# Integrable and chaotic dynamics of spins coupled to an optical cavity

## TL;DR

This paper explores the transition between integrable and chaotic dynamics in a class of all-to-all coupled spin models realized with atoms in an optical cavity, revealing new integrable structures and their disappearance in the classical limit.

## Contribution

It introduces a novel integrable structure with higher-body conserved charges in spin models coupled to optical cavities, and analyzes the transition to chaos in the large-spin limit.

## Key findings

- Identifies parameter regimes for integrable and chaotic dynamics.
- Discovers a new integrable structure with higher-body conserved charges.
- Shows the disappearance of integrability in the large-spin classical limit.

## Abstract

We show that a class of random all-to-all spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram due to the pairwise separable nature of the couplings. By controlling the experimental parameters, one can tune between integrable and chaotic dynamics on the one hand, and between classical and quantum regimes on the other hand. For two special values of a spin-anisotropy parameter, the model exhibits rational-Gaudin type integrability and it is characterized by an extensive set of spin-bilinear integrals of motion, independent of the spin size. More generically, we find a novel integrable structure with conserved charges that are not purely bilinear. Instead, they develop `dressing tails' of higher-body terms, reminiscent of the dressed local integrals of motion found in Many-Body Localized phases. Surprisingly, this new type of integrable dynamics found in finite-size spin-1/2 systems disappears in the large-$S$ limit, giving way to classical chaos. We identify parameter regimes for characterizing these different dynamical behaviors in realistic experiments, in light of the limitations set by cavity dissipation.