Exceptional Points in a Non-Markovian Anti-Parity-Time Symmetric System

Round 1

Reviewer 1 Report

The authors are investigating the existence of exceptional points in an optical system with APT symmetry and time-delayed evolution (non-Markovian). In addition to presenting numerical results, the authors provide an analytical treatment within specific approximations, strengthening the article. The results indicate the potential for controlling solution regimes that can be accessed by varying coupling and time-delay.

Systems with PT (APT) symmetry have received considerable attention, mainly due to the possibility of experimental realizations.

I believe the article presents valuable results and deserves publication.

Some comments to the authors:

• Page 1: "much interest in the firld´´ verify the misprint

 • Page 2: In the last paragraph, "physical separation between the lasers. which was comparable..." verify the extra dot.

 • Page 5: "pictured in fig. 3.2a insert," I believe it should be fig. 2.2a.

 • Page 9: The authors commented, "Should we show this proof?" It appears that it was a comment between the authors themselves at an earlier stage of the article. I would suggest that including a small appendix with the proof might be useful.

Parte superior do formulário

Parte inferior do formulário

Author Response

We thank the referee for a thorough reading of the paper, the comments, and the recommendation for publication in Photonics.  Below we address the specific comments:

1. We have corrected the spelling of field.
2. Removed the extra period in the sentence mentioned by the referee.
3. Corrected fig. 3.2a to fig. 2a.
4. Removed the comment which the referee correctly surmised was a communication between the authors. We have now provided the proof in the text of the manuscript itself, which is contained in the red font text and Eq. 21-24.

Reviewer 2 Report

This manuscript reports on a theoretical investigation of the exceptional point landscape in a non-Markovian anti-parity-time symmetric system. More precisely, the authors, inspired by recent studies with coupled lasers, consider a pair of mutually delayed-coupled gain systems and provide analytical results for the occurence of exceptional points depending on the mutual coupling strength and the delay time (which is the novel aspect of the study). Qualitatively new regimes are observed as compared to the no-delay situation, which open for further studies of non-Markovian anti-parity-time symmetric systems.

The manuscript is in general well-presented, with clear analytical derivations. The findings are original and relevant for fundamental and practical studies of the abovementioned systems, which are currently investigated in many different physical platforms. The paper would have been stronger in my opinion if the (or at least some) consequences for the dynamics of the system in the novel regimes had been presented, but I can also accept that such aspects are treated in future studies. I would thus give a positive recommendation for publication in Photonics.


Minor remarks:
-many typos are still present in the text, e.g.
"field of optics", 1st paragraph, p. 1
missing point at the end of 2nd paragraph, p. 1
"Should we show this proof?" 2nd paragraph, p. 9
"are parallel", caption Fig. 8

-isn't a minus sign missing in the r.h.s. of eq. 12?

-should the text for the subtitle be "(b) after breaking" rather than "before breaking" in the captions of Figs. 4 and 5?

-The formatting of the captions of these figures, as well as Figs. 7, 8, 9, with subtitles and caption is a bit messy.

-In the introduction the authors write that EPs have been "shown to occur in coupled microwave cavities [22]". They have also been observed in many other systems, so citing a recent review is probably appropriate here.

Author Response

We are grateful to the referee for the very positive comments and the recommendation to publish in Photonics.  We appreciate the careful reading and identifying the typos.  Below we address the specific comments:

1. We have corrected the spelling of “field” which was pointed out by another reviewer also.
2. We replaced the comma at the end of 2^nd para on 1^st page with a period.
3. Removed the “Should we show this proof” remark which was a communication between authors that slipped into the submitted manuscript. We have now provided the proof in the text of the manuscript itself, which is contained in the red font text and Eq. 21-24.
4. Modified second sentence in Fig. 8b caption to read “When the magnitude is equal to 1, the eigenvectors become parallel to each other”.
5. The referee is correct – a negative sign has been added to the rhs of Eq. 12. Note that the rest of the analysis is correct.
6. captions for 4b and 5b have been changed to “after breaking”.
7. We have revised the caption for figs. 4-8.
8. Citations to two recent review articles EPs in photonic systems have been added (Ref. 29, 30).

Reviewer 3 Report

In this manuscript ‘Exceptional points in a non-Markovian anti-parity-time symmetric system’, the authors focus on the Markovian and non-Markovian property of anti-parity-time-reversal (APT) systems, realized by a pair of time-delay coupled semiconductor lasers (SCLs). By tuning the time-delay and coupling of the SCLs, the authors find different EPs.

It is an interesting work and non-Hermitian physics is a hot and frontier area which attracts increasing attentions. I believe that the formulas and figures are able to support the author’s conclusion. Therefore, I approve this article should be accepted by ‘Photonics’ in general, while the authors should notice and consider the following points:

1)    In the first paragraph, the authors claimed that parity (x → −x, p → −p) and time-reversal (t → −t). I agree with the parity, but time-reversal has effects that x → x, i → −I (not the simple t to -t). [It is not required to cite, but please refer the EPL, 123 (2018) 40002]

2)    I don’t find the APT Hamiltonian in the paper.

3)    To enlarge the readers’ vision of APT quantum system, the authors should reference some other properties, such as quantum simulation of APT systems using linear combination of unities, information flow in APT systems, etc.

It is fine. 

Author Response

We thank the referee for the positive recommendation and have made modification as per suggestions:

1. We have added that the time operator changes i to -i.
2. The APT Hamiltonian is not explicitly mentioned in the paper but is represented by Eq. 1 in the manuscript. As we mention in the manuscript, this work is motivated by the experimental implementation of a non-Markovian APT system described in Ref. 48.  In such classical systems, the complex ‘i’ in front of the time derivative of the wave function is missing, as indicated in Refs 48 and 50, and so the Hamiltonian in Eq. 1 of the manuscript becomes a APT Hamiltonian.  We have discussed this in some detail in the text below Fig. 1 of the manuscript.  The emphasis of the paper is on non-Markovianity and the general conclusions of the manuscript are valid for both non-Markovian PT and APT systems.
3. A citation to a recent article on information flow in anti-PT systems has been added (Ref. 25).