Unified error bounds for perturbations of non-Markovian open quantum systems in Gaussian environments

We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.

💡 Research Summary

This paper addresses the quantitative error analysis of non‑Markovian open quantum systems interacting with Gaussian environments under a very general Liouvillian dynamics. The authors extend the earlier work of Mascherpa et al. (Phys. Rev. Lett. 118, 100401, 2017), which provided a Grönwall‑type error bound for unitary system‑bath evolution, by deriving a substantially tighter bound that applies to both unitary and non‑unitary (Lindblad‑type) dynamics, and to both bosonic and fermionic baths.

The setting is as follows. The total Hilbert space is a tensor product 𝓗 = 𝓗_E ⊗ 𝓗_S, with the system space 𝓗_S finite‑dimensional and the environment 𝓗_E possibly infinite‑dimensional (e.g., a bosonic or fermionic Fock space). The dynamics of the total density operator ρ(t) is generated by a strongly continuous one‑parameter semigroup e^{Lt}, where the Liouvillian L = L_0 + L_{SE} splits into a free part L_0 (acting separately on system and environment) and an interaction part L_{SE}. The interaction is assumed to be separable, L_{SE} = Σ_{α=1}^N E_α ⊗ S_α, with each system operator S_α bounded (norm normalized to 1) and each environment operator E_α satisfying Wick’s theorem, i.e., the environment is Gaussian. Consequently, the entire influence of the bath on the reduced system dynamics is encoded in the set of two‑point bath correlation functions (BCFs) C_{α,α′}(t−t′).

The central question is: if the BCFs are perturbed by a small amount ΔC(t−t′), how does the expectation value O_S(t)=Tr