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Probing nonthermal density fluctuations in the onedimensional Bose gas
by Jacopo De Nardis, Miłosz Panfil, Andrea Gambassi, Leticia F. Cugliandolo, Robert Konik, Laura Foini
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Submission summary
Authors (as registered SciPost users):  Jacopo De Nardis · Laura Foini · Milosz Panfil 
Submission information  

Preprint Link:  http://arxiv.org/abs/1704.06649v2 (pdf) 
Date submitted:  20170519 02:00 
Submitted by:  De Nardis, Jacopo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Experimental, Theoretical 
Abstract
Quantum integrable models display a rich variety of nonthermal excited states with unusual properties. The most common way to probe them is by performing a quantum quench, i.e., by letting a manybody initial state unitarily evolve with an integrable Hamiltonian. At late times, these systems are locally described by a generalized Gibbs ensemble with as many effective temperatures as their local conserved quantities. The experimental measurement of this macroscopic number of temperatures remains elusive. Here we show that they can be obtained for the Bose gas in one spatial dimension by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuationdissipation theorem that we provide. Our procedure allows us to completely reconstruct the stationary state of a quantum integrable system from stateoftheart experimental observations.
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Reports on this Submission
Anonymous Report 2 on 2017616 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1704.06649v2, delivered 20170616, doi: 10.21468/SciPost.Report.166
Strengths
1. interesting idea on a timely topic
2. combines expertise from two different fields (TBA on one side, fluctuationdissipation theorems on the other side)
3. experimental relevance
Weaknesses
1. the paper could be clearer and more concise (see suggestions for improvement below)
Report
The authors focus on a macrostate of the LiebLiniger model characterized by its occupation function $\vartheta (\lambda)$.
They show that the dynamical structure factor $S(k,\omega)$ at small $k$ is related to $\vartheta (\lambda)$ by a fluctuationdissipation theorem, and they support their claim with numerical results.
This result is very appealing, and there is no doubt that it deserves to be published. I think, however, that the way
the result is presented could be improved, and made more concise. Below are some suggestions that the authors may want to take into account.
Requested changes
1. Sections 1 and 2 are too long, in my opinion. Section 1 gives a general introduction and, in page 3,
gives an overview (with words, no formulas) of what is going to be done later in the paper. Then section 2 sounds
like a second introduction, with a rather long discussion of GGE that finally arrives at the fact
that the authors deal with a macrostate characterized by $\vartheta (\lambda)$. The latter point is, as far as
I can see, the only thing that is really needed in the paper, so the detours of section 2 (about various
equivalent ways of writing a GGE) seem superfluous to me. Then section 2 ends with a second discussion
(still in words, and no formulas) of what is going to be done in the paper. This is redundant.
I suggest that sections 1 and 2 be fused together, and significantly reduced.
I understand that, with their section 2, the authors want to claim that they have a "general framework" that is later "applied to the LiebLiniger model".
But this way of presenting their result makes the paper weaker, and more confusing, because:
(i) the authors do not say much about other models anyway, and their "general framework", instead of being a fluctuationdissipation theorem valid for all integrable models (as one would expect when reading the introduction), merely consists in writing the parametrization (6) for $\vartheta (\lambda)$
(ii) the authors seem to imply that, once the LiebLiniger case is understood, generalization to other integrable models is presumably straightforward. I'm happy with that argument, but then there is no need for an entire section about the "general case".
2. The reader has to wait for section 3.4 to finally get a taste of the physical reason why the occupation number should be
related to the structure factor. This is too late. Section 3.4 is much more instructive than the
sections 1 and 2 combined: it clearly explains the physical idea that is at the heart of the whole paper. So, it
should appear much earlier; in my opinion, this section should be in the introduction.
3. Finally, I am a bit confused by the treatment of the harmonic trap in appendix D. The authors point out two important questions that come to mind: (i) since the quantity of interest is the structure factor $S(k,\omega)$ at small $k$, it is sensitive to longrange correlations, which are more sensitive to the trap than shortrange ones, and (ii) at finite $\gamma$, harmonic confinement breaks integrability.
I am not convinced by the two answers that they provide. This is not an obstacle to publication, because this is merely a nonquantitative discussion hidden in an appendix, and the paper otherwise contains many precise quantitative results. Nevertheless, maybe the authors could clarify a bit the following points:
(i) the authors argue that "under actual reported experimental conditions", the difference between the DSF in a trap and in the translationinvariant case is relatively small. But is that only for a particular set of parameters? Or is a more quantitative criterion that says when this works? Then after that sentence, the authors turn to a calculation with LDA. But what is the justification for using LDA here? Is it because the macrostate in which $\left< \rho(x,t) \rho(0,0) \right>$ is evaluated is analogous to a hightemperature state, with shortrange correlations only? If so, is there an estimate of the correlation length that could be compared to the scale on which the density varies, that would justify the use of LDA?
(ii) The authors claim that there might be a separation of time scales $\tau_{break} \gg \tau_{rel}$. Is it possible to give an estimate of these two time scales?
Anonymous Report 1 on 2017613 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1704.06649v2, delivered 20170613, doi: 10.21468/SciPost.Report.164
Strengths
1. timely topic of generally high relevance in the field
2. relation to experimental setup
3. nontrivial advancement of field
Weaknesses
1. presentation can be slightly improved at some points (see below)
2. discussion of expectation values appearing in the correlation functions is not completely transparent (see below)
Report
The authors study integrable systems after a quantum quench and here focus on the stationary state at late times, which is known to be described by a GGE. Specifically they show how to relate the Lagrange multipliers appearing in the GGE to the fluctuationdissipation ratio and thus to measurable correlation functions. They first consider general integrable systems (with one type of particles) and then show for the specific example of the LiebLiniger gas how their approach works in practice.
Requested changes
1. The definition (1) is not completely transparent when it comes to the time dependence of the different ingredients. Specifically, the subindex t refers to the expectation value in the timeevolved state at time t. Thus this quantity seems to probe the full time evolution, not just the stationary state. Also, the integration over t' also includes times before the quench (t'<0) as well as after this time (t'>t). Is the quantity defined in (1) really measurable and useful? Please clarify this point.
Furthermore, it is not clear which state the expectation values in Sec. 3.4 refer to. Is it the GE? This should be clarified before (22).
Also, how do I have to take the order of limits? For example, the text above (34) mentions S in the stationary state, which I interpret as taking t\to\infty first. The text after (34) as well as (1) give the opposite impression.
Thus the discussion around (1), in Secs. 3.4, 3.5 and 4 as well as App. A should be clarified.
Some other remarks:
2. The abstract states "The experimental measurement of this macroscopic number of temperatures remains elusive". I have two remarks here: (i) How does your statement relate to the results reported in Ref. 9? (ii) Why would you expect an integrable system and thus the GGE to be observable in experiments at all, given that integrability is for sure broken in real systems?
3. I read the statement after (7) such that one has to solve a coupled set of equations. Is this correct? Maybe clarify this in the text.
4. The wording after (31) can be improved.
5. Is it clear that the longtime limit of S is identical to the time averaged object (37)? Maybe you can justify this better?
6. In Fig. 6 the result for k=0.2 k_F is invisible.
7. Are the oscillations in Fig. 7 also an artefact of the numerical evaluation of S? The caption doesn't say anything about them.