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

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