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Statistical Mixture of Kaleidoscope States Interacting with a Two-Level Atom: Entropy and Purification

Photonics 2023, 10(2), 150; https://doi.org/10.3390/photonics10020150

by Jorge A. Anaya-Contreras^1,†

, Arturo Zúñiga-Segundo^1,†

and Héctor M. Moya-Cessa^2,*,†

Reviewer 1: Anonymous

Reviewer 2:

M K Tavassoly

Reviewer 3: Anonymous

Reviewer 4:

Watson Kuo

Reviewer 5: Anonymous

Photonics 2023, 10(2), 150; https://doi.org/10.3390/photonics10020150

Submission received: 19 October 2022 / Revised: 4 January 2023 / Accepted: 30 January 2023 / Published: 31 January 2023

(This article belongs to the Special Issue Lasers and Dynamic of Systems)

Round 1

Reviewer 1 Report

Upon reading this manuscript I was searching for both motivations and physical insights. I think this is missing in both cases. In the end it is a rather straightforward analytical derivation, but the results, in my opinion, do not provide any new physics/understanding. The derivation in itself does not motivate publication. This is especially so since the paper is a direct application of their earlier work [8]. It is in fact much of a copy of that work (now we added Wigner plots) but for a different initial state, and even worse is that some sentences are direct copies of that work. I thereby cannot support publication of this manuscript.

Author Response

Upon reading this manuscript I was searching for both motivations and physical insights. I think this is missing in both cases. In the end it is a rather straightforward analytical derivation, but the results, in my opinion, do not provide any new physics/understanding.
ANSWER: We thank the reviewer for the detailed comments made to our manuscript.
The derivation in itself does not motivate publication. This is especially so since the paper is a direct application of their earlier work [8]. It is in fact much of a copy of that work (now we added Wigner plots) but for a different initial state, and even worse is that some sentences are direct copies of that work. I thereby cannot support publication of this manuscript.
ANSWER: We have modified the manuscript to make it clear that it is different from our previously published work.

Reviewer 2 Report

Report on the paper entitled: Kaleidoscope states as a statistical mixture in the Jaynes-Cummings model: Entropy and purification

By: Anaya-Contreras et al,

Submitted to: MDPI,

The authors studied physical features of the interaction of mixed kaleidoscope states (particular statistical mixtures of coherent states) with two-level atoms using the JC model. They evaluated the von Neumann entropy of the field using the virtual atom method previously introduced by the authors. A purification of the initial state is achieved. In this region, the Wigner function that resembles Schrodinger’s cats is obtained.

The subject of the paper is sound for the quantum optics community and is of interest for nowadays researches of the field. Therefore, I can recommend its publication, however, after making the following revisions.

1- For the readers to follow the contents of the paper, I advise the authors to add, at least in an appendix, the details of the calculations of Eqs. (8-10), (15-18), and (21-23).

2- The evolution of the field entropy of Kaleidoscope-States as a function of the scaled time and different values of statistical mixture of coherent states $n = 1, 2, 4, 8 and 16, with $\alpha = 6.0$ is presented. I advise to check and present for another mean value of photons, i.e., $\alpha$, say, 1, or 2. Then they interpret their results with the previous conclusions come from the present figure in the paper for $\alpha=6$.

3- Why, while the authors outlined the atomic entropy, they did not pay their attention to this issue, and no numerical calculations is presented. I advise to do this, too.

4- Some details about the calculation procedures of the Wigner function, the used relations as well as the procedures are needed.

6- The bibliography is poor, so I recommend to add a few related papers such as, Eur. Phys. J. D 58, 147–151 (2010); Phys. Rev. B 76, 045317; Physical Review A 93 (2), 022327; Quantum Rep. 2022, 4(1), 22-35.

Author Response

1- For the readers to follow the contents of the paper, I advise the authors to add, at least in an appendix, the details of the calculations of Eqs. (8-10), (15-18), and (21-23).
ANSWER: We have added an appendix to detail the calculations you commented.

3- Why, while the authors outlined the atomic entropy, they did not pay their attention to this issue, and no numerical calculations is presented. I advise to do this, too.
ANSWER: We have also added a figure of the atomic entropy.

4- Some details about the calculation procedures of the Wigner function, the used relations as well as the procedures are needed.
ANSWER: We have added some calculations to obtain the Wigner function.

5- I did not understand that how the authors connected the results of the Wigner plots with the cat states, since there are many field state for which the Wigner function may be negative. More explanations are needed to demonstrate their claim.
ANSWER: We have added comments to connect the oscillations of the entropy with the Wigner function and multi-component Schroedinger cats.

Reviewer 3 Report

The paper makes an effort to calculate the von Neumann entropy dynamics of the Jaynes-Cummings model with help of the virtual atom method. The photonic field is prepared in a statistical mixture of coherent states and the atom is in the excitation state.
It is argued that a purification of the initial state is hinted by oscillations, where the Wigner function of the field resembles Schrodinger’s cats.

The paper contains many imprecise and bewildering statements and terminologies. For example, the concepts of the mixed state and the superposition state in the introduction are confusing; Is the "Kaleidoscope states" a specialized terminology for Eq.(1)? What is the meaning of "purification"? The the von Neumann entropy is a quantity characterizing the entanglemnt entropy. In the introduction, Araki-Lieb inequality is mentioned but not introduced and used in the main text. The introduction lacks enough investigation of relevant work and detailed support materials. Some physical notations are quite puzzling. For example, after Eq.(1), it reads, "and $\omega^k$ the nth root of unity".

Overall, the paper is short of innovation, just repeating the theory of Ref.[5] and Ref.[7]. There lacks a physical interpretation and insight of the a global minimum. I do not recommend thisi work to be published in MDPI.

Author Response

The paper contains many imprecise and bewildering statements and terminologies. For example, the concepts of the mixed state and the superposition state in the introduction are confusing; Is the "Kaleidoscope states" a specialized terminology for Eq.(1)? What is the meaning of "purification"? The the von Neumann entropy is a quantity characterizing the entanglemnt entropy. In the introduction, Araki-Lieb inequality is mentioned but not introduced and used in the main text. The introduction lacks enough investigation of relevant work and detailed support materials. Some physical notations are quite puzzling. For example, after Eq.(1), it reads, "and $\omega^k$ the nth root of unity".
ANSWER: We have rewritten many paragraphs to clarify the point raised by the reviewer. We have added an equation plus and explanation of the Araki-Lieb inequality to clarify how it is used.

Overall, the paper is short of innovation, just repeating the theory of Ref.[5] and Ref.[7]. There lacks a physical interpretation and insight of the a global minimum. I do not recommend thisi work to be published in MDPI.
ANSWER: We have enlarged the manuscript in order to give a physical interpretation.

Reviewer 4 Report

This paper presents a method to calculate the entropy of a kaleidoscope state coupled to a two-level atom. The field entropy shows a global minimum at \lambda*t~ 19 if n>1, indicating an enhancement in purity introduced by the atom-light interaction. Here are the comments and questions for this work:

-In the introduction, many recent works on the interaction between a cat state and atom(natural or artificial ones) are missing.

-The Sec. Kaleidoscope states is very lengthy and mostly provides the basicunderstanding of the kaleidoscope. The most part can be considered to be moved to appendix.

-What determine the time to get a minimum entropy? Could the author give a simple picture to understand the existence of such a minimum? It seems that the minimum time are the same for all kaleidoscope state n. We only see calculated results without much discussion in physics.

-It is difficult to see the connection between Eq. (8) and Eq. (15).

-The expressions given by Eq. (16) and (17) easily create confusion. In particular the range of k for each equation is not given directly following the equation.

-The field entropy quickly goes up to a maximum at a small time, \lambda*t=1~2. What is the Wigner function of the Kaleidoscope state at this moment?

-Are the curves in Fig. 1 shifted upward with some value?

-There is no loss in the model. Could such a "purification" still exist with finite relaxation in atom?

-In line 87. is "mya" the typo of "may"?

Author Response

-In the introduction, many recent works on the interaction between a cat state and atom(natural or artificial ones) are missing.
ANSWER: We have enlarged the number of references to account for this.

-The Sec. Kaleidoscope states is very lengthy and mostly provides the basic understanding of the kaleidoscope. The most part can be considered to be moved to appendix.
ANSWER: We have added the appendix as suggested by the reviewer.

-It is difficult to see the connection between Eq. (8) and Eq. (15).
ANSWER: We have rewritten greatly the manuscript in order to clarify this point.

-The expressions given by Eq. (16) and (17) easily create confusion. In particular the range of k for each equation is not given directly following the equation.
ANSWER: We have rewritten greatly the manuscript in order to clarify this point.

-The field entropy quickly goes up to a maximum at a small time, \lambda*t=1~2. What is the Wigner function of the Kaleidoscope state at this moment?
ANSWER: We have added a figure of the Wigner function for such time.

-Are the curves in Fig. 1 shifted upward with some value?
ANSWER: This point has been explained in the text and the figure caption.

-There is no loss in the model. Could such a "purification" still exist with finite relaxation in atom?
ANSWER: We have not investigated this point. However we do not believe that such purification would exist as the non-classicality risen by the transfer of coherence from the atom to the field would be rapidly lost.

-In line 87. is "mya" the typo of "may"?
ANSWER: We have corrected the typo.

Reviewer 5 Report

1. Why is there a special interest in the evolution of this particular state?

2. I have not found any relevant physical discussion of the obtained result. This makes the manuscript seem to be an exercise in quantum optics, rather than a research paper.

3. The authors claim that the maximum purity is attained at certain time, which depends on the parameters of the initial state. As far as I understand this result was obtained numerically. Is it possible to estimate it analytically?

Technical issues:

2. What is the explicit form of the transformation between the representations Eq.8 and Eq.2

3. I don´t think that Eqs. 24 really matter, since they are direct consequences of Eqs. 16-18.

I do not recommend the manuscript for publication in its current form, but it can be reconsidered if the authors take into account the aforementioned suggestions.

Author Response

In the present submission the authors numerically study the interaction between a particular mixed state of the quantized field and a single two-level atom in the framework of the JCM. The manuscript is free from basic errors; however, several questions arise:
ANSWER: We thank the reviewer for the detailed comments made to our manuscript.
1. Why is there a special interest in the evolution of this particular state?
ANSWER: Being a mixture of states we were looking for how purification could take place. Furthermore, one could think of passing several atoms each one lending its coherence to the field to produce greater purification. However, it was not the point of the present manuscript to do this.
2. I have not found any relevant physical discussion of the obtained result. This makes the manuscript seem to be an exercise in quantum optics, rather than a research paper.
ANSWER: We have greatly enlarged the manuscript in order to enphasizi the relevan results.

Technical issues:

1. The notation in Eq. 4 is quite unfortunate: on the left hand side you have an abstract state, while on the right hand side the vectors in the linear space appear. The same is true in Eq. 11, where the matrix elements of the density matrix in some basis should appear on the left hand side.
ANSWER: We use a common notation that allows operators to be written as matrices. For instance in the 2x2 case, the Pauli spin operator |e><e|-|g><g| is written by a 2x2 matrix with elements 1,0,0,-1.

2. What is the explicit form of the transformation between the representations Eq.8 and Eq.2
ANSWER: We have enlarged the manuscrript to make clear several of such connections. In particular we have added 2 appendices.

3. I don´t think that Eqs. 24 really matter, since they are direct consequences of Eqs. 16-18.
ANSWER: We agree.

I do not recommend the manuscript for publication in its current form, but it can be reconsidered if the authors take into account the aforementioned suggestions.
ANSWER: We have made many changes in the manuscript. We hope that you find the new version suitable for publication.

Round 2

Reviewer 1 Report

In the revised vesrion the authors have added substantial text and also some new derivations. However, I don't think the added material make the analyzes much more informative. The manuscript is still much of a presentation of numerical and analytical results lacking deeper physical insight. I thereby cannot recommend publication.

Author Response

There is a notable difference between this study and the first work (now reference [13]). In order to highlight these differences, in this study we again consider the Jaynes-Cummings model, for the case of a mixture of two coherent states in the field state, prepared in the excited state, as in reference [13].
In reference [13], we calculate the von Neumann entropy with the eigenvalues of the density operator associated with a four-level virtual atom, defined in equation (32) of reference [13]. This matrix is not diagonal in 2X2 blocks, as is the case with our new density defined in equation (16). Although we treated the same mixture of two coherent states, kaleidoscope states allowed us to do this reduction in the entropy calculation, and we show it by obtaining the same results, shown for n=2, as in reference [13].

Reviewer 2 Report

Report on the paper:

Statistical mixture of Kaleidoscope states interacting with a two-level atom: Entropy and purification,

by: Anaya-Contreras et al.

The authors investigates on some new features of the interaction of a mixture of coherent states with two-level atoms in the JCM model. The von Neumann entropy of the field is calculated by which a state of purity greater than the initial state has been characterized. Also, a negative Wigner function has been obtained showing a multiple Schrödinger cat state.

The subject of the paper is enough sound and the paper is well written, so, ignoring some typo errors, it deserves publication. However, before this, I have a minor comment which should be clarified by the authors. It is stated that, “As two coherent states are sufficiently apart when α ≈ 2, they may be considered orthogonal. As is well-known for two coherent state, if for instance, |\alpha - \beta| is enough large they can be considered orthogonal, what the authors mean by their statement in above. Is α ≈ 2 the difference between the labels of two coherent states? If so, it seems that it is not enough, and if this is a label for a single coherent state what orthogonality means?

Thanks to the authors which presented all their calculation in detail in the Appendixes.

Author Response

We thank the reviewer for his/her comments. We have added the following paragraph relating to orthogonality>

They may be considered orthogonal as $\langle \alpha|-\alpha\rangle=\exp(-2|\alpha|^2)$ and as $\alpha$ becomes larger the exponential approaches zero.

Reviewer 3 Report

The authors have considered the comments and made substantial expansion to the manuscript. Based on these efforts, I will recommend this paper for publication if the authors could improve the English a bit. Some sentences are quite obscure, for example, the last sentence in the added paragrapha.

Author Response

We have improved the English.

Reviewer 4 Report

The authors have made substantial revision to improve the readability.

Some minor correction are needed.

Eq.(3), missing a ">" symbol in the 2nd term, RHS.

on page 6: "In order to ask the above question we calculated the field Wigner function" . do you mean "answer the above question" ?

The writing for the newly appended paragraph can be improved. Such as line 74-87, 97-101. 113-122, 126-132.

Author Response

We have corrected the points you have raised.

Reviewer 5 Report

The authors have formally answered my comments. However, the discussion of the physics behind the calculations is still deficient. The conclusions are formal and do not provide any new, interesting insight and should be significantly improved.

Author Response

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