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

The Bloch-Redfield equation is a common tool for studying the evolution of qubit systems weakly coupled to the 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 a 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 Di Vincenzo D. P. and Loss D., Phys. Rev. B, 71 (2005) 035318. At the same time, the associated dephasing is not affected and is described by the Born-Markov approximation.