Thermodynamics of adiabatic quantum pumping in quantum dots

The paper addresses an interesting question:
how thermodynamic quantities (work, heat) accumulated in cyclic thermodynamic processes correlate with a pumping effect through that cycle?

From the methodology side, the question is: How is it possible to address this question near the quasistatic limit?

The paper provides a synergetic link to the beautiful Green function methods of Ref.10 to address these questions.

While the calculations are new, the conceptual novelty of this paper is not emphasized. Rather than discussing the physics and its significance, the paper looks like a bunch of technical calculations.

As explained above (Strengths and Weaknesses), the paper touches upon a very interesting topic, but I believe it is not appealing at this stage to physics oriented readers. Yet I believe this paper can be very useful for the readers and should be reconsidered if the presentation will be considerably improved by the authors. Also, there are many typos both in text / equations that should be fully fixed (some examples: therecent --> the recent, arguments t,t' should be reversed between Eqs.8,9; using g for tunneling and for Green functions is confusing, Eq.13 subscripts do not match, Eq.16 r-->R ...).

Section 4 starts with providing expressions for work in the adiabatic limit. In this limit, there no dissipated heat. Therefore the presence of the quasistatic heat exchange, Eq.48 is confusion, since this heat is not dissipated in the reservoir, instead it is the entropy change of the system itself. Indeed, in this section, the (reversible) work equals the free energy.

When going more step beyond, when a finite dissipated heat is expected, the authors refer to Ref.10 but do not explain the connection between thermodynamics and the Green function. The authors write: "We first connect the thermodynamic quantities to the non-equilibrium Green’s functions (Appendix B)." But Appendix B only gives something else, the expectation value of the tunneling term. This is the essential part of the calculation so it should be well described.

Relevant references that were not cited are:
Phys. Rev. Lett. 96, 166802 (2006) in the context of interactions, and specifically Phys. Rev. Lett. 124, 150603 (2020) in the context of thermodynamics.

It would be interesting if the authors can comment on the possibility to compute stochastic thermodynamic quantities (like work distribution function) near the quasistatic limit using their methods.

1. The authors have developed a consistent thermodynamic description of a model: a single-level quantum dot connected to two fermionic baths.
2. By using the non-equilibrium Green function, they have computed the thermodynamic quantities and transport properties.

1. The present version seems to be a collection of different quantities computed for a specific model. A physical insight/motivation is lacking.
2. What is the big banner of this work? Are the results valid for the specific model or some physical intuition can be extracted for other models which can make the present work versatile?
3. What about experimental insights of their work?
4. In Eq-32, they have divided the charge fluctuation into two parts. What are the physical significances of those parts?
5. What are the significant advancements of this work compared to other earlier works (Refs. 5-12)? A clear distinction would be very helpful.
6. What is the physical mechanism for generating the noise?
7. Which one is the shot noise contribution and which one is the thermal noise contribution and how to see those?
8. The figure captions contain very small amount of information. It is hard to follow the qualitative picture (besides the quantitative values) from those.

Owing to the above-mentioned points, the present version is not suitable for publication in SciPost Physics. However, a detailed discussion of all the points, listed in the “Weaknesses” may make the manuscript suitable for SciPost physics.

A detailed discussion of all the points, listed in the “Weaknesses”.

1. The authors present a well-controlled calculation of thermodynamic quantities for several pumping cycles, using the systematic gradient expansion of non-equilibrium Green's functions.

1. The work is very incremental: all tools and concepts were developed earlier (e.g., in Refs. [9,10]), and the present work represents their straightforward application to a specific case of quantum pumps with non-interacting electrons.

2. Physical results are not clearly formulated. The authors have calculated several quantities, some of which are obviously zero in the steady state (those characterising the change in the central island's state over one pumping cycle), and some are not (like work and heat during the cycle). What are we supposed to learn from this?

3. In general, the authors present many calculations but little discussion of their physical meaning.

Based on the above arguments, I would not recommend the present version of the manuscript for publication in any journal. However, after some revision, the manuscript may become suitable for publication in SciPost Physics Core. Below are specific issues that should be addressed.

1. In the introduction, the authors mention "a self-consistent thermodynamic analysis of the operation of a quantum pump". I don't see what is self-consistent here. The authors perform an explicit calculation via gradient expansion.

2. In Eqs. (6) and (7) there is some confusion between operators and their averages. What state is the average taken over? Why is the charge in Eq.(6) is defined over one cycle, while its noise in Eq.(7) instead involves some unspecified observation time T_m?

3. The authors define different contributions to the noise in Eqs. (32)-(34), and then they speak about thermal and shot noise, but they never explain which is which and why.

4. In all cases, the geometric (zero-order) part of the work is found to be zero. Isn't it just the property of any isothermal cycle?

5. For the peristaltic cycle, the authors' finding that the charge noise vanishes in the charge quantisation limit is rather obvious, since in the limit of infinite energy excursion, it is certainly filled from one reservoir and certainly emptied into the other one, so there is no room for any fluctuations. However, for the triangular cycle the situation is much less clear to me. To start with, the authors do not specify the temperature at which their calculation is done, so I assume it is zero. Then, during 1/3 of the cycle the level is coupled to both reservoirs and if epsilon_0 is at the Fermi level, an arbitrary charge can pass from one side to the other for a given realisation of the cycle, so I don't see why the noise should vanish in this limit. From Fig. 12 it is not clear if the noise vanishes or not. And all this the authors comment by just one phrase "The current noise has the usual behaviour" after Eq. (96). So, does the noise vanish or not, and why? What is the role of finite temperature? Besides, I do not see why the limit epsilon_0 -> 0 is called quantisation limit, since 1/2 is not really quantisation.