Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time ${\tau }_E$ in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic $N$-particle systems. We first show how the growth of OTOCs up to ${\tau }_E=(1/\lambda )\log N$ is related to the Lyapunov exponent $\lambda$ of the corresponding chaotic mean-field dynamics in the semiclassical large-$N$ limit. Beyond ${\tau }_E$, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit $\hslash \to 0$ for fixed $N$, including quantum-chaotic single- and few-particle systems.

• Received 17 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.124101