Prethermalization in coupled one-dimensional quantum gases

Dear Referee,

thank you for feedback and comments on our manuscript. In the following, we would like to address the specific raised points in your report.

1) After eq 16 say that "$Q_0$ is defined below" or the like for more readability

Thank you for this comment. We have modified the text below eq. 16.

2) After eq 33: perhaps also say that we need the operation of dressing to be invertible to conclude that we must have vanishing of $Q_0$.

We have added the comment.

3) Just before eq 34: say that this is indeed a necessary condition as all other factors in eq 29 are strictly positive (on the set eqs 36,37)

We have expanded the discussion around eq. (34) by stating explicitly that the bare integral can vanish as a whole or because the integrand is zero. In the latter case, because the other factors are strictly positive, this leads to a necessary condition of eq. 29.

4) Page 12 "energy momentum-energy conservation laws"

Corrected.

5) Equation 58 is valid to what order in $1/c$? First order only? All orders? Please clarify.

It is valid up to and including $1/c^2$. See eq. (62) of the new version.

6) Solution to eq 59 (now eq. 68): it is stronger than that, as what we want is to find $f_1$ and $f_2$ that are independent of the state (as in the value of conserved quantities on the state, eq 49, the functions $f_i(\lambda)$ do not depend on the state). The state-independent $f_i$ must satisfy eq 59 for all states. It is very different from eq 56, where $\epsilon_i(\lambda)$ are determined by (characterise) the state. The problem appears to be very different. The special cases consider, where dressing disappear, are ok, but the full set of solutions will be different it seems.

We agree with the referee and we emphasise now the difference between the equations for the stationary state and for the charges in the text. Furthermore, we now explicitly show that the equation for the conserved charges can be still solved in the strongly interacting case that we dub the deformed Tonks-Girardeau gas. The deformed Tonks-Girardea gas includes $1/c$ corrections to the Tonks-Girardeau gas up to and including order $1/c^2$. In this case we find that $f_i$ depend on the state only through the total density of particles in each tube - a quantity that is an invariant of the dynamics. Furthermore, in Appendix E, we show that it is impossible to find such conserved charges beyond the deformed Tonks-Girardeau case.

7) Eq 62 is not a conserved charge, it is the average value of the conserved charge in the state characterised by $\rho_{p,i}(\lambda)$ (GGE of the unperturbed system). Eq 64 is unclear: what is the time T? What is the specific dynamics? It in fact does not make sense, as the average value of the conserved charge should only depend on the state, and should not have explicit time dependence (there is no time, there is just a state). Do the authors want to take $T\rightarrow \infty$? Is the idea that $I_{i,j}$ is then a function of the state, specifically the time-averaged value of $I_{i,j}(t)$ starting form that state? Please clarify this. If so, then indeed $I_j$ is a function of the state only, and this would make more sense.

Upon further studies of our numerical construction of the charges, we have decided to abandon it for now. We have found that observing the minimisation of the fluctuations is not a sufficiently sensitive condition to define a conserved charge. This is something to which we plan to return, but for the present work, we have decided to shift our attention to the analytical results available at large $c$.

We would like to thank the referee for the comments on the numerical construction and replying to the referee comment, indeed we meant to take the $T\rightarrow \infty$ limit.

8) On this point, the authors said that locality of $I_j$ would be clear. But then, in order for this to be so, one should interpret eq 62 more "literally" as an operator equation, not just an equation between average. Just an equation between averages in GGEs (of the unperturbed system) would not guarantee that $I_j$ is local as an operator; there are many nonlocal operators that have the same expectation values as local operators in all GGEs. Then this should be made clear, perhaps with hatted symbols, and in this case $I_{i,j}$ is not given by eq 63 (which is just its average). But then, this would be a problem, as the definition of the operator $I_j$ would involve explicitly not only the charges $I_{i,j}$, but also denominators that depend explicitly on the state. We cannot define intrinsically a charge with coefficients that depend on the state (we can take linear combinations of conserved quantities with state-dependent coefficients, e.g. in subtracting the average times the identity operator, but then this is not intrinsic). Thus, eq 62 as an operator equation does not make sense.

The construction of the charges for the deformed Tonks-Girardaeu case allows us to address this point directly. The charges have then coefficients that depend on the state only through gas densities which are conserved by the evolution. Thus, working in the Hilbert space with fixed number of particles, these charges correspond to linear operators. On the other hand, going beyond the deformed Tonks-Girardeau gas, as we now understand, invalidates the whole construction because we are unable to solve for functions $f_{i,j}$ and define $I_{i,j}$. We then agree with the referee that the old eq. 62 would not make sense at the operatorial level for generic values of $c$. However again, limiting ourselves to the large $c$ regime of the deformed Tonks-Girardeau gas, avoids this issue.

9) how does one minimise the fluctuations (sentence after eq 65)? Why does on do so? How is the condition of non-zero norm of the operator translate to eq 66? Given that only averages of charges in states were defined, how does one get to an operator norm condition?

The minimization was achieved by solving a linear problem for the coefficients and we were using it as a criteria to validate that the proposed quantity is a conserved charge. We were working only at the level of averages not trying to relate directly the resulting objects with operators conserved at the quantum mechanical level. We, in any case, have now abandoned the numerical construction as we found it was not sufficiently sensitive to indicate whether or not a conserved quantity was actually present.

10) Eq 68: I don't understand this. Naively this seems to be try from eq 67 with eq 62. However, in eq 62 each denominator seems to be a function of both $\rho_{p,1}$ and $\rho_{p,2}$, because the time evolution involves coupling between them (unless I am missing something); and also perhaps the coefficients because of the minimisation of fluctuations.

In the present version of the manuscript we have limited the GGE construction to the deformed Tonks-Girardeau gas where the expressions for the conserved charges are explicit. In the Hilbert space with fixed number of particles they are linear operators. This allow us to construct the GGE for a coupled system in an analogous way for a single Lieb-Liniger gas. The main difference occuring for the coupled system is a renormalization of the the chemical potential that occurs because the conserved charges have coefficients depending on the density of the particles.

11) Then, I am rather lost on the rest of that section. In particular, in eq 70 there is $f_{i,j}(\lambda)$, the one-particle contribution of nontrivial conserved quantities. But if the source term of the pseudo energy has this form, then the average of the nontrivial conserved quantities should be linear in $\rho_{p,j}$. But I concluded above that this couldn't be the case from the definition eq 62. So this seems to be inconsistent.

This tension is now resolved by limiting the construction of the GGE to the deformed Tonks-Girardeau gas. The conserved charges of the coupled system are given then by the usual ultra-local conserved charges multiplied by a particle number dependent prefactors. Working in the Hilbert space of fixed particle number, their eigenvalues on a multi-particle state are linear in $\rho_{p, j}$.

Dear Referee,

Thank you for feedback and comments on our manuscript. In the following, we would like to address the specific points raised in your report.

1) This is a fully theoretical work, using a hydrodynamic framework based on the Boltzmann collision integral approach.

In our approach, we do not rely on hydrodynamic approximation. The system is homogeneous and the Boltzmann equation can be systematically derived starting from the Fermi's Golden Rule (FGR), which is the lowest order of quantum time-dependent perturbation theory. Thus, it is different from the phenomenological hydrodynamic approach based on the gradient expansion and the existence of conserved charges. We use the standard FGR method (widely used in atomic physics and optics) to a situation of interacting many-body system.

2) The main reason is that in this work there is no independent numerical result supporting the analytical predictions (by "independent" I mean some ab-initio numerical simulation, using for instance tensor-network techniques). Of course, I understand that producing numerics for this model might be very challenging or even impossible. Still, without an independent numerical test it is difficult to quantitatively estimate the impact of the approximations made in the hydrodynamic setting.

Indeed, a reliable numerical check for such systems is a very difficult task. Homogeneous, non-equilibrium initial states that we consider are hard to prepare in tensor-network simulations especially that we are considering a continuum system, which brings additional challenges from the perspective of numerical simulations. Instead, our calculations are based on standard time-dependent perturbation theory truncated at the level of FGR. This is a systematic and established method, and there are no reasons, why it should break down, assuming that inter-tube interactions are sufficiently weak.

Furthermore, in a recent preprint co-authored by one of us (Ref. 51 in the manuscript), a similar prethermalization phenomena was observed in a fully quantum dynamics of the XXZ spin chains forming a ladder. This additionally secures that the results of our work are not peculiar to the Lieb-Liniger model and are not an artefact of the approximations that we perform.

3) Accordingly, in my opinion the main value of this work relies in its technical, mathematical aspects. They are certainly impressive, but perhaps of interest for a more specialized audience.

Apart from technical aspects involving properties of the Boltzmann collision integral, we emphasize that our work identifies a concrete and interesting physical phenomenon, i.e. existence of prethermalization plateaux in the relaxation dynamics. As was pointed out by the Second Referee, the setup of long-range coupled Lieb-Liniger gases is an important problem from the theoretical point of view. What is more, it was also studied in recent ultracold gases experiments (see Refs. 19 and 20 in our manuscript).

4) I also have a curiosity: it seems to me that the model is integrable in the limit opposite to that studied by the authors, namely when $A(x)=\delta(x)$ (after regularization). In this case, it seems to me the model can be mapped onto a Yang-Gaudin Hamiltonian. Is this true? Do the authors expect that this observation can be used to study the dynamics in the opposite limit of short-range interactions?

This is true and very interesting observation. This situation was studied in the spin chain ladder in the aforementioned preprint [51]. The results also show the prethermalization plateau. We also note that the Yang-Gaudin model requires inter- and intra-tube interactions to be identical. This makes it less relevant for the experiments with cold atoms.

5) In conclusions, I would recommend publication of the draft in Scipost Physics Core.

After the feedback from the Referees, we still believe that our manuscript deserves publication in SciPost Physics. Both the setup and uncovered phenomenon are interesting for general audience. Most of the technical aspects of the work are related to identification of the conserved charges. On the other hand, the most important part (existence of a prethermalization plateau) only involves the Boltzmann equation formalism and the Thermodynamic Bethe Ansatz. The latter is a standard tool, as illustrated by its ubiquity in generalized hydrodynamical computations, to study thermodynamics and dynamics of low-dimensional many-body systems.