Quasiclassical theory of out-of-time-ordered correlators

TL;DR

This paper develops a quasiclassical formalism for out-of-time-ordered correlators (OTOCs) using semiclassical propagators, revealing limitations of classical approximations and emphasizing the importance of quantum effects in chaotic quantum systems.

Contribution

It introduces a quasiclassical approach to OTOCs derived from the van Vleck-Gutzwiller propagator, highlighting its validity at short times and its limitations at longer times due to quantum effects.

Findings

Quasiclassical OTOCs match Wigner-Moyal results at short times.

Classical saturation values underestimate quantum thresholds.

Quantum contributions dominate beyond Ehrenfest time.

Abstract

Out-of-time-ordered correlators (OTOCs), defined via the squared commutator of a time-evolving and a stationary operator, represent observables that provide useful indicators for chaos and the scrambling of information in complex quantum systems. Here we present a quasiclassical formalism of OTOCs, which is obtained from the semiclassical van Vleck-Gutzwiller propagator through the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner-Moyal formalism, i.e., OTOCs are classically described via the square of the Poisson bracket between the two involved observables, thus giving rise to an exponential growth in a chaotic regime. For long times, for which the semiclassical framework is, in principle, still valid, the diagonal approximation yields an asymptotic saturation value for the quasiclassical OTOC under the assumption of fully developed classical chaos. However, numerical simulations, carried out within chaotic few-site Bose-Hubbard systems in the absence and presence of periodic driving, demonstrate that this saturation value strongly underestimates the actual threshold value of the quantum OTOC, which is normally attained after the Ehrenfest time. This indicates that nondiagonal and hence genuinely quantum contributions, thus exceeding the framework of the quasiclassical description, are primarily responsible for describing OTOCs beyond the short-time regime.