Approximate but Accurate Quantum Dynamics from the Mori Formalism: I. Nonequilibrium Dynamics

TL;DR

This paper introduces a unified formalism combining Nakajima-Zwanzig and Mori theories for nonequilibrium quantum dynamics, using a Dyson expansion and auxiliary kernels, demonstrated on the spin-boson model with quasi-classical Ehrenfest dynamics.

Contribution

It develops a self-consistent equation for the memory kernel that simplifies nonequilibrium quantum dynamics calculations, integrating different projection operators and analyzing their effects.

Findings

Memory kernels can be short-lived under certain conditions.

The approach improves accuracy over standard semi-classical methods.

Different projection operators influence the properties of the memory kernels.

Abstract

We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics.