang, G. K.-L. Chan, and L. Lin, Quasi-Lindblad pseudomode theory for open quantum systems, Phys. Rev. B 110, 195148 (2024).
  36. N. Lambert, S. Ahmed, M. Cirio, and F. Nori, Modelling the ultra-strongly coupled spin-boson model with unphysical modes, Nat. Commun. 10, 3721 (2019).
  37. W. T. Strunz, L. Diósi, and N. Gisin, Open system dynamics with non-Markovian quantum trajectories, Phys. Rev. Lett. 82, 1801 (1999).
  38. D. Suess, A. Eisfeld, and W. T. Strunz, Hierarchy of stochastic pure states for open quantum system dynamics, Phys. Rev. Lett. 113, 150403 (2014).
  39. R. Hartmann and W. T. Strunz, Exact open quantum system dynamics using the hierarchy of pure states (HOPS), J. Chem. Theory Comput. 13, 5834 (2017).
  40. R. Hartmann and W. T. Strunz, Open quantum system response from the hierarchy of pure states, J. Phys. Chem. A 125, 7066 (2021).
  41. T. Gera, L. Chen, A. Eisfeld, J. R. Reimers, E. J. Taffet, and D. I. G. B. Raccah, Simulating optical linear absorption for mesoscale molecular aggregates: An adaptive hierarchy of pure states approach, J. Chem. Phys. 158, 174103 (2023).
  42. P.-P. Zhang and A. Eisfeld, Non-perturbative calculation of two-dimensional spectra using the stochastic hierarchy of pure states, J. Phys. Chem. Lett. 7, 4488 (2016).
  43. V. Link, W. T. Strunz, and K. Luoma, Non-Markovian quantum dynamics in a squeezed reservoir, Entropy 24, 352 (2022).
  44. X. Gao, J. Ren, A. Eisfeld, and Z. Shuai, Non-Markovian stochastic Schrödinger equation: Matrix-product-state approach to the hierarchy of pure states, Phys. Rev. A 105, L030202 (2022).
  45. C. Gardiner and C. Savage, A multimode quantum theory of a degenerate parametric amplifier in a cavity, Opt. Commun. 50, 173 (1984).
  46. M. J. Collett and C. W. Gardiner, Squeezing of intracavity and traveling-wave light fields produced in parametric amplification, Phys. Rev. A 30, 1386 (1984).
  47. V. Boettcher, R. Hartmann, K. Beyer, and W. T. Strunz, Dynamics of a strongly coupled quantum heat engine—Computing bath observables from the hierarchy of pure states, J. Chem. Phys. 160, 094108 (2024).
  48. D. Suess, W. T. Strunz, and A. Eisfeld, Hierarchical equations for open system dynamics in fermionic and bosonic environments, J. Stat. Phys. 159, 1408 (2015).
  49. P. Menczel, K. Funo, M. Cirio, N. Lambert, and F. Nori, Non-Hermitian pseudomodes for strongly coupled open quantum systems: Unravelings, correlations, and thermodynamics, Phys. Rev. Res. 6, 033237 (2024).
  50. W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold, Non-Markovian dissipative semiclassical dynamics, Phys. Rev. Lett. 100, 230402 (2008).
  51. H.-P. Breuer, B. Kappler, and F. Petruccione, Stochastic wave-function method for non-Markovian quantum master equations, Phys. Rev. A 59, 1633 (1999).
  52. A. Gilchrist, C. W. Gardiner, and P. D. Drummond, Positive P representation: Application and validity, Phys. Rev. A 55, 3014 (1997).
  53. P. Deuar, A. Ferrier, M. Matuszewski, G. Orso, and M. H. Szymańska, Fully quantum scalable description of driven-dissipative lattice models, PRX Quantum 2, 010319 (2021).
  54. S. Chuchurka, V. Sukharnikov, A. Benediktovitch, and N. Rohringer, Stochastic modeling of superfluorescence in compact systems, Phys. Rev. A 110, 053703 (2024).
  55. S. Chuchurka, A. Benediktovitch, Å. Krušič, A. Halavanau, and N. Rohringer, Stochastic modeling of x-ray superfluorescence, Phys. Rev. A 109, 033725 (2024).
  56. S. Chuchurka, V. Sukharnikov, and N. Rohringer, Hermitian stochastic methodology for x-ray superfluorescence, Phys. Rev. A 109, 063705 (2024).
  57. D. Manzano, A short introduction to the Lindblad master equation, AIP Adv. 10, 025106 (2020).
  58. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  59. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  60. N. Gisin and I. C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen. 25, 5677 (1992).
  61. D. Gatarek and N. Gisin, Continuous quantum jumps and infinite-dimensional stochastic equations, J. Math. Phys. 32, 2152 (1991).
  62. C. Rackauckas and Q. Nie, DifferentialEquations.jl–A performant and feature-rich ecosystem for solving differential equations in Julia, J. Open Res. Software 5, 15 (2017).
  63. C. W. Gardiner, Inhibition of atomic phase decays by squeezed light: A direct effect of squeezing, Phys. Rev. Lett. 56, 1917 (1986).
  64. 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).
  65. A. S. Parkins and C. W. Gardiner, Effect of finite-bandwidth squeezing on inhibition of atomic-phase decays, Phys. Rev. A 37, 3867 (1988).
  66. H. Ritsch and P. Zoller, Systems driven by colored squeezed noise: The atomic absorption spectrum, Phys. Rev. A 38, 4657 (1988).
  67. G. Adam, Density matrix elements and moments for generalized Gaussian state fields, J. Mod. Opt. 42, 1311 (1995).
  68. A. Ablimit, F.-H. Ren, R.-H. He, Y.-Y. Xie, and Z.-M. Wang, Effects of non-Markovian squeezed bath on the dynamics of open systems, Physica A 630, 129251 (2023).
  69. Y.-Y. Xie and Z.-M. Wang, Enhancement of adiabaticity via combined control in open quantum systems, APL Quantum 2, 016129 (2025).
  70. A. Ablimit, A. Abliz, Y.-S. Chen, and Z.-M. Wang, Non-Markovian dynamics of open quantum systems in squeezed thermal baths, Quantum Inf. Proc. 24, 102 (2025).
  71. K. E. Dorfman, F. Schlawin, and S. Mukamel, Nonlinear optical signals and spectroscopy with quantum light, Rev. Mod. Phys. 88, 045008 (2016).
  72. C. Sánchez Muñoz, G. Frascella, and F. Schlawin, Quantum metrology of two-photon absorption, Phys. Rev. Res. 3, 033250 (2021).
  73. S. Panahiyan, C. S. Muñoz, M. V. Chekhova, and F. Schlawin, Nonlinear interferometry for quantum-enhanced measurements of multiphoton absorption, Phys. Rev. Lett. 130, 203604 (2023).
  74. F. Schlawin and M. Gessner, Theory of quantum-enhanced stimulated Raman scattering, arXiv:2502.19344 .
  75. F. Schlawin, D. M. Kennes, and M. A. Sentef, Cavity quantum materials, Appl. Phys. Rev. 9, 011312 (2022).

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Phys. Rev. Research 8, 013123– Published 3 February, 2026
  • Received 17 November 2025
  • Accepted 13 January 2026
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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. Open access publication funded by Max Planck Society.

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