# Superflow decay in a toroidal Bose gas: The effect of quantum and thermal fluctuations

### Submission summary

 As Contributors: Ashton Bradley · Zain Mehdi Arxiv Link: https://arxiv.org/abs/2105.03154v2 (pdf) Date submitted: 2021-08-10 05:39 Submitted by: Mehdi, Zain Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Quantum Physics Approaches: Theoretical, Computational

### Abstract

We theoretically investigate the stochastic decay of persistent currents in a toroidal ultracold atomic superfluid caused by a perturbing barrier. Specifically, we perform detailed three-dimensional simulations to model the experiment of Kumar et al. in [Phys. Rev. A 95 021602 (2017)], which observed a strong temperature dependence in the timescale of superflow decay in an ultracold Bose gas. Our ab initio numerical approach exploits a classical-field framework that includes thermal fluctuations due to interactions between the superfluid and a thermal cloud, as well as the intrinsic quantum fluctuations of the Bose gas. In the low-temperature regime, our simulations provide a quantitative description of the experimental decay timescales. At higher temperatures, our simulations give decay timescales that range over the same orders of magnitude observed in the experiment, however, there are some quantitative discrepancies. In particular, we find a much larger perturbing barrier strength is required to simulate a particular decay timescale (between ${\sim}0.15\mu$ and ${\sim}0.5\mu$), as compared to the experiment. We rule out imprecise estimation of simulation parameters, systematic errors in experimental barrier calibration, and shot-to-shot atom number fluctuations as causes of the discrepancy. However, our model does not account for technical noise on the trapping lasers, which may have enhanced the superflow decay in the experiment. For the intermediate temperatures studied in the experiment, we also observe some discrepancy in the sensitivity of the decay timescale to small changes in the barrier height, which may be due to the breakdown of our model's validity in this regime.

###### Current status:
Has been resubmitted

### List of changes

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CHANGES TO MANUSCRIPT
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Introduction
Extended discussion of previous theoretical work in the introduction.
Slight re-wording of introduction to clarify our approach.
2 Details of the experiment
Corrected description of experimental measurement of winding number.
3.2 SPGPE Theory
4.1 Qualitative features
Added slight clarifications to the discussion.
Corrected Figure 4 caption - normalisation of density plot was to set peak density to 1, not integrated density to 1.
4.2 Quantitative features
Added sentence relating choice of simulation timescales to experimental data.
5.1.2 Truncated Wigner Approximation
Extended discussion and added Ref. [91] to clarify why the breakdown of Truncated Wigner approximation is not expected to be an issue at higher temperatures.

### Submission & Refereeing History

#### Published as SciPost Phys. 11, 080 (2021)

Resubmission 2105.03154v3 on 17 September 2021

Resubmission 2105.03154v2 on 10 August 2021
Submission scipost_202105_00011v1 on 7 May 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-9-2 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2105.03154v2, delivered 2021-09-02, doi: 10.21468/SciPost.Report.3477

### Strengths

(1)First beyond mean-field results about the experiments by Kumar et al.
(2)Partially reproduce the experimental results.
(3)Comprehensive discussions about the inconsistency between the theory and  the experiments.

### Weaknesses

(1)Presence of the limitation of validity of the stochastic projected Gross-Pitaevskii equation (SPGPE) formalism
(2)No discussion about the effects of the inelastic collisions (see below).

### Report

This is the first referee report on the manuscript “Superflow decay in a toroidal Bose gas: The effect of quantum and thermal fluctuations” by Z. Mehdi et al., which is submitted to SciPost Physics.
The authors investigated the decay of superflow of the Bose-Einstein condensates (BECs) confined in a toroidal trap.
This work analyzes the experiment by Kumar et al.
Although the experiments observed the decay of superflow at finite temperature, the origin of this remains unclear.
The authors tried to reproduce the experimental results using the SPGPE formalism, which includes the quantum and thermal fluctuations. They succeeded in reproducing the experimental results for the lowest temperature case.
However, there are significant discrepancies between the theory and experiment in other parameter regions.
The authors discussed the discrepancy from various viewpoints, for example, the validity of the theory and uncharacterized experimental effects, and so on.

I think that this manuscript is well written and seems to be scientifically sound.
This work will be a good guide for future researches.
However, I have some questions and comments.
In my opinion, this manuscript can be published in SciPost Physics after the following questions and comments are clarified.

### Requested changes

(1)The authors mentioned the effects of the atom loss in reply to report 1.
I think this is an important point because the timescale of the experiments by Kumar et al. is long compared with typical cold atom experiments.
The authors should add the descriptions of the particle loss in the discussion part.

(2)The authors showed that the decay of superflow is sensitive to the strength of the barrier height.
This suggests that the low-density region near the potential barrier is essential for describing the decay of superflow.
Is it possible to treat this region correctly by the SPGPE?
The SPGPE is justified when the occupation number is large enough.
Naively thinking, the occupation number is small in the low-density region in the LDA sense.

(3)Because the present system is isolated, the thermal reservoir can be regarded as thermally excited atoms in the ring trap.
The thermally excited atoms feel the potential barrier. Consequently, the density of the reservoir (excited atoms) decreases near the potential barrier.
I think that the effects of non-uniformity of the reservoir should affect the noise term.
On the other hand, in the SPGPE theory, the noise term is a complex Gaussian, which is spatially uncorrelated.
I guess that the effects of the non-uniformity may change the results at least quantitatively.

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

### Author:  Zain Mehdi  on 2021-09-17  [id 1763]

(in reply to Report 1 on 2021-09-02)
Category:
remark

We thank the new referee for their report and are pleased that they recommend the publication of our manuscript in SciPost, conditional on our satisfactory response to their queries and comments. We address these in detail below.

(1) "The authors mentioned the effects of the atom loss in reply to report 1. I think this is an important point because the timescale of the experiments by Kumar et al. is long compared with typical cold atom experiments. The authors should add the descriptions of the particle loss in the discussion part."

We have extended our analysis in Section 5.2 (final paragraph of Section 5.2.2) to include a discussion on the effect of atom loss on our results and the discrepancy observed with the experiment.

(2) "The authors showed that the decay of superflow is sensitive to the strength of the barrier height. This suggests that the low-density region near the potential barrier is essential for describing the decay of superflow. Is it possible to treat this region correctly by the SPGPE? The SPGPE is justified when the occupation number is large enough. Naively thinking, the occupation number is small in the low-density region in the LDA sense."

For systems such as the one considered in this work, the LDA is a very poor guide as to the validity of the SPGPE. It is not required that the density at each spatial location is high, so long as the occupation condition is met for each of the single-particle modes contained in the low-energy (C) region. This is the reason for using the (approximate) single-particle basis of the toroid as our numerical grid (see Appendix A) - our explicit inclusion of a projector in this basis ensures this condition is consistently satisfied. As an example, the SPGPE provides an excellent treatment of finite-temperature vortex dynamics despite the density going to zero at the vortex cores - see, for example, [PRA 88 063620 (2013)].

We have clarified the above point in Section 3.2 of the manuscript.

(3)"Because the present system is isolated, the thermal reservoir can be regarded as thermally excited atoms in the ring trap. The thermally excited atoms feel the potential barrier. Consequently, the density of the reservoir (excited atoms) decreases near the potential barrier. I think that the effects of non-uniformity of the reservoir should affect the noise term. On the other hand, in the SPGPE theory, the noise term is a complex Gaussian, which is spatially uncorrelated. I guess that the effects of the non-uniformity may change the results at least quantitatively."

To be precise, a large fraction of the thermal cloud is included within the c-field region, which contains not only the condensate mode but all modes with significant average occupation. Therefore in our simulations, many of the thermal atoms are non-uniformly distributed by both the trapping potential and the perturbing barrier.

In our simulations, the energy cutoff is set sufficiently high such that the reservoir, comprised of atoms with energy greater than the energy cutoff, is unaffected by the perturbing barrier. This is a distinct advantage of the SPGPE; provided the perturbing potential is within the support of the c-field region (representable by the mode subspace), the damping amplitude is unchanged for peak barrier strengths smaller than ⅔ of the cutoff value [PRA 77 033616 (2008)]. In the simulations performed in this work, this condition is satisfied (all barrier heights considered are a fraction of the chemical potential), and thus the homogeneity of the damping amplitude (gamma) is justified.

Additionally, as the projector is explicitly included in the single-particle basis of the system (as described earlier), the noise term is not spatially uncorrelated as suggested. This can be seen by inspection of Eq 24 in Appendix A, where the noise term is explicitly constructed in the single-particle basis of the toroid, and thus has strong spatial correlations.

We have added remarks clarifying the above in Section 3.5 of the manuscript.