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However, the numerical estimates seem to increase for larger values of ω ; in particular, for the case ω = 1, the result for exponent γ is significatively larger than γ = 1 / 4. 

## B. Decoherence factor for the first order ESQPT 

For the case ω ̸ = 0, the Hamiltonian considered produce, in addition to the continuous ESQPT studied in the preceding subsection, a first order ESQPT at energy E (1) c . This critical energy can be estimated calculating the local minima in the energy surface H ( φ, ξ ), as it is shown in Fig. 4. Inserting this value in Eq. (17) a critical coupling λ (1) c is obtained. For the case α = 1 / 2 and ω = 1 $_{/}$√$_{2 the first order EQSPT is obtained at}$ λ (1) c ≈ 1 . 05. 

In Fig. 8 we show the exact result for the decoherence factor | r ( t ) | for α = 1 / 2, ω = 1 $_{/}$√$_{2,}$ and three different values of λ around λ = λ (1) c ≈ 1 . 05. The most significative result is that 

