Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

Round 1

Reviewer 1 Report

In this paper, the authors claim to find 1D quantum dark soliton solutions corresponding to classical dark solitons of Gross-Pitaevskii equation. The authors obtain the single dark soliton with a non-zero winding number in terms of the Bethe ansatz formulation and compare to the known classical solutions. They also discuss double dark-solitons. The paper is carefully written and quality of the figures are sufficient. The whole discussions are proper and transparent. This paper will be useful for readers who is interested in classical-quantum correspondence of soliton models.  

The analysis in the present paper relies on several prior well-established methods/knowledges, and the originality of the current authors seems not so apparent. This work may be regarded as a kind of extension/application of the previous known methods.

Though the analysis is less original, the results and discussions are reasonable and certainly reliable ones. Therefore, I think it is well worth to publish.

Before moving to the publication process, if possible it is better to incorporate following issues for readers benefit.

(i) The authors should give some comments for dependency of the position locus of the double quantum solitons to the results. If obtained solutions are truly some kind of solitons, it should involve the essential information.

(ii) In some figures, the comments are not enough for our understanding. For example, in Figure 3 the authors do not mention which equation is used for evaluation of the density profiles (red lines). Also, in Figures 4, 6, 7, 9, 10, some oscillating behavior are repeatedly observed but no comments exist for them. I hope the authors add the explanations.

(iii) I think one of the new insights of this paper is construction of the quantum state for the case of nonzero winding number Eq.(28). The validity of the prescription (the summation shifting) should be elaborated in more detail.

(iv) The authors should mention a novel feature in their obtained quantum solitons, if exists.

(v) There are some typos in the draft. For example, the miss-spelling of name “Schrödinger” is not justified for any physicists.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

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Report of the Referee – universe-1483664 / Kayo Kinjo,

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This manuscript presents the quantum dark solitons in the 1D Bose gas from single to double dark-solitons, where quantum double dark-solitons by constructing corresponding quantum states in 2 the Lieb-Liniger model for the one-dimensional Bose gas. They expect that the Gross-Pitaevskii equation should play a central role in the long distance mean-field behavior of the 1D Bose gas, where they introduce novel quantum states of a single dark soliton with a nonzero winding number. They show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N - 1) particle states, while for elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions in the weak coupling regime. They show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. After they assigned the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, they obtain double dark-solitons of distinct narrow notches with different depths, then suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

However, they did not consider the interaction depending time and space case induced Feshbach resonance in experiments, for example, dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential [Phys. Rev. Lett. 94, 050402 (2005)]. They also did not consider two dimension case and new exact solutions, for example, quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity [Phys. Rev. A 81, 025604 (2010)], matter rogue wave in Bose-Einstein condensates with attractive atomic interaction [European Physical Journal D 64, 473 (2011)], exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates [Phys. Rev. A 72, 033611 (2005)].

I can recommend the revised version of this manuscript for publication in Universe after authors corrected it.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

In this manuscript provide a lengthy presentation that accomplishes two goals.  First, they review many of the results on quantum soliton solutions from the literature and compare with the GP approximation results.  This provides some welcome context for theoretical physicists unfamiliar with this literature.  Second, they derive and explore some new results for multi-soliton solutions in the repulsive case, as well as some new results with non-zero winding number.  The paper appears to be on sound technical footing and provides some interesting results of interest to researchers in this field and working with systems such as BECs.  I recommend publication.  Below are a few comments for the authors to consider:

-On p.4, Eq.(7) I could not identify what is meant by S_N in one of the summations.  Please clarify the notation in this equation. 

-In Fig. 2, I could not find for what value of nonlinearity c these results were for.

-The small oscillations that appears in the exact quantum solutions (first appearing in Fig. 4 but later as well), and which don’t appear in the mean field approximations, were interesting and new to me.   Some physical intuition for these and some comment on whether any experimental systems are in a regime where these should be visible would be helpful.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Referee report on the article entitled “Quantum dark solitons in the 1D Bose gas: From single to double dark-solitons” by Kayo Kinjo, Eriko Kaminishi, Takashi Mori, Jun Sato, Rina Kanamoto and Tetsuo Deguchi.  
 
In the present work, the authors construct quantum states characterizing single and a pair of dark (effectively grey) solitary waves. Their analysis is based on suitable superpositions of the eigenstates  containing a single (yrast states) or a pair of particle-hole excitations of the Lieb-Liniger  model. A comparison of the predictions of their method with others already reported in the literature for quantum dark solitons is also provided and differences/similarities are exposed at the level of different observables, e.g. density and phase profiles of the ensuing nonlinear excitations. The same holds also regarding the results obtained from the widely used Gross-Pitaevskii equation which are discussed for different interaction regimes and atom numbers. Other interesting findings of this work include the derivation of the solution of elliptic multiple dark solitons and the generalization to the case of free fermions. The manuscript shows a plethora of technical details on the above-mentioned derivations. 

Overall, I find the results of this work interesting providing also an overview of some of the different methods that have been used to describe correlation properties of nonlinear excitation and in particular here the so-called quantum dark solitons. I also believe that the present study might be potentially helpful for upcoming investigations in this topic. Moreover, the manuscript is written clearly, at least mostly,  and the reader is able to follow. However, to my opinion, there are still some places which can be improved in the current version in order the manuscript to be on the one hand more self contained and also provide more concrete statements. Also, I believe that appropriate citation is missing at several places both regarding mean-field phenomena but most importantly the list of works for quantum solitons is certainly incomplete (below some examples are given but these are only limited ones). In this sense, I have a number of comments and questions regarding the presentation and the interpretation of the presented results. Therefore, if the authors are able to address my concerns by performing revisions according to the following recommendations/suggestions, the present work will be certainly suitable for publication in the Universe. Let me list below my suggestions for improvement: 

1) I would strongly recommend the authors to clearly comment in their Introduction the basic characteristics of quantum dark solitons and provide some additional relevant references. At the moment the reason why the quantum dark solitons are intriguing objects is not clear to the general audience. For instance, a fundamental property of quantum dark solitons is the filling of the dark soliton notch which stems from the presence of quantum depletion in the condensate. This property is in turn related to the quantum dispersion of the dark soliton position. In fact, the impact of correlations on more composite objects such as quantum dark-bright and dark-antidark solitons has been reported in some papers. Moreover, quantum dark solitons in fermionic ensembles have been investigated in some papers. Please include relevant comments. These are some characteristic references along with others that need certainly to be included in the work.                                                                                                                         

2) Moreover, I would strongly suggest the authors to explicitly mention in their introduction which results of the present work constitute new ones and which are reviewed.   

3) On page 2, third paragraph, in the statement “the density profile of a single grey soliton appears in the quantum measurement process” I would suggest also to include reference Phys. Rev. A 98, 013632 (2018) where the same conclusion has been drawn by relying on the emulation of the single-shot experimental process based on beyond mean-field simulations. In particular, the observation made in the aforementioned work renders the argument of the authors stronger since first it was done with a completely different approach and also in the presence of an external trap. 


4) In the abstract, the term double dark solitons is not clear. I would suggest to either explain that a soliton pair is meant or rephrase accordingly. 

5) On page 2, third paragraph, I would encourage the authors to briefly explain what is meant by the terminology elliptic soliton. This notation is not clear to a broad audience.  

6) The sentence in the lines 84-86 is not understandable to me. Please rephrase accordingly. 

7) In the Lieb-Liniger model it is common to use the so-called Lieb-Liniger parameter for characterizing the strength of the interactions. Then, one can naturally argue about weak and strong interactions. Why the authors do not adopt this notation here? 

8) In Figure 2 the density profiles of quantum solitons essentially coincide with those of classical dark solitons if I understand correctly. In principle I would not expect such a result but rather that the quantum depletion will always lead to a less black profile. Why this is not the case here?


9) Are the densities profiles shown, for instance, in figures 2 and 3 normalized to the particle number? Please clarify. 

10) Figures 4 and 5 present profiles of the square amplitude  and phase, among others, for c=10 and 100. I would expect that in these cases the 1D gas is strongly interacting and actually enters the fermionization regime. If this is correct, then how can the authors justify the validity of their treatment? Please elaborate. 


11) It is not clear to me (see section 4.3.2) whether the deviations of the double quantum dark solitons from the mean-field solutions are larger compared to the single soliton case. Please clarify. Along the same lines, can the authors make a prediction whether such deviations are maximised in the case of soliton lattices? 

12) On line 527, the equation number is missing. 

13) On line 534, I would suggest to include references New J. Phys. 19, 123012 (2017) and Phys. Rev. A 98, 013632 (2018) where the beyond mean-field dynamics of quantum dark solitons has been investigated.  

14) Finally, I wonder whether the method presented by the authors can be extended to dynamical situations since then an appropriate superposition of the many-body wave function needs to be considered during the time-evolution? Can the authors comment on this issue?  

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

The authors have adequately addressed the comments and suggestions raised in my previous referee report and included the corresponding changes in their revised manuscript. For instance, several conceptual issues have been clarified and further explanations are provided. Due to the above-mentioned reasons, I recommend the manuscript for publication in Universe in its current version.