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

We greatly appreciate the suggestions made by the referees, and we believe the presentation of our results has improved considerably based on the feedback provided in their reports. We hope that the revised manuscript is deemed suitable for publication.

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Changes based on Report 1:
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- Sec. 1, par. 2, sentence 2: added "...as these two phenomena are both characterised by the emergence of system-scale eddies."

- Sec. 2.3: paragraph 2 added to describe validity of c-field methods.

- Sec. 3.4 has been restructured in order to include the additional analysis suggested by the referee, and the corresponding data is shown in the new Fig. 4 and the new Fig. 6(b) (including its inset; the caption of Fig. 6 has been updated accordingly). An entirely new Sec. 3.4.1 has been added to describe the additional data, while the previous discussion has been split over what is now Secs. 3.4.2 and 3.4.3. While the text in Sec. 3.4.2 is mostly unchanged from the previous version, Sec. 3.4.3 has been largely rewritten to accommodate the newly added results. A second paragraph has also been added to the beginning of Sec. 3.4 to summarise the subsequent subsections.

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Changes based on Report 2:
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- Sec. 1, par. 4. Replaced sentences:
"The unitary projected Gross–Pitaevskii equation (PGPE) that conserves both the energy and the normalisation of the classical field is used for describing the dynamics of the Bose gas. We systematically vary the initial condensate temperature by sampling microstates using the stochastic projected Gross–Pitaevskii equation (SPGPE) and determine the resulting effect on the turbulent dynamics for an ensemble of statistically equivalent vortex distributions."
with:
"Briefly, this uses the Gross-Pitaevskii equation to simulate the dynamics of not only the condensate, but also the low-energy thermal fluctuations of the field. We simulate the grand canonical stochastic projected Gross–Pitaevskii equation (SPGPE), describing the classical field coupled to a bath, to generate initial thermal ensembles. We imprint vortices on these microstates to form an ensemble of vortex distributions, and determine the resulting effect of the finite temperature on the turbulent vortex dynamics by integrating the microcanoncial projected Gross–Pitaevskii equation (PGPE) that conserves both energy and normalisation of the classical field."

- In the text following Eq. (3), the factor of dt has been moved to the end of the expression for the non-zero moment of the driving noise.

- Sec. 2.2, final sentence: added "in order to avoid the detection of numerical 'ghost' vortices in regions of low classical field density" (and associated reference)

- Sec. 3.2, par. 1: the description of the three vortex temperature regimes has been formatted as a list, and each item has been expanded to include additional details. The footnote has been incorporated into the main text.

- Sec. 3.4, par. 1 has been reworded as per the referee's request. It now reads:
"Despite the complexity of 2D QT, it has been demonstrated that the dynamics can in many cases be characterised in terms of statistically steady distributions that are only weakly dependent on the microscopic details of the system. This is a general and powerful approach to understanding the evolution of a system out of equilibrium, and allows its characterisation in terms of far from equilibrium universality classes [56,57]."

- Sec 3.4.2, final paragraph: the final sentence, which previously read:
"It is predicted that the system would eventually return to the Gaussian fixed point; although the time required to do so may approach infinity as T->0"
has now been changed to:
"It is predicted that the system would eventually return to the Gaussian fixed point, because vortex–phonon interactions give rise to gradual vortex diusion [61], which serves to break up vortex clusters and encourage vortex–antivortex annihilation. However, it is possible that the time required for the system to cross over to the Gaussian fixed point may approach infinity as T->0."

- Sec. 4, final paragraph: the question "does the system eventually revert to the Gaussian fixed point by annihilating all vortices?" has been changed to: "does the system eventually thermalise by reverting to the Gaussian fixed point and annihilating all vortices, as predicted?"
Immediately after this, the following sentence has been added:
"Although the vortices should gradually diffuse toward the fluid boundary due to interactions with phonons, perhaps this effect is not strong enough to overcome the topological protection of the vortices."

- Sec. 4, final paragraph: Four new sentences have been added to the end of the section:
"Some of these questions for wave turbulence in three-dimensional homogeneous Bose gas have been addressed in a recent experiment by Navon et al. [66], in which they observed the establishment of a direct energy cascade. Similar experiments could be performed in two-dimensional systems [25,26,28]. In the future it will be interesting to address these questions for vortex turbulence in two-dimensions using the numerical methods utilised here. Three-dimensional simulations pose a larger numerical challenge, but could be addressed using supercomputer simulations of the GPE as utilised in Ref. [67]"

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Other changes:
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- Added references [23, 41, 58, 61, 66, 67].

- Removed the final sentence from the abstract. It previously read: "The cross-over between these two dynamical behaviours is found to occur at earlier times with increasing condensate temperature."

Sec. 1:
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- Par. 3, sentence 2: added the words "incompressible kinetic" before the word "energy".

- Par. 4 has been split into two paragraphs. Added the words "In our simulations" to the beginning of the new paragraph 5.

- Final paragraph: based on changes made as a result of Report 1, we have replaced the sentence:
"The time-dependence of the nearest-neighbour vortex spacing provides evidence of a cross-over from evolution towards an anomalous non-thermal fixed point to a thermal fixed point."
to:
"Evidence is also provided for universal scaling in our simulations, and based on this we are able to interpret the dynamics as evolving towards either a thermal or non-thermal fixed point, depending on the temperature of the system."

Sec. 2.1:
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- Par. 2, sentence 3: removed factor of 10^-5 in front of definition of the unit T_o. The numerical temperature had previously been inadvertantly divided by a factor of N=10^5 (the normalisation of the classical field). This has now been corrected, and as a result the factor of 10^-5 has been removed. This change does not affect any other temperature values quoted in the manuscript.

- Par. 2: new sentence added at the end of the paragraph to express the numerical units in terms of experimentally realistic quantities.

- Par. 3, sentence 2: the words "above the cutoff" have been added.

- Par. 4, sentence 1: the words "damping parameter" have been replaced with "growth coefficient", and the word "although" has been removed from the beginning of the comment in parantheses.

- Par. 4, sentence 2: we have changed:
"the temperature is set to a constant value in the range 0 < T/T_o <~ 3 throughout each simulation"
to:
"the temperature is set to one of three values, T/T_o =~ {0.9, 1.8, 2.7}, for each simulation ensemble."
Immediately following this, we have added the sentence:
"Note that we also perform a T = 0 simulation, but we do not need to evolve the SPGPE (3) to find the initial state."

- Par. 4, sentence 4: the words "the condensate fraction vanishes in equilibrium" have been replaced with
"the condensate fraction of the classical field vanishes in equilibrium while holding the chemical potential constant".
Following this sentence, we have added:
"Note that this is different to the typical scenario in which the total number of atoms is kept constant as the temperature is varied."

- Par. 6, first sentence: the word "extracted" has been replaced with "determined".

- Final paragraph: immediately after defining the function \chi(r), the word "captures" has been replaced with "approximates".

Sec. 2.2
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- First sentence: added "(corresponding to t~3.8s for the physical parameters chosen in Sec. 2.1)"

Sec. 2.3
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- The two paragraphs that previously comprised Sec. 2.3 have been joined into a single paragraph, and following "\Delta x ~ \xi_o / 3.5", we have added:
"which ensures that the vortex cores are accurately represented by the numerics. This choice leads to a value for the wavevector k_cut of the projector."

Sec. 3.1
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- First sentence: corrected the reference to "Fig. 1(d)-(e)" to "Fig. 1(d)-(f)".

Sec. 3.2
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- Par. 1, first sentence: the words "incompressible kinetic" have been added before "energy".

- Par. 1, sentence 2 has been added:
"The inverse temperature is defined as \beta = 1 / k_b \partial S / \partial E", where E and S are the entropy and energy of the vortex gas, respectively."

- Par. 1, immediately following the list: the words "extract this" have been replaced with "determine the".

- Par. 2, sentence 1: the word "extracted" has been replaced with "determined".

- Par. 2, sentence 2: the words "vortex gas evaporatively heats up" have been replaced with "evaporative heating of the vortex gas proceeds"

Sec. 3.3
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- Par. 1, first sentence: the words "occasionally also" have been reversed to "also occasionally".

- Fig. 3 caption, first sentence: "decay curves" has been replaced with "as a function of time".

Sec. 3.4.2
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- Par. 1, sentence 1: the words "Previously, Groszek et al. identified related scale invariant behaviour" have been changed to:
"Previously, Groszek et al. identified scale invariant behaviour in decaying quantum turbulence".

- Par. 1, sentence 2: the words "the decaying quantum turbulence rapidly approaches" have been replaced with "the vortex configuration rapidly approaches".