  The left-handedness was demonstrated by the simulation with a three-level
quantum system in an E3+r -dopped ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAFN) optical
fiber. And the left-handedness can be regulated by the incoherent pumping
field. Our scheme may provide a solid candidate other than the coherent atomic
vapour for left-handedness, and may extend the application of the
rare-earth-ion-doped optical fiber in metamaterials and of the incoherent
pumping light field in quantum optics.
