Engineering Quadrupolar Large-Spin Models with Multilevel ⁸⁷Sr in a Cavity

Cavity QED platforms offer a powerful playground for engineering collective light–matter interactions, with applications ranging from metrologically useful entanglement to emergent dynamical phases reminiscent of those predicted in quenched superconductors. Yet most implementations effectively reduce atoms to two-level systems, severely restricting the family of many-body Hamiltonians that can be explored. In this poster, we propose leveraging the full F=9/2 hyperfine manifold of ⁸⁷Sr to realize large-spin physics with quadrupole–quadrupole interactions, interactions that simply do not exist for spin-1/2. By coupling these internal states to an optical cavity and external laser fields, we engineer effective spin-9/2 Hamiltonians with a wide variety of interactions. We show that tuning the initial state and cavity parameters unlocks qualitatively distinct dynamical regimes: from familiar one- and two-axis twisting when the dynamics is effectively restricted to two levels, to genuinely multilevel processes including pair creation and correlated hopping in synthetic dimensions. This framework expands the scope of cavity QED toward richer large-spin quantum matter and suggests new routes to generating spin-squeezed and other highly nonclassical states.