Estimation of equilibration time scales from nested fraction approximations

TL;DR

This paper introduces a new approximation method for estimating the equilibration time of quantum systems using nested fraction approximations, matching the accuracy of advanced numerical techniques with minimal input data.

Contribution

It proposes a novel approximation scheme based on nested fraction approximations to estimate quantum equilibration times, demonstrating practical advantages and accuracy.

Findings

Accurately estimates equilibration time with minimal data

Mathematically equivalent to existing recursive methods

Validated across various relaxation dynamics examples

Abstract

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function, and use it to estimate the equilibration time $\tau$ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of $\tau$, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a new practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.