# Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest time-scale

## TL;DR

This paper investigates the growth of out-of-time-order correlators in bipartite quantum systems, revealing two distinct phases of scrambling: an initial intra-system exponential growth and a universal interaction-driven saturation, modeled by random matrices.

## Contribution

It introduces a two-phase model of quantum chaos in bipartite systems, extending understanding beyond the Ehrenfest time with a solvable random matrix approach.

## Key findings

- Initial phase shows intra-subsystem exponential growth independent of interaction.
- Second phase exhibits universal exponential approach to saturation described by a random matrix model.
- Participation ratio indicates delocalization and mixing during the two phases.

## Abstract

Out-of-time-order correlators (OTOC), vigorously being explored as a measure of quantum chaos and information scrambling, is studied here in the natural and simplest multi-particle context of bipartite systems. We show that two strongly chaotic and weakly interacting subsystems display two distinct phases in the growth of OTOC.The first is dominated by intra-subsystem scrambling, when an exponential growth with a positive Lyapunov exponent is observed till the Ehrenfest time. This phase is essentially independent of the interaction, while the second phase is an interaction dominated exponential approach to saturation that is universal and described by a random matrix model. This simple random matrix model of weakly interacting strongly chaotic bipartite systems, previously employed for studying entanglement and spectral transitions, is approximately analytically solvable for its OTOC. The example of two coupled kicked rotors is used to demonstrate the two phases, and the extent to which the random matrix model is applicable. That the two phases correspond to delocalization in the subsystems followed by inter-subsystem mixing is seen via the participation ratio in phase-space. We also point out that the second, universal, phase alone exists when the observables are in a sense already scrambled. Thus, while the post-Ehrenfest time OTOC growth is in general not well-understood, the case of strongly chaotic and weakly coupled systems presents an, perhaps important, exception.