which represent further sources of defects. Indeed, by using the Bogoliubov transformation, we can rotate the Master equation in this frame. This allows us to write down the dynamical equation for $\langle1_{k}|\rho_{k}|1_{k}\rangle$. We find $$\tfrac{\mathrm{d}}{\mathrm{d}t} \langle1_{k}|\rho_{k}|1_{k}\rangle
  = \kappa \, \langle0|\rho_{k}|0\rangle \, f(\Gamma,k),$$ with $$\label{eq:scaling_function}
  [f(\Gamma,k)]^{-1} = 1 \! + \!
  \bigg( \dfrac{\Gamma - \cos k + \sqrt{1 \! + \! \Gamma^2 \! - \! 2\Gamma \cos k}}{\sin^2 k} \bigg)^2$$ for the specific choice of $L_{n} = c^{\dagger}_{n}$. In the adiabatic regime where the KZ scaling argument holds and for small dissipation, the density of defects is much smaller than $1$, so that $\langle0|\rho_{k}|0\rangle$ can be approximated by its initial value $1$. Note that, since the density of defects $\mathcal{N}$ is written as $\mathcal{N} = \sum_{k} \gamma^{\dagger}_{k}\gamma_{k}$ in the Bogoliubov basis, excitations of the form $|1_{k}\rangle$ only contribute to the positive values of $k$, while excitations due to coherent dynamics $|1_{k} 1_{-k}\rangle$ contribute to both, $k$ and $-k$. Following this, the incoherent part of density of defects can be estimated according to $$\mathcal{N}_{\text{inc}} = \dfrac{\kappa}{L} \sum_{k>0}\int_{-\infty}^{0} \mathrm{d}t\ f[\Gamma(t),k]
  = \dfrac{1}{2} \kappa \, \tau,$$ where the last equality has been obtained after a change of variables from $t$ to $\Gamma(t) = -t/\tau$, 