Exact out-of-equilibrium steady states in the semiclassical limit of the interacting Bose gas

We thank the three referees for their positive comments on our manuscript and for supporting its publication on SciPost Physics. The referees requested some clarifications and put forward some suggestions to further improve our paper: hereafter, we separately reply to each referee describing the changes we made to meet their requests.

Referee 1: We thank the first referee for her/his positive comments. We clarify the issues questioned by the referee using the same numbering used in her/his report. 1. This is indeed an interesting question and comparisons can be made. First, let us point out some caveats: the GS of the non interacting theory in the classical case is nothing else than the constant field phi(x)=constant. No fluctuations neither inhomogeneities are present and the dynamics is easily solved, showing persistent oscillations without any time-relaxation. This pathological limit is due to the absence of statistical fluctuations in the initial state while quantum fluctuations are suppressed in the semiclassical limit. In view of this pathological behavior, we chose not to focus on the dynamics of such an initial state, nevertheless our method can be still applied to compute the GGE root density (even though the system, strictly speaking, does not show any relaxation), which results in a semicircle law. This root density provides sensible predictions for time-averaged quantities. This finding is in agreement with the weakly interacting limit of the quantum result. In the improved version of our manuscript, we explicitly discuss this connection at the end of Appendix D.2. 2. We added a new plot with the requested comparison between GGEs and thermal states. This figure is Fig. 6 in our resubmitted manuscript. Note that the free parameters of the thermal states (beta and mu) are fixed from the inital state. In particular, as it is now discussed in the caption of Fig. 6., beta is unchanged while mu is adjusted to reproduce the initial densities. 3. We thank the referee for pointing out these typos which we promptly corrected.

Referee 2: We thank the referee for appreciating both our results as well as the presentation of our findings. Here, we provide a detailed reply to the referee’s concerns, following the same numeration of the report: 1. Within the quantum case, the determination of the post quench GGEs passes through either computing the charge expectation values (which can be troublesome as we discuss in the manuscript) or through the quench action approach, where one uses the exact computation of certain overlaps among the initial state and the post quench eigenbasis. However, the computation of these overlaps is only possible for very specific initial states. In this respect, a method valid for general states is still lacking. We added the relevant references as the referee suggests. 2. One expects classical physics where statistical fluctuations dominate on quantum ones: in this respect, quantum density matrices that smoothly populate a large part of the Hilbert space can be arguably well approximated by classical physics within a statistical approach. As we extensively discussed, this is the natural conclusion for thermal states with high temperatures. As the referee points out, this is not the only situation where the NLS equation emerges as a description of a quantum model: this is for example the case of the Bose-Einstein condensate. However, the setup is rather different: even though it is true that a single-mode (the condensate) is macroscopically populated, no thermal fluctuations are present, since the system is described by a pure state and not by a density matrix. This results in a purely deterministic dynamics for the classical equation: while an extensive literature has been dedicated to this interesting problem, the relaxation in the GGE sense requires a notion of statistical averaging and thus is not of direct applicability in this case. On the other hand, using classical thermal states to initialize the system, ensures the emergence of a GGE thanks to the averaging on the initial conditions.  3. As the referee correctly understood, we use the quantum TBA as a convenient way to determine the classical TBA, as well as other analytical results presented in the paper. Of course, once the semiclassical limit has been taken, the final result is expressed only in terms of classical objects, thus the classical TBA must be used. In the resubmitted version of our manuscript, we clarify this point at the beginning of the section “TBA in the semiclassical regime”. 4. We use hbar as the small parameter to study the semiclassical limit for the sake of clarity and probably with a more theoretically-oriented taste, but the semiclassical limit can be equivalently reached acting on experimentally-tunable quantities. More precisely, the limit on thermal states can be achieved keeping fixed the chemical potential and taking large temperature and weak coupling, while the product of the latter two is kept fixed. We already pointed out this fact in the previous version of our manuscript, but for the sake of clarity we stress it more in the revised version: at the end of Sec. 3 we wrote a short paragraph explaining how the semiclassical limit can be attained in the lab. We also point out that the quench we explicitly consider in our work is from a non-interacting thermal state to a finite value of the interaction. Therefore, we are changing the interaction and not the chemical potential (which would have led to trivial dynamics) as the referee commented. Provided the fields are correctly normalized, the final classical interaction can always be set equal to 1. At the end of Sec. 5, we now clarify more the connection between the quantum and classical quench, explaining which quantum quenches are approximated by our classical analysis. 5. As we extensively discuss in the manuscript, the physics of the LL model is expected to be classical in the weak coupling/high temperature regime. This has been studied in detail in arXiv:2003.11833 (now included in our reference list) for thermal states, with a systematic inclusion of quantum corrections on the pure classical result. Our work provides a first step in a semiclassical treatment of out-of-equilibrium setups: corrections to the classical approximation are surely a compelling question which we left for future investigations.

We thank the referee for having pointed out some typos which we promptly corrected.

Referee 3: We thank Referee 3 for her/his appreciation of our results, albeit she/he suggests a change in our presentation style. First, we would like to point out that both the other referees explicitly express their appreciation for the presentation (Referee1: “The paper however is well written and can be understood by a large audience. The introduction is clear and the paper is pedagogical.” ; Referee 2: “The manuscript is very well-written with an overall very good presentation.”). We understand that our work contains a certain amount of technical aspects, which are interesting by themselves. Our choice was however to combine such aspects within a discussion of the physical interpretation of both the semiclassical limits involved and the behavior of the out-of-equilibrium dynamics. We chose a more pedagogical exposition that grants accessibility to a broader audience, with a long introduction clarifying the messages of the paper. In the new version, we modify the introduction accommodating for a clearer list of the paper’s content, in such a way the reader can be guided through the presentation to the result she/he is interested in. Those readers who are less interested in the technical details can easily skip some sections and jump directly to the results and the comparison with numerical simulations. On the contrary, an expert reader can quickly move to the technical parts, having already clear in mind the general framework. Let us stress that the results about the density moments and FCS in the classical regime are among our more important results, therefore we believe that they must have a central role in our exposition and they cannot be relegated to appendixes.

We thank the referee for pointing out some typos that have been corrected.

1. A detailed "organization of the paper" section has been added to the introduction to better guide the reader through the main results of the paper.
2. We added the comparison between the analytical results of the quench in the quantum Lieb-Liniger model and our semiclassical approach in Appendix D.3.
3. We compare now the GGEs root density against thermal distributions in Fig. 6.
4. References to the initial-value problem in quantum quenches have been added.
5. The discussion about how to realize the semiclassical limit with experimentally-tunable quantities has been added at the end of Section 4.