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The origin of the period$2T/7$ quasibreathing in diskshaped GrossPitaevskii breathers
by J. Torrents, V. Dunjko, M. Gonchenko, G. E. Astrakharchik, M. Olshanii
Submission summary
As Contributors:  Vanja Dunjko · Maxim Olshanii 
Preprint link:  scipost_202108_00053v1 
Date submitted:  20210820 22:55 
Submitted by:  Dunjko, Vanja 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We address the origins of the quasiperiodic breathing observed in [Phys. Rev.\ X vol. 9, 021035 (2019)] in diskshaped harmonically trapped twodimensional Bose condensates, where the quasiperiod $T_{\text{quasibreathing}}\sim$~$2T/7$ and $T$ is the period of the harmonic trap. We show that, due to an unexplained coincidence, the first instance of the collapse of the hydrodynamic description, at $t^{*} = \arctan(\sqrt{2})/(2\pi) T \approx T/7$, emerges as a `skillful impostor' of the quasibreathing halfperiod $T_{\text{quasibreathing}}/2$. At the time $t^{*}$, the velocity field almost vanishes, supporting the requisite timereversal invariance. We find that this phenomenon persists for scaleinvariant gases in all spatial dimensions, being exact in one dimension and, likely, approximate in all others. In $\bm{d}$ dimensions, the quasibreathing halfperiod assumes the form $T_{\text{quasibreathing}}/2 \equiv t^{*} = \arctan(\sqrt{d})/(2\pi) T$. Remaining unresolved is the origin of the period$2T$ breathing, reported in the same experiment.
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Reports on this Submission
Anonymous Report 2 on 2021108 (Invited Report)
Strengths
1 The manuscript provides an elegant theoretical explanation of some aspects of the important experiment of the Jean Dalibard group.
2 The analytical approach employed for the study of the dynamics of an initially discontinuous density profile is highly original.
3 The manuscript provides intriguing results that should inspire further studies.
Weaknesses
1 The manuscript does not discuss differences between the numerical GrossPitaevskii solutions and the hydrodynamic solutions.
Report
The manuscript theoretically addresses an important problem posed by the puzzling experimental work of the Paris group. It (analytically) delivers a quite general, sort of surprising, result for the breathing "halfperiod", which in two spatial dimensions, surprisingly well, reproduces the experimental measurement. It is rather doubtful that such a result could be obtained with less sophisticated analytical techniques. I think that it definitely deserves publication in a highimpact journal such as SciPost after consideration of the remarks from the Requested Changes sections.
Requested changes
1The lefthand side of (2) does not match the expression proportional to $C$. I suppose the definition of $C$ needs to be corrected. The brackets, $\{\cdots\}$, look awkward in (2), etc.
2There is velocity field missing in one place in (5).
3If (8) satisfies (6), after setting $\omega=0$ and $\delta t=t$, only when $d=1$ , then how is (8) found for an arbitrary $d$? Where does it come from? What fixes its form?
4 Below (11): $0+\to0^+$.
5 Should there be "$\text{for} \ r>R_\text{outer}(t)$" in one of the expressions in (12)?
6 What does the statement "the velocity field at $t^*$ is nearly zero" mean (remark right below Statement 2)? In what sense it is nearly zero?
7 I am not sure whether $N$, in the equation for $\sqrt{\langle r^2\rangle(t)}$, is somewhere defined.
8 Could it be explained why singlevaluedness of the velocity field leads to the proof of Statement 5?
9 Should there be $\bar{R}_\text{inner}(t)$ instead of $R_\text{inner}(t)$ in the expression for $1/r$, the one below (23)?
10 It is unclear to me why the empty space, the $\omega t\gtrsim1$ region, is displayed in Fig. 1. Can analytical solutions be put on the plot as, e.g., the dashed line? Change "(34)(28)" in the caption to, e.g., "(28,34)".
11 Should it be said in Statement 7 that (34) holds when $\mu(n)\sim n^\nu$? Similar question applies to Statement 8.
12 Should one explicitely define $V_\text{outer}(t)$ as $dR_\text{outer}(t)/dt$?
13 Figs. 2 and 3: the "dotted curves" seem to me to be rather "dashed" than "dotted".
14 Fig. 3: why the blue, GrossPitaevskii results, are missing? Smaller range of the vertical axis could lead to better presentation of the results.
15 Can GrossPitaevskii equation be written down? Can one comment on how its simulations are carried out (e.g. write explicitly what initial conditions for the numerics are employed)? Are there some specific parameters used for the GrossPitaevskiibased simulations?
Anonymous Report 1 on 2021927 (Invited Report)
Report
The manuscript by Torrents et al. addresses the origin of the 2T/7 quasiperiod of the diskshaped breather that was observed in trapped twodimensional BEC by Dalibard’s group. In this paper, the authors have brilliantly argued that at a certain time t*, the velocity field almost vanishes, which naturally leads to a 2T/7 quasiperiod. The key step of their argument is to focus on the trajectories of the singularities of the density and velocity fields (R_{inner} and R_{outer}) instead of the whole solution to the hydrodynamic equations. By doing this, they are able to extract the exact trajectories of both R_{inner} and R_{outer} and hence prove that the velocity field at these two points vanishes at t*. Since this work solves an experimental puzzle and the method is quite innovative, I think this paper can be published in Scipost.
Requested changes
Minor comments:
1．In eq.(5), a $\mathbf{v}$ is missed on the L.H.S. of the Euler’s equation.
2．Last paragraph on page 12, 2/7 should be around 2.86, instead of 2.96.
3．The converse of statement 1 (If the velocity field vanishes at T/2, then T is a period of the breather.) should also be true. Even though it is not directly related to the following argument in the paper, it will be nice that the authors add it to statement 1 to make things complete.