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