Quantum many-body thermal machines enabled by atom-atom correlations

We would like to thank the Referee #2 for their insightful comments and suggestions. Below we provide our detailed responses and describe the changes made to the manuscript. (A copy of the revised version of the manuscript where all changes are highlighted in blue text -- for ease of navigation -- can be accessed at https://people.smp.uq.edu.au/KarenKheruntsyan/pubs/Engine_v5.pdf.)

1) The work done by this engine can indeed also be calculated from the interaction energy <H_int>, which is simply the product of the interaction strength \chi and the total correlation function \overline{G_2}. However, we would like to dissect the Referee's comment that "it is also valid to say [that] without changing the interactions the engine will not work also according to Eq.4." It is important here to be precise in what we mean by "interactions" -- interaction strength \chi (or its dimensionless counterpart \gamma) or the mean interaction energy <H_int>? If we mean the interaction strength \chi, then without changing the interaction strength the engine will indeed not work, because this is an interaction quench engine which operates between two different values of \chi. However, if the referee is referring to the mean interaction energy, then we can propose a counterexample in which the interaction strength \chi changes, the interaction energy also changes, yet the engine still does not work, because the correlation functions do not change as a result. The counter-example that we have in mind is in the regime of mean-field approximation. In this approximation, the correlation function g2(0)=1 always, yet the mean filed interaction energy <H_int>=0.5\chiN\rhog2(0) can change simply because of the change in the interaction strength \chi. Despite this, the engine does not produce work according to Eq. 5 (Eq. 4 in the old version) because g2_h(0)=g2_c(0)=1, even though <H_int>_h is not equal to <H_int>_c.

See also our response to p. 3) below.

We finally emphasise that correlations are not determined solely by interactions, but also by quantum statistics (bosons versus fermions) and thermal fluctuations; at sufficiently high temperatures the effect of interactions and quantum statistics can be swamped by thermal fluctuations rendering the correlations to be thermal in nature. Accordingly, we don't view interactions and correlations as synonyms. For a given interaction strength, the mean interaction energy can be smaller or larger depending on the correlations (which will depend on the temperature as well); the reverse is also true, i.e. the same magnitude of correlations can be achieved at different interactions strengths (as can be seen e.g. from Fig. 6c), again resulting in different mean interaction energies.

2) We thank the referee for this suggestion: we now moved Fig. 5b from Appendix D to the main text. We have also included a brief identification of different physical regimes that we refer to as I-VI.

3) Practically speaking, yes, the work output of our proposed engine would be measured solely by the g2 function (plus the knowledge of the interaction strength $\chi_c$ and $\chi_h$) as can be seen from Eq. (4). Together these two quantities give the interaction energy of the system, which is sufficient for determining W in a sudden interaction quench. On the other hand, for efficiency \eta, Eq. (4), one would also need to know the total energy of the system, <H_c> and <H_h>, which we note includes the kinetic energy of the gas in addition to the interaction energy. We did not state that "the maximum net work is found not by maximizing the difference in the g2 function but rather through the difference in interaction strength"; instead our claim was that " the magnitude of net work is governed more strongly by the difference in the interaction strengths..." than the difference in the correlations because the difference in correlations is bounded by 2 (g2_max=2, g2_min=0), whereas the difference in the interaction strength can, in principle, be arbitrarily large. We agree though that, for small change in g2, minimising the measurement error of g2 will be critical. At the same time, we note that measuring the density alone cannot give us any information about the interaction energy (and hence about the total energy of the system and the performance of the engine) because the interaction energy is proportional to g2 and cannot in general be deduced from the density, except in the mean field approximation.

Minor points:

4) We are not sure what the Referee means by "underperformance" (of the refrigerator). At the same time we note that processes that might heat up or cool down the reservoirs are not accounted for in our analysis as we assume an idealized case of infinitely large reservoirs at constant temperatures; their only role is to equilibrate the working fluid to the same temperature when it is brought into thermal contact with the reservoir.

5) Chen et al. themselves refer to the relevant strokes of their engine as isentropic, so we are simply following the terminology used by the authors of Ref. [30].

6) We have now clarified on page 3 that "The working fluid starts in some non-equilibrium state $\textbf{A}$ (corresponding to the final state of the previous stroke) and is left to equilibrate with the reservoir..."

7) Yes; we have now fixed this typo.

We are grateful to Referee #1 for their careful reading of the manuscript and providing valuable feedback. The referee raised some reservations about some of the points, which we address below. (A copy of the revised version of the manuscript where all changes are highlighted in blue text -- for ease of navigation -- can be accessed at https://people.smp.uq.edu.au/KarenKheruntsyan/pubs/Engine_v5.pdf.)

1) The question raised here is similar to the question 1 raised by Referee #2. Our response is equally applicable here. To reiterate, in the sudden interaction quench proposed in this work, the work output is determined by the difference in correlation functions, as can be seen through Eqs. (3) and (5). Even though the mean interaction energy is proportional to the correlation function, the magnitude of the correlation function is not determined solely by the interaction strength \chi, but depends also on the interplay of quantum statistical effects of exchange interaction (bosons versus fermions) and thermal fluctuations. See also our arguments in response to the question 1 of Referee #2.

Regarding the question of whether a similar engine could operate with a classical working fluid and classical correlations, we believe that the answer here is yes.

2) These are indeed very good points:

• For a sudden quench of the interaction strength, the continuous lines in Fig. 1 connecting the points B-C and D-A are meaningless as the system does not pass through those intermediate points as in a typical quasi-static stroke. In fact, there is no way to represent the passage from B to C and from D to A by some continuous lines in this sudden (instantaneous) interaction quench engine, where only the difference in the energies of the system at the end points of work strokes can be defined and calculated. To reflect on this issue, we modified Fig. 1 and connected the points B-C and D-A by dashed lines, which gave us an opportunity to expand the figure caption and explain that this is not a "typical" engine cycle -- along the lines of referee's comment.
  We would like to add here that we believe it is a beautiful aspect of this sudden quench engine cycle that its overall performance can be evaluated solely from the knowledge of energies at just two equilibrium points, B and D.

• We thank the referee for this insightful comment. We have now added a brief discussion of this point in the paragraph following Eq. (5).

• We have added a discussion and comparison of our sudden quench results against the ideal (maximum possible efficiency) Otto cycle in Appendix G.

3) We thank the referee for noticing this typo; we have now fixed it in the figure label. The same inequality in the relevant part of the text was actually correct.

4) We are grateful to the referee for these suggestions (we were not aware of these papers). We have now cited all these three suggested papers. Additionally, we added two more citations [4,5] to the list of recent review articles [1-5] on quantum thermodynamics.

We are grateful to Referee #1 for their careful reading of the manuscript and providing valuable feedback. The referee raised some reservations about some of the points, which we address below. (A copy of the revised version of the manuscript where all changes are highlighted in blue text -- for ease of navigation -- can be accessed at https://people.smp.uq.edu.au/KarenKheruntsyan/pubs/Engine_v5.pdf.)

1) The question raised here is similar to the question 1 raised by Referee #2. Our response is equally applicable here. To reiterate, in the sudden interaction quench proposed in this work, the work output is determined by the difference in correlation functions, as can be seen through Eqs. (3) and (5). Even though the mean interaction energy is proportional to the correlation function, the magnitude of the correlation function is not determined solely by the interaction strength \chi, but depends also on the interplay of quantum statistical effects of exchange interaction (bosons versus fermions) and thermal fluctuations. See also our arguments in response to the question 1 of Referee #2.

Regarding the question of whether a similar engine could operate with a classical working fluid and classical correlations, we believe that the answer here is yes.

2) These are indeed very good points:

For a sudden quench of the interaction strength, the continuous lines in Fig. 1 connecting the points B-C and D-A are meaningless as the system does not pass through those intermediate points as in a typical quasi-static stroke. In fact, there is no way to represent the passage from B to C and from D to A by some continuous lines in this sudden (instantaneous) interaction quench engine, where only the difference in the energies of the system at the end points of work strokes can be defined and calculated. To reflect on this issue, we modified Fig. 1 and connected the points B-C and D-A by dashed lines, which gave us an opportunity to expand the figure caption and explain that this is not a "typical" engine cycle -- along the lines of referee's comment.
We would like to add here that we believe it is a beautiful aspect of this sudden quench engine cycle that its overall performance can be evaluated solely from the knowledge of energies at just two equilibrium points, B and D.

We thank the referee for this insightful comment. We have now added a brief discussion of this point in the paragraph following Eq. (5).

We have added a discussion and comparison of our sudden quench results against the ideal (maximum possible efficiency) Otto cycle in Appendix G.

3) We thank the referee for noticing this typo; we have now fixed it in the figure label. The same inequality in the relevant part of the text was actually correct.

4) We are grateful to the referee for these suggestions (we were not aware of these papers). We have now cited all these three suggested papers. Additionally, we added two more citations [4,5] to the list of recent review articles [1-5] on quantum thermodynamics.