0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1  0  5  10  15  20  25 | r(t) | t =1 =2 =3 =4 =6 FIG. 2: | r ( t ) | for α = 0 and five different values of λ . In all cases N = 10000. the envelope of | r ( t ) | and the period of its oscillating part. Making use of Eq. (7) for α = 0 we obtain precisely λ$_{c}$ = 2, the value at which the coherence of the system is completely lost. Therefore, the existence of an ESQPT in the environment has a strong influence on the decoherence that it induces in the central system. We can summarize this result with the following conjecture: If the system-environment coupling drives the environment to the critical energy E$_{c}$ of a continuous ESQPT, the decoherence induced in the coupled qubit is maximal . Let us check this conjecture for different values of α < α$_{c}$ = 4 / 5 . In Fig. 3 we show how the decoherence process changes around the critical value λ$_{c}$ for different values of N and α . As a representative quantity, we plot r$_{max}$ ( α ) which is the value of | r ( t ) | at the second maximum (the first maximum is | r (0) | = 1 in all cases). We can extract two main conclusions from Fig. 3. First, r$_{max}$ ( λ ) is minimum at λ ≈ λ$_{c}$ , as given in Eq. (7) — the four cases plotted in the figure correspond to λ$_{c}$ = 0 . 25 , 0 . 5 , 1 . 0 , and 2 . 0. Second, the behavior of this quantity is smooth and independent of the environment size N , except in 6