FIG. 5: (Color online) | r ( t ) | for α = 1 / 2 and five different values of λ for two selections of the coupling system-environment parameter ω , ω = 0 on the left and ω = 1 $_{/}$√$_{2 on the right. In all}$ cases N = 1000. Solid (black) lines correspond to the exact solution, and dashed (red) lines to the TDA calculation. of the first order ESQPT on the decoherence factor will be discussed in the next subsection. In Fig. 5 we show the results for the decoherence factor for N = 1000 bosons and α = 1 / 2, for two ω values, ω = 0 (left panels) and ω = 1 $_{/}$√$_{2 (rigth panels). The objective}$ of this figure is to show the effect of the continuous ESQPT on the decoherence factor. Solving Eq. (17) for the value of α = 1 / 2, the continuous ESQPT ( E (2) c = 0) takes place at λ (2) c = 0 . 75 for ω = 0 (left panel) and at λ (2) c = 1 . 17 for ω = 1 $_{/}$√$_{2 (right panel). Several}$ features deserve to be discussed. First of all, we can see that the TDA calculation works pretty well for small and large values of λ . In particular, the shape of the envelope, which remains unaffected by the increase of λ for λ ≫ λ (2) c , is very well described by the TDA calculation (see panels for λ = 4 and λ = 8 in Fig. 5). Since this approximation mainly 17 0 0.5 1 0 0.5 1 0 0.5 1 | r(t) | 0 0.5 1 0 5 10 15 20 t 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 | r(t) | 0 0.5 1 0 5 10 15 20 t 0 0.5 1 (a) ω = 0 (b) ω = 1 $_{/}$√$_{2}$