generated by the dissipation. The above mentioned competition is thus explained in terms of an incoherent summation of the two (independent) contributions.

![Optimal excess energy $\varepsilon_{\rm opt}$ (orange squares) and corresponding annealing time $\tau_{\rm opt}$ (violet diamonds), as a function of the dissipation strength $\kappa$. Numerical data (symbols) are obtained using the same parameters as in Fig. \[fig:ExcT\_FreeF-Pump\], and nicely follow a power-law behaviour (dashed lines) with slope $1/3$ and $-2/3$, respectively.[]{data-label="fig:OptT_FreeF-Pump"}](OptT_FreeF-Pump.pdf){width="\columnwidth"}

Scaling of the optimal point {#sec:ScalOpt}
----------------------------

We start from the observation that, after the annealing procedure, the final state of the closed system can be easily written as a Bogoliubov state where excitations are provided by pairs of quasiparticles with equal and opposite momenta [@Dziarmaga_2005]: $$|\psi(t_{\rm fin})\rangle = \prod_{k>0} (\alpha_{k} + \beta_{k} \gamma^{\dagger}_{k}\gamma^{\dagger}_{-k}) |0\rangle.$$ Here $|0\rangle$ indicates the Bogoliubov vacuum corresponding to the final ground state of $H(0)$, $\alpha_k$ and $\beta_k$ are complex amplitudes, while the momentum $k$ can take $L/2$ positive values from 0 to $\pi$ (see App. \[app:TraslInv\_U\] for details).

In the dissipative case, we will not only have those doubly excited states $|1_k, 1_{-k}\rangle = \gamma_k^\dagger \gamma_{-k}^\dagger |0\rangle$, but also singly excited states such as $|1_{k}\rangle = \gamma_k^\dagger |0\rangle$ and $|1_{-k}\rangle = \gamma_{-k}^\dagger |0\rangle$, 