  Bloch-Redfield equation is a common tool for studying evolution of qubit
systems weakly coupled to environment. We investigate the accuracy of the Born
approximation underlying this equation. We find that the high order terms in
the perturbative expansion contain accumulating divergences that make
straightforward Born approximation inappropriate. We develop diagrammatic
technique to formulate, and solve the improved self-consistent Born
approximation. This more accurate treatment reveals an exponential time
dependent prefactor in the non-Markovian contribution dominating the qubit
long-time relaxation found in Phys. Rev. B 71, 035318 (2005). At the same time,
the associated dephasing is not affected and is described by the Born-Markov
approximation.
