  We present a series of cosmological magnetohydrodynamic (MHD) simulations
that simultaneously follow the formation of a galaxy cluster and evolution of
magnetic fields ejected by an Active Galactic Nucleus (AGN). Specifically, we
investigate the influence of both the epoch of AGN (z $\sim$ 3-0.5) and the AGN
energy ($\sim$ 3 $\times$ 10$^{57}$ - 2 $\times$ 10$^{60}$ ergs)on the final
magnetic field distribution in a relatively massive cluster (M$_{vir}$
$\sim$10$^{15}$ M$_\odot$). We find that as long as the AGN magnetic fields are
ejected before the major mergers in the cluster formation history, magnetic
fields can be transported throughout the cluster and can be further amplified
by the intra-cluster medium (ICM) turbulence cause by hierarchical mergers
during the cluster formation process. The total magnetic energy in the cluster
can reach $\sim$ $10^{61}$ ergs, with micro Gauss fields distributed over
$\sim$ Mpc scale. The amplification of the total magnetic energy by the ICM
turbulence can be significant, up to $\sim$1000 times in some cases. Therefore
even weak magnetic fields from AGNs can be used to magnetize the cluster to the
observed level. The final magnetic energy in the ICM is determined by the ICM
turbulent energy, with a weak dependence on the AGN injection energy. We
discuss the properties of magnetic fields throughout the cluster and the
synthetic Faraday rotation measure maps they produce. We also show that high
spatial resolution over most of the magnetic regions of the cluster is very
important to capture the small scale dynamo process and maintain the magnetic
field structure in our simulations.
