and observing that the summation over $k>0$ after the integral over $\Gamma$ yields a constant factor $L/2$.

Assuming now that the mechanisms of defect generation due to KZ and due to dissipation are unrelated [@Dutta_2016], we have: $$\mathcal{N} \sim \mathcal{N}_{\text{KZ}} + \mathcal{N}_\text{inc}
  = \dfrac{1}{2\pi \sqrt{2}} \, \tau^{-1/2} + \dfrac{1}{2} \kappa \, \tau.$$ From this expression for the total density of defects, the optimal annealing time minimizing the defects production can be thus estimated by the condition $\partial_\tau \mathcal{N}(\tau) |_{\tau_{\rm opt}} = 0$. A direct calculation gives $$\tau_{\rm opt} = \left( \dfrac{1}{2\pi \sqrt{2}}\right)^{2/3} \kappa^{-2/3},
  \label{eq:TauOpt}$$ with a corresponding density of defects $$\mathcal{N}_{\rm opt} = \mathcal{N}(\tau_{\rm opt}) = \dfrac{3}{2} \left( \dfrac{1}{2\pi \sqrt{2}}\right)^{2/3} \kappa^{1/3} .
  \label{eq:NOpt}$$

The predictions given by these equations are in nice agreement with our numerical data shown in Fig. \[fig:OptT\_FreeF-Pump\], keeping in mind that $\varepsilon = 2 \mathcal{N}$.

To further highlight the role of the dissipation during the annealing procedure, we have also analysed the excess energy at the end of the annealing, after subtracting the corresponding excess energy in the absence of dissipation: $$\Delta(\tau) = \varepsilon(\kappa,\tau) - \varepsilon(\kappa=0,\tau).
  \label{eq:Delta_def}$$ Note that, in order to properly define the quantity $\Delta$, we have manifested in Eq.  the $\kappa$-dependence of $\varepsilon$. 