A novel approach to accounting for correlations in evolution over time of an open quantum system

TL;DR

This paper introduces a new projection operator method to incorporate initial correlations into the evolution equations of open quantum systems, providing explicit calculations and demonstrating the decay of initial correlation effects over time.

Contribution

A novel projection operator transforms the Nakajima--Zwanzig equation into a homogeneous form that includes initial correlations without approximations, applicable to open quantum systems.

Findings

Explicit calculation of all terms affecting a quantum oscillator's evolution.

Initial correlations influence short- and intermediate-time dynamics.

Long-time behavior recovers Lindblad form, showing decay of initial correlations.

Abstract

A projection operator is introduced, which exactly transforms the inhomogeneous Nakajima--Zwanzig generalized master equation for the relevant part of a system +bath statistical operator, containing the inhomogeneous irrelevant term comprising the initial corrrelations, into the homogeneous equation accounting for initial correlations in the kernel governing its evolution. No "molecular chaos"-like approximation has been used. The obtained equation is equivalent to completely closed (homogeneous) equation for the statistical operator of a system of interest interacting with a bath. In the Born approximation (weak system-bath interaction) this equation can be presented as the time-local Redfield-like equation with additional terms caused by initial correlations. As an application, a quantum oscillator, interacting with a Boson field and driven from a Gibbs initial equilibrium system+bath state by an external force, is considered. All terms determining the oscillator evolution over time are explicitly calculated at all timescales. It is shown, how the initial correlations influence the evolution process. It is also demonstrated, that at the large timescale this influence vanishes, and the evolution equation for the quantum oscillator statistical operator acquires the Lindblad form..