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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_00024v2 (pdf) |
| Date submitted: | 2022-01-25 20:07 |
| Submitted by: | Salasnich, Luca |
| Submitted to: | SciPost Physics Core |
| 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.
Author comments upon resubmission
I thank the Referee for the useful comments and suggestions.
In the new version of the paper I tried 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 also explained the relevant role of the modulational instability (of the Bogoliubov elementary excitations) which triggers the spontaneous symmetry breaking.
The changes related to the suggestions and requests of Referee 1 (and 2) are in blue.
Response to Referee 3
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 better discussed the absence of beyond-mean-field corrections in the model: 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 emphasized this relevant issue: the Bose system I am considering has a finite size because it is confined in a finite elliptical ring.
The changes related to the suggestions and requests of Referee 3 are in red.
List of changes
1. In the Introduction I added a comment about the modulational instability citing a new reference, Ref. [11].
2. At the end of Section III there is now a phrase about the fact that Eq. (7) is reliable only in the
weak-coupling and strong-transverse-confinement regime.
3. At the end of Section IV there is a more detailed explanation about the determination of the plot of \kappa vs s. The key idea is to use \phi as dummy variable.
4. At the beginning of Section VI there is a discussion of the equivalence, in our specific problem, of quantum phase transition, spontaneous symmetry breaking, and modulational instability.
5. After Eq. (29) I included another discussion about the modulational instability, quoting a new reference, Ref. [23]. Moreover, it has been added a sentence about the dependence of \gamma_c with respect to \epsilon.
6. At the end of Section VI there are some arguments about the validity of Eqs. (27) and (29) for our finite-size system.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-1-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202109_00024v2, delivered 2022-01-28, doi: 10.21468/SciPost.Report.4260
Report
This is a beautiful work considering the effect of curvature in the NPSE derived by the author and collaborators years ago. Specifically, when a quantum matter-wave is embedded in a vent waveguide under tight-transverse confinement, it is possible to integrate out the transverse degrees of freedom and reduce the dimensionality of the description. Curvature effects can be captured by potential terms along the arc length. In the simplest case, one finds a potential proportional to the curvature square. The generalization of these results to the NLSE was studied earlier on by others, such [6,7]. In parallel, the NPSE was introduced as an improvement for the dimensional reduction of the NLSE in tight waveguides in the absence of curvature. From that point of view, the current study is a natural step forward.
The work is nicely written but the bibliography does not make justice to the abundant literature on curvature-induced effects. Much of this is indeed focused on the linear case, but this is still a reference limit in the present study. I encourage the author to put the work into a broader context, making a minor revision, citing the pioneering and beautiful early works
Switkes, E., Russel, E. L. & Skinner, J. L. Kinetic energy and path curvature in bound state systems. J. Chem. Phys. 67, 3061 (1977).
da Costa, R. C. T. Quantum mechanics of a constrained particle. Phys. Rev. A 23, 1982 (1981).
da Costa, R. C. T. Constraints in quantum mechanics. Phys. Rev. A 25, 2893 (1982).
Goldstone, J. & Jaffe, R. L. Bound states in twisting tubes. Phys. Rev. B 45, 14100 (1992).
In the math-phys literature, other authors like Bracken and Exner devoted a great deal of attention to the linear case
Clark, I. J. & Bracken, A. J. Effective potentials of quantum strip waveguides and their dependence upon torsion. J. Phys. A: Math. Gen. 29, 339 (1996).
Clark, I. J. More on effective potentials of quantum strip waveguides. J. Phys. A: Math. Gen. 31, 2103 (1998).
Exner, P. & Seba, P. Bound states in curved quantum waveguides. J. Math. Phys. 30, 2574 (1989).
Exner, P. & Vugalter, S. A. On the number of particles that a curved quantum waveguide can bind. J. Math. Phys. 40, 4630 (1999).
I believe there is even a book by Exner on the topic. Here it is:
Quantum Waveguides. Pavel Exner, Hynek Kovařík (2015)
With an eye on applications, it is my impression that curvature-induced effects still play a subdominant role in BEC/atomtronics given the scales involved. One can get a feeling considering the length scales required for the da CIP/Costa potential to affect the dynamics. Still, that is a purely technological issue that should not prevent this kind of study.
If the author is having fun with this, an interesting prospect is the inclusion of torsion effects that have been much less studied, to the best of my knowledge.
Beyond that, I think the manuscript makes a valuable contribution that is nicely elaborated and worth publishing.
Report
The revision of the paper is adequate. Th resubmitted paper may be recommended for the publication.
Author: Luca Salasnich on 2022-01-28 [id 2127]
(in reply to Report 1 on 2022-01-26)I thank the Referee who thinks that the revision is adequate.
Author: Luca Salasnich on 2022-01-28 [id 2126]
(in reply to Report 2 on 2022-01-28)Thanks a lot for the nice Report. I will be delighted to improve the paper by adding and discussing the relevant references suggested by the Referee.