The effect of atom losses on the distribution of rapidities in the one-dimensional Bose gas

1. Non-trivial quantitative predictions and exact results
2. New method developed to solve difficult problem
3. The topic is timely
4. The draft reads very clearly

No weaknesses

In this manuscript the authors study the effects of atom losses on the quasi-stationary description for a 1D Bose gas with point-wise interactions. The topic is of manifest experimental interest, and certainly timely from the theoretical point of view. Furthermore, the draft is well-written and (despite the necessarily technical discussions) it is easy to follow.

The problem under study is difficult, because the integrability of the model makes conventional hydrodynamic descriptions not applicable, while the large number of particles make exact integrability-based calculations unfeasible. On top of that, atom losses explicitly break the unitarity of the evolution, which introduces additional complications. Despite all of this, the authors manage to provide highly non-trivial quantitative (and even analytic) predictions, which I believe is quite impressive. Finally, the method developed opens the way to further generalizations and applications to other settings.

For these reasons, I believe the draft clearly deserves publication. However, I have a few comment and questions that I list below.

1. First, the authors do not comment on one natural numerical check which appears to be available. Namely, after Eq. (2) they show that the integral of the function F[\rho](k) must be equal to K*n*(g_K). For these, there exist exact formulas in the thermodynamic limit, so that one has an analytic prediction for this number, given \rho. Thus, one could integrate numerically the functions F[\rho](k) obtained via Monte Carlo simulations, and compare against the exact thermodynamic result. Note that for K>1, this expectation value is zero in the Tonks-Girardeau limit, so this would yield a non-trivial test only for finite values of the interactions.

Did the authors perform this check? In any case, I think this would be a valuable check, and maybe the authors could comment on the numerical difference obtained, as this would give us a quantitative measure of, say, finite-size effects or other sources of numerical inaccuracy.

2. Do the authors have an intuition regarding the double-peak shape of the function F[\rho]? Could this be expected? Is this particular of thermal states? (as compared to more generic GGEs)

3. The authors say that the computation of F[\rho] takes~ 10 hours. On the other hand, if I understand correctly, Figure 4. requires at least 6 such subsequent computations. Does this mean that the computation time for this plot is ~60 hours? (this is just a curiosity, but the author might comment on this on the draft if they think it is relevant)

4. In general, two- and three-body losses will happen at the same time, meaning that the Lindbladian in Eq. (1) will contain several terms, each corresponding to a different K. What would be the differences to treat this case, and how much more difficult would this be in practice?

1: Relevance and interest of subject.
2: Exact formulation of evolution equation for rapidity distribution
3: Exact derivation of kernel (sum over form factors)
4: Exact results in two limits, one of which exhibits non trivial effects
of atom losses with potential experimental consequences.

1: the derivation of \rho(k) after Eqn 10 (free boson case) should be clarified.

This paper aims at describing the evolution of a Bose gas (Lieb Liniger gas) taking into account atom losses, a highly relevant issue in the perspective of describing as accurately as possible the dynamics of a physical system of cold atoms. Hence the relevance and general interest of the problem.
The integrable properties of the model requires the use of the full rapidity distribution to encapsulate the thermodynamical properties of the system.
The authors propose an evolution equation for the rapidity distribution, characterized by a functional kernel. They then derive an exact expression for this kernel, and rexpress it in terms of sum over form factors of the Bethe states, a key object in most studies of integrable systems and measure of their observables. Numerical evaluations are proposed, and two exact resummations are identified, respectively ideal Bose gas and Tonks Girardeau limit. In both cases exact results for the the rapisity distribution are obtained, with remarkable, non trivial properties in the hard-core repulsion limit.
It is undoubtedly a very interesting and very relevant paper. It is written in a rather terse and dense way, but all relevant elements are present to allow for a full (if not always easy) follow-up of the reasoning and computations.

I have some remarks and clarifications to ask for, however, which I think should be adressed to allow for final publication of this paper.

Point 1: could the authors slightly expand what they mean by "typicality of eigenstates" p4 column 2 ? and why it fails in this case ?

Point 2: I am quite confused by the derivation of the rapidity distribution after Eqn 10:
-The variable \mu is not defined, to the best of my understanding;
-The comment about a "rescaled" BE distribution is not clear;
- I am confused by the supposedly equivalent conditions stated as " T << mu (i.e. T >> n²)" ?? there must be a misprint somewhere.

Finally there is a clear misprint in (7) where \psi(0)^K should obviously read \Psi(0)^K in the 1st line