Anomalous external-magnetic-field dependence of dephasing in a spin bath

We theoretically investigate the dephasing of a central spin-1 model. An interesting mechanism of spin decoherence is found with this model, namely {\em hyperfine mediated spectral diffusion}. This mechanism contains both the features of dipolar interactions induced spectral diffusion and hyperfine mediated interactions. We also find an anomalous magnetic field dependence of decoherence, which is caused by the competition between crystal field splitting and Zeeman splitting of the central spin. As the external magnetic field increases, in the V type level structure regime, the decoherence rate becomes much stronger; while in the cascade type level structure regime, just like localized electron spin in quantum dots, the decoherence rate becomes much weaker.