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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/2110.10588v3 (pdf) |
| Date submitted: | 2022-02-18 11:58 |
| 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 thank the referees for their insight and for helping us in making our work more complete and understandable. This new version of the manuscript includes several additions and changes to address the points raised by the referees.
We replied directly to the two reports and, given the impossibility of uploading a pdf with the resubmission, we included in our replies a pdf of the manuscript, in which the main changes are highlighted in red.
The main addition is an Appendix in which different kinds of superradiant scattering that occur in this system are demonstrated. This is intended to back up our conclusion of the superradiant nature of some of the instabilities we observe, whose discussion in the main text was also largely rewritten. Moreover, we updated some Figures to show more details and expanded the discussion of some key points, as requested by the first referee.
We also improved the exposition in several points, in particular those that we believe caused some misunderstanding with the second referee about the purpose of our comparisons with results from classical hydrodynamics.
We believe that these modifications, listed below, make our discussion clearer, display new interesting physics and reinforce our conclusions. We hence ask you to consider our paper for publication in SciPost Physics.
Best regards,
The authors
List of changes
A version of the manuscript in which these changes are highlighted in red was attached to the responses to the reports.
- We rephrased the parts of the introduction that were unclear
- We added two panels to Figure 1 showing the y dependence of the velocity, with a comparison with the hyperbolic tangent profile
- On page 5 we added a footnote to explain the seeding of the instabilities in the GPE computations
- We added a paragraph at the beginning of Section 4.1, better explaining the rationale of out comparison with the hydrodynamic results. We also added a second textbook reference for those results.
- We added some details to the discussion before Eq. (14) and a footnote explaining that the form in which we take the fluctuation field does not depend on the physical properties of our superfluid system.
- Because of its increased length, we separated Section 4.2 in two subsections
- In Section 4.2.1 we improved the discussion of SRI and added references to the Appendix, were additional computations displaying superradiant scattering of wavepackets are shown
- Section 4.2.2 was rewritten to address the points the referee raised about our inference on the infinite size system. Figure 5 was updated to show how this inference was made.
- Figure 6 was remade to include a panel showing the exponential growth of the unstable modes, from which the red circles shown in the Ly=120 panel Figure 5 are extracted. The mismatch between the diagonalization and the time-dependent calculations pointed out by one referee is now discussed.
-The superradiant nature of RI is now better discussed in Section 4.2.2, with references to the new Appendix A2.
-An appendix was added in which different kinds of superradiant scattering at the basis of SRI and RI are shown through additional numerical computations.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2022-3-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2110.10588v3, delivered 2022-03-10, doi: 10.21468/SciPost.Report.4657
Report
The authors have improved the presentation of the paper and have clarified numerous points, while also making the presentation of the figures clearer. I am happy with the revised paper and recommend publishing.
Requested changes
As a minor comment, I found the added text in the caption of Fig. 1 made the following sentence, starting with “The first case shows the KHI behaviour…”, hard to follow. It is somewhat unclear what is referred to here by the authors in this sentence and the next. It would be clearer to refer to plots directly, i.e. “The upper plots shows the KHI behaviour…”.
Report #1 by Anonymous (Referee 4) on 2022-2-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2110.10588v3, delivered 2022-02-21, doi: 10.21468/SciPost.Report.4483
Report
I cannot find a good reason why authors do not compare their numerical result with the analytic dispersion of the KHI. To ensure the reliability of numerical calculation by the comparison is crucial to judge the scientific quality of the current work. Since numerical results of the Bogoliubov equations with complex eigenvalues are normally quite complicated, it is difficult in general to give a proper physical interpretation to them, as in the current case. Additionally, high-precision calculation is required for diagonalizing the Bogoliubov equation because a small error can change the dispersion drastically in the case with complex eigenvalues. For example, the Nambu-Goldstone mode, which always exists in the presence of vortices and are localized at the vortex core, can be sensitive to a numerical error causing a unphysical complex mode.
The previous work by Baggaley et.al. is based on the Gross-Pitaevskii equations while the current work is on the Bogoliubov equations. In this situation, the comparison with the dispersion of the KHI is the fundamental step to do first for establishing the reliability before investigating more advanced, complicated situations. The simplest way to establish the reliability is to compare the numerical result with an analytic one if we have, and we have it; the numerical result can be compared with the dispersion of the KHI in fluid dynamics. Essentially, it does not matter which theory should be compared, the point vortex model or the model introduced by authors, if authors could make a proper introduction for it.
Finally, I am still confused with the formulation of the Bogoliubov transformation introduced by authors. The difference of the sign (plus or minus) in the front of the coefficient V_K is not essential. The transformation should be formulated with a matrix in Eq. (7). Or, does it include a matrix implicitly? This is the point I wanted to ask authors. I know the formulation, \dleta psi=U-(+)V^* and \delta psi^*=U^*-(+)V, but I am not familiar with the Bogoliubov "formulaiton" of the form, \dleta \psi =U_K and \delta\psi^*=V_K, introduced by authors.
Author: Luca Giacomelli on 2022-03-30 [id 2341]
(in reply to Report 1 on 2022-02-21)We must say that we find it difficult to respond to the first referee. Their present objections on the comparison with the hydrodynamic results seem to us to be in contrast with the ones in their previous report.
They require a comparison of our numerical results with an analytical prediction, but such an analytical prediction is not available for our system. The analytical results from hydrodynamics we are reporting are not applicable to the present case, for the multiple reasons that also the first referee addressed in their first report (e.g. the shear layer in hydrodynamics is not irrotational). And, as we also explained in the response to the previous report, we are not using them as a good model of the physical situation under study.
We moreover feel that their concerns of the reliability of the results of our numerical diagonalizations are generic and do not point to a problem in particular. We agree that numerical diagonalizations in presence of complex-frequency modes can be delicate, however we performed many checks on our numerical results. As is good practice, for the diagonalizations as for the other codes, we checked the stability of our results with respect to variations of the numerical parameters (e.g. the spatial discretization and the integration step). Moreover we compared the results of the diagonalizations with the ones of time evolutions of the Bogoliubov problem, that are usually more robust. An example of such a comparison is visible in the $L_y=120\xi$ panel of Figure 5, where we compare the instability rates obtained via diagonalization with the ones extracted from a time evolution with a noisy initial condition. This shows that the complex-frequency modes we find are not spurious ones. A further check of consistency can be seen in the right panel Figure 5, where one can see that the real parts of the SRI unstable modes fall well inside the analytical red region for an infinite system.
Note that we had already done similar consistency checks with analogous codes for different physical setups in our previous publications, for instance in Ref.[21], where the instability rates for single vortices obtained with diagonalizations were confirmed with time-dependent simulations. This gives us full confidence on the reliability of our numerical calculations. To make this point fully clear to the reader, we added a footnote in Section 3 commenting on these consistency checks.
About the formulation of the Bogoliubov problem, we now understood what is the question of the referee and we thank them for pointing out this subtle point of a widely used approach. The formulation we are using is the one presented in detail in reference [27] (section 6.1.1), that we cite just before using it. By taking $\delta\Psi$ and $\delta\Psi^*$ as independent variables one recovers the linearity of the equations for the fluctuation field, that otherwise mix the field and its complex conjugate. This allows to obtain the spectrum via diagonalization of the Bogoliubov matrix of equation (9). This is a convenient mathematical treatment of the problem that however doubles the dimension of the space of the fluctuation field. In fact a mode $(U,V)$ with frequency $\omega$ has always a partner $(V^*,U^*)$ with frequency $-\omega^*$. After the diagonalization, to recover the physical fluctuation field one has to impose the conjugation relation between $\delta\Psi$ and $\delta\Psi^*$ (that were initially taken as independent). To this purpose it is enough to consider the sum of a pair of partner modes, so that $(\delta\Psi, \delta\Psi^*) = (U,V)+(V^*,U^*)=(U+V^*,V+U^*)$, which indeed satisfies the conjugation condition. This is the combination we take for example to obtain the density fluctuations shown for example in Figure 6. We added a footnote before equation (7) to explain this.