Thermodynamics of adiabatic quantum pumping in quantum dots

The authors would like to thank the Reviewer for their appreciation of our article.

We would like to thank the Reviewer for their useful comments. In the revised version, we have sought to address all the points raised in the Report. In particular:
**Comment 1 ** : "The leads L and R are sometimes labelled by the index i, sometimes by the index alpha. Could the authors please choose one and use it everywhere? (I would prefer alpha.)"
**Reply** : The labelling of the leads has been changed to $\alpha=L,R$ along all the article.
**Comment 2** :" In Eq. (10) f is not defined at all, and S is defined only by words which is a bit ambiguous. This happened because the derivation of Eq.(10) was moved to Appendix A (which was probably a good idea in general). But there is no reference to Appendix A around Eq. (10) in the revised version. Citing Appendix A here would be helpful."
**Reply**: The definition of the Fermi function has been included after Eq. (10) of the current version and a reference to Appendix A has been added.
**Comment 3**:"After Eq.(24), it is beta that is inverse temperature, rather than 1/beta, isn't it?"
**Reply**: The mistake has been corrected in the new version.
**Comment 4**: " Just before Eq.(26), Omega is not defined. My brave guess would be that $\Omega = 2 \pi / T_0$ and not the grand-canonical potential introduced a few paragraphs later. Please define."
**Reply**: The definition has been included in the new version.
**Comment 5**:"Eq.(29) is a complete disaster: the integration is over t, in the exponential there is Omega and no t, and A has no argument at all."
**Reply**: In Eq. (29) there is a missing t in the exponential. Of course, A depends on t and this has been pointed out in the new version.
**Comment 6**:" After Eq.(63), delta appears without definition. In the original version, it was mentioned that introducing delta was necessary in order to avoid divergences due to the breakdown of adiabatic approximation. That was a very useful remark, but for some reason it disappeared from the revised version. May the authors please put it back?"
**Reply**: A definition of $\delta\Gamma$ has been included in Sec. 5.1, coinciding with the Reviewer's guess. The remark about the breakdown of the adiabatic approximation has been put back where it was and the normalized regularization parameter has been renamed to $\xi=\delta/\Gamma$.
**Comment 7**:"In Sec. 5.1 deltaGamma is not defined, neither in Fig.1, nor in the text. We can only guess that $\Gamma_L = (\Gamma + \delta\Gamma) / 2$ and $\Gamma_R = (\Gamma - \delta\Gamma) / 2$ during the peristaltic cycle. And in Fig.1 the barely readable and not very useful numbers go in the order 1-2-4-3 - what is this supposed to mean? The initial and final positions of the dot level are sometimes called $\epsilon_1$ and $\epsilon_2$, and sometimes $+\epsilon_0$ and $-\epsilon_0$. In addition, epsilon also denotes the ratio delta/Gamma. I beg the authors to clean up all this mess."
**Reply**: The numbers in Fig. 1 have been put in clockwise ascending order and the endpoints in the $\epsilon_d$ axis have been set to $-\epsilon_0$ and $\epsilon_0$.
**Comment 8**: " Figures 2 and 7 would better be plotted in log scale, otherwise most of the plot (with $x_0 > 20$) is not very informative."
**Reply**: The suggestion of the Reviewer has been accepted and the log scale has been introduced in Figg. 2 and 7.
**Comment 9**:" I don't understand how is Eq.(70) "signalling the presence of a quantization rule for the work rate". I don't see any quantization here. This expression is valid only for constant level velocity; if the level position changed by some other law, the work would probably be proportional to some typical value of the level velocity, but there is no natural quantity which would play the role of the work quantum. The work per cycle is just finite, that's all I can see here."
**Reply**: We agree that this concept can be confusing in this framework. We signal that the results for the work indicate that this quantity is finite and proportional to the charge pumped if we investigate the case of constant velocity.

We thank the Reviewer for their positive feedback on the article.