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Effective theory for matter in non-perturbative cavity QED

by Juan Román-Roche, David Zueco

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Submission summary

Authors (as registered SciPost users): Juan Román-Roche
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-04-19
Date submitted: 2022-03-09 09:00
Submitted by: Román-Roche, Juan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


Starting from a general material system of $N$ particles coupled to a cavity, we use a coherent-state path integral formulation to produce a non-perturbative effective theory for the material degrees of freedom. We tackle the effects of image charges, the $A^2$ term and a multimode arbitrary-geometry cavity. The resulting (non-local) action has the photonic degrees of freedom replaced by an effective position-dependent interaction between the particles. In the large-$N$ limit, we discuss how the theory can be cast into an effective Hamiltonian where the cavity induced interactions are made explicit. The theory is applicable, beyond cavity QED, to any system where bulk material is linearly coupled to a diagonalizable bosonic bath. We highlight the differences of the theory with other well-known methods and numerically study its finite-size scaling on the Dicke model. Finally, we showcase its descriptive power with three examples: photon condensation, the 2D free electron gas in a cavity and the modification of magnetic interactions between molecular spins; recovering, condensing and extending some recent results in the literature.

Author comments upon resubmission

Dear Editor,

I am resubmitting a revised version of the manuscript “Effective theory for matter in non-perturbative cavity QED”. In this revised manuscript, we have made changes to address the referees’ comments and suggestions, which have resulted in the addition of a new section to the paper, as well as more minor changes.

The referees criticised the lack of novelty in the previous version of the manuscript, which resulted in your editorial recommendation for SciPost Physics Lecture Notes. We respect the decision, and would feel honoured to be featured in SciPost Physics Lecture Notes, as we acknowledge the pedagogical caliber of its publications. However, we find that appearing as a Lecture Note would mislead potential readers of the manuscript into thinking that it contains nothing new, that it just reviews well-known concepts. While the idea of an effective action dates back to the works of Feynman and Vernon on the influence functional, its application to the field of cavity qed materials, ending up with an exact effective Hamiltonian valid for all light-matter-coupling regimes, is definitely novel. Moreover, the new section is dedicated to testing the validity of the theory, highlighting its differences against other effective theories, which is something that the referees pointed out the previous version lacked. With this concern in mind, we would like to ask you and the referees to reconsider your recommendation for SciPost Physics Lecture Notes.

Detailed responses to each referee report are provided separately via the “Reply to the above Report” functionality on the website.
An overview of all the changes in the updated manuscript is provided below.

Juan Román-Roche

List of changes

- In the first line below Eq. 1, corrected typo on the commutation relations
- In the third line of page 4 corrected duplicate 'of'.
- Minor changes to the Abstract, Introduction, Summary of main results and Conclusion to accommodate the new section.
- Minor change to Section III A to clarify the consequences of the approximation made in Eq. (13)
- Minor change to Section VII to include citation.
- Added paragraph to Section IV C following Comment 9 of Referee 1.

Published as SciPost Phys. Lect. Notes 50 (2022)

Reports on this Submission

Anonymous Report 2 on 2022-3-31 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2110.07632v4, delivered 2022-03-31, doi: 10.21468/SciPost.Report.4836


I appreciate the time and effort the authors took to improve the manuscript and answer my initial report. The authors responded to my initial criticism mainly by adding a section in the revised version of the manuscript (section V) where they compare the effective theory presented in this work with other known effective theories. To this end they use the paradigmatic example of the Dicke model and compare their effective Hamiltonian with other effective Hamiltonians obtained via i) a Polaron transformation and ii) a Schrieffer-Wolff transformation. This novel section is a valuable addition to the manuscript and it also clarifies my original question why the presented theory can be called "non-perturbative".

While the new section increased the quality of the manuscript, it still contributes mainly to the pedagogical aspect of the manuscript. As I already pointed out in my initial report the "result" presented in this work is the model derivation. I still think that applying the introduced methodology to a particular problem, which cannot be understood by other (e.g. perturbative) effective theories would be necessary to fulfill the criteria for publication in SciPost Physics.

Therefore, I stay with my original assessment of this manuscript and I think it is not suitable to be published in SciPost Physics.

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Anonymous Report 1 on 2022-3-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2110.07632v4, delivered 2022-03-30, doi: 10.21468/SciPost.Report.4825


I appreciate the detailed answers provided by the authors, however they were not able to change my initial recommendation. I thus re-affirm that this article does not fully meet the criteria to be published in SciPost Physics. Perhaps I can still suggest it for SciPost Lecture notes, since it covers in quite details the derivation of effective Light-Matter interactions and it reports several existing examples.

My rejection recommendation is mainly motivated by the fact that the authors provided only minor changes to the manuscript, while it should be drastically changed to fulfill the acceptance criteria.

Here I want to address the answers provided by the authors poit by point:
1) If all the results in this article were already known then it is more a review or a lecture note rather than an article containing new original results.
Moreover the new paragraph about the Dicke model provided by the authors is a quite trivial example. It doesn't bring that much extra information/understanding and instead it makes the article longer and less fluid to read.
2) Basically the same comment as in 1)
3) The authors didn't really resolve my doubts and the role of their assumptions is still unclear.
4) What I meant in my comment was: if I have a magnet and I put it in a cavity (and nothing happens) do I have a photon condensate? Because in that case, due to the magnetostatic field the expectation value of "a" is different from zero in equilibrium. Then I'd say that using the term "photon condensation" in this context is rather misleading.
5) Here I meant that if I think the cavity as the boundary of the system and I take the cavity volume to infinity the effect of the boundary should be quite negligible, even if the cavity is fully filled. Then electromagnetism still plays a role but it is not anymore "cavity QED" in the sense that mode quantization doesn't play any role.
6) I kind agree with the authors. However it seems to me that what they say is simply "do the calculation correctly without mistakes". Still it is not clear whether all their assumptions and approximations respect gauge invariance.
7) The Dicke model is quite easy and perhaps trivial example that could hide features of the author's effective treatment which are critical in other less trivial systems. I do not think that their example gives really any useful information. I think I expected a deeper analysis on the general model rather than a single example with one of the simplest toy model on the market.
8) I agree with the authors that one introduce cut-offs to cure divergences. It would be interesting to see a proper calculation on this side and to check which kind of physical predictions may arise from there, for instance regarding the Lamb-shift.
9) The authors added a vague sentence about observation. I do not think it adds nothing more than was already written and thus it doesn't solve my doubt.

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