We study multilevel fermions in an optical lattice described by the Hubbard model with on-site $\mathrm{SU}\left(n\right)$-symmetric interactions. We show that in an appropriate parameter regime this system can be mapped onto a spin model with all-to-all $\mathrm{SU}\left(n\right)$-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 $\mathrm{SU}\left(n\right)$ 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 intraspin coherences. Our predictions are readily testable in current experiments with ultracold alkaline-earth-metal(-like) atoms.

• Received 27 September 2021
• Revised 26 January 2022
• Accepted 27 January 2022

DOI:https://doi.org/10.1103/PhysRevA.105.023326