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

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.

Figures (11)

• Figure 1: (a) Sketch of the experimental setup. Bosons are placed on $L$ sites in an optical cavity. Each site has $n_j$ particles, which can be in $N=3$ different modes, visualized as a Hanoi tower. The allowed transitions involve two particles at a time, and respect the total magnetization, defined as the number of particles on all columns 1 minus the ones on columns 3, Eq. \ref{['eq:M']}. The transitions can be either intra-site (b) or inter-site (c).
• Figure 2: Fragmentation of the Hamiltonian into block-diagonal sectors, according to the successive application of the symmetries: on-site particle number conservation (gray), permutations (blue) and total magnetization (red). If the number of particles $n_j$ on each site is the same (top left gray box), then the Hilbert space is decomposed according to the irreps of the symmetric group $\mathrm{S}_L$, Eq. \ref{['eq:Schur-Weyl']}, that are labeled by Young diagrams $\lambda$. Finally, the magnetization constraint splits every intermediate (blue) sector into finer (red) sectors according to Eq. \ref{['eq:decomposition_Hilbert_perm_mag']}. If instead the number of particles is different on each site (other two gray boxes), only the magnetization constraint needs to be applied, Eq. \ref{['eq:decomposition_Hilbert_noperm']}. The permutation symmetry, not respected by the states, remains manifest in the isomorphicity between sectors that have the same occupation numbers, but arranged in a different order.
• Figure 3: An example fully symmetric SU$(3)_\mathrm{atoms}$ representation for $L=4$ sites with $n=1$ particle each. The partition into magnetization sectors corresponds to fixing the isospin quantum number $V_3$.
• Figure 4: Comparison of Wigner-Dyson (dashed black lines) vs Poissonian (dotted black lines) level statistics in the shallow and deep portions of the Hilbert space. (a) Poissonian statistics, associated with non-chaotic behavior, is found when considering systems with $n_j \equiv n >1$ particles on each site, in the permutation fully-symmetric sector (one-row Young diagram). (b) The same happens for the permutation fully-antisymmetric sectors, irrespective of the choice of sites, occupation number, and magnetization. (c) Fixing one particle per site $n_j \equiv 1$ and the magnetization $M=0$, chaos appears only for non-planar representations of SU(3), i.e. $D(p,q)$ with both $p>0$ and $q>0$. This result can be understood from the quantum-classical correspondence, as detailed in Sec. \ref{['sec:classical']}. The spectral statistics for $q=0$ is not Poissonian, but displays the rigidity typical of harmonic oscillators (rather than chaotic behaviour). (d) We generically find the absence of chaotic behaviour also when the permutation symmetry is broken by a choice of site-dependent number of bosons $n_j$. In all panels, the parameters of the Hamiltonian, Eq. \ref{['eq:H_b']}, read $g_1=1.7$, $g_2=1$ and $h=1$.
• Figure 5: Example of a non-degenerate SU(3) representation D(3,0), with a magnetization sector highlighted. Each "link" between two states corresponds to the application of an annihilation/creation operator, which for SU(3) is a bosonic bilinear. Any two neighboring states in a fixed magnetization sector are connected by two such links. This sketch showcases how states within each magnetization sector have a fixed value of the $V_3$ isospin and different values of the $Y'$ hypercharge.