## In Fig we present the absolute value of

In order to have a rough analytical estimation of decoherence times, we consider that RG7112 the qubit is solely coupled in the longitudinal direction, and that there are no anomalous and dissipation terms in the master equation. This means that for the moment we neglect the effect of the tunnelling term (proportional to both transverse directions) in the main system Hamiltonian. Thus, we may follow the result given in Refs. [12] and [24] for the purely dephasing model. There, decoherence time in the high temperature approximation can be estimated as tD∼2/(kBTγ0)tD∼2/(kBTγ0)[17], which does not depend on the frequency cutoff Λ. Considering parameters used in Fig. 3, one can estimate decoherence time to be tD∼1ΔtD∼1Δ, in good agreement with the corresponding plots in Fig. 3. For the ohmic case at zero temperature, the decoherence scales as tD≥2/(γ0Λπ)tD≥2/(γ0Λπ) for times Λt≥1Λt≥1. In this case, decoherence is delayed as γ0γ0 decreases. This is a very long bound for decoherence time, especially when the longitudinal coupling constant is very small. Indeed, this reflects the fact that the contribution of all the coefficients in the master equation (anomalous diffusion and dissipation coefficients) are important in the limit of zero temperature, as can be seen in Fig. 3.