Uniform process tensor approach for the calculation of multi-time correlation functions of non-Markovian open systems

The process tensor framework to open quantum systems provides the most general description of multi-time correlations in non-Markovian quantum dynamics. A compressed representation of a process tensor in terms of matrix product operators (MPO) can be used for numerically exact calculations of multi-time correlation functions in systems strongly coupled to a non-Markovian reservoir. We show here that the numerical scaling for computing multi-dimensional spectra can be significantly improved using a time-translation invariant MPO representation of the process tensor obtained from the uniform time-evolving matrix product operator (uniTEMPO) method. In particular, this approach provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution. We calculate linear and 2D electronic spectra for an example system and discuss the performance and numerical scaling of our simulations.

💡 Research Summary

The paper presents a novel computational framework for evaluating multi‑time correlation functions of non‑Markovian open quantum systems by exploiting the uniform time‑evolving matrix product operator (uniTEMPO) method. Traditional approaches such as PT‑TEMPO generate process tensors within a finite time window, which obscures the inherent time‑translation invariance of stationary baths. In contrast, uniTEMPO leverages the stationarity of the bath correlation function α(t‑s) to construct an infinite tensor‑network representation that yields a time‑independent MPO for the influence functional. This MPO, denoted Q, acts on an enlarged space of dimension d²·χ, where d is the system Hilbert space dimension and χ is the auxiliary bond dimension automatically determined by a user‑specified accuracy. By diagonalising Q (Q = Σ_k q_k |q_k⟩⟨q_k|) the authors express the propagator for any time interval as a sum of exponentials q_k^{Nτ}=e^{λ_k τ}, with λ_k = –i ω_k – γ_k separating into real frequencies ω_k and positive damping rates γ_k. Consequently, half‑sided Fourier transforms of the time intervals replace the exponential factors by simple complex Lorentzian functions (i(ω−ω_k)−γ_k)^{-1}. This spectral representation enables direct computation of linear absorption spectra and two‑dimensional electronic spectra in the frequency domain without any explicit real‑time propagation, dramatically improving numerical scaling. The formalism is applied to a three‑level model (ground state plus two excited states) where only the excited states couple to a bosonic bath with an Ohmic‑type spectral density J(ω)=2α ω e^{-ω/ω_c}. Various regimes of reorganization energy λ and electronic coupling Ω are explored, reproducing the linear and 2D spectra previously obtained with more costly methods. Benchmarking shows that the uniTEMPO‑based approach reduces memory consumption and CPU time by orders of magnitude while maintaining exactness within the prescribed tolerance. The authors conclude that the time‑translation‑invariant MPO representation provides a powerful and versatile tool for simulating multi‑time response functions in a broad class of non‑Markovian systems, with potential extensions to optimal control, real‑time spectroscopy, and many‑body dynamical mean‑field calculations.