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Nonlocal field theory of quasiparticle scattering in dipolar BoseEinstein 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  

Preprint Link:  scipost_202208_00063v1 (pdf) 
Date accepted:  20221102 
Date submitted:  20220823 19:52 
Submitted by:  Fischer, Uwe R. 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the propagation of quasiparticle excitations in a dipolar BoseEinstein 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 Bogoliubovde 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 contactinteraction 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 nonmonotonic 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
{\it Strengths: 1First 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 .}
$\Longrightarrow$ We appreciate the Referee's assessment of our ms main goal: To explore implications and ramifications of roton phenomena on wave scattering in condensates. Indeed, in connection to the Referee's report, we note that this work in part of a project aiming to study analogue gravity phenomena in dipolar condensates, from which the current ms intends to lay down requisites which are naturally convoluted due to the (longrange) dipolar interactions in inhomogeneous configurations. Also, besides the approximation we proposed to the dipolar kernel in section III D, all of the results and equations in the ms, including the hypersingular integral equation, are in essence only the BdG equation applied to the arguably simplest case of a sound barrier in a dipolar quasi1D condensate. In this sense, we believe that all descriptions of analytic solutions of the BdG equation in similar configurations must share the same level of complexity we found while manipulating the BdG equation.
Furthermore, although the question of numberconservation in condensates is wellknown in the literature and can be incorporated to our work, the equations from our first report were not added to the ms because we do not discuss quasiparticle quantization, a necessary step to study numberconservation in interacting condensates. We discuss in the below each of the weaknesses appointed by the Referee and the corresponding ms amendments. We also revised the ms thoroughly in order to improve its overall readability.
{\it Weakness (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 longrange/anisotropic interactions?}
$\Longrightarrow$ Our point of view is that the nomenclature ``mode'' should not be used in connection to the existence of complex wave vector solutions to the dispersion relation, for the latter admit no physical interpretation in term of quasiparticles, i.e., they, alone, do not represent solutions to the BdG equation {\it per se}. We note, however, that real wave vector solutions, at least in configurations similar to the one we considered in which an asymptotic regime exists (far from the barrier), do admit an interpretation in terms of quasiparticles of the asymptotic system. The interpretation of each component appearing in a quasiparticle is thus artificial and model dependent. Furthermore, the existence of evanescent channels in the quasiparticles of our system is not a peculiarity of the dipolar interactions, as they appear also in contactonly inhomogeneous condensates. In the system we considered the novelty regarding evanescent channels is their increased number, which from a physical perspective is an indication that near the barrier the transient regime caused by the barrier existence is distinct when dipolar interactions are present, as expected. We believe that because of its nuances the study of near boundary effects in dipolar condensates deserves a separate study. We added a discussion regarding the evanescent channel possible implications to the end of section III B.
{\it Weakness (2): I think the nonanalyticity 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: $3\cos^2 (\theta) 1$. 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.}
$\Longrightarrow$ We thank the Referee for pointing it out that we missed a motivation for studying the analytical properties of the dipolar kernel in section II A, although it is said analytical kernels lead to simplifications. The reason for switching from the configuration space to Fourier space is that the scattering problem is better treated in this way because the waves sent towards the interface are parametrized by their wave vectors. We amended section II A accordingly in order to improve clarity.
{\it Weakness (3): What is the significance of the ansatz (16) and the accompanying equations (17) and (18)? Is this standard or is it something radical?}
$\Longrightarrow$ As explained in the first paragraph of section III C, solutions to the BdG equations far from the barrier, i.e., given by the Bogoliubov dispersion relation, are not valid close to the barrier in contrast to the local condensate case. Equations (17) and (18) extend the plane wave solutions of the Bogoliubov dispersion relation to be valid everywhere but at the barrier, i.e., substitution of either Eq.~(17) or Eq.~(18) into the BdG equation gives zero everywhere but at the barrier. Finally, the ansatz (16) combines these extended local solutions in the same fashion as it is done for local condensates in order to build the quasiparticles. In this sense, our results follow from the realization that it is possible to build local solutions of the BdG equation from the asymptotically valid solutions to the dispersion relation which can then be combined in the ansatz (16).
{\it Weakness (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?}
$\Longrightarrow$ Eq.~(19) is the BdG equation for our dipolar condensate configuration written in an alternative form which is useful for numerical simulations and already incorporates analytical asymptotic properties necessary to study scattering phenomena. Equation (19) does not appear for contactonly condensates, because all the $\Lambda$ coefficients are trivially zero. Moreover, as explained in the item 1, the $\Lambda$ functions measure the weight of the evanescent channels coming from dipolar kernel, which have no physical interpretation outside the quasiparticles. We amended the introduction by stating explicitly that the hypersingular integral equation {\it is} the BdG equation written in another form.
{\it Weakness (5): Section III D on approximate solutions confuses me, especially the 2N solutions. What is Fig 4 telling us?}
$\Longrightarrow$ Figure 4 depicts the solutions of the Bogoliubov dispersion relation for the approximated dipolar kernel: The blue circles, characterized by a complex wave vectors $k=k_{r}+ik_{i}$. As shown in subsection III C, there is an intrinsic continuum of evanescent channels associated to the dipolar interactions bound to the barrier. The approximated kernel represents a discretization scheme of this continuum in terms of extra solutions to the dispersion relation given by the blue circles on the imaginary axes of Fig.~4. In the limit of $\mathcal{N}\rightarrow\infty$ and $\Delta q\rightarrow 0$, the number of solutions on the imaginary axes tend to the continuum of the exact kernel. We stress that each evanescent channel has no physical interpretation outside the quasiparticle. We amended the caption of Fig.~4
and revised the text of Section III D to improve clarity.
{\it Weakness (6): Discussion of the physical implications of the results shown in Figs 5,6,7,8,9 would be useful.}
$\Longrightarrow$ We thank the Referee for this suggestion, which we believe increases the clarity of the presentation. The reflectance/transmittance discussed in section V is the dipolar condensate analogue of electromagnetic wave scattering at an interface. We added a new sentence to the first paragraph of section V explaining the significance of the results that follow.
{\it Weakness (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.}
$\Longrightarrow$ As discussed in item 4, the hypersingular equation (19) is the BdG equation written for our particular dipolar condensate, and its relevance for the present case is that singular integral equations are generically harder to solve numerically. In particular, no universal method exists which can be used for obtaining the solutions of such equations, which is just a reminder of how intricate is the BdG equation for this simple condensate configuration. We have experimented different numerical methods, and only the spline discretization produced reliable solutions.
Published as SciPost Phys. Core 6, 003 (2023)
Reports on this Submission
Report #3 by Anonymous (Referee 7) on 20221018 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202208_00063v1, delivered 20221018, doi: 10.21468/SciPost.Report.5918
Strengths
1) The authors have clarified some queries of the previous referee reports and I think the manuscript is stronger as a result.
2) The paper presents a serious and hard calculation of the scattering properties of a dipolar BEC with a sound barrier. I admire the fact that this calculation could be pushed through for this complicated system.
Weaknesses
1) I find this to be quite an exotic problem, at least in terms of current experiments and theory; the results are quite technical and detailed in nature rather than being qualitatively new or surprising.
2) The main results seem to be the identification of a continuum of evanescent modes and some `peculiar' scattering properties although there is little in the way of physical interpretation or consequences that would make me say "experiments must be done to search for these effects" admirable though the calculation is.
Report
Without more physical intuition I find it hard to appreciate and understand this paper fully. In my opinion this manuscript falls in the range of "Addressing an important problem in the field using appropriate methods with an abovethenorm degree of originality" rather than "Detailing a groundbreaking theoretical/experimental/computational discovery". I think this is perfectly respectable paper that should be published in SciPost Physics Core but I do not think it meets the criteria for SciPost Physics.
Report #1 by Anonymous (Referee 5) on 2022927 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202208_00063v1, delivered 20220927, doi: 10.21468/SciPost.Report.5778
Report
Compared with their previous manuscript, the authors have added clarifications of the physical interpretation of evanescent modes near the barrier. While the article is technically sound, it is dealing with the solution of the linearized GrossPitaevskii equation in the vicinity of the barrier in the presence of a dipolar interaction using singular integral equation methods in a highly idealized case.
The criterion for new pathway or synergistic link between different research areas is not ment since singular integral equation techniques have been in use in manybody theory at least since P. Nozieres' work on the XRay absorption edge problem [see for instance J. Phys. (Paris) 32 p. 913 (1971)]. The criterion for a breakthrough on a longstanding research block or groundbreaking discovery is not met since the difficulties in the treatment of linearized GrossPitaevskii equations, even in the presence of long range interaction are not conceptual but technical in the sense that there is no ambiguity on the form of the linearized equations to be solved, but obtaining the correct ground state density in the presence of a barrier and solving the resulting linearized equation demand some numerical analysis and computer programming effort.
Given the above criteria, although the work deserves being published, I cannot justify publication in Scipost Physics. However, it seems appropriate for Scipost Core.