Kibble-Zurek (KZ) scaling [@Zurek_2005; @Dziarmaga_2005] $$\varepsilon(\tau) \sim 1/ \tau^\gamma \quad \mbox{with} \quad \gamma=1/2 ,$$ which can be obtained by the knowledge of the Ising critical exponents associated to the phase transition at $\Gamma_{\rm c}=1$ across which the system is driven. A finite dissipation $\kappa>0$ induces a competition between the KZ mechanism of defect generation due to the crossing of a gapless point (which is progressively reduced, with increasing annealing time $\tau$), and the production of defects generated by the incoherent driving itself. Such competition clearly emerges in Fig. \[fig:ExcT\_FreeF-Pump\] as a non-monotonic behaviour, which generates an optimal working point for the annealing procedure in the presence of dissipation.

Let us now have a closer look at the non-monotonicity, and focus on the optimal (minimal) value $\varepsilon_{\rm opt}$ reached by the excess energy, and on the corresponding annealing time $\tau_{\rm opt}$. Figure \[fig:OptT\_FreeF-Pump\] displays how such quantities depend on $\kappa$. Our numerical data nicely agree with a power-law behaviour over more than two decades of $\kappa$ values, such that $\varepsilon_{\rm opt} \sim \kappa^{1/3}$ and $\tau_{\rm opt} \sim \kappa^{-2/3}$. Below we show that this behaviour can be easily predicted by assuming that the KZ production of defects is totally independent of that 