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

### Submission summary

 As Contributors: Jianhua Zeng Arxiv Link: https://arxiv.org/abs/2109.05515v2 (pdf) Date submitted: 2021-10-13 08:45 Submitted by: Zeng, Jianhua Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

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

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

### List of changes

Point 1: "We find a new exact solution as bright solitons..."
"...exact solution has not been reported before for purely quadratic nonlinear model (without harmonic trap)."

Reply: "We find a new exact solution as bright solitons..." have been revisited as “We find an exact solution as bright solitons (which can be derived from the 1D quantum droplets by ignoring the mean-field term [28, 43, 44])”.
This statement "...exact solution has not been reported before for purely quadratic nonlinear model (without harmonic trap)." has been deleted.

Point 2: The relation between the chemical potential and the atom number is also known analytically. The shapes of LHY solitons for different values of the chemical potential are obviously self-similar. Checking these results numerically (Fig.1) does not merit publication.

Reply: We think it is necessary to show the shapes of 1D LHY fluids, since they are different from the previous obtained exact solutions for 1D quantum droplets in Ref. [28, 43, 44] where the mean-field cubic term keeps always, while here in our model we just consider the LHY term.

Point 3: I repeat that I do not understand the red dashed line in Fig.1 and the statement on the "very limited stability interval for fundamental modes". There is a problem either with validity or with interpretation. The authors basically argue that the soliton can be unstable in free space and that there is a stability threshold (since they speak about a "stability interval"). That the soliton is stable has been shown in Ref.[44] and the absence of a threshold follows from the property of self-similarity. Figure 4f (in the new version) requires more explanations. In what sense does it show that the soliton is unstable?

Reply: To facilitate the reading, the statement on the "very limited stability interval for fundamental modes" concerning the red dashed line in Fig.1 has been revisited as “where includes also the linear-stability results expressed as maximal real values of eigenvalues $\lambda_R$ vs $\mu$ , it is seen from the latter that the fundamental modes have a very limited stability interval, recalling that the solutions are stable only when $\lambda_R=0$.”
We stress once again that the stability property of 1D LHY fluids is very different to that of 1D quantum droplets. In evaluating the stability property of LHY fluids, the stability is first checked in linear stability analysis method and further confirmed in direct perturbed simulations, and both methods can match well; and we find that the linear stability analysis results for the fundamental modes have a very limited interval for $\lambda_R=0$. While for the 1D quantum droplets, the focusing and defocusing competing nonlinearities could help to improve the stability of their modes.
Concerning Figure 4f , we state that “an unstable and decaying localized mode (the corresponding number of atoms reduces) like the one in Fig. 4(f) could not maintain good movement since the loss of coherence of a localized mode (soliton).”, thus the soliton is unstable since the total number of atoms diminishes during the evolution.

### Submission & Refereeing History

Resubmission 2109.05515v2 on 13 October 2021
Resubmission 2109.05515v1 on 14 September 2021
Submission scipost_202107_00032v1 on 20 July 2021

## Reports on this Submission

### Strengths

1. Figure 6 is interesting and provides some physical intuition for the collision physics.

### Weaknesses

1. No clear discussion of main advances over previous published works on the subject where the vanishing mean field limit can be directly ascertained by sending a parameter to zero in equations.
2. Limited physical motivation provided for the reported lack of stability for the fundamental modes.
3. Discovery of an exact solution for the fundamental mode of LHY quantum fluids is still reported in the abstract and conclusions.

### Report

While the authors have significantly toned down claims in earlier versions of a new exact solution for the profile of Lee-Huang-Yang droplets, it still appears in the abstract and conclusion, i.e. "An exact solution is found for fundamental LHY fluids" and "An exact solution for fundamental mode of LHY fluids without any external trap was derived". As pointed out by the second referee, Eq. (9) is a parametric limit of a previously derived expression in Ref. [28] (Eq. (16)).

In my opinion, the authors have also failed to satisfactorily address the concerns of the second referee around the stability of the fundamental mode. The authors claim that "the stability property of 1D LHY fluids is very different to that of 1D quantum droplets" but don't back this up in the manuscript with any specific reasoning. For example, can it be shown how this stability is recovered perturbatively in the prefactor of the quartic term? I am also somewhat concerned by the two kinks in the dashed line in Figure 1. What is the origin of these? Their presence could indicate an issue or instability in the solution.

I also have similar concerns to the previous referees on readability. While the authors have made improvements, there are still residual issues, for example:

1. what is the yellow dashed line in Figure 1?
2. Below Eq. (13) there is a sentence on :"shape's comparison" between numerical and approximate solutions. I don't see such a comparison in the figure.
3. The inequality \mu > -1.5 seems flipped to me, I see a discrepancy for smaller values of \mu. This sentence should be clarified.
4. The organization of panels in Figure 4 is very confusing. Perhaps an additional labelling scheme for trap vs. no trap is needed.

• validity: low
• significance: low
• originality: low
• clarity: low
• formatting: reasonable
• grammar: below threshold

### Report

I am not convinced by the author's response. They want to publish the article at any cost. I am pretty sure that there is a problem with validity and they refuse to check it seriously. I do not buy the argument that the point of vanishing mean field is special in terms of stability.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -