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Long-time exponential decompositions of non-Markovian Gaussian baths

Zhen Huang1,2,3, Zhiyan Ding4, Ke Wang4, Jason Kaye2,3, Xiantao Li5, and Lin Lin6,7,8,*

  • *Contact author: linlin@math.berkeley.edu
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Phys. Rev. Research 8, 043053 – Published 15 October, 2026

DOI: https://doi.org/10.1103/7765-s8lk

Abstract

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval [0,T] by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time T, inverse temperature β, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of T, achieving time-uniform complexity. The T dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the main bottleneck for long-time simulation is the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems and for Markovian embeddings of classical generalized Langevin equations with memory kernels.

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Phys. Rev. Research 8, 043053– Published 15 October, 2026
  • Received 5 April 2026
  • Accepted 11 September 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.

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