Coarse graining a non-Markovian collisional model

The dynamics of systems subjected to noise is called Markovian in the absence of memory effects, i.e., when its immediate future depends only on its present. Time correlations in the noise source may generate non-Markovian effects that, sometimes, can be erased by appropriately coarse graining the time evolution of the system. In general, the coarse-graining time t(CG) is taken to be much larger than the correlation time tau but no direct relation between them is established. Here we analytically obtain a relation between tCG and tau for the dynamics of a qubit subjected to a time-correlated environment. Our results can be applied in principle to any distribution of the environmental correlations and can be tested through a collisional model where the qubit sequentially interacts with correlated qutrits.