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

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