Engineering infinite-range SU($n$) interactions with spin-orbit-coupled fermions in an optical lattice

We study multilevel fermions in an optical lattice described by the Hubbard model with on site SU($n$)-symmetric interactions. We show that in an appropriate parameter regime this system can be mapped onto a spin model with all-to-all SU($n$)-symmetric couplings. Raman pulses that address internal spin states modify the atomic dispersion relation and induce spin-orbit coupling, which can act as a synthetic inhomogeneous magnetic field that competes with the SU($n$) exchange interactions. We investigate the mean-field dynamical phase diagram of the resulting model as a function of $n$ and different initial configurations that are accessible with Raman pulses. Consistent with previous studies for $n=2$, we find that for some initial states the spin model exhibits two distinct dynamical phases that obey simple scaling relations with $n$. Moreover, for $n>2$ we find that dynamical behavior can be highly sensitive to initial intra-spin coherences. Our predictions are readily testable in current experiments with ultracold alkaline-earth(-like) atoms.