Effective non-Markovian description of a system interacting with a bath

TL;DR

This paper develops an effective description for a harmonic system interacting with a bath modeled as a chain of oscillators, enabling simplified analysis with quantifiable error bounds.

Contribution

It introduces a method to approximate the exact dynamics with a shorter chain and provides explicit error bounds and relationships between error, timescale, and modes.

Findings

Explicit effective dynamics expression derived

Error bounds for truncated chain approximation established

Relationship between error, timescale, and number of modes demonstrated

Abstract

We study a harmonic system coupled to chain of first neighbor interacting oscillators. After deriving the exact dynamics of the system, we prove that one can effectively describe the exact dynamics by considering a suitable shorter chain. We provide the explicit expression for such an effective dynamics and we provide an upper bound on the error one makes considering it instead of the dynamics of the full chain. We eventually prove how error, timescale and number of modes in the truncated chain are related.