Finite-time Lyapunov fluctuations and the upper bound of classical and quantum out-of-time-ordered expansion rate exponents

TL;DR

This paper investigates the relationship between the exponential growth rate of out-of-time-ordered correlators and classical Lyapunov exponents in chaotic maps, revealing that fluctuations of finite-time Lyapunov exponents determine the upper bound and exact values of the growth rate.

Contribution

It provides a novel analytical framework linking OTOC growth rates to finite-time Lyapunov exponent fluctuations, including explicit formulas and numerical validation for classical and quantum maps.

Findings

OTOC growth rate equals Lyapunov exponent plus fluctuation term.

Jensen's inequality bounds the growth rate from above.

Numerical results show fluctuation contribution is approximately ln(√2).

Abstract

This Letter demonstrates for chaotic maps (logistic, classical and quantum standard maps (SMs)) that the exponential growth rate ($\Lambda$) of the out-of-time-ordered four-point correlator (OTOC) is equal to the classical Lyapunov exponent ($\lambda$) \textit{plus} fluctuations ($\Delta^{\mbox{\tiny (fluc)}}$) of the one-step finite-time Lyapunov exponents (FTLEs). Jensen's inequality provides the upper bound $\lambda\le\Lambda$ for the considered systems. Equality is restored with $\Lambda = \lambda + \Delta^{\mbox{\tiny (fluc)}}$, where $\Delta^{\mbox{\tiny (fluc)}}$ is quantified by $k$-higher-order cumulants of the FTLEs. Exact expressions for $\Lambda$ are derived and numerical results using $k = 20$ furnish $\Delta^{\mbox{\tiny (fluc)}} \sim\ln{(\sqrt{2})}$ for \textit{all maps} (large kicking intensities in the SMs).