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

We thank the referees for their useful and relevant comments. Below we reply to their points and we list the changes to the manuscript.

REFEREE 1:

We very much thank Referee 1 for his/her appreciation of our work. Please find below the answers to his/her comments.

1. We have merged Sections 1 and 2 and we think that now the paper has improved in readability.

And while we focus on the Lieb-Liniger model in this paper, our approach is completely general. It has now been shown for a wide range of models, both those with diagonal and non-diagonal scattering, that there is a simple relation between the small momentum limit of response functions and \vartheta (or its generalization), the function that encodes the GGE (Refs. 49 and 50). This is the key relation in our paper. That it exists allows us to conclude that the program proposed in this paper will work for integrable models beyond Lieb-Liniger.

1. Following the referee's suggestion, we have included some of the content of Section 3.4 in the Introduction. Nonetheless we have left Section 3.4 in the text to allow us to give additional technical details.

2. (i) The referee writes

“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?”

We have made such statement with a chosen a set of parameters that are used in real experiments with cold atoms in optical traps, see ref [43,44]. The main point is that in real experiment the ratio \omega_{trap}/\omega is small.

The referee continues,

“Is it because the macrostate in which $\langle\rho(x,t)\rho(0,0)\rangle$ 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?”

Yes, the GGE state is typically a finite temperature-like state with a finite correlation length. We have commented on this in the text. However the value of this correlation length depend on the quench protocol and no general statement can be given.

(ii) We have given an estimate of these time scales in the text. They are given by the inverse of the trapping frequency and the inverse of the Fermi energy of the system respectively.

REFEREE 2:

We very much thank Referee 2 for his interest in our work. We answer his/her questions in order:

1. We have moved Eq. (1) to Eq. (3) to only define the dynamical structure factor in the steady state, where the limit of large times has been already taken, so that the integration over time can be extended from -infinity to +infinity. More details can now be found in Section 4.

The referee writes,

“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).”

The expectation value in Eq. (22) is on a generic stationary state, while the later is restricted to a thermal state and in Sec 3.5 to a GGE state. We have clarified this in the text.

The referee continues,

“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.”

The order of limits is always first t to infinity (late times after the quench) and than later t’ to infinity. We have commented on this in Sec. 4 and we have removed the ambiguity from the Introduction.

Some other remarks:

1. (i) In Ref. 9 it was studied a particular non-equilibrium setting where a weakly interacting gas (in the semi-classical regime) was split in two. There a GGE with two effective temperature proved to be good enough to reproduce the long time limit of local observables. We here show a protocol to measure a generic number of effective temperatures that manifest themselves when a strongly interacting gas is quenched.

(ii) Numerous experimental results, including Ref. [9], have shown that even in real settings, where integrability is nominally broken, non-thermal steady states nonetheless can be observed, at least in the time scale accessible to experiments. In the past years it has been shown [4,9][26-31] that in the presence of small integrability breaking terms in the Hamiltonian, integrability is formally not broken at intermediate time scales and a GGE emerges. Then after some crossover time, the integrability-breaking perturbations start to play a role and, typically, the system relax towards a canonical Gibbs Ensemble. We have commented on this in Appendix D.

1. “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”

We moved this part to Sec 3.5, however the referee’s question remains still appropriate. Given functions $\beta(\lambda)$ and $\omega(\lambda)$, ${bar{\mu}}$ the chemical potential controls the total density $n=N/L$. If we want to fix the density and determine the chemical potential we have to indeed solve a set of coupled equations (13) and (15) with $\vartheta(\lambda)$ given by Eq. (32) [the old Eq. (7)]. We have added a comment in the text.

1. We have rewritten the part after Eq. (31).

2. For typical finite-size systems with no extensive degeneracies, the time average is equivalent to the diagonal ensemble average, which is equivalent to the GGE ensemble average, see Eq. (43) and (44). These are well established facts, see Ref. [10] for example.

3. The two sets of data are indeed indistinguishable on the scale of the plot. We have added a comment in the caption of Fig. 6.

4. Yes, the oscillations are also a feature of the numerics. We added this comment inside the caption of Fig 7.