## SciPost Submission Page

# Dynamics of Hot Bose-Einstein Condensates: stochastic Ehrenfest relations for number and energy damping

### by Rob G. McDonald, Peter S. Barnett, Fradom Atayee, Ashton S. Bradley

#### This is not the current version.

### Submission summary

As Contributors: | Ashton Bradley |

Arxiv Link: | https://arxiv.org/abs/1908.05809v1 |

Date submitted: | 2019-08-19 |

Submitted by: | Bradley, Ashton |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

Describing high-temperature Bose gases poses a long-standing theoretical challenge. We present exact stochastic Ehrenfest relations for the stochastic projected Gross-Pitaevskii 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 two-time correlators are in close agreement with simulations of a harmonically trapped prolate system. The formalism lays the foundation to analytically explore experimentally accessible hot Bose-Einstein condensates.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2019-10-22 Invited Report

### Report

This is an interesting piece of work from one of the centres of finite-temperature 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 well-suited for the low-temperature 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 Gross-Pitaevskii equation is obtained by taking the functional derivative of the Gross-Pitaevskii 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 many-body quantum-field-theory 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 single-body quantities, i.e., R and P are the centre-of 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 single-body Schrodinger equation, when that's not really what is going on, due to the system under consideration being a many-body 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 single-body system. The outcomes are not incorrect, but there is some potential for confusion.

### Anonymous Report 1 on 2019-9-25 Invited Report

### Report

In the present paper the authors discuss Ehrenfest relations of the SPGPE. The latter is a well-established 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 low-energy 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 high-energy 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 draw-back of this high-energy 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 high-energy 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.