  We study the decoherence dynamics of a qubit coupled to a quantum two-level
system (TLS) in addition to its weak coupling to a background environment. We
analyze the different regimes of behaviour that arise as the values of the
different parameters are varied. We classify those regimes as two weak-coupling
regimes, which differ by the relation between the qubit and TLS decoherence
times, and a strong-coupling one. We also find analytic expressions describing
the decoherence rates in the weak-coupling regimes, and we verify numerically
that those expressions have a rather wide range of validity. Along with
obtaining the above-mentioned results, we address the questions of qubit-TLS
entanglement and the additivity of multiple TLS contributions. We also discuss
the transition from weak to strong coupling as the parameters are varied, and
we numerically determine the location of the boundary between the two regimes.
