# 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_00053v2 Date accepted: 2022-02-02 Date submitted: 2021-11-16 04:38 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.

Published as SciPost Phys. 12, 092 (2022)

### List of changes

1. Fixed Eq. (2) to address Referee 2's requested change 1, namely, we removed the proportionality constant $C$ and instead used the proportionality symbol $\propto$. Also, we removed the braces.

2. Corrected a typo in the second equation in Eq. (5), namely the missing velocity field after the del operator on the left-hand side. This was Referee 1's requested change 1 and Referee 2's requested change 2.

3. The power-law equation of state is now numbered as Eq. (8).

4. After Eq. 8 (formerly 7), we clarified that the Damski-Chandrasekhar shock wave solution is approximate and valid at the initial stages of propagation.

5. Right before Eq. (12) (formerly 11), we explained that $V_{\text{inner}}(t) \equiv \dot{R}_{\text{inner}}(t)$ and
$V_{\text{outer}}(t) \equiv \dot{R}_{\text{outer}}(t)$. This addresses Referee 2's requested change 12.

6. Right after Eq. (12) (formerly 11), fixed the notation of the limit of delta t, as reqested by Referee 2's requested change 4.

7. After Eq. (12), we added an explanation of why Eq. (9) (formerly 8) is an approximate solution of Eq. (6) for d>1, as reqested by Referee 2's requested change 3.

8. In Eq. (12) (formerly 12), in the last equation, corrected 'for $R_{\text{outer}}$' to read 'for $r>R_{\text{outer}}$'. This was Referee 2's requested change 5.

9. After Eq. (17) (formerly 16), we added an explanation that $N$, the number of particles, is a volume integral over $n(r,t)$. This addresses Referee 2's requested change 7.

10. We added a substantial amount of detail to the proof of Statemet 5, that the single-valuedness of the velocity field implies that the velocity at the origin is zero. This addresses Referee 2's requested change 8.

11. In the expression for $1/r$, below Eq. (24) (formerly 23), we replaced all appeances of $R_{\text{inner}}(t)$ by $\bar{R}_{\text{inner}}(t)$. This addresses Referee 2's requested change 9.

12. In the caption to to Fig. 1, we corrected the way we refer to Eqs. (29) and (35) (formerly 28 and 34). This addresses one part of Referee 2's requested change 10.

13. At the end of the caption to Fig. 1, we added an explanation of why the hydrodynamical equations cannot be propagated past $t^{∗}$. This addresses the other part of Referee 2's requested change 10.

14. In the text of Statement 7, we clarified that the statement holds for power-law equations of state. This addresses one part of Referee 2's requested change 11.

15. In the text of Statement 8, we also clarified that the statement holds for power-law equations of state. This addresses the other part of Referee 2's requested change 11.

16. After the second displayed equation in Sec. 12, we added a statement explaining that the relevant velocity scale to compare the velocity field with is the initial speed of sound. This addresses Referee 2's requested change 6.

17. Near the end of the page where Sec. 12 begins, we corrected $2T/7 = 0.296\ldots \times T$ to $2T/7 = 0.286\ldots \times T$. This addresses Referee 1's requested change 2.

18. In the caption of Fig. 2, we changed the description of lines from 'dotted' and 'dashed' to 'short-dashed' and 'long-dashed'. This addresses Referee 2's requested change 13.

19. At the end of Sec. 12, we wrote down the Gross-Pitaevskii equation, with an explanation of the relevant parameters. We emphasized that the healing length and the number of particles were chosen to be close to the values they had in the ENS experiment [Phys. Rev. X vol. 9, 021035 (2019)]. This addresses Referee 2's requested change 15.

20. At the end of Sec. 12, we added a sentence explaining why we decided against attempting to introduce a quantum-mechanical analogue of the local velocity. This addresses Referee 2's requested change 14.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 092 (2022)

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Resubmission scipost_202108_00053v2 on 16 November 2021
Submission scipost_202108_00053v1 on 20 August 2021

## Reports on this Submission

### Report

I am satisfied with the revision and support for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Maxim Olshanii  on 2022-01-03  [id 2062]

(in reply to Report 2 on 2021-12-12)
Category:
remark

We are grateful to the Referee 2 for all the help with the manuscript.

### Report

It is my impression that the points raised in my first report have been adequately taken into account in the revised manuscript. I have no further "urgent" comments.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Maxim Olshanii  on 2022-01-03  [id 2063]

(in reply to Report 1 on 2021-11-21)
Category:
remark

Without all the hard work by the Referee 1, this article would not happen: thank you.