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 is equal to the classical Lyapunov exponent (λ) plus fluctuations (Δ^{(fluc)}) of the one-step finite-time Lyapunov exponents (FTLEs). Jensen's inequality provides the upper bound λ≤Λ for the considered systems. Equality is restored with Λ=λ+Δ^{(fluc)}, where Δ^{(fluc)} is quantified by k-higher-order cumulants of the (covariant) FTLEs. Exact expressions for Λ are derived and numerical results using k=20 furnish Δ^{(fluc)}∼ln(sqrt[2]) for all maps (large kicking intensities in the SMs).