The deep Hilbert space of all-to-all interacting SU(3) atoms: from quantum to classical

TL;DR

This paper explores the complex structure of the Hilbert space in a multilevel all-to-all interacting atomic system, revealing how symmetry sectors influence the emergence of chaos and regularity in quantum dynamics.

Contribution

It provides a detailed analysis of Hilbert space fragmentation and symmetry sectors in a 3-level Tavis-Cummings model, linking quantum chaos with classical behavior.

Findings

Hilbert space fragments into multiple dynamical sectors.

Some sectors exhibit chaotic dynamics, others regular.

Classical limit involves many symmetry sectors beyond the symmetric subspace.

Abstract

We study the emergence of chaos in multilevel atoms with all-to-all interactions, inspired by cavity QED. Focusing on a 3-level Tavis-Cummings model in a far detuned limit, we detail its deep Hilbert space structure -- i.e. we enumerate all distinct dynamical sectors, beyond the totally symmetric subspace -- by using the Schur-Weyl duality, which is applicable thanks to the permutation symmetry in the all-to-all Hamiltonian. Strong Hilbert space fragmentation ensues from the non-abelian nature of the symmetry, with some sectors displaying regular dynamics and others being chaotic. We uncover that many permutation symmetry sectors contribute to the dynamics in the classical limit, in addition to the commonly studied totally symmetric subspace. To elucidate the dynamical responses in each of the symmetry sectors, we propose a semiclassical description in terms of spin coherent states, which is also able to explain the origin of chaotic or regular dynamics with a simple geometrical argument. Our work contributes to the study of the quantum-classical correspondence in chaotic systems, and uncovers a rich structure in multilevel all-to-all interacting models.