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Dynamics of Hot BoseEinstein Condensates: stochastic Ehrenfest relations for number and energy damping
by Rob G. McDonald, Peter S. Barnett, Fradom Atayee, Ashton S. Bradley
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Submission summary
Authors (as registered SciPost users):  Ashton Bradley 
Submission information  

Preprint Link:  https://arxiv.org/abs/1908.05809v1 (pdf) 
Date submitted:  20190819 02:00 
Submitted by:  Bradley, Ashton 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Describing hightemperature Bose gases poses a longstanding theoretical challenge. We present exact stochastic Ehrenfest relations for the stochastic projected GrossPitaevskii equation, including both number and energy damping mechanisms, and all projector terms that arise from the energy cutoff separating system from reservoir. Analytic solutions for the center of mass position, momentum, and their twotime correlators are in close agreement with simulations of a harmonically trapped prolate system. The formalism lays the foundation to analytically explore experimentally accessible hot BoseEinstein condensates.
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Reports on this Submission
Anonymous Report 2 on 20191022
(Invited Report) Cite as: Anonymous, Report on arXiv:1908.05809v1, delivered 20191022, doi: 10.21468/SciPost.Report.1250
Report
This is an interesting piece of work from one of the centres of finitetemperature BEC (and related many body system) theory. The work and conclusions are basically fully sound, and my comments are essentially to do with paying some more careful attention to how the problem is set up. On a more minor note there is the odd bit of careless spelling/grammar/formatting, and the huge chunk of white space at the bottom of page 11 is fairly unattractive! I will put my numbered comments int he next box below.
Requested changes
1. Describing the ZNG approach as wellsuited for the lowtemperature regime; I think the authors need to be a bit more specific as to what they mean by this regime. I would say ZNG is well suited to regimes where there is both significant condensate and significant thermal cloud  I would not consider it to be, for example, especially well suited to describing the very low temperature regime where depletion from the condensate is due primarily to quantum fluctuations or dynamics, and my understanding is that it is quite well suited to describing e.g. excitation spectra at temperatures considered to be significant enough to have a noticeable effect  witness the early calculations of Jackson and Zaremba in describing the JILA experiment in the earlier days of BEC.
2. "the expectation of momentum", e.g., to my mind makes little grammatical sense relative to "the expectation value of momentum"
3. I don't think it is right to say "the GrossPitaevskii equation is obtained by taking the functional derivative of the GrossPitaevskii Hamiltonian". Something like "can be generated from" would be better.
4. As a notational note, starting in equation 5, and very frequently thereafter, ket notation is used to describe what are effectively field modes. The authors may find this to be a convenience, however it should not be forgotten that these do not, in a direct sense, describe the state vector of the system. And to describe the state vector of the system in occupation number representation (as must necessarily be assumed when working in a manybody quantumfieldtheory formalism), it is a ket notation that is customarily used. Although this is sort of obvious to someone relatively experienced, a certain lack of notational care of this kind can cause considerable confusion.
5. I'm not sure that I'm happy with the definitions given of R, P and L. As described later they are effectively singlebody quantities, i.e., R and P are the centreof mass position and momentum of the many body system, and I'm tempted to suggest some defining equations to pin down exactly what these quantities are.
6. A little bit more motivation as to why the basis (which is in principle arbitrary  except for the fact that it is cut off) would be desirable (below Eq. (11))
7. Analagous should be spelled analogous (below Eq. (17)).
8. Below Eq. (23), I think the discussion of conservation laws could be clarified. The point seems to be that the way the projector is defined means that any conservation laws observed in the "total" state space/system must also be observed within the subspace described by C  therefore it is necessary, or at least highly desirable, that this should be respected in any numerical implementation
9. In the same place: "quantities may exist" (not "exists").
10. The subscript "dB" (for de Broglie), or anything else where the subscript is a word or an abbreviation of words) should really be formatted in roman (not maths italic) type. "1D" is another example)
11. Eq. (31), what seem to be Fourier transforms are not defined as such.
12. Re. Eq. (41): while I understand what the authors are trying to say, and think it basically sound, they are using a notation which looks in a sense self evidently like an expectation value for a singlebody Schrodinger equation, when that's not really what is going on, due to the system under consideration being a manybody system. In first =quantised form $\hat{A}=\sum{k=1}^{N}\hat{A}_{k}$ where the \hat{A}_{k} are operators corresponding to a particular observable for a particular particle. This is also a case where what are effectively field modes are used in a way which makes them look like basis states for a singlebody system. The outcomes are not incorrect, but there is some potential for confusion.
Anonymous Report 1 on 2019925 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1908.05809v1, delivered 20190925, doi: 10.21468/SciPost.Report.1199
Report
In the present paper the authors discuss Ehrenfest relations of the SPGPE. The latter is a wellestablished method to describe thermal effects in a BEC of ultracold atoms. In its present form I do not recommend publication of this paper for the following reasons:
1) The paper is way too long and should much more focus on the new results, i.e., the Ehrenfest relations, the authors have obtained. There existst already a large amount of literature on the SPGPE an his does not require to be reproduced here.
2) It is not fully clear to me to what equilibrium the noises that are used by the authors lead. I have the impression that the equilibrium is the classical equilibrium that is valid for the lowenergy physics, but not the quantum equilibrium that is needed at high energies (the thermal cloud). I think that it is very important that the authors discuss this equilibrium of the SPGPE so that the readers can judge its validity. At present the authors do not discuss this point at all.
3) The use of a highenergy reservoir in the SPGPE, is reasonable if one is interested in condensate formation, for which this method was originally developed. However, the authors now apply this method for collective modes and in particular the Kohn mode. The results for the Kohn mode immediately show the drawback of this highenergy reservoir, because the Kohn theorem is no longer satisfied and the Kohn modes becomes damped. This is a serious problem, because in typical ultracold atom experiments the Kohn theorem is actually satisfied and experimentalists actually use it to measure the trap frequencies. So this clearly seriously restricts the use of this method to situations where the Kohn theorem is violated, which is exceptional. Moreover, the Kohn theorem will then be violated for other reasons and not because of a highenergy reservoir. This therefore leads to the question what precisely the purpose of this paper is as the theory does not seem immediately applicable to real experiments.