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Thermodynamics of adiabatic quantum pumping in quantum dots
by Daniele Nello, Alessandro Silva
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Submission summary
| Authors (as registered SciPost users): | Daniele Nello · Alessandro Silva |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2306.08621v1 (pdf) |
| Date submitted: | 2023-06-16 15:09 |
| Submitted by: | Nello, Daniele |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider adiabatic quantum pumping through a resonant level model, a single-level quantum dot connected to two fermionic leads. We develop a consistent thermodynamic description of this model accounting for the variation of the energy level of the dot and the tunnelling rates with the thermal baths. We study various examples of pumping cycles computing the relevant thermodynamic quantities, such as the entropy produced and the dissipated power. We then compare these quantities with the transport properties of the system. Among other results, we find that the entropy production rate vanishes in the charge quantization limit while the dissipated power is quantized in the same limit.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 2) on 2023-11-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.08621v1, delivered 2023-11-14, doi: 10.21468/SciPost.Report.8115
Strengths
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.
Weaknesses
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.
Report
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.
Report #2 by Anonymous (Referee 3) on 2023-10-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.08621v1, delivered 2023-10-26, doi: 10.21468/SciPost.Report.8001
Strengths
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.
Weaknesses
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.
Report
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.
Requested changes
A detailed discussion of all the points, listed in the “Weaknesses”.
Author: Daniele Nello on 2024-05-23 [id 4509]
(in reply to Report 2 on 2023-10-26)
We thank Referee 2 for reviewing this manuscript. We revised extensively the manuscript reorganizing it completely and improving the presentation paying attention to physical interpretation. In the following, we will respond to the points raised. 1) In the revised version the physical implications of the results are discussed, and the presentation improved through a thorough reorganization of the manuscript. 2) We consider only this specific model: a single-level quantum dot coupled to two fermionic baths. However, the analysis can be extended as well to multi-level quantum dots and the initial assumptions of equal temperature and chemical potential can as well be relaxed. However, the results provided in this paper provide interesting insights into the phenomenon of adiabatic pumping and can inspire studies in other contexts (for example Thouless pumps). We include a mention of this in the Conclusion and Outlooks section. 3) The adiabatic pumping in the context of quantum dots has already been studied experimentally. An experimental study of the theoretical results of this article is within the possibilities offered by the available experimental platforms, though probably it would be easier to measure the heat dissipated and change in internal energy while deducing the work through the first principle, contrary to what happens theoretically. 4) We concede that this point needs to be further clarified. The two terms of the mentioned equation are the two contributions of Ref. [25] in the new version. The distinction has no physical ground, the only difference between them is that one is quadratic and the other one is quartic in the scattering matrix elements. They both contribute to the thermal part of the noise and the 'shot' noise. Their distinction is explained in greater detail in a new section of the Appendix, Appendix F. 5) The present work is a generalization of the approach of Ref. [11] to the specific framework previously described. The aspects of novelty with respect to the previous literature are the comparison between the transport properties and the thermodynamic quantities and the study of the quantization limits regarding the latter. 6) Since we are considering the zero-temperature limit, the physical mechanism responsible for the noise is the quantum fluctuations of the tunnelling process between the dot and the thermal baths. A phrase has been added in Sec. 3.2 to explain this key concept. 7) In the article we distinguish between the thermal noise and the 'shot' noise and we explain the difference between the two and classify the two contributions in the adiabatic expansion. 8) In the new version of the article, the presentation of the figures is improved. Relevant comments about the physical interpretation of them are added in the course of the whole article.
Report #1 by Anonymous (Referee 1) on 2023-9-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.08621v1, delivered 2023-09-10, doi: 10.21468/SciPost.Report.7791
Strengths
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.
Weaknesses
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.
Report
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.
Author: Daniele Nello on 2024-05-23 [id 4508]
(in reply to Report 1 on 2023-09-10)
We would like to thank Referee 1 for reviewing our manuscript and for the precious comments. We have reviewed extensively the manuscript taking into account all recommendations of the referees. In particular, we agree that our work represents an extension of the tools and the concepts developed in Refs [11] accounting for the change in the level-leads coupling, making it applicable to quantum pumping. What we develop is a self-contained (not self-consistent) thermodynamic description of pumping, bridging these results to those related to the transport properties. A detailed comment about the aspects of novelty of this article with respect to the previous literature has been added in the Introduction in the revised version of this manuscript. Moreover, we completely reorganized the manuscript rewriting entire sections to improve the presentation.
Responding to the specific points raised in the report: 1) Here "self-consistent" has been replaced with "self-contained" meaning the adherence to the laws of thermodynamics. We have calculated all the relevant quantities, not only the ones necessary for our analysis. 2) Indeed there is a misplacement of expectation values in formulas (6) and (8), which has been corrected in the new version of the article. We specify that the average indicated has to be intended as the expectation value w.r.t. the states of the system and has not to be intended as an average over the period. The definition of the noise, with reference to the measuring time $T_m$, has been explained in detail. 3) In the new version of the article a clear distinction between the various contributions is made on the basis of the low-temperature behaviour. Their difference is explained properly and the classification is performed in a detailed way, in particular in a new section of the Appendix (Appendix F). Notice that the calculation of the noise with the gradient expansion presented previously is incomplete and lacks a shot-noise term first order in the pumping frequency that has an important role in the third example presented. 4) Exactly, since there is no difference in temperature and chemical potential between the two baths, the equilibrium quantities integrated over a cycle must be 0. This is not surprising, it is a trivial consequence of this fact, as well as a sanity check for the correctness of our results. A comment is added in Sec. 4. 5) The example of the cycle with fractional charge pumped is indeed confusing and the previous calculation (and interpretation) was indeed incorrect due to the lack of one term in the noise calculation. We are grateful to the referee for pointing out the discrepancies and forcing us to reconsider the results. The shot noise term at first order in the frequency, lacking in the previous analysis, gives rise to a finite noise in the "quantization" limit and we can see that the limiting value is equal to $1/4$ as expected. A proper explanation of how to derive this term and why it is not present in the gradient expansion is presented in Appendix E.
Author: Daniele Nello on 2024-05-23 [id 4510]
(in reply to Report 3 on 2023-11-14)We thank Referee 3 for reviewing this manuscript which has been extensively revised as a result of the comments. In the following, we will summarize the changes made. In the new version of the article, we have corrected the typos pointed out, and revised thoroughly the presentation improving its clarity and reorganizing it. We also added comments about the physical interpretation of our results.
Regarding the more specific points raised: 1)The adiabatic limit in this context means that the driving period is large. The implications of this assumption are described in detail in Appendix D. In Section IV, where the equilibrium quantities are computed, it is not true that the heat exchange rate must be vanishing. The concept of adiabatic process in classical thermodynamics differs from the same in the context of quantum mechanics. In the context of classical thermodynamics, this would be more properly called "quasi-static limit". Further comments on how to describe the heat exchange have been added at the end of page 11, with appropriate references 2) We have stressed the link between thermodynamic quantities and Green's functions in the new version, as it is indeed one of the key concepts of this article. The link with Appendix B is appropriate as there we compute all the relevant Green's functions. The computation of the expectation value of the coupling term of the Hamiltonian has been moved to a separate section of Appendix, section C, in order to avoid confusion. 3) These references have been included, we thank the reviewer for mentioning them. 4) This is beyond the scope of this article. We have included a discussion in the Conclusions, as this is relevant for the future prospects.