Effect of a gap on the decoherence of a qubit

We revisit the problem of the decoherence and relaxation of a central spin coupled to a bath of conduction electrons. We consider both metallic and semiconducting baths to study the effect of a gap in the bath density of states (DOS) on the time evolution of the density matrix of the central spin. We use two weak coupling approximation schemes to study the decoherence. At low temperatures, though the temperature dependence of the decoherence rate in the case of a metallic bath is the same irrespective of the details of the bath, the same is not true for the semiconducting bath. We also calculate the relaxation and decoherence rates as a function of external magnetic fields applied both on the central spin and the bath. We find that in the presence of the gap, there exists a certain regime of fields, for which surprisingly, the metallic bath has lower rates of relaxation and decoherence than the semiconducting bath.