Interplay of Kelvin-Helmholtz and superradiant instabilities of an array of quantized vortices in a two-dimensional Bose-Einstein condensate

• We thank the referee for their insight. In the following we respond to each of their remarks, that we think mostly stem from a misinterpretation of our aims when using results from hydrodynamics. We modified the text so to make the rationale of our comparisons more clear. We include here a pdf of the manuscript where the changes and additions are highlighted in red.

(1) Point vortex model Authors try to analyze the stationary solution of an array of quantized vortices in terms of the theory of KHI in an unconventional formalism. This system should be considered more fundamentally as a vortex street (not Karman’ vortex street). It is also known that the system is unstable against a transverse shift of vortices (see some textbook on hydrodynamics). Then, I would like to ask authors to compare the accuracy of their numerical analysis with the result of the point vortex model by removing the trapping potential. I think the situation considered is too complicated to understand such a fundamental aspect of the system.

We are sorry but we did not manage to find stability analyses of the vortex street you mention in the hydrodynamics textbooks we consulted. If you could give us some direct references we would be interested in looking at them. For the point of the present work however we feel that additional comparisons with hydrodynamic models are not needed. Our aim was to further characterize what was identified as KHI in [12] ([18] in the new manuscript), before dealing with the other instabilities we identified. We feel that the comparison with finite-width shear layers already shows a good parallelism with what is commonly called KHI. The main point we want to highlight is that the hydrodynamics results we report are not meant as being a good description of our system, it is just a reference model for a \textit{different system} that we take as a comparison for our numerical results. We tried to better highlight this in the text. We think it is interesting that many features of KHI are present also in our very different physical system.

(2) Width of the shear layer The smallest length scale, which determine hydrodynamics in a superfluid is the healing length $\xi$. However, the width $\delta_v$ defined by Eq. (3) is no such a restriction. The interpretation of the width is crucial to characterize the fundamental property of this system because it is used in Eq. (11) for justifying the numerical analysis in Fig. 2. A vortex street looks like a velocity shear layer if it is viewed on the length scale much longer than the distance between neighboring vortices. Therefore, I can agree with authors expectation that theory of KHI is applicable to this system under some condition. Concretely speaking, it is natural to expect that KHI would be realized when the wave number $k_H$ of the unstable mode is much smaller than $n_{vort}$. However, the value of $k_H$, considered by authors, seems to be similar to $n_{vort}\sim Mv/hbar$ as is mentioned in the last paragraph of Sec. 4.1.

Again we wish to stress that we are not using Eq. (11) (now eq. (12)) to \textit{justify} our numerical results, we are just taking the hydrodynamic model as a comparison to see which feature survive the very different setting. We agree that we consider values of the momenta of the perturbations comparable to the healing length, and that also the shear layer width is comparable to that. This in fact makes the hydrodynamic model not a good model for our system, that however we investigate directly by numerical means, and we take the hydrodynamics results as a comparison of what is usually understood as KHI.

Although I think this is the most fundamental aspect of this system to be examined first, authors start with unconventional formulation of KHI with no kind explanation. For example, the result of Eq. (10) looks quite different from the conventional result obtained by considering the stability of interface tension wave (I do not have the book of Ref [1]). However, there is no explanation about the difference in the manuscript. Furthermore, authors continues to extend the theory for the case with a finite width of the shear layer. The velocity field is not potential flow (vortex free) inside the shear layer in classical fluid, but it is fully potential flow in superfluids. How to connect the two systems? If authors try to apply approximately, what is the condition that can be safely applied?

We do not believe that the dispersion relation of the KHI we report is unconventional, since we also found it in other books. As an example we added a second book as a reference for the classical hydrodynamics results we mention (the one by Drazin and Reid [2]) where this result is also obtained in the introductory chapter. We added some specifications to give some missing information about the system for which that result is derived, that is an incompressible inviscid fluid of constant density without gravity. For what concerns the final questions we agree that the flow is not a potential one in the hydrodynamic model, while it surely is in the superfluid we are considering. But again, let us stress that we are not trying to apply that model to our system.

(3) Interpretation of the results in Fig. 2. As authors suggest in the text, there could be the localized modes at the shear layer or vortices and the acoustic mode (phonons) propagating in the bulk fluid. Although a lot of branches are plotted for the real part in Fig. 2, there is no explanation about which branch corresponds to the localized or bulk modes. Thereby, the authors’ assertion that the resonance occurs between the bulk and localized modes lacks persuasiveness. This is contrast to the explanation on the resonance and the superradiance due to the dynamical instability of a multiply quantized vortex in superfluid in Ref. [14], or originally and more quantitatively by Takeuchi, et. al., J. Phys. Soc. Jpn. 87, 023601 (2018).}

In Figure 2 the superradiant physics of the localized modes does not come significantly into play, since it matters for supersonic velocities and large systems, that we investigate in Section 4.2.2. In that context the presence of a localized mode is visible in Figure 6, and also in the newly added Appendix A2, where it can be also seen that negative-energy localized modes exist, very similarly to the vortex case (for which we added also the work you mentioned as a reference [15]). About the plots of Figure 2, we think it is not really possible to point at modes and say which is localized and which is not, since the output of the diagonalization are modes of the whole system, in which resonances between localized and phononic modes appear as a single mode.

The style of the plot also makes it difficult for readers to understand. The difficulty might come from the introduction of Eq. (12) without enough explanation. Where does this relation come from? The relation of momentum and wave number can be complicated in the Bogoliubov analysis because of the complicated property of the norm in the presence of unstable mode with complex eigenvalue. This is related to the form of the Bovoliubov transformation and its normalization condition. The corresponding form of Eq. (6) is different from the conventional transformation, i.e., $\delta\psi =u-v^*$, which make readers confused more.

The relation between hydrodynamic and Bloch momentum is a technical point depending on how one chooses to take the fluctuation field. This is what we explain in the discussion on page 9, that we tried to improve. This is not altered by the Bogoliubov dispersion, and in fact would be present also if we considered the so-called hydrodynamic approximation for fluctuations, that removes dispersion. It is a matter of convention and all the properties of the system, including dispersion and the presence of dynamical instabilities, come after and independently of this choice. Also, we do no think that our convention for the Bogoliubov transformation is unusual, this is the one used for example in the book by Pitaevskij and Stringari (ref. [24] section 5.6), that differs for a minus with respect to the one you wrote. This is also not related to out choice of the Bloch momentum. We added footnote 2 in page 9 to highlight these facts and try to avoid this confusion.

I am not also satisfied with the fact that the “adjustment” is not so small for the fitting curve in the leftmost panel of Fig. 2. In such a difficult situation, I am sorry that I could not read the content of Sec. 4.2 seriously.}

As we already pointed out, that curve is not a theoretical justification of our results, it just serves as a comparison with a model for a different system in which the KHI is usually studied. Moreover, the content of Section 4.2 does not mention those hydrodynamic models anymore, the analysis is purely based on our solutions of the Bogoliubov problem.

(4) Previous works on KHI Baggaley and Parker revealed the nonequilibrium dynamics for the case with small and large number of vortices in a very similar situation of single-component superfluids in Ref. [12]. Since authors investigate a very similar situation, they should explain distinction between the result of Ref. [12] and their result more clearly.}

Our work starts from the results of Baggaley and Parker. We first characterized in a different way the KHI they observed and we then proceeded to consider higher velocity differences, in which novel physical mechanisms emerge that, to our knowledge, were not reported before. In ref. [12] (now [18]) only velocity differences below $2 c_s$ were considered. Also, the drift of vortices at smaller velocities was not reported. We tried to make more clear the comparison with that work in the introduction.

Additionally, although authors give attention to the radiation of acoustic excitations propagating away in the bulk fluid for the supersonic case, a very similar phenomenon has been observed for KHI in immiscible binary BECs by Kokubo et. al. Phys. Rev. A 104, 023312 (2021). According to this paper, the acoustic excitations are shock waves forming Mach-cone-like structure. A similar pattern is also observed in the rightmost panel in Fig. 1. This is just a nonlinear effect although authors try to describe such acoustic radiation as superradiance in terms of the linear analysis based on the Bogoliubov theory. How to explain this inconsistency?

We think there is no inconsistency here. The Mach-cone-like structure of that work may well be something very similar to what we observe (we were not aware of that work, that we added to the references of previous works on the KHI). At least in our case however that is not a shock wave, since it emerges as small amplitude fluctuations. This is also confirmed by the fact that those features can be captured by the linear Bogoliubov theory. It is hence firstly a linear phenomenon, since it involves acoustic radiation, that only becomes nonlinear when the amplitude of the unstable modes become large enough.

Attachment:

manuscript_changes_EgFryJ4.pdf

We thank the referee for the careful reading of our manuscript and for the positive assessment of our results. In the following we reply point by point to their remarks, indicating the changes we made to the manuscript to address the issues they raised. We attach a pdf of the new manuscript with the main changes highlighted in red.

Major comments

1. The term superradiant was not intended as a descriptive terminology, the SRI stems from superradiant processes that occur in the system. We concluded this from the spectra, since the dynamical instabilities emerge from resonance of opposite-normed modes that are phononic propagating waves. However we probably have relied too much on our intuition in showing this. To back up our conclusions we added an Appendix (A1), with appropriate references in the main text, in which we report the result of numerical computations that show the occurrence of amplified scattering processes, for frequencies and Bloch momenta at which we found SRI. Our simulations involve the scattering of wavepackets in a system large enough that they do not interact with the boundaries. These simulations show that these sizable amplifications are stable processes in an infinite system, while they inevitably lead to SRI in a finite system.

2. We improved the discussion of the inference of the spectrum in the infinite system. In particular we included in the right plot of Figure 5 a plot of all the dynamically unstable modes of the large-but-finite system, that clearly distinguishes the instabilities that disappear for an unbound system from those that remain.

3. We changed Figure 6 to include a panel showing the exponential growth of (some of) the momentum components of the unstable growing perturbation. For what concerns the superradiant nature of the RI, we updated the discussion and, as for the SRI, we added in Appendix A2 a scattering numerical experiment in which we show how the RI is associated to superradiant scattering given by the existence of a negative-energy mode localized around the array of vortices. Differently from the SRI case, here the amplified scattering cannot occur without starting the associated instability. The mechanism is completely analogous to the one we illustrated for multiply quantized vortices in [20].

4. What we meant with good agreement is that the measured instability rates for the setup mimicking an infinite system are in correspondence and comparable to the ones obtained by diagonalizing a large but finite system. An exact match is actually not to be expected and the mismatch in the dominant instability bubble can be ascribed to a finite size effect. Comparing with the panel for $L_y=60\xi$ one can see that the dominant bubble is higher in that case. The dominant instability is hence still a decreasing function of the system size, and a further decrease in the $L_y\to\infty$ limit is to be expected. We improved the discussion about this point.

Additional comments 5-6. We improved the first two paragraphs 7. We listed directly the relevant references 8. We added more details about that comparison. The parallelism is further explained in Section 4.2, when dealing with the RI. 9. We added two panels to Figure 1, showing the velocity profile obtained numerically and a superimposed plot of the hyperbolic-tangent profile. 10. Me modified the term. 11. We added a footnote commenting on this. The way in which the shear layer is created via a time-dependent potential implies that fluctuations are present in the system. Also, as you pointed out, numerical noise is a further seed for instabilities. 12-15. We defined the undefined symbols, uniformed the notation to $\Delta v$ and corrected the wrong indication for the red line.

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manuscript_changes.pdf