  We propose for the first time an experimentally feasible scheme to disclose
the noncommutative effects induced by a light-induced non-Abelian gauge
structure with trapped ions. Under an appropriate configuration, a true
non-Abelian gauge potential naturally arises in connection with the geometric
phase associated with two degenerated dark states in a four-state atomic system
interacting with three pulsed laser fields. We show that the population in
atomic state at the end of a composed path formed by two closed loops $C_1$ and
$C_2$ in the parameter space can be significantly different from the composed
counter-ordered path. This population difference is directly induced by the
noncommutative feature of non-Abelian geometric phases and can be detected
unambiguously with current technology.
