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One-dimensional purely Lee-Huang-Yang fluids dominated by quantum fluctuations in two-component Bose-Einstein condensates

by Xiuye Liu, Jianhua Zeng

Submission summary

As Contributors: Jianhua Zeng
Arxiv Link: (pdf)
Date submitted: 2021-09-14 13:34
Submitted by: Zeng, Jianhua
Submitted to: SciPost Physics
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Quantum Physics
Approach: Theoretical


Lee-Huang-Yang (LHY) fluids are an exotic quantum matter emerged in a Bose-Bose mixture where the mean-field interactions, interspecies attraction $(g_{12})$ and intraspecies repulsive $(g_{11}, g_{22})$, are tuned to cancel completely when $g_{12}=-\sqrt{g_{11}g_{22}}$ and atom number $N_2=\sqrt{g_{11}/g_{22}}N_1$, and as such the fluids are purely dominated by beyond mean-field (quantum many-body) effect -- quantum fluctuations.Three-dimensional LHY fluids were proposed in 2018 and demonstrated by the same group from Denmark in recent ultracold atoms experiments [T. G. Skov,et al., Phys. Rev. Lett. 126, 230404], while their low-dimensional counterparts remain mysterious even in theory. Herein, we derive the Gross-Pitaevskii equation of one-dimensional LHY quantum fluids in two-component Bose-Einstein condensates, and reveal the formation, properties, and dynamics of matter-wave structures therein. An exact solution is found for fundamental LHY fluids. Considering a harmonic trap, approximate analytical results are obtained based on variational approximation, and higher-order nonlinear localized modes with nonzero nodes $\Bbbk=1$ and $2$ are constructed numerically. Stability regions of all the LHY nonlinear localized modes are identified by linear-stability analysis and direct perturbed numerical simulations. Movements and oscillations of single localized mode, and collisions between two modes, under the influence of different incident momenta are also studied in dynamical evolutions. The predicted results are available to quantum-gas experiments, providing a new insight into LHY physics in low-dimensional settings.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

We thank the two anonymous reviewers for reviewing our work and providing very useful comments which help to improve the quality of this paper.

List of changes

Reply to the First Reviewer—scipost_202107_00032v1

Point 1: In particular, I surmise that there are typos in Eqs. (1) and (3), as it seems strange that the second terms in those expressions do not include any density of the condensate.
Reply to this point: We think this is a misunderstanding. Actually, there are not typos in Eqs. (1) and (3), the density of the condensate is included in parameter “$c_{\pm}$” which is given in Eq. (2).

Point 2: By the way, what is meant by the "momentum integration" of Eq. (1)?
Reply to this point: The meaning of “momentum integration” is to integrate in momentum space, just like the ones usually do in real space to reduce the 3D space to the quasi-one-dimensional case in tackling Bose-Einstein condensate. Such term is not coined by us, and has been used in a seminal work by Petrov and Astrakharchik in Ref. [28], which we have cited to describe such term in the first version.

Point 3: Particularly, the statement that Eq. (9) is a completely new solution seems disputable, as it is a special case of a more general solution (16) for the "droplets", reported in Ref. [28].
Reply to this point: This comment is relevant. We have revised the corresponding expression and cited that paper. Our new description is attached here:” While based on the previous work for 1D quantum droplets in quadratic-cubic model in Ref. [28], discarding the cubic term in where can lead to an exact solution for our model Eq. (8) in the absence of harmonic trap”.

Point 4: Another inaccuracy is the reference to Fig. 5(b) as the formation of "two matter-wave breathers", while in reality one observes merger of colliding droplets into a single breather in this figure.
Reply to this point: This comment is relevant too. We have replaced the “two matter-wave breathers” by “one matter-wave breather”.

Point 5: Lastly, it is recommended to present results for collisions between the "droplets" in a more systematic form. In particular, it is relevant to find a critical collision velocity separating quasi-elastic collisions and the merger.
Reply to this point: This suggestion is very relevant. We have calculated all the possible cases and produced a new figure, which is the current Fig. 6 in this revised version; and in the section before conclusion, we add a paragraph to describe this figure, which is attached as follows:
“In Fig. \ref{fig6}, we have summed the collision scenarios of two LHY fluids with identical atom number $N_1=N_2=N$ in the absence of trap, expressed as a function of incident momentum $k_0$ and $N$, emphasizing the separation of merger and quasi-elastic collisions.”

Reply to the Second Reviewer
Point 1: The analytic droplet solution has been obtained in Ref.[28] in the general case, including the point of vanishing mean field.
Reply to this point: This comment is relevant. We have revised the corresponding expression and cited that paper. Our new description is attached here:” While based on the previous work for 1D quantum droplets in quadratic-cubic model in Ref. [28], discarding the cubic term in where can lead to an exact solution for our model Eq. (8) in the absence of harmonic trap”.

Point 2: The stability analysis (Bogoliubov modes) of the ground-state droplet solution in free space has been performed by Tylutki et al., Phys. Rev. A 101, 051601 (2020). The droplet is found stable. I thus do not understand the authors' conclusion on "very limited stability interval for fundamental modes" in this case. Accordingly, I also do not understand Fig.4d. The validity is questionable.
Reply to this point: Actually, our model is a purely quadratic attractive nonlinear model, very different from the quantum droplets with competing nonlinear models like the 1D case with attractive quadratic and repulsive cubic nonlinear terms (as in the mentioned reference, which has been cited as Ref. [44]), thus the stability property is different. In quadratic nonlinear model, the nonlinear localization effect cannot be simply duplicated from the cubic (Kerr) model either. To clarify this, our conclusion and the Fig.4d are appropriate.

Point 3: Figures 4b, 4c, 4e, and 4f seem to show trivial center-of-mass dipole oscillations in the harmonic trap. I see no monopole mode there. I thus doubt that this figure is useful.
Reply to this point: For the cases in Figs. 4b, 4c, 4e, and 4f, the oscillation was induced by initial momentum $k_0$, making the fundamental modes and higher-order ones out of equilibrium (but not evolve themselves into new possible localized modes), while they are still trapped by harmonic trap; such effect has been explained well in the original draft. Such effect can in a certain imply the robustness of the finding localized modes, and is thus useful, we keep it in this revised version too.
Concerning the second point about the monopole mode, we thank this reviewer for reading our draft very carefully and for stressing this misunderstanding point made by us. In this revised version, we improve our understanding and expressions related to the above subfigures. Since the monopole oscillations and the associated monopole modes are described by some very professional ways and which are out of our specialty, and we thus let them open; to this, we rewritten such expressions at the last sentence before talking about collision of two LHY fluids, where we previously stated that “We notice that the oscillations of 1D LHY fluids presented here, briefly speaking, resemble the monopole oscillations of a 3D LHY fluid confined in a spherical potential observed recently in a mixture of BECs Refs. [69] and [70].”, such expressions are deleted right now, and at the end of the Conclusion part, we have added a sentence to clarify this, which is stated as “We notice from Refs. [69] and [70] that the monopole breathing mode is the signature of LHY fluids, which deserves to be revealed in future studies in low-dimensional settings.”

Point 4: There is a significant overlap between this manuscript and the work of Astrakharchik and Malomed, Phys. Rev. A 98, 013631 (2018) who studied collisions between droplets in a slightly more systematic manner.
Reply to this point: The collisions between two LHY fluids indeed resemble some similarities as those for 1D quantum droplets in Astrakharchik and Malomed, Phys. Rev. A 98, 013631 (2018), which is cited as Ref. [43]. A flat-top profile is a signature of the quantum droplets, provided by the competing attractive quadratic and repulsive cubic nonlinear terms. To provide dynamical properties of the finding fluids, such collisions could tell the rich dynamics during colliding evolution, while the physical model for LHY fluids considered here keeps only the attractive quadratic term, which is radically different from the 1D quantum droplets presented in Ref. [43].

Point 5: There are many inconsistencies. Some examples are the absence of "yellow dashed line" in Fig.2d, no "shapes' comparison" in Figs.2a and 2b, no labels a, b, c, and d in Fig. 3, etc. The authors should have carefully read their draft before submitting.
Reply to this point: We thank this reviewer for reading our draft very carefully and pointing out inconsistent expressions. In this revised version, "yellow dashed line" in Fig. 2d has been corrected as “pink dashed line”; "shapes' comparison in Figs. 2a and 2b" were replaced by "shapes' comparison between Figs. 2a and 2b"; and the labels a, b, c, and d have been added to Fig. 3.

Point 6: The potential impact of the paper could be improved if the authors better explain physical motivation for studying higher-order modes. It would also help if they discuss stability of these modes in a more systematic manner. Are there stability thresholds for excited modes in Fig. 3? If yes, what is their physical meaning?
Reply to this point: The higher-order modes could tell more details about the robustness and rich dynamical properties of the underlying physical model. We have added four subfigures, denoted as Figs. 4b, 4c, 4g and 4h, to show the stability and instability of these higher-order modes in the current version. The stability thresholds can be seen from the maximal real values of the eigenvalues (perturbation growth rate, red dashed line) λR in Fig. 3, the higher-order modes are stable only when λR=0. The stability thresholds tell us the excitation of stable higher-order modes can be possible only within some certain value of chemical potential, and such condition changes with the variation of harmonic trap.

Submission & Refereeing History

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Resubmission 2109.05515v1 on 14 September 2021

Reports on this Submission

Anonymous Report 1 on 2021-9-14 (Invited Report)


The revised manuscript adequately addresses all the points from the original review. Thus, strengths remain the same as before, to which an essentyially clearer presentation is added.


Weakness mentioned in the original review have been fixed in the revised manuscript.


I think the revised manuscript meets all acceptance conditions.

  • validity: -
  • significance: -
  • originality: -
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