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Bloch-Redfield equation is a common tool for studying evolution of qubitsystems weakly coupled to environment. We investigate the accuracy of the Bornapproximation underlying this equation. We find that the high order terms inthe perturbative expansion contain accumulating divergences that makestraightforward Born approximation inappropriate. We develop diagrammatictechnique to formulate, and solve the improved self-consistent Bornapproximation. This more accurate treatment reveals an exponential timedependent prefactor in the non-Markovian contribution dominating the qubitlong-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-Markovapproximation.

Non-Makovian decoherence of a two-level system weakly coupled to a bosonic bath