# The origin of the period-$2T/7$ quasi-breathing in disk-shaped Gross-Pitaevskii breathers

### Submission summary

 As Contributors: Vanja Dunjko · Maxim Olshanii Preprint link: scipost_202108_00053v1 Date submitted: 2021-08-20 22:55 Submitted by: Dunjko, Vanja Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Approaches: Theoretical, Computational

### Abstract

We address the origins of the quasi-periodic breathing observed in [Phys. Rev.\ X vol. 9, 021035 (2019)] in disk-shaped harmonically trapped two-dimensional Bose condensates, where the quasi-period $T_{\text{quasi-breathing}}\sim$~$2T/7$ 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 quasi-breathing half-period $T_{\text{quasi-breathing}}/2$. At the time $t^{*}$, the velocity field almost vanishes, supporting the requisite time-reversal invariance. We find that this phenomenon persists for scale-invariant gases in all spatial dimensions, being exact in one dimension and, likely, approximate in all others. In $\bm{d}$ dimensions, the quasi-breathing half-period assumes the form $T_{\text{quasi-breathing}}/2 \equiv t^{*} = \arctan(\sqrt{d})/(2\pi) T$. Remaining unresolved is the origin of the period-$2T$ breathing, reported in the same experiment.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202108_00053v1 on 20 August 2021

## Reports on this Submission

### 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 Gross-Pitaevskii 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 "half-period", 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 high-impact journal such as SciPost after consideration of the remarks from the Requested Changes sections.

### Requested changes

1-The left-hand 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.

2-There is velocity field missing in one place in (5).

3-If (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 single-valuedness 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, Gross-Pitaevskii results, are missing? Smaller range of the vertical axis could lead to better presentation of the results.

15- Can Gross-Pitaevskii 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 Gross-Pitaevskii-based simulations?

• validity: high
• significance: high
• originality: top
• clarity: high
• formatting: good
• grammar: excellent

### Report

The manuscript by Torrents et al. addresses the origin of the 2T/7 quasi-period of the disk-shaped breather that was observed in trapped two-dimensional 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 quasi-period. 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

1．In eq.(5), a $\mathbf{v}$ is missed on the L.H.S. of the Euler’s equation.