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Interplay of Kelvin-Helmholtz and superradiant instabilities of an array of quantized vortices in a two-dimensional Bose--Einstein condensate
by Luca Giacomelli, Iacopo Carusotto
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Submission summary
| Authors (as registered SciPost users): | Iacopo Carusotto · Luca Giacomelli |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2110.10588v4 (pdf) |
| Date submitted: | 2022-03-31 10:11 |
| Submitted by: | Giacomelli, Luca |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the various physical mechanisms that underlie the dynamical instability of a quantized vortex array at the interface between two counter-propagating superflows in a two-dimensional Bose--Einstein condensate. Instabilities of markedly different nature are found to dominate in different flow velocity regimes. For moderate velocities where the two flows are subsonic, the vortex lattice displays a quantized version of the hydrodynamic Kelvin--Helmholtz instability (KHI), with the vortices rolling up and co-rotating. For supersonic flow velocities, the oscillation involved in the KHI can resonantly couple to acoustic excitations propagating away in the bulk fluid on both sides. This makes the KHI rate to be effectively suppressed and other mechanisms to dominate: For finite and relatively small systems along the transverse direction, the instability involves a repeated superradiant scattering of sound waves off the vortex lattice; for transversally unbound systems, a radiative instability dominates, leading to the simultaneous growth of a localized wave along the vortex lattice and of acoustic excitations propagating away in the bulk. Finally, for slow velocities, where the KHI rate is intrinsically slow, another instability associated to the rigid lateral displacement of the vortex lattice due to the vicinity of the system's boundary is found to dominate.
Author comments upon resubmission
we resubmit the present article as you requested, with minor modifications to better explain the points raised by the First referee.
We kindly ask you to consider this new version of the manuscript for publication in SciPost Physics.
Sincerely,
The authors
List of changes
- We modified the caption of Figure 1 to make more clear which panels we are referring to, as suggested by the Second referee
-We added a footnote before equation (7) to explain the formulation of the Bogoliubov problem we are using
- We added another footnote in Section 3 to comment on the numerical checks we performed
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-4-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2110.10588v4, delivered 2022-04-16, doi: 10.21468/SciPost.Report.4932
Report
It seems that there is a gap between me and authors. To close the gap efficiently, I would like to point out only the two problems below, (i) and (ii), that have been already suggested in my first report. I will accept the revised manuscript for the publication if authors could answer to the problems clearly. In the previous report, I did make the criticism about the numerical computation but it is just the secondary problem. Thus, I will not mention any more about it for avoiding further unproductive communication.
(i) Authors should clarify the applicable condition of the hydrodynamic theory.
Authors make an artificial fitting of the numerical data with the hydrodynamic prediction of Eq. (13) in Fig. 2. There is no argument on how the way of the fitting is reasonable in the current situation. In my first report, I suggested that the fitting is not good. However, authors have made no reasonable answer for this problem. The fitting curve in Fig. 2 misleads readers into believing the numerical result are well-described by the hydrodynamic theory.
From the title of this manuscript, it is natural for reviewers to regard Kelvin-Helmholtz instability (KHI) as an important subject to be judged scientifically. At this stage, I have no judgement material without response from authors on the above problem. It is quite unclear how the wave number of k_H~0.6/\delta, as is mentioned just below Eq. (13), is reasonable for applying the hydrodynamic theory to the current system. Why is the theory reasonable despite a fact that the wavelength is comparable to the size of a quantized vortex? Additionally, a Bogoliubov excitation may have a single-particle behavior for such a large wave number because of the kinetic energy term in the Hamiltonian, where the hydrodynamic approximation can break down. The unclearness comes partly from the confusion of the Bogoliubov transformation and it will be discussed in the latter part of this report.
In this situation, we could not consider the hydrodynamic theory as “a reference model for a different system that we take as a comparison for our numerical results” as was stated by authors in the previous response. The unclearness might come from my misunderstanding of the hydrodynamic theory because I could not access the literatures [1,2]. I would be happy if authors could explain the point of the model more clearly and the connection with the current system quantitatively.
(ii) The form of the Bogoliubov transformation is confusing.
The Bogoliubov transformation and the Bogoliubov equation is originally introduced by Bogoliubov. The form of Eq. (9) is consistent with the original one but Eq. (5) with Eq. (7) is not. Since the consistency of Eq. (9) is the same as the original one, the numerical result in the manuscript could be accepted itself. On the other hand, the inconsistency from Eq. (7) can make an essential difference in giving the physical interpretation.
The Bogoliubov transformation is originally introduced to diagonalize the Hamiltonian (or the energy functional). The Bogoliubov equation is derived for this purpose and the problem is reduced to the eigenvalue problem. The linear combination of terms with exponents, e^{iwt} and e^{-iwt} with the eigenfrequency w, is crucial to achieve the diagonalization of the Hamiltonian; the form should be \delta\Psi= u+(-)v^*, where u \propto e^{iwt} and v \propto e^{iwt}. By inserting this formula into the original Hamiltonian, one obtains the contribution of the fluctuation to the energy as the form of Eq. (11).
The diagonalization of the Hamiltonian is essential to consider the fluctuation as elementary excitations with eigen energy. This idea is also crucial to this system because the dynamical instability is induced by a coupling between positive- and negative-energy excitations (modes). I wonder whether the fluctuation of Eq. (5) with Eq. (7) reproduces the contribution of Eq. (11) by inserting authors’ formula, \delta\Psi =U and \delta\Psi^* =U^*, into the energy functional or the Hamiltonian. I am sorry that I know only the widely used approach, which is historically used in the problem of dynamical instability in Bose-Einstein condensates, and I did not follow all the content of Ref. [27]. But, I am not sure the physical reason why \delta\psi and \delta\psi^* can be considered as independent for describing elementary excitations. They state just that “the functions \delta \psi and \delta \psi^* being now considered as independent” according to Ref. [27]. Even if they could succeed the diagonalization, I am not sure whether the formalism is consistent with the description of the perturbation theory for the dynamical instability, which derives a coupling between positive- and negative-energy modes as was revealed quantitatively in the problem of the splitting instability in Ref. [20]?
In addition to the formula of the excitation energy, I wonder the definition of the (pseudo-)momentum of excitation in their formalism. In the widely used approach, the momentum is defined as \hbar k N with the norm N=\int dxdy(|U|^2-|V|^2) and the wave number k of the excitation. This is computed as the contribution of the fluctuation to the total momentum of the condensate. This point could influence the interpretation of Eq. (14) and I want authors to clarify this problem too.
Finally, note that the problem of (ii), except for the problem of momentum just above, seems to be resolved only by replacing the formula of Eq. (7) by the widely used form of the traditional Bogoliubov transformation. This is because the mathematical treatment of U and V below Eq. (7) looks the same as the widely used one.
Report #1 by Anonymous (Referee 2) on 2022-4-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2110.10588v4, delivered 2022-04-14, doi: 10.21468/SciPost.Report.4922
Report
Dear Professor Davis,
I have carefully read the latest revision of the manuscript, before viewing the earlier reports. My overall impression of this manuscript is that it does open a new pathway in an existing research direction, with clear potential for multipronged follow-up work. I find the results on superradiant scattering of phonons from an array of vortices particularly interesting and worth publishing in SciPost Physics. In addition, the authors have provided detailed and satisfactory responses to all previous referee reports. Moreover, I found the added Appendices that clarify the physics of the superradiant scattering particularly welcome and helpful.
I suspect that the identity of the Referees 1 and 2 have been swapped between the first and second round of refereeing and I found it somewhat challenging to follow some points raised in the Anonymous Report 2 on 2022-2-17 and the Anonymous Report 1 on 2022-2-21. For instance, the request to compare the presented results with a point vortex model seems unjustified in this case where the most interesting new results are inherent to phonon radiation which cannot be modeled using point vortex models that assume complete absence of phonons. Many other points seemed to hinge on technicalities that do not affect the value of the presented key findings.
In summary, after reading the manuscript I would have recommended acceptance. After reading the referee reports and the authors' responses, I am not persuaded to change my original opinion and am therefore recommending to accept this manuscript for publication in SciPost Physics.
I would also like to share a few additional comments that are sufficiently minor that I can trust the authors' judgement on whether or not they require action or not.
1. perhaps the weakest point of this manuscript is how it is placed in the broader physics context as the referencing to previous works is quite scarce. For instance, it would be useful to link present work to the recent experimental observation of quantum Kelvin-Helmholz instability in a Bose gas
https://www.nature.com/articles/s41586-021-04170-2
2. in the first line of Chapter 2, the reference to a pancake shape made me incorrectly think of circular disk condensates. Perhaps the word pancake is not necessary here at all or at least it seems unnecessary to use italics to emphasize this word.
3. Looking at the equation (9) made me wonder if the -D operator on the lower diagonal of the matrix should be complex conjugated since it contains the -iK\partial_x operator?
4. In the context of Fig.2, the caption states that energetic instabilities begin to be present at the crossing of Dv = 2.0 c_s. However, there is a very large gap between Dv = 1.26 c_s and 2.42 c_s and already in the Dv = 1.26 c_s case there is one negative energy, positive norm, eigenstate.
5. on page 11, I think I am missing the point of the paragraph starting with "Notice...". The authors seem to highlight a difference between two systems based on the fact that they satisfy the same criterion Dv > 2 c_s?
6. in the context of [20,21], the instability of a doubly charged vortex was identified in https://doi.org/10.1103/PhysRevA.59.1533, experimentally observed in https://doi.org/10.1103/PhysRevLett.89.190403, and discussed as a resonance between positive and negative energy eigenstates for instance in
https://doi.org/10.1103/PhysRevA.68.023611
7. on page 16 in the context of [28], the mechanism of image vortices driving the bulk vortex motion seems to be equally well present in both uniform and harmonically trapped condensates with a boundary, as well as in the drifting case discussed in this work. In the harmonically trapped case the density gradient effect and the image vortex effect have been shown to be almost equally strong, https://doi.org/10.1103/PhysRevA.97.023617
Author: Luca Giacomelli on 2022-07-25 [id 2683]
(in reply to Report 1 on 2022-04-14)Here we attach the new version of the manuscript, with changes highlighted in red.
Attachment:
Author: Luca Giacomelli on 2022-07-22 [id 2677]
(in reply to Report 1 on 2022-04-14)
We thank the referee for their careful reading of our work and for their positive assessment of our results. In the following we reply to their comments, that we kept into account with some modifications of the manuscript.
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We thank the referee for pointing out this recent work. We think that we connected our results with previous works on the KHI in superfluids in the introduction, where we cite the works we are aware of (i.e. [13-19]). We added reference [15] and the article the referee mentioned, commenting the connection with the rest of the literature.
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The referee is right, the term can give rise to some confusion. We removed the word pancake, and we now refer only to a BEC tightly confined in one direction.
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In equation (9) there is no complex conjugate on the lower diagonal element. This is due to the fact that this expression of the Bogoliubov problem was obtained by substituting equation (7) in the Bogoliubov problem featuring only real terms in the diagonal elements. The fact that $e^{iKx}$ is a global factor for the spinor implies that the term $-iK\partial_x$ will appear without conjugation in the lower diagonal elements.
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The referee is right, strictly speaking the statement in our caption was incorrect. What we meant is that at the threshold $\Delta v=2c_s$ negative energy phononic waves begin to be present due to Landau instability. The negative energy states visible at the edge of the Brillouin zone are instead due to the dynamics of the single vortices and do not fall in the general picture we discussed associated to Fig. 2. They are instead responsible for the drift instability. Moreover, the threshold is not exactly $2c_s$ because the system is finite along $y$ and hence the phononic modes discrete. Thus the two parts of the spectrum begin to merge only at $\Delta v\simeq 2.42c_s$ and the spectra for $1.26c_s<\Delta v<2.42c_s$ are not particularly different from the first. We modified the caption and the discussion below Fig. 2 to make these points clearer.
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With that comment we wanted to highlight a difference in the interpretation of negative-energy modes between the translationally invariant case one has in the absence of quantized vortices, and the present case. Reading that section again, we however noticed that this paragraph is not optimally placed and we have taken the opportunity of moving it to a footnote earlier in the section and slightly rephrasing it.
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Regarding the instability of multiply charged vortices, we are aware that the literature on this topic is extremely large. However, it mostly deals with trapped condensates in which the role of the trap is unclear. On this basis we prefer to restrict our citations to works specifically dealing with vortices in unbound condensates that are most relevant for the physics we are discussing in our manuscript.
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Thank you for pointing out that work, we were not aware of it. We changed our comment on the precession of vortices in trapped condensates and included that reference.
Author: Luca Giacomelli on 2022-07-25 [id 2682]
(in reply to Report 2 on 2022-04-16)We thank the referee for trying to close the gap between us. In the following we reply to the two main points they raised. We kept them into account with further minor modifications of the manuscript, to make our aims and our notation clearer. These changes are visible in red in the version of the manuscript we attach here. We however believe that these points stem from a mutual misunderstanding, are sufficiently minor ones and do not affect the main results on superradiant phenomena, that are presented in the rest of our work.
(i) Applicability of the hydrodynamic theory
What we did with the hydrodynamic prediction is not a fitting, rather a qualitative comparison. What we found interesting and wanted to show is that similar spectral features are shared by the two models, even though, as the referee correctly pointed out, the hydrodynamic model cannot take into account the presence of vortices and the single-particle nature of Bogoliubov excitations. As a side note the wavenumber $k_H\sim0.6/\delta$ is actually never reached since, as we argue in Section 4.1, the Brillouin zone is always smaller than this value.
The similarity of the spectral features in our opinion further justifies the name KHI for the leading instability we find in this regime, also considered the fact that similar instabilities have been called KHI in the literature simply from the nonlinear behavior of the system, without considering the spectral properties (e.g. in [19]).
We did not want to mislead readers into believing that numerical results are well described by the hydrodynamic theory. To make this clearer we modified the discussion below equation (13).
Anyway, we would like to stress once again that our discussion of the KHI is simply a further characterization at the spectral level of the instability that was first studied in [19] with the use of the Gross--Pitaevskii equation, and that the core of our new results are the superradiant instabilities we characterize in the following of the article, where no reference to the hydrodynamic model is made any more.
(ii) The form of the Bogoliubov transformation
First of all we would like to stress that the Bogoliubov problem we used is equivalent to the one the referee is referring to, as is discussed in [29]. All numerical data are obtained using equation (9), that seems to be familiar to the referee, so that our numerical diagonalization of the problem correctly diagonalizes the quadratic Hamiltonian and all the expected properties of the Bogoliubov modes hold.
We believe that the confusion may come from our notation in equation (7), that is strictly speaking not correct. The components of the spinor are $U_K$ and $V_K$ only after the diagonalization. To solve this confusion we changed equations (7) and (9) by introducing $\eta_K$ and $\chi_K$ to label the two components of the Bogoliubov spinor and reserved $U_K$ and $V_K$ to indicate the amplitudes of the eigenmodes, as indicated before equation (11) and in footnote 2 in Section 3, where the connection with the usual expression for the fluctuation field is made.
For what concerns the definition of momentum, the quantities $K$ and $k_H$ we introduce are actually not the momenta of the excitations but their wavenumbers. In particular $K$ is the Bloch wavenumber introduced in equation (7) and $k_H$ is the wavenumber of excitations along the shear layer in the hydrodynamic treatment. We changed the word \textit{momentum} to \textit{wavenumber} where we used it to refer to these quantities.
We would like to comment that we feel that this kind of approaches have been widely used in the literature and we think that this article is not the place to present these points more in detail, since references such as [29] are available.
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Anonymous on 2022-08-02 [id 2705]
(in reply to Luca Giacomelli on 2022-07-25 [id 2682])The problem of (ii) has not been solved yet essentially. I can accept other revisions.
As mentioned, the interpretation of the dynamical instability as a resonance of positive- and negative-energy modes has been formulated by starting from the widely used formulation of \delta \psi =u+(-)v^* [more concretely for the current problem, \delta \psi =e^{iKx}(u+(-)v^*)]. I have not known literatures that formulate with the form of Eq. (7). Authors should show at least some references where such formulation is described clearly, although I think it is impossible.
I understand that this formulation is essential to connect the considered phenomenon with the superradiance. As I implied in the last report, the problem will be “resolved only by replacing the formula of Eq. (7) by the widely used form of the traditional Bogoliubov transformation.”
I found a typo in the added footnote on page 6;
\delta \psi=…+V^*e^{+iwt}
---> \delta \psi=…+V^*e^{+iw^*t}
This correction is also necessary for the formulation mentioned above.
Anonymous on 2022-09-02 [id 2786]
(in reply to Anonymous Comment on 2022-08-02 [id 2705])We thank the referee for pointing out the typo in the footnote, that we are going to correct. We are sorry that the referee feels we did not address their concerns. We however think that we did and that the discussion and the references we give in the article are sufficient to make this point clear.
As it is currently written, equation (7) is not a Bogoliubov transformation, but a definition of the spinor components. The trasformation is obtained with the diagonalization of equation (9): the resulting eigenvectors have as components the usual Bogoliubov amplitudes, that are connected to $\delta\psi$ with the usual Bogoliubov transformation written by the referee, as discussed in the footnote 2.
As we already pointed out, such an approach is explained in Section 6 of ref [29], where, after taking $\delta\psi$ and $\delta\psi^*$ as independent, the usual Bogoliubov transformation is obtained by expanding on the eigenmodes' spinors. See in particular equation (6.33) at page 69 of the journal version or equation (251) at page 72 of the arXiv version (arxiv.org/abs/cond-mat/0105058).
The Bogoliubov transformation we are performing is hence exactly the same the referee wrote and there is hence no difference in terms of spectral properties such as dynamical instabilities and positive and negative norm modes. Moreover, apart from the overall formulation, our calculations are performed with equation (9), that is very well known in the literature.