# Classical and quantum chaos in a three-mode bosonic system

## TL;DR

This paper explores the connection between classical chaos and quantum signatures in a three-mode bosonic system, demonstrating how classical dynamics influence quantum level statistics and out-of-time-ordered correlators, with potential experimental realizations.

## Contribution

It introduces a tunable three-mode bosonic model linking classical chaos with quantum signatures, including level statistics and OTOCs, and proposes an experimental setup with Bose-Einstein condensates.

## Key findings

- Classical chaos affects quantum level statistics, changing from Poissonian to Wigner-Dyson.
- OTOCs differ significantly between regular and chaotic classical regions.
- Semi-classical simulations accurately capture short-time quantum dynamics.

## Abstract

We study the dynamics of a three-mode bosonic system with mode-changing interactions. For large mode occupations the short-time dynamics is well described by classical mean-field equations allowing us to study chaotic dynamics in the classical system and its signatures in the corresponding quantum dynamics. By introducing a symmetry-breaking term we tune the classical dynamics from integrable to strongly chaotic which we demonstrate by calculating Poincar\'e sections and Lyapunov exponents. The corresponding quantum system features level statistics that change from Poissonian in the integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of out-of-time-ordered correlators (OTOCs), specifically the squared commutator, for initial states located in regular and chaotic regions of the classical mixed phase space and find marked differences between the two cases. The short-time behavior is well captured by semi-classical truncated Wigner simulations directly relating these features to properties of the underlying classical mean field dynamics. We discuss a possible experimental realization of this model system in a Bose-Einstein condensate of rubidium atoms which allows reversing the sign of the Hamiltonian required for measuring OTOCs experimentally.