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Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates
by Caio C. Holanda Ribeiro, Uwe R. Fischer
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Submission summary
| Authors (as registered SciPost users): | Uwe R. Fischer |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2111.14153v2 (pdf) |
| Date submitted: | 2022-02-11 13:25 |
| Submitted by: | Fischer, Uwe R. |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider the propagation of quasiparticle excitations in a dipolar Bose-Einstein condensate, and derive a nonlocal field theory of quasiparticle scattering at a stepwise inhomogeneity of the sound speed, obtained by tuning the contact coupling part of the interaction on one side of the barrier. To solve this problem $ab$ $initio$, i.e., without prior assumptions on the form of the solutions, we reformulate the dipolar Bogoliubov-de Gennes equation as a singular integral equation. The latter is of a $novel$ $hypersingular$ type, in having a kernel which is hypersingular at only two isolated points. Deriving its solution, we show that the integral equation reveals a continuum of evanescent channels at the sound barrier which is absent for a purely contact-interaction condensate. We furthermore demonstrate that by performing a discrete approximation for the kernel, one achieves an excellent solution accuracy for already a moderate number of discretization steps. Finally, we show that the non-monotonic nature of the system dispersion, corresponding to the emergence of a roton minimum in the excitation spectrum, results in peculiar features of the transmission and reflection at the sound barrier which are nonexistent for contact interactions.
Author comments upon resubmission
List of changes
The changes we performed are already detailed in the response letter to the Referee.
In summary, they are
On the Referee point 1: With reference to the claim of the Referee that we performed an approximation by neglecting density gradients, which is not the case.
Answer: We reworked section IIIA, in which we now explicitly demonstrate the
order parameter stationary solution for our homogeneous density condensate, by amending the manuscript with the details of obtaining Eq. (8), showing how the external potential should be chosen in the experiment such that the condensate background is homogeneous.
On the Referee point 2: The alleged problem related to particle number conservation.
We addressed this by adding a paragraph added after Eq. (27), explaining that the conservation law we have is not due to the U(1)-symmetry related particle conservation of the full theory.
On the Referee point 3: This is in reality point 1 reiterated, so there is no need of an additional change. Our development relies on no approximation besides the Bogoliubov expansion to linearize the Gross-Pitaevskii equation and study its perturbations, and thus Eq. (8) is exact for the condensate with constant particle density we assumed.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022-6-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2111.14153v2, delivered 2022-06-21, doi: 10.21468/SciPost.Report.5265
Strengths
1-First paper I am aware of to consider effect of roton in dispersion relation to problems involving inhomogeneous speed of sound.
2- Nice discussion of conserved currents and related issues for BdG problem given in response to another referee comment. It might be useful to include that in the paper .
Weaknesses
1-I found this paper hard to read and there was not much intuition/explanation provided of the physics involved. What is the significance of the evanescent modes? Why do they particularly appear in the presence of long-range/anisotropic interactions?
2-I think the non-analyticity of the kernel could be better explained. I note that equation (3) shows that the dipolar interaction in momentum space is better behaved than in real space since it is purely angular: 3Cos^2 (theta) -1. This seems easier to deal with than the 1/r^3 divergence in coordinate space. Please explain why you go into the complex plane to search out the poles.
3-What is the significance of the ansatz (16) and the accompanying equations (17) and (18)? Is this standard or is it something radical?
4-Eq (19) is also mysterious to me and it would be useful for it to be explained. How unusual is this equation? (In comparison to contact case, say.) What, physically, are the solutions Lambda in this equation? Are they propagators?
5- Section III D on approximate solutions confuses me, especially the 2N solutions. What is Fig 4 telling us?
6- Discussion of the physical implications of the results shown in Figs 5,6,7,8,9 would be useful.
7- the term hypersingular is mentioned in the abstract and the introduction but I am not sure what the significance of this is for the current problem and how special or unusual it is.
Report
Speaking as someone who has worked both on dipolar BECs and dabbled in analogue Hawking radiation, I still found this paper hard to understand even though it is ``in my area''. The work is a generalization to the case of dipolar interactions of work on sound barriers in standard BECs with contact interaction and is related to analogue Hawking radiation experiments. The authors make some interesting points since the roton minimum in the excitation spectrum allows for waves with strange properties and is probably correct, but for my taste not enough physical intuition is given on the results (and hence I find them hard to appreciate). I did not find the presentation very clear and for that reason I would not recommend publication in SciPost Physics. I think it meets the criteria for publishing as a standard article (in the Core journal) . I think the basic premise is interesting and there may well be other groups who follow up on this work
Requested changes
See section on "weaknesses" for suggested changes.
Anonymous Report 1 on 2022-2-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2111.14153v2, delivered 2022-02-23, doi: 10.21468/SciPost.Report.4502
Strengths
1) The authors propose a general method to consider the effect of long range interactions in linearized Gross-Pitaevskii equations. They are able to show that the solution takes the form of plane waves plus evanescent waves and derive the scattering coefficients in the presence of a roton minimum in the dispersion at the left of the defect.
2) They have also introduced a numerical approach relying on the discretization of the Fourier transform of the long range interaction potential, turning the branch cut into a sum over discrete poles.
Weaknesses
1) The authors are choosing the external potential in Eq. (10) in such manner that the Gross-Pitaevskii equation (9) has a uniform solution for a particular density. This allows plane/evanescent wave solutions to the linearized equation (12) discussed in the rest of the manuscript, but the treatment applies only at the special density such that $U(x)+ng_c(x)$ is a constant. In the case of a more general potential, the average density $n(x)$ will vary around the defect over a length of order the healing length. This is likely to hamper the construction of the asymptotic states starting from Eq. (11).
2) It appears that in Eq. (B2), the integral over q' has to be taken in principal value due to the presence of the pole at q'=q. The contribution of the pole is $\bar{G}$ in Eqs. (20) and (B2). Eq. (B8) is also a relation between principal values. This should be stated explicitly since the extra delta functions contribution that would arise from regularizing asymetrically Eq. (B8) by adding a positive infinitesimal imaginary part to $q_\pm$ would spoil the boundary conditions (21) and (22).
Report
The authors have clarified their choice of external potential and the meaning of the current conservation in Eq. (27). While the solution is limited to a particular choice of potential to ensure a constant density in the time independent solution, the solution for the linearized mode is valid and should be applicable to other models of integrodifferential equations with translationally invariant Kernel and inhomogeneous boundary conditions.
I would thus incline to recommend publication of the manuscript.
Author: Uwe R. Fischer on 2022-02-26 [id 2247]
(in reply to Report 1 on 2022-02-23)We thank the Referee for the careful reading of the manuscript, and for providing an insightful report.
We respond to the two weakness points as follows.
Weakness 1: We thank the Referee for raising this issue, and we agree: The solutions we constructed are accurate for the particular constant density profile we analyzed, and sound barriers produced by continuous variations of the system density are likely to produce quantitatively different results. We believe the generalization of our technique to condensates with continuously varying parameters density and coupling deserves a dedicated analysis, and we amended the manuscript conclusion by mentioning the corresponding further perspective.
Weakness 2: We thank the Referee for noting this aspect in our manuscript. Indeed, the Cauchy principal value is to be taken for the integral wrt $q'$ in Eq. (B2), and this was already noted before in the main text after Eq. (19). We amended the manuscript to state that also explicitly after Eq. (B2).