Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons

Dear Referee,

We thank you for careful reading and for the positive opinion. Concerning the comments:

1. Given the qualitative similarity of the GPE and LLGPE linear dispersions shown in Fig. 2 and the respective discussion at the end of Sect. 3, it appears desirable to also have a plot of the respective sound velocities as functions of gamma as well as of the deviations between the exact and the approximated dispersions (maybe as insets). Is the apparent agreement between LLGPE and LL of the same quality in the low- and high-momentum limits? (The dispersion formula suggests identity at high p and dependence on the speed of sound at low momenta.

We fully agree with this comment. We have added a next figure (Fig.~3 in the current version), with two panels. In the left panel, we show the speeds of sound: the exact result, equivalent to LLGPE and an outcome of linearization of the GPE. In the right panel, we show relative deviations between excitation energy obtained from the LLGPE versus the exact result of the LL model. Indeed, the relative error for high momenta and low is tiny. Significant deviations are present for the intermediate momenta.

2.

When I came across Fig. 3 the question of how the widths depend on gamma immediately occurred to me. This is, partly, the subject of Fig. 6 and Sect. 5. Maybe pull the direct presentation of Fig. 6 to the front, into 4.2?

We have followed this advice. Indeed, we have discussed the widths already in Sec. 4.2. 3.

In the discussion of the left panel of Fig. 4, the authors point to the agreement between the LLGPE excitation energy and the LL type-II dispersion. They may want to briefly summarize the LL dispersions somewhere, also discussing the momenta where the type-II dispersion goes down again and touches zero at the "umklapp" momentum?

To address the comment (of both referees), we decided to give more details about the type-II excitations already in the introduction. To this aim, we improved the graphical abstract and described it in the text. We use more familiar names of these excitations (hole and particle excitations) already in the abstract. 4.

The functions $G_2$ and $G_3$ shown in Fig. 5 are not defined properly, i.e., it is unclear for which arguments they have been evaluated, given that they are defined by Eq. (28). In particular, assuming that the $G_2$ function is evaluated at two different positions, why can I compare it with the LLGPE density in the upper right panel? What would help in this context is to explain in some more detail how I should imagine the soliton solution to come out of the LL eigenstates. The spectral distributions (spectral functions) of the LL excitations calculated by Brand and Cherny show that for large gamma it is hard to distinguish type I and type II excitations, as the full band in between the respective energies is equally filled for a given momentum k while for smaller gamma, still larger than 1, the low-energy solutions are predominantly of type II. Is there any insight on relations between the properties of these excitations and the soliton excitations of the LLGPE?

We thank the Referee for these comments. Firstly, we clarified relations between the GPE /LLGPE solitons and the type-II excitations, adding two paragraphs of explanations to the text. The solitons should appear out of the type-II (solitonic) excitations in the course of the particle measurement. The measurement is understood in a broader sense, and it may be performed spontaneously, for instance, by particle losses (as discussed in Phys. Rev. Research 2, 033368 (2020)). The other typical way is just absorption imaging, which measures a high-order correlation function. On the other hand, the low weight of solitons in the spectral function for large $\gamma$ indicate that other methods, like the Bragg scattering, may be insensitive to solitons. 5.

In the final sentence of Sect. 4, the authors write that it "is alarming in the context of the hydrodynamical origin of the LLGPE" that "the LLGPE soliton is close to [the] $G_2$ of the fermionized gas". Could this be elaborated on a bit further? Where do I see this proximity/being close? Maybe, the scales in Fig. 5, lower right panel, should be expanded in an inset?

In the very last stage of the manuscript preparation, we introduced a bug in the figure. We are very sorry for this. Now, the figure is corrected. We hope that it is clear now that $G_2$ coincides with the LLGPE soliton density. Then $G_3$, with two density dips is of course totaly different. 6.

The authors refer to the formation of quantum droplets in a 1D system which have been the subject of intense research recently. For the less specialized reader, it appears desirable to have a slightly more detailed explanation of why the present research supports the theory used for describing quantum droplets. I understand that the NLSE used in Ref. 24 is of the same type as that used here, in the sense that the nonlinearity is at least similar, with a $|\phi|^6$ factor multiplying a non-polynomial function. In addition, however, the droplet theory also includes a non-local dipole-dipole interaction.

Indeed, this remark was solely related to Ref. [24]. Droplets found in [24] arise due to the competition between the local and dipolar forces AND the contribution of the beyond mean-field effects to the energy coming from strong local interactions. The role of the latter corrections is crucial for droplet formation. In this manuscript, we wanted to test such beyond mean-field corrections (also beyond Bogoliubov). Moreover,  in [24], we proposed the equation referring to the local density approximation. This manuscript's purpose was to understand the equation better, find appropriate interpretations, and benchmark it with an exact solution. We changed the Conclusions to indicate precisely the LLGPE scope and its relation to the work about the mentioned quantum droplet.

First, it should be clarified to what extent the two models are equivalent to each other (e.g., what would happen if one would approximate the $e_LL$ function in the same manner as it was done in Ref. 24). Secondly, it should be said, to what extent probing the solitary wave solutions of the LLGPE in the present ms. supports the specific model used for droplet formation. Are the solitons here of direct relevance for the droplet solutions there? Do they have the same kind of properties? And if yes, which specifically are relevant?

We had done such benchmarks way before submitting this manuscript. The model used in [24] is qualitatively but not quantitatively correct. We found the comparison, however, not appropriate for the core part of this manuscript. In the present version, we have commented on the relations and added comparisons to an appendix. The only role of solitons in this respect is in the discussion of the validity of the LLGPE. The hydrodynamical equations underlying the LLGPE proved useful for long-wave structures like phonons and limited applicability for short-wave ones like narrow solitons. Concerning the extension of the LLGPE with the dipolar interaction -- the droplet solution and its low-energy excitations should be in the validity range of the hydrodynamical approach. Still, we are trying to use more sophisticated methods, like DMRG, to check numerically the validity of solutions of the LLGPE extended by non-local interactions. That study is, however, beyond the scope of this manuscript.

7.

Minor points (typos etc): - footnote 2: it used rough - Sect. 4, 3rd word - p.11, 2nd par, l. 6: hexagons->pentagons - p.11, last sentence. - Fig. 5: legend boxes overlay each other. - Fig. 5, Caption, l.2: dyspersion - Fig. 5, Caption and figure: NI: do the authors mean "weakly interacting"? - Sect. 4, 3rd last line: soliton.Actually - Sect. 5, l.6: larger that - Sect. 5, 1st par, 2nd last line: around gamma=1/2: the authors mean "gamma ~ 1", I presume (the crossing is at gamma~pi). - Sect. 6, l.3: fundaments->foundations?

We thank you for your very careful reading of our manuscript and are sorry for so many misprints. Concerning: "Fig. 5, Caption and figure: NI: do the authors mean "weakly interacting" The result is practically the same for very weakly interacting gas (with the healing length larger than the size of the box $L$) and for the ideal gas. Although there are no solitons in the limit $g\to0$, soliton-like states exist. We discuss this paradox in Phys. Rev. A 97, 063617 (2018). We used results for the ideal gas, knowing that they do not differ from the weakly interacting system. We have corrected the rest of the typos.

We thank you once again for reviewing our manuscript and providing the useful recommendations,

Jakub Kopyciński Maciej Łebek Maciej Marciniak Wojciech Górecki Rafał Ołdziejewski Krzysztof Pawłowski

Dear Referee,

We thank you for careful reading and for the positive opinion. Concerning the comments: 1.

In the abstract the meaning of the so-called type-I and type-II excitations should be briefly clarified. Which are these excitations and what are their differences? Do they refer to particle and hole excitations related to phonons and solitons respectively? Also, an example needs to be provided for the long- and short-wave structures.

We have followed the comment of the Referee and clarified the meaning of the elementary excitations in the abstract of our work. With this respect, we have also changed the graphical abstract and the introduction.

2.

I would strongly encourage the authors to describe in the Introduction in more detail the major results of their study. Currently, the description is very short and it is focused on the topics that will be discussed and not the actual new results. Moreover, since the topic of quantum dark solitons is a main topic of interest in the main text I believe that it deserves to be mentioned even briefly in the Introduction. Here, relevant references besides the ones by Sacha should be included such as earlier studies where some of the authors participated but also from other groups (see e.g. Phys. Rev. A 98, 013632 (2018), New J. Phys. 19, 073004 (2017)). This way, the reader can be directed to them and have an overview both for the homogeneous and for the trapped cases.

The main goal of our work is the analysis of the generalized non-linear equation describing point-like interacting one-dimensional Bose gas for any interaction strength. We address the applicability of the equation and its strong and weak points. The main findings of our work are the following: +Coherent derivation of the LLGPE +Explicit analysis of the particle excitations by linearization of the equation and finding agreement between the LLGPE and the Lieb-Liniger model +Analysis of LLGPE dark solitons with respect to the GPE and the yrast states from the LL model +Analysis of the LLGPE limitations: there are such solutions of the LLGPE that are beyond the equation's validity range. This includes narrow solitons. One thus cannot now judge whether solitons in the strongly interacting 1D gas exist or do not exist. As suggested by the Referee, we have extended the description of the enlisted results in the Introduction. We thank the Referee for their valuable comment. According to their second remark, we have also added the discussion of quantum dark solitons in the first paragraph of the Introduction. From the perspective of the content of our work, the essential part of the body of the exhaustive literature on quantum dark solitons that should be introduced in the manuscript pertains to the correspondence between the effective single-particle Gross-Pitaevski equation for one-dimensional untrapped repulsive gas and the underlying many-body model given by the Lieb-Liniger model. We have added the relevant references providing a better overview of the topic than before.
The references suggested by the Referee study different, important problems concerning solitons with the ML-MCTDHB method. A question arises whether the LLGPE could be also used for investigating an impurity moving in a cold gas or the dynamics of dark-in-bright solitons. Although answering that goes beyond the scope of the current manuscript, this intriguing prospect warrants further study.

3.

In a related note a mentioning on the fact that droplets in one-dimension have been investigated already by using ab-initio approaches such as the quantum monte-carlo is missing. This should be certainly rectified and relevant references in one-dimension such as Phys. Rev. A 102, 023318 (2020), Phys. Rev. Lett. 122, 105302 (2019) and arXiv:2108.00727 need to be included if the authors want to keep the claims regarding the Lee-Huang-Yang energy correction. Currently only references regarding higher dimensions are provided while the focus of the present analysis is in one-dimension.

We thank the Referee for this remark. Indeed, we overlooked citing the relevant papers on 1D systems in the introduction. We have added suitable references. 4.

In the Introduction, the sentence “The figures of merit are their dispersion relations and ...” reads awkwardly. Please rephrase.

We thank the Referee for the meticulous reading of our manuscript. We have corrected the sentence.

5.

What is the physical interpretation of the pressure term given in Equation (4)? Please elaborate.

The role of pressure is analogous to that in classical hydrodynamic equations. To give the reader some intuition about the quantum effects occurring here, we have added after equation (17) a clear interpretation of the origin of this expression in the limit $\gamma\to\infty$, where fermionization occurs. The pressure may be thus seen as fermion degeneracy pressure.

6.

After equation (7) it is stated that the Gross-Pitaevskii theory is not justified for strongly repulsive atoms. I do not entirely agree with this statement. I think it should be complemented and mention that also in the weakly interaction regime the Gross-Pitaevskii theory can fail due to the presence of quantum correlations. There are several demonstrations of this fact especially regarding the dynamics of cold gases.

Indeed, the statement was too strong. We have rephrased the paragraph to stress both the importance of the GPE and its limitation

7.

At the end of equation (10) the “,” should be replaced by “.”

Fixed.

8.

The last terms appearing in equations (13) and (17) and representing beyond Gross-Pitaevskii corrections (if I understand correctly) as well as their role should be clearly explicated. Why these terms operate beyond a macroscopically occupied orbital and for which order/type of correlations do they account for? This is a central finding of the present work and should be clearly communicated to the reader.

This term is taken directly from the exact results of the Lieb-Liniger model. It represents the energy in a small volume in all for any interaction strength. Should we expand it in $\gamma$ around $\gamma\approx 0$, then the zeroth-order would be equivalent to the Gross-Pitaevskii energy. The following term would correspond to LHY correction, and the subsequent terms would mean higher-order corrections. On the other hand for large $\gamma$, one can expand $\mu_{\rm LL}$ in $1/\gamma$, but we do not know how to interpret the expansion terms physically. In particular in the case of strong contact interaction, there is no orbital occupied by almost all atoms, as assumed in the GPE equation. Still, the equation we used to describe the gas density and its velocity can be valid even in this regime.

9.

After equation (17) it is stated that equation (13) coincides with the equation proposed in Ref. [15] in the limit of $\gamma \ll 1$. I suggest to explicitly state some more details here regarding the equation provided in Ref. [15] in order the corresponding description to be accessible to a broad audience.

We agree with the Referee that a short discussion about the limit $\gamma \to \infty$ (we assume a typo in the comment above), including the equation, would be here in place. 10.

At the beginning of Section 4 the word “apperas” should read “appears”.

Fixed

11.

In Section 4.1 the case of a soliton in a Tonks-Girardeau gas is discussed among others. What is the value of the interaction strength g where this strongly coupled bosonic system is approached? It is also very important that the properties of the black solitons in this regime will be mentioned. Do they remain the same as in the weakly interacting case? Please elaborate at least briefly and also provide some relevant references if any to this topic such that the interested reader can be directed.

In section 4.1, we have added information that the Tonks-Girardeau limit is approached for $\gamma\geq 100$. At present, we briefly mention black soliton properties and have added a proper reference.

12.

On page 11, the sentence “We examine closer a relation between these excitations studying ...” is not understandable to me. Please rephrase.

We have removed the sentence and modified the subsequent paragraph accordingly. 13.

The legend of figure 5 is not displayed properly. Please fix this issue.

This problem occurred due to improper figure rendering. It has been already fixed.

14.

In the right panels of figure 5 the two- and three-body correlation functions are shown. It can be seen that in the non-interacting case both the two- and three-body correlations do not become zero at the center of soliton. However, in the Tonks regime there is a clear correlation hole probably due to fermionization. I wonder whether the finite value of $G_2$ and $G_3$ at the center of the soliton in the non-interacting case is related to the fact that quantum dark solitons have a filled core due to presence of correlation effects. Please comment. If this is not the case how the basic properties of quantum black solitons can be discerned within this approach. Here I mean properties such as the filled core of the dark soliton which is a manifestation of the presence of condensate depletion and in turn related for instance to the dispersion of the dark soliton’s position. These processes are already discussed in the already cited references and the ones suggested in question 3. A relevant discussion is necessary.

Indeed, the quantum effects are responsible for the fluctuations of the soliton centre in interacting gas, and, after averaging, the gas density increases inside the solitonic core. This effect was nicely presented in Phys. Rev. A 66, 043620 (2002). However, as shown in Phys. Rev. A 97, 063617 (2018), this effect does not emerge in the limit of vanishing interactions. Therefore, the origin of the effect shown in Fig. 5 is different. We simply presented only the $G_2$ and $G_3$ functions. In the higher-order correlation functions $G_m$ in the limit $m\to\infty$, the density will reach exactly zero (as rigorously proven in Phys. Rev. A 97, 063617 (2018)). 15.

Along the same lines, can the authors comment on the dynamical evolution of usual mean-field black soliton embedded in the LLGPE approach as an initial condition?

A usual mean-field soliton is narrower than the LLGPE soliton. In the dynamics of the system initiated in such a state, the shock waves appear. We have added examples of such dynamics in appendices, for $\gamma = 1$ and $\gamma = 100$.

16.

In the first paragraph of page 13 it is commented that higher order correlation functions have m local minima and not a single one as the common soliton structure. Is this observation a manifestation of the fact that quantum solitons may exhibit a splitting process or a dispersive behavior due to correlations? Please clarify.

We have shown that the $G_2$ function practically overlaps with the LLGPE soliton. Then $G_m$ will probably look like a collection of $m-1$ such solitons. Our main conclusion from the last plot is that the relationship between LLGPE and the quantum soliton remains a puzzle. The width of the soliton in the strongly interacting gas is of the order of the interparticle distance. Therefore such a state is beyond the validity regime of a hydrodynamical approach. It means that our LLGPE solitons and the solitons presented in Kolomensky et al. may not have a real, physical counterpart in the many-body quantum system.

17.

I would suggest to carefully check the entire manuscript for typos and grammar errors beyond the ones mentioned above. This is certainly needed and will be definitely beneficial for the reader.

We have reviewed the manuscript.

We thank you once again for reviewing our manuscript and providing the useful recommendations,

Jakub Kopyciński Maciej Łebek Maciej Marciniak Wojciech Górecki Rafał Ołdziejewski Krzysztof Pawłowski