Probing non-thermal density fluctuations in the one-dimensional Bose gas

1. interesting idea on a timely topic
2. combines expertise from two different fields (TBA on one side, fluctuation-dissipation theorems on the other side)
3. experimental relevance

1. the paper could be clearer and more concise (see suggestions for improvement below)

The authors focus on a macrostate of the Lieb-Liniger 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 fluctuation-dissipation 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.

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 Lieb-Liniger 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 fluctuation-dissipation 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 Lieb-Liniger 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 long-range correlations, which are more sensitive to the trap than short-range 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 non-quantitative 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 translation-invariant 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 high-temperature state, with short-range 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?

1. timely topic of generally high relevance in the field
2. relation to experimental setup
3. non-trivial advancement of field

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)

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 fluctuation-dissipation 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 Lieb-Liniger gas how their approach works in practice.

1. The definition (1) is not completely transparent when it comes to the time dependence of the different ingredients. Specifically, the sub-index t refers to the expectation value in the time-evolved 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 long-time 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.