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Bose-Einstein condensate in an elliptical waveguide
by Luca Salasnich
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Submission summary
| Authors (as registered SciPost users): | Luca Salasnich |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202109_00024v1 (pdf) |
| Date submitted: | 2021-09-18 16:40 |
| Submitted by: | Salasnich, Luca |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Phenomenological |
Abstract
We investigate the effects of spatial curvature for an atomic Bose-Einstein condensate confined in an elliptical waveguide. The system is well described by an effective 1D Gross-Pitaevskii equation with a quantum-curvature potential, which has the shape of a double-well but crucially depends on the eccentricity of the ellipse. The ground state of the system displays a quantum phase transition from a two-peak configuration to a one-peak configuration at a critical attractive interaction strength. In correspondence of this phase transition the superfluid fraction strongly reduces and goes to zero for a sufficiently attractive Bose-Bose interaction.
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Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2021-11-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202109_00024v1, delivered 2021-11-04, doi: 10.21468/SciPost.Report.3796
Report
This paper presents a theoretical study of a Bose-Einstein condensate confined in a quasi-uni-dimensional elliptic wave guide. The ellipticity of the guide induces an effective potential which is a periodic double well. The ground state is determined by numerically solving the mean field Gross-Pitaevskii equation. A quantum phase transition is identified in which most of the condensate spontaneously locates into one of the two identical minima of the potential.
Although the setting studied is elegant and may enable to study interesting aspects of the interplay between Bose condensation, superfluidity and an additional spontaneous symmetry breaking, I do not consider that this work presents enough new and interesting material.
(1) In particular, the fact that the author uses only a mean field approach is, according to me, a drawback, considering that beyond mean field corrections have already been addressed long ago in similar settings (Refs. [12] and [13] for instance).
(2) Also, I am skeptical concerning the use of Leggett's formula for determining the superfluid fraction in the absence of interatomic repulsion (i.e. when gamma=<0).
2a. It is known for instance that a uniform non interacting BEC is not superfluid, whereas Fig. 4 seems to indicate that the system is 100% supersolid when gamma=0 (i.e., in the absence of interaction).
2b. In the same line: at T=0 and gamma=0 the uniform system reaches a quantum tricritical point and the existence of a superfluid phase for negative s-wave scattering length depends on the contribution of higher order terms [see e.g., Zwerger, J. Stat. Mech., 103104 (2019)].
For these reasons, I think that the results presented in Fig. 4 should be considered with caution, or at least require a more extended discussion.
Considering the comments (1) and (2) above, I do not believe that the quality of this work warrants publication in SciPost Physics.
Report #2 by Anonymous (Referee 4) on 2021-11-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202109_00024v1, delivered 2021-11-04, doi: 10.21468/SciPost.Report.3795
Report
The paper reports sufficiently interesting results which may draw interest of readers, and are recommended for the publication, provided that some amendments will be made, as summarized below.
A technical comment is that the underlying model is not formulated in a sufficiently clear form. Namely, in Eq. (7) the effective potential is written in terms of \kappa, which is defined in Eq. (15) as a function of \phi, but \phi is not defined as a function of coordinate s. Rather, s is defined as a function of \phi by Eqs. (12) and (13). Taken together, these implicit definitions seem confusing.
An essential comment is about what is called quantum phase transitions in the manuscript. First, in the case of the zero eccentricity, this transition is nothing else but the commonly known onset of the modulational instability in the NLS equation with periodic boundary conditions, in the absence of an external potential. It seems somewhat strange that MI is not mentioned, and the commonly known threshold for the onset of MI on the circular ring is not referred to. The main result, reported in the paper, viz., the transition to the single density peak in the elliptic ring, i.e., spontaneous symmetry breaking of the ground state in the dual-well potential, is quite interesting, but it will be relevant too to state how the ellipticity affects the onset of the MI on the ring.
Author: Luca Salasnich on 2021-11-21 [id 1963]
(in reply to Report 2 on 2021-11-04)
I thank the Referee. In the new version of the paper I shall try to better explain how to calculate the
effective potential: the plot \kappa vs s is obtained calculating \kappa vs \phi and also s vs \phi.
\phi is a dummy variable here. Unfortunately, an explicit analytical formula of \kappa as a function of s is not available. In the new version I shall also explain the relevant role of the modulational instability (of the Bogoliubov elementary exacitations) which triggers the spontaneous symmetry breaking.
Report #1 by Anonymous (Referee 5) on 2021-11-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202109_00024v1, delivered 2021-11-04, doi: 10.21468/SciPost.Report.3794
Report
The paper reports sufficiently interesting results which may draw interest of readers, and are recommended for the publication, provided that some amendments will be made, as summarized below.
A technical comment is that the underlying model is not formulated in a sufficiently clear form. Namely, in Eq. (7) the effective potential is written in terms of \kappa, which is defined in Eq. (15) as a function of \phi, but \phi is not defined as a function of coordinate s. Rather, s is defined as a function of \phi by Eqs. (12) and (13). Taken together, these implicit definitions seem confusing.
An essential comment is about what is called quantum phase transitions in the manuscript. First, in the case of the zero eccentricity, this transition is nothing else but the commonly known onset of the modulational instability in the NLS equation with periodic boundary conditions, in the absence of an external potential. It seems somewhat strange that MI is not mentioned, and the commonly known threshold for the onset of MI on the circular ring is not referred to. The main result, reported in the paper, viz., the transition to the single density peak in the elliptic ring, i.e., spontaneous symmetry breaking of the ground state in the dual-well potential, is quite interesting, but it will be relevant too to state how the ellipticity affects the onset of the MI on the ring.
Author: Luca Salasnich on 2021-11-21 [id 1964]
(in reply to Report 3 on 2021-11-04)I thank the referee who thinks that the setting studied is elegant and may enable to study interesting aspects of the interplay between BEC and superfluidity. In the new version of the manuscript I shall discuss the role of beyond-mean-field corrections (also in the case \gamma=0): the calculations are obtained in a regime where both beyond-mean-field and transverse- size effects are very small. Regarding the absence of superfluidity for \gamma = 0, this is true in the thermodynamic limit. However, in a ring there is a finite energy gap between the ground state and the first excited state also for \gamma = 0. In the new version of the paper I shall discuss this relevant issue related to the fact that the Bose system I am considering has a finite size because it is confined in a finite elliptical ring.