Variational approach to multi-time correlation functions

Roger Balian

A variational determination of multi-time correlation functions

Hermann Haken, Wolfgang Weidlich

Roger Balian

R. Dümcke

Matias Bundgaard-Nielsen, Jake Iles-Smith

While the quantum regression theorem (QRT) is the standard tool for computing multi-time correlation functions in open quantum systems, it relies on system-bath separability and an environment that remains in equilibrium, assumptions that are violated once dynamical correlations develop. Using the projection operator formalism, we derive an extension to the QRT that explicitly incorporates these correlation-induced corrections, enabling us to capture multi-time correlation functions in non-Markovian regimes. We apply this framework to the variational polaron master equation for the spin-boson model in ohmic and super-ohmic regimes, where the polaron transformation mixes system-bath degrees of freedom to produce a non-thermal effective environment. Benchmarking against numerically exact tensor-network simulations, we demonstrate quantitative agreement for single- and two-time observables, including linear-response spectra, even at strong coupling. Our approach broadens the reach of analytic master equations to multi-time observables in strong-coupling and non-Markovian regimes, where environmental memory effects and system-bath correlations are crucial.

Yiying Yan, Yang Zhao

ABSTRACT Markovian pseudomode models are shown to accurately describe non‐Markovian and non‐perturbative reduced dynamics and multi‐time correlation functions under certain constraints. However, it is challenging to solve the Lindblad master equation for such models when a number of pseudomodes are involved. A variational approach based on multiple Davydov states is used to solve the master equation of pseudomode models for the multi‐mode Jaynes–Cummings (JC) model, and the spin‐boson system across four distinct configurations, enabling computation of the two‐time correlation function that incorporates bath phonon occupation dynamics. In parallel, calculations of the unitary evolution of full Hamiltonians are chosen as benchmark references. Benchmark comparisons demonstrate that the variational approach for Markovian pseudomode models successfully predicts both the single‐time and two‐time expectations within the multi‐mode JC model. In contrast, in the spin‐boson model, the variational simulations for the pseudomode system and the full Hamiltonian yield inconsistent bath occupation numbers in a non‐perturbative strong coupling regime despite that these two kinds of simulations predict almost identical reduced dynamics, resulting from a fitting artefact in the pseudomode model. Our findings offer insights into the applicability of the non‐unitary variational approach on the simulation of two‐time correlation functions within the pseudomode framework.