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We consider the simulation of non-abelian gauge potentials in ultracold atomsystems with atom-field interaction in the $\Lambda$ configuration where twointernal states of an atom are coupled to a third common one with a detuning.We find the simulated non-abelian gauge potentials can have the same structuresas those simulated in the tripod configuration if we parameterize Rabifrequencies properly, which means we can design spin-orbit coupling simulationschemes based on those proposed in the tripod configuration. We show thesimulated spin-orbit coupling in the $\Lambda$ configuration can only be of aform similar to $p_{x}\sigma_{y}$ even when the Rabi frequencies are not muchsmaller than the detuning.

Synthetic spin-orbit coupling in ultracold $\Lambda$-type atoms