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Cavity Induced Many-Body Polaritons: diamagnetic interactions, localization and coherent transfer
by V. Rokaj, S. I. Mistakidis, H. R. Sadeghpour
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Submission summary
| Authors (as registered SciPost users): | Simeon Mistakidis · Vasil Rokaj · H. R. Sadeghpour |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2207.03436v1 (pdf) |
| Date submitted: | 2022-07-08 17:00 |
| Submitted by: | Rokaj, Vasil |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Cavity quantum electrodynamics provides an ideal platform to engineer and control light-matter interactions with polariton quasiparticles. In this work, we investigate a many-body system of interacting particles in a harmonic trap coupled to a homogeneous quantum cavity field. The many-body system couples collectively to the cavity field, through its center of mass, and collective polariton states emerge. Due to the collective coupling, the cavity field mediates pairwise dipole-dipole interactions and enhances the effective mass of the particles. This leads to the localization of the center of mass which becomes maximal when light and matter are on resonance. The light-matter interaction modifies also the photonic properties of the system and the polariton ground state is populated with virtual, thermal (bunched) photons. In addition, the necessity of the diamagnetic $\mathbf{A}^2$ term for the stability of the many-body system is discussed, as in its absence a superradiant ground state instability occurs. We demonstrate the emergence of coherent transfer of polaritonic population under an external magnetic field by monitoring the underlying Landau-Zener probability.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022-11-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2207.03436v1, delivered 2022-11-11, doi: 10.21468/SciPost.Report.6119
Strengths
1- Well-written and clear.
2- Sufficiently detailed for the non-specialist
Weaknesses
1- No many-body physics (misleading title and abstract)
2- Trivial for the most part
3- Narrow in scope
Report
The article studies the coupling of the otherwise free centre-of-mass of an $N$-body system with a cavity field. The title and the abstract suggest otherwise. Since the field is homogeneous and matter is trapped harmonically, it is obvious that centre-of-mass and relative coordinates decouple. The authors correctly mention (citing a future, unfinished publication), that interactions will play a role nonetheless (this reminds me of Tarruell's group's latest article on interacting gauge fields, where something alike is engineered). However, this article only deals with a somewhat trivial one-body system. The article is exceedingly long, and the results obtained, while valid and interesting, deserve publication albeit in a different journal. I do not think the article meets the standards of quality of Scipost Physics.
I am also a bit confused regarding the choice of centre-of-mass coordinate with a $1/\sqrt{N}$ instead of $1/N$ factor. Much more confusing is the definition of the relative coordinates with respect to the particle labeled "1" with a $1/\sqrt{N}$ instead of a unit factor.
Other than that, the paper is nicely written and is enjoyable to read. I hope the authors will agree with this reviewer that the paper perhaps belongs elsewhere (a solid, good journal like Physical Review A would be a perfect place for it) .
Requested changes
1- Please change the title of the paper to avoid misleading the reader.
2- Redefine the centre-of-mass and relative coordinates to align with usual standards.
Anonymous Report 1 on 2022-10-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2207.03436v1, delivered 2022-10-09, doi: 10.21468/SciPost.Report.5852
Strengths
1. Exact solution of polariton problem of N particles in a harmonic trap coupled to a cavity
Weaknesses
1. Not clear that the Hamiltonian can be realized
2. The polariton states are decoupled from particle-particle interactions, so no many-body physics
3. Results are overstated
Report
The manuscript by Rokaj et al. investigates a cavity QED setup where they have many particles in a harmonic trap coupled to a homogeneous cavity light field. Using an exact treatment of the model system, they demonstrate that collective polaritons emerge, where the light-field is coupled to the center of mass (CM) mode of the matter component. They investigate the properties of the resulting polariton states (hybrid states of light and matter), including their modified trapping potential.
While I was initially excited to read this paper based on the abstract, the further I delved into it, the more I was disappointed. The authors make strong claims about exactly solving a many-body system. However, in the end the photon field is completely decoupled from the relative motion of the particles, and therefore from any matter interaction effects. The localization described by the authors is simply a modification of the CM trap frequency, and it does not modify any of the properties of the many-body system, such as its density. Likewise, the 'induced' interaction between the particles is just a rewritten harmonic oscillator potential for the CM. Thus, the claims of the paper appear overstated. Likewise, the paper is probably overly long.
As such, I do not see that this paper satisfies any of the Scipost Physics expectations, and I do not recommend publication in Scipost Physics.
Requested changes
1) Below the Hamiltonian Eq. (1) it is stated that "for charged particles, g_0 is in units of the electromagnetic charge". I found this confusing, does this imply that the Hamiltonian is valid beyond the case of charged particles? I would not think so.
2) How would one in practice create a harmonic potential for charged particles, such as electrons or cold ions?
3) It is well-known that in a harmonic trap, the center of mass (CM) motion decouples from the relative motion. While I think one would have to go far back to find an original reference on this, this is for instance argued in the context of the hyperspherical formalism in Phys. Rev. A 74, 053604. Therefore, it should come as no surprise that the cavity field is completely decoupled from the particle-particle interactions (which depend on relative positions), as the authors argue in section 2, and the corresponding discussion could be made clearer. In this sense, the setup is very similar to e.g. that of Ref. [64] (by the same first author) where the same Hamiltonian was investigated without a confining potential, and where the same decoupling takes place.
4) The authors make multiple claims about this being a many-body interacting system coupled to light. However, since the interactions only depend on the relative positions of the particles, the presence of particle-particle interactions is completely irrelevant for any of the results presented in this work. Therefore, I disagree with the terminology 'many-body' which is usually used to describe interacting systems of many particles, and I think the authors should tone down some of these claims, including changing the title to remove the terminology 'many-body'.
5) Furthermore, since only the CM mode is affected by the coupling to light, the actual density of the system is not changed by this coupling. This should be made clear in Section 4 where density and CM density are often used interchangeably, and where the reader might get the wrong impression.
6) The claim that the particles are 'localized' appears overstated, since the particles are already confined in the harmonic trap. It is simply that the effective mass of the center of mass is modified due to the light-matter coupling, and the 'localization' would not occur if the particles were not already trapped. In this sense, the 'localization' is very different from that found in Ref. [88] (as referenced on page 12), which considered many-body localization in the presence of disorder.
7) The authors also derive an effective interaction between the matter particles. However, this is simply a rewritten form of the modification of the harmonic oscillator potential for the CM mode. Since $R^2$ contains terms like $r_i \cdot r_j$ for the i'th and j'th particles, this looks like an infinite-ranged dipole-dipole type interaction. In some sense, this trades a simple problem, that of a cavity mode coupled to the CM, for a complicated problem, that of N matter particles interacting via a complicated interaction. Therefore, it is not clear that there is any scenario in which it is useful to write the $R^2$ term in this manner. It certainly does not lead to any physics that is not already described by the cavity coupling to the CM mode.
There have also been many previous works on mediated interactions in cavities, and trying to engineer these such that they become non-trivial. For instance, I would refer the authors to PRX 8, 011002 (2018) and references therein.
8) I do not understand the claim that the photon occupation in the polariton ground state corresponds to 'virtual' photons? I have superposition of photons and CM motion, and there is a real occupation of photons in the ground state, i.e. the photon annihilation operator acting on the ground state is non-zero, $a|g.s.>\neq 0$. Due to the $A^2$ term in the Hamiltonian, we also have $a^\dagger|g.s.>\neq 0$, is that what the authors mean by 'virtual'?
9) Likewise, the authors never consider temperature, so how can they have thermal photons? Yes, thermal photons are bunched, but this does not imply that bunched photons are necessarily thermal. The fact that photons become bunched has previously been argued in a very similar model, see ACS Photonics 2017, 4, 9, 2345.
10) The authors have not defined the parameters $m$ and $\Omega$ that appear in the Hamiltonian.
Author: Vasil Rokaj on 2022-12-22 [id 3176]
(in reply to Report 1 on 2022-10-09)
We would like to thank the referee for taking the time and putting the effort to review our manuscript. In the revised manuscript, we have taken all comments and concerns into consideration and we have accordingly modified the manuscript. Thus, we hope that the referee supports our manuscript for publication in Scipost Physics.
To ease reading, in the link below, we additionally provide a pdf file with a detailed reply to all the points raised by both referees and the requested changes. Further, in the reply letter we have kept the statements of the referees in italics. Also we provide a list of the most important changes we have made as well as a version of the manuscript with the changes in red.
https://drive.google.com/file/d/113EdH-C1qQg9_EXLq4DtPVD_EF_HEmIK/view?usp=share_link
Before we proceed, we would like to take the chance and highlight some of the key findings of our work that have not been discussed elsewhere. Our paper provides analytical insights to strong and ultrastrong light-matter interactions for the emerging field of cavity QED materials and addresses several important points which have emerged due to experimental observations, both in condensed matter and molecular systems coupled to cavities (see Refs.[2,3, 25, 61] in the manuscript). Many of the open questions in this field relate to the impact of polariton formation in the ground state of a many-particle system and in this work we take the first steps towards addressing them. For example:
1)Can the cavity field modify the ground state density of matter? [See Sec.4.3]
2)Are there collective phenomena in the ground state? [See Sec.4.3]
3) How the resonance between light and matter manifests in the quasiparticle properties? [See Sec.4.1]
4)What are the induced interactions in the matter subsystem due to the light field and the matter-mediated correlations between photons? [See Secs.4.2 and 5 respectively]
5)How can the polaritons be controlled with external probes? [See Sec.7]
Reply to the referee's overall report
We hope that by clarifying the main issues raised, i.e.: (i) that the cavity field modifies the ground state density of the many-body system, and (ii) that the cavity mediates pairwise long-range interactions among the particles, the referee supports our manuscript for publication in Scipost Physics.
The referee in her/his assessment is correct that the cavity field is decoupled from the relative motion of the particles. However, from this point, the referee reaches the conclusion that the cavity is decoupled from any matter interaction effects and does not modify any of the properties of the many-body system, such as its density. In contrast, our calculations presented in the revision demonstrate that the cavity mediates an enhanced localization into the density of the matter subsystem and modifies the density.
To render our claim transparent and demonstrate the effect of the cavity-induced interactions ($V_{\textrm{cav}}$ see Eq.(16) in the main text) for the ground state of the many-particle system, we have now performed one-dimensional simulations using the mean-field approximation, in the case of no Coulomb interaction and for trapped bosons spanning particle numbers ($N$) from 10-$10^4$ particles. We computed the deviations between the ground state density of the bosons with and without the cavity-induced interactions at resonance ($\gamma_2=1$) which are shown in Fig.~5(a) [in the manuscript] for various particle numbers. The coupled system's density increases around the trap center and decreases at the edges which clearly indicates an enhanced localization tendency. This result demonstrates without ambiguity that the cavity modifies the ground state density of the matter subsystem, a result can be certainly amplified with a multimode cavity.
Importantly, these modifications get amplified for larger number of particles as it can be deduced by inspecting the maximal density deviations occurring at $x=0$ with respect to $N$, see Fig.~5(b)[in the manuscript]. This observation implies that the enhanced localization of the matter system is a collective, Dicke-type phenomenon taking place in the ground state. This an important finding as it generalizes Dicke collectivity from being an excited state phenomenon in the spontaneous emission [see Ref.[72] in the main text] to a ground state phenomenon as well. To the best of our knowledge such a collective ground state behavior has not been demonstrated before.
In the revised version of our manuscript we have added Sec. 4.3 where we present the cavity-induced modifications in the ground state density of the system. We hope that with these additional computations the referee can reconsider his/her conclusion that the cavity does not modify any of the properties of the many-body system, such as its density and recommends now our paper for publication in Scipost Physics.
Reply to the requested changes
1) The referee is correct that the minimal-coupling Hamiltonian in Eq.(1) corresponds to charged particles coupled to the electromagnetic field. It was not our intention to imply that the Hamiltonian in Eq.(1) (as it is) is also applicable to neutral particles, like atoms for example. We wanted to make clear that $g_0$ has units of electromagnetic charge. For electrons $g_0=e$, while for ions $g_0$ it will be an integer multiple of $e$. We have rephrased now this particular sentence in the revised version of our manuscript to avoid any further confusion. It is important to note however, that the minimal-coupling Hamiltonian in Eq.(1) can be generalized for the interaction also of atoms with the electromagnetic field, by including both positive and negative charges, which then constitute a neutral atom.
2) The experimental realization of harmonic trapping potentials for charged particles is well-known and was achieved several decades ago [see W. Paul Rev. Mod. Phys. 62, 531 (1990)]. These traps are known as quadrapole ion traps or simply Paul traps. Further, Paul traps have been used widely in cavity QED for the manipulation of individual ions with photons [see D. J. Wineland Rev. Mod. Phys. 85, 1103 (2013)]. Thus, our theoretical proposal to investigate charged systems in harmonic traps coupled to a cavity is based on well established experimental techniques and we believe it is experimentally realizable.
3) It is indeed well-known that in a harmonic trap, the center of mass (CM) decouples from the relative motion of the particles. However, for charged particles in a harmonic trap coupled to a quantized cavity field this fact, to the best of our knowledge has not been demonstrated before. In Ref. [64], Phys. Rev. Research 4, 013012 (2022), Coulomb interaction between the particles was not taken into account and there was no external trap. In Ref.[64] the Sommerfeld model of the free electron gas coupled to a cavity was solved, without the Coulomb interaction. The Coulomb interaction was considered then only as a perturbation to the exact solution. As a matter of fact the separation of the CM from the relative motion, under coupling to the cavity was not demonstrated rigorously. Only a comment was made about it in section IV. Thus, we believe this is an important message.
We would like to add, that it is not so important whether this result is surprising or not, but what are actually the physical consequences of the fact that the CM couples to the photon field. The most important physical consequence is that in the presence of the harmonic trap, due to the coupling of the CM to the cavity we have the emergence of collective polariton states. This phenomenon was not shown in Ref.[64] and it is one of the key results of this work. Such polariton states do not show up for the case of the free electron gas coupled to the cavity because the Hilbert space of the interacting system consists of product states between the electrons and the photons and there is only an effective dressing of the electrons via the photons, but no hybridization between light and matter and as a consequence no light-matter correlations. The solution for the free electron gas coupled to the cavity is provided in Appendix B where one can infer the differences to the solutions of Ref. [64].
4) We understand the referee's perspective, that in AMO and condensed matter physics the term ``many-body'' refers to interacting particle systems where the interactions depend on the relative distances among the particles. In the original version of the manuscript we borrowed the standard terminology from AMO and condensed matter physics to describe the cavity-induced interactions $V_{\textrm{cav}}(\mathbf{r}_i,\mathbf{r}_j)$. After, considering the referee's point we agree that the cavity-induced phenomena are more appropriately described as collective. For that purpose we have now adopted the title of the manuscript to Cavity Induced Collective Behavior in the Polaritonic Ground State. Also we have modified the overall presentation of the manuscript and interpretation of our results in order to bring in the perspective of cavity-induced collective phenomena and interactions, without using many-body terminology. We believe this improves the manuscript considerably and we thank the referee for raising this point.
5) It is of course true that the density of the system and CM density profile are not the same. In the manuscript we have been careful and we have not used these terms interchangeably. The CM is a collective degree of freedom, as it depends on the position of all the particles. Thus, through the CM, the cavity field affects the full system and modifies the ground state density of the system. The modification of the density of the matter subsystem by the cavity field is shown in Fig.5 [in the manuscript]. Please see our reply in before the Reply to the requested changes for a detailed discussion. In the revised version of our manuscript we have added Sec. 4.3 where we present the cavity-induced modifications in the ground state density of the system.
6) The referee is correct that the particles are indeed localized in the harmonic potential. What the cavity field does to the system is to enhance the localization through the coupling to the CM. To clarify this phenomenon we have replaced in the revised version of our manuscript the term cavity induced localization to cavity enhanced localization. This indeed describes more appropriately the effect of the cavity field on our system.
In addition, the referee is accurate that the in Ref.[88] the disorder is a key ingredient for the many-body localization. This indeed makes the setting of Ref.[88] quite different from ours. To avoid any such confusion we have now removed from the manuscript Ref.[88] in order to avoid any confusion or misinterpretation of our results. We thank the referee for pointing this out.
7) As suggested by the referee, the problem of $N$ particles in the harmonic trap interacting with a cavity (even without the Coulomb interaction) is an inherently difficult problem because all the particles couple collectively to the cavity field and an analytic solution of the full system is not possible. The key observation in our paper, is the fact that the light-matter processes induced by a homogeneous cavity field occur at the CM of the matter subsystem allowing us to obtain analytical insights for the emergent polariton states. Additionally, our analysis reveals that the localization of the CM density profile is caused by the cavity and can be understood using a purely matter Hamiltonian without having to treat also the photons. Then, by transforming back to the ``lab frame'' we can identify the actual mediated interactions by the cavity field. However, still the relative part of the system needs to be solved [see Appendix A, in the revised manuscript, for $\hat{H}_{\textrm{rel}}$] in order to understand and characterize the behavior of the full system.
Most importantly, by deriving the effective matter Hamiltonian in the lab frame we can make use of available ab initio methods such as Quantum Monte Carlo and multiconfigurational approaches, in order to obtain the exact many-body ground-state of the matter subsystem without treating explicitly the photonic Hilbert space. We remark that tackling a many-body system coupled to a cavity while accounting for photon-matter correlations is a great challenge and currently exact many-body methods do not exist. In this sense, the effective matter Hamiltonian becomes important because it provides the possibility to treat and access the ground-state of a many-body system in a trap coupled to a cavity using the existing state-of-the-art methods.
To demonstrate that this is indeed possible we have now implemented the cavity-induced interactions $V_{\textrm{cav}}(\mathbf{r}_i,\mathbf{r}_j)$ and performed numerical computations using the mean-field approximation for large number of particles. The cavity-induced modifications in the ground state density is shown in Fig.5 in the manuscript. Please see also our reply before the Reply to the requested changes where we provide a detailed discussion. In the revised version of our manuscript we have added Sec. 4.3 where we present the cavity-induced modifications in the ground state density of the system.
8) This is an interesting point that the referee raises. So let us clarify what we mean by virtual photons. The photons in the ground state are called virtual because these photons are not freely propagating but they are actually bound in the polaritonic ground state. Further, the ground state photons cannot decay because there is no lower energy state in the system. Thus, as long as the system is considered to be closed these photons cannot escape and what they do is to actually mediate interactions among the particles. This can be understood in our work from the cavity induced potential $V_{\textrm{cav}}(\mathbf{r}_i,\mathbf{r}_j)$ derived in Eq.(28). This is in analogy to the virtual photons considered in Feynman diagrams which mediate the Coulomb force. However, in a lossy cavity the photons in the polaritonic ground-state will eventually leak out of the cavity and will become measurable. On this regard the photons in the polaritonic ground state differ from the standard virtual photons in Feynman diagrams which cannot be measured. This is an important difference and we agree with referee that the term virtual ground state photons becomes then misleading. Thus, in the revised version of our manuscript we refrain from using this term such that no further confusion arises.
9) What was meant in the manuscript by the term thermal photons was the fact that the photons in the polariton ground state follow super-Poissonian statistics. Thermal photons obey super-Poissonian statistics as well, and consequently one would not be able to actually tell the difference between the photons in the polariton ground state and actual thermal photons by measuring their statistics. The referee however is correct, that since we do not consider temperature in our system this leads to confusion or misinterpretations. In the revised version we no longer use the term thermal photons but rather super-Poissonian and bunched which are appropriate in our setting. We thank the referee for raising this point and for bringing to our attention the paper by Garziano et al ACS Photonics 2017, 4, 9, 2345 where they see similar effects of bunching due to ultrastrong light-matter coupling. We mention now in the revised version of our manuscript the paper by Garziano et.al.
10) The parameters $m$ and $\Omega$ are defined in Fig.1 where the setting of our problem is set. However, following the referee's suggestion, in the revised manuscript we have now included the definitions for $m$ and $\Omega$ also below Eq.(1) were the Hamiltonian of the system is described.
Author: Vasil Rokaj on 2022-12-22 [id 3175]
(in reply to Report 2 on 2022-11-11)We would like to thank the referee for taking the time and putting the effort to review our manuscript. In the revised manuscript, we have taken all comments and concerns into consideration and we have accordingly modified the manuscript. Thus, we hope that the referee supports our manuscript for publication in Scipost Physics.
To ease reading, in the link below, we additionally provide a pdf file with a detailed reply to all the points raised by both referees and the requested changes. Further, in the reply letter we have kept the statements of the referees in italics. Also we provide a list of the most important changes we have made as well as a version of the manuscript with the changes in red.
https://drive.google.com/file/d/113EdH-C1qQg9_EXLq4DtPVD_EF_HEmIK/view?usp=share_link
Before we proceed, we would like to take the chance and highlight some of the key findings of our work that have not been discussed elsewhere. Our paper provides analytical insights to strong and ultrastrong light-matter interactions for the emerging field of cavity QED materials and addresses several important points which have emerged due to experimental observations, both in condensed matter and molecular systems coupled to cavities (see Refs.[2,3, 25, 61] in the manuscript). Many of the open questions in this field relate to the impact of polariton formation in the ground state of a many-particle system and in this work we take the first steps towards addressing them. For example:
1) Can the cavity field modify the ground state density of matter? [See Sec.4.3]
2) Are there collective phenomena in the ground state? [See Sec.4.3]
3)How the resonance between light and matter manifests in the quasiparticle properties? [See Sec.4.1]
4) What are the induced interactions in the matter subsystem due to the light field and the matter-mediated correlations between photons? [See Secs.4.2 and 5 respectively]
5) How can the polaritons be controlled with external probes? [See Sec.7]
Reply to the referee's overall report
We are glad that the referee finds our manuscript ``\textit{well-written, enjoyable and the results obtained valid and interesting}''. However, we would like to emphasize that the system under consideration is not a trivial one-body problem. Our model consists of $N$ particles in a harmonic trap coupled to a cavity field, which couples collectively through the center of mass (CM) of the $N$ particles, leading to the emergence of collective polariton states. The polariton formation has fundamental implications for the many-particle system. One of our key results is that the cavity field renormalizes the scalar trapping potential and induces pairwise long-range interactions which are described by $V_{\textrm{cav}}(\mathbf{r}_i,\mathbf{r}_j)$ in Eq.(16) of the main text.
To demonstrate the effect of the cavity-induced interactions ($V_{\textrm{cav}}$ see Eq.(16)) for the ground state of the many-particle system, we have added Sec. 4.3 where we perform mean-field simulations, in the case of no Coulomb interaction for trapped bosons ranging from 10-$10^4$ particles. Calculating the deviations between the ground state density of the bosons with and without the cavity-induced interactions at resonance ($\gamma_2=1$) [see Fig.~5(a) in the manuscript] for for varying particle number we are able to identify an enhanced localization tendency. This result clearly demonstrates that the cavity alters the ground state density of the matter subsystem.
Importantly, we explicate that the above-described modifications satisfy a power law dependence with respect to the particle number, which depends on the strength of the light-matter interaction. For weak coupling we have a linear behavior, $\sim N$, while for strong light-matter coupling a square root, $\sim \sqrt{N}$, as it is shown in Fig. 5(b) in the manuscript. This implies that we have a collective, Dicke-type phenomenon in the ground state density, thus generalizing Dicke collectivity from being an excited state phenomenon in the spontaneous emission [see Ref.[72] in the main text] to a ground state phenomenon as well. To the best of our knowledge such a collective ground state behavior has not been reported elsewhere.
We hope that the referee will find the revised version of our manuscript substantially improved and clearly conveying the important new results which show the non-trivial, collective character of the cavity-induced effects such that he/she finds it now suitable for publication in SciPost Physics.
Reply to the requested changes
1) We have changed the title of the paper to ``Cavity Induced Collective Behavior in the Polaritonic Ground State''. We believe this title is more appropriate for describing the properties of our system and highlights collectivity, which is a key aspect of our work. We thank the referee for raising this point.
2) The center-of-mass (CM) and relative distance coordinates defined in Eq.(3) are merely a symmetrically scaled version of the coordinates that referee mentions in her/his report by merely a multiplicative prefactor. We define the CM and relative coordinates in such a symmetric fashion with respect to the $1/\sqrt{N}$ for mathematical convenience, as it is done for example for the two-particle problem in Ref.[80] (in the manuscript). These coordinates are equivalent to the ones the referee has in mind. They are canonical, independent and facilitate the separation between the CM and the relative coordinates. Since these coordinates are mathematically and physically equivalent we find that redefining our coordinates is not necessary. We have added a brief comment and a citation to Ref.[80] in the revised manuscript before defining the CM frame in Eq.(3), which highlights the symmetric definition of our coordinates.