# Decaying quantum turbulence in a two-dimensional Bose-Einstein condensate at finite temperature

### Submission summary

 As Contributors: Matthew Davis · Andrew Groszek Arxiv Link: https://arxiv.org/abs/1903.05528v1 Date submitted: 2019-03-14 Submitted by: Groszek, Andrew Submitted to: SciPost Physics Domain(s): Theor. & Comp. Subject area: Atomic, Molecular and Optical Physics - Theory

### Abstract

We numerically model decaying quantum turbulence in two-dimensional disk-shaped Bose-Einstein condensates, and investigate the effects of finite temperature on the turbulent dynamics. We prepare initial states with a range of condensate temperatures, and imprint equal numbers of vortices and antivortices at randomly chosen positions throughout the fluid. The initial states are then subjected to unitary time-evolution within the c-field methodology. For the lowest condensate temperatures, the results of the zero temperature Gross-Pitaevskii theory are reproduced, whereby vortex evaporative heating leads to the formation of Onsager vortex clusters characterised by a negative absolute vortex temperature. At higher condensate temperatures the dissipative effects due to vortex-phonon interactions tend to drive the vortex gas towards positive vortex temperatures dominated by the presence of vortex dipoles. We associate these two behaviours with the system evolving toward an anomalous non-thermal fixed point, or a Gaussian thermal fixed point, respectively. The cross-over between these two dynamical behaviours is found to occur at earlier times with increasing condensate temperature.

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

### Submission & Refereeing History

Submission 1903.05528v1 on 14 March 2019

## Reports on this Submission

### Strengths

1. The paper is well written and was a straightforward read for someone who works in a neighboring field, i.e., vortex dynamics of quantum fluids in three-dimensions. In my notes, I have at the top of page 5 that I was *enjoying* the read.

2. The narrative was highly logical, I walked away with a clear understanding of what the research program seeks to accomplish.

3. The references were timely and appropriate and I feel that I could *work* with/through this manuscript.

### Weaknesses

1. Some of the technical terms/ideas could be unpacked a bit more. Due to the writing quality, I was able to follow the discussion without issue. However, b/c of the high quality I was left wanting more information since I expect that it would have been illuminating.

### Report

Nice paper. Readable for a non-expert. It is a new result and novel to researchers in the field since it supports a mature line of research asking fundamental questions about vortex sourced dynamics in two-dimensions. For this reason, it is both interesting and important.

The quality of writing is quite high and the graphics, in the color pdf, were useful and not too difficult to decode/understand. It should be noted that in black and white (print), much of the meaning is lost and one can really only big features/trends.

The sanity checks are clear/appropriate and their results pass them.

That said, I do have some suggestions that could improve the document, which I discuss below.

### Requested changes

1. The inline nonzero moment defined just after Eq. (3) is confusing since the romanized d seems out of place. If the \hbars from Eq. (3) were written with the dt terms, then the reader would quickly understand the notation associated with the inline equation. You could also write the dt/\hbar after the delta.

2. The reader could be helped with the inclusion of a sentence in the introduction just following the two sentences about PGPE and SPGPE. At this point in my read, I now know your tools and your focus. However, if I am not familiar with the tools, then I won't really see the contrast between them and the classical field approach. A summarizing sentence would set the reader up for the discussion of your methodology.

3. I understand why 2.2 and 2.3 are separated out from 2.1 but as single paragraphs, they dangle. Perhaps it would be better to just incorporate them?

4. In 2.2, is there a reason for working with the region defined by 95% of the radius? I didn't find intuition with a quick skim backward. I have several ideas about why one might/should do this but it would be good to know the author's perspective.

5. The start of 3.2 and its footnote was dense/technical. Consider reformatting into a trichotomized list. The technical matters could be unpacked more but in the end, the reader needs to walk away with the three regimes. This gets a little lost in the succinct but technical discussion which left me feeling split between re-reading the technical statements or pushing forward with the trichotomy in mind.

6. I found the first sentence of 3.4 awkward and felt that it could be clearer with a re-organization of content to get rid of several parenthetical statements.

7. In the second to last paragraph of section 3, the authors state that the system may eventually return to the Gaussian fixed point. Why does this argument exist? It seems to me that clusters might have a similar sort of topological protection and so I could imagine that they may not totally annihilate each other, at least at zero temperature as the authors note. This idea seems to split over this paragraph and two others in section 4. As it is an important and interesting question, perhaps a more focused discussion would be better. Alternatively, the authors could add on a sentence that hints at the forces at play that would prevent long-time protection of the clusters and the openness of this question.

8. I like the rapid-fire question list at the end of the conclusion. It really helps punctuate their program/results. However, since I was enjoying the read, I wanted to hear about the author's opinions about whether the methods of the paper are enough to address these questions. If not, then what changes/hurdle do they expect? Completing in this way would keep me interested in their next work or becoming more acquainted with this/their research program.

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

### Strengths

1-Numerical investigation of finite-temperature effects on Onsager vortex formation in two-dimensional Bose-Einstein condensates.
2-Numerical investigation of the turbulent decay from the perspective of the rate equation of quantized vortex number.

### Weaknesses

1-Validity of the stochastic projected Gross-Pitaevskii equation for the preparation of initial states.
2-Choice of the cutoff parameter in the projected GP equation.
3-Insufficient discussions on the relation between turbulent decay in the numerical simulations and nonthermal fixed points.

### Report

The authors consider an interesting topic on Onsager vortex formation in two-dimensional (2D) Bose-Einstein condensates (BECs) captured by a cylindrical uniform trap. In their previous study, the authors reported that the Onsager vortex (negative temperature state) can indeed be formed in 2D BECs if the evaporative vortex cooling works well. The condition for the cooling was discussed in terms of vortex interactions and trapping-potential configurations. As a result, they found that the cylindrical uniform trap was a good platform for the Onsager vortex formation.
This paper is a sequel to the above previous works. The authors have systematically studied the Onsager vortex formation in “finite-temperature” 2D BECs by using the stochastic projected Gross-Pitaeskii (SPGP) equation for preparation of initial states and the projected GP (PGP) equation for the time evolution. In the previous work, the initial state is prepared by randomly distributed vortices, and they do not consider finite-temperature effects. The present work takes the effects into consideration by preparing the initial states with the stochastic projected GP equation. Then, investigating turbulent decay from the initial state with various temperatures, they uncover that the finite-temperature fluctuations can disturb the evaporative vortex cooling and the Onsager vortex cannot be formed in the high temperature case even if the uniform trap is used. Furthermore, time evolution of the vortex number is numerically calculated, and they discuss nonthermal fixed points (NTFPs).
Their finding for the finite-temperature effects is very useful for experimental observation of Onsager vortex, and the discussion on the NTFP is interesting since recent experiments actually succeeded in observing universal relaxation for NTFPs. However, the following two reasons prevent me from recommending the publication in the present manuscript.

Reason1) Validity of the SPGP equation and the PGP equation
The SPGP is valid if the temperature is higher than half-$T_{\rm c}$. Here, $T_{\rm c}$ is the BEC transition temperature. For example, the review paper [Advances in Physics 57 (5), 363-455 (2008)] describes this point. However, their numerical simulations consider a low temperature region $0< T/T_{\rm c}<0.3$, so that their use of the SPGP cannot be justified. Also, both SPGP and PGP methods introduce a cutoff parameter, but they have not given any discussions on how to determine it. The authors should address the issues and describe them in the revised version.

Reason2) Discussion on the relation between turbulent decay and NTFPs.
Usually, universal relaxation of NTFPs can be characterized by dynamical scaling of spatial two-pint correlation functions [Phys. Rev. Lett. 101, 041603 (2008)]. The nice review is given in [J. Berges, arXiv:1503.02907 (2015)]. Despite the importance of the scaling, the authors do not show time evolution of the correlation functions at all and there are no discussions on the dynamical scaling. In my opinion, the present results are insufficient for judging whether the turbulent decay is close to the Gaussian or anomalous nonthermal fixed points. I recommend the authors to confirm the dynamical scaling and estimate the power exponents.

If the authors satisfactory address these issues, I recommend the publication.

The followings are minor questions, which may improve the manuscript.

1-Relation between Onsager vortex formation and inverse cascades
In the introductory part, Onsager vortex formation and inverse cascades are mentioned. However, I cannot understand a relation between two concepts in the 2D GP model in the present introduction. It would be nice to clarify it.

2-Two rate equations for vortex number
Is it possible to fit the numerical data using Eq.(5)? If possible, we cannot judge which rate equations are correct. How do you answer it?

3-One-body particle loss
Realistic ultra-cold gases suffer from one-body particle loss. Does it disturb the evaporative vortex cooling?

### Requested changes

1-Discuss the validity of the SPGP and PG equations.
2-Need to show time evolution of correlation functions, check the dynamical scaling, and calculate the power exponents.

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