Applying the time-dependent variational principle of Balian and Vénéroni, we derive variational approximations for multi-time correlation functions in characterized by a density operator equal to a Gaussian density matrix. Then, we study the more realistic situation where only a few expectation values are given at the initial time and we perform an optimization with respect to the initial state. We calculate explicitly the two-time correlation functions with two and four field operators at equilibrium in the symmetric phase.
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.
Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.