〈 i, e $^{2}$| H$_{i}$ | i, e $^{2}$〉 = 1 (1 + β 2 $_{i}$) 2 ( (2 c + 4 d ) β 2 i + 4 eβ 3 i + 2 fβ 4 i ) + 2 1 + β 2 i ( a + c + f − 2 bβ$_{i}$ ) + 2 c (1 − β 2 $_{i}$) 2 (1 + β 2 $_{i}$) $_{2}$( N − 2) + 8( N − 2) (1 + β 2 $_{i}$) $_{2}$( fβ 2 i − 2 dβ 2 i + e ( β$_{i}$ − β 3 $_{i}$)) + ( a + c + f ) β 2 i 1 + β 2 i ( N − 2) + 2 b β$_{i}$ 1 + β 2 i ( N − 2) + ( N − 2)( N − 3) (1 + β 2 $_{i}$) 2 ( c + 2 d ) β 2 i + 2 eβ 3 i + fβ 4 i ) . (47) 16 All the information about the decoherence process induced by the environment (5) in the central qubit is encoded in the decoherence factor (4). As mentioned above, for the Hamiltonian we are using there is always a continuous ESQPT independently of the value of ω . In addition, for ω = 0 there also appears a first order ESQPT. In this subsection we will analyze the effect of the continuous ESQPT on the decoherence factor, while the effect In this section we present the main features of evolution of the system described by (1) under the influence of the environment given by (5). A brief report of the relationship between the decoherence in the qubit and the excited state quantum phase transitions in the environment was given in [9]. Here, we extend the analysis, comparing the numerical results with the Tamm-Dankoff approximation, and also facing the case of ω = 0, that was not considered in [9]. As two paradigmatic cases, we deal with the cases α = 1 / 2, and ω = 0 and ω = 1 $_{/}$√$_{2. Different choices for the defining parameters of the model give rise to the}$ same qualitative results. Inserting (37), (38) and (47) into (46) we arrive to the final form of the decoherence factor r ( t ) within the extended TDA approximation. In this case, a semi-quantitative analysis as the previous one cannot be easily done. A comparison with exact numerical calculations is performed in next section. IV. RESULTS A. Decoherence factor for the Continuous ESQPT