Systematic Perturbation Theory for Dynamical Coarse-Graining

TL;DR

This paper extends the dynamical coarse-graining method to higher orders, ensuring positivity preservation and accurate short-time, non-Markovian dynamics, especially for simple bath structures and small systems.

Contribution

It systematically develops higher-order dynamical coarse-graining, maintaining positivity and capturing non-Markovian effects accurately.

Findings

Unconditionally preserves positivity up to second order.

Accurately captures short-time non-Markovian dynamics.

Shows remarkable agreement with exact solutions for complex systems.

Abstract

We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix -- even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics, i.e., it is suitable to analyze non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.