Entanglement transitions in SU(1, 1) quantum dynamics: applications to Bose-Einstein condensates and periodically driven coupled oscillators

The Authors investigate the problem of bosonic coherent state dynamics under $\mathrm{SU}(1,1)$ dynamics in two physical systems, namely a Bose-Einstein condensate (BEC) and a pair of periodically driven harmonic oscillators. In both cases, they identify different phases that can be tuned by changing some parameters, and they diagnose it by studying the proliferation of excitations and entanglement. The main novelties of the work are 1) the calculation of momentum-space entanglement entropy and negativity of the BEC, and 2) the full study of the Floquet dynamics of the two oscillators under the alternating evolution of two Hamiltonians. Overall, the topic of the paper is interesting, and the results are presented clearly with step-by-step derivations of the main equations, making them relatively easy to follow.

The main weakness of the manuscript is that it is not written in perfect English, and it contains multiple grammar mistakes. This does not prohibit the comprehension of the text, but the quality of writing does not currently meet the standards for a publication.

All in all, I believe the problem under investigation is interesting and the results are convincing. For this reason, I am willing to recommend this article for publication after major changes have been made. In particular, I expect the Authors to significantly improve the quality of the text, as well as to address some questions and observations that I put forward in the remainder of this report. In addition, I believe the title can be misleading, as the nomenclature ``Entanglement transitions'' more commonly refers to measurement-induced phase transitions in the literature, and I thus suggest to update it.

Here are some questions that I would like the Authors to answer.

1- The Hamiltonian of Eq.~(36) describes a BEC with contact interactions in free space. In realistic experimental implementations, the condensate must be constrained by a trap, leading to the addition of an external potential. This changes multiple properties of the BEC. Can the Authors comment on whether or not they expect their result to apply also to a condensate in a trap? Is there a limit in which the trapped condensate Hamiltonian reduces to the free-space one considered in the manuscript? I understand that the formalism used in the article is probably no longer applicable in absence of translational invariance, but I believe that it is important to contextualize the problem in a realistic situation, especially considering that the Authors claim the system is experimentally feasible.
2- The entanglement entropy considered in Sec.~4 is evaluated in momentum space between modes with opposite momenta. Is it possible to access this quantity experimentally? If so, I believe that adding appropriate references would benefit their discussion, and would also highlight that their calculation of $S_\mathbf{k}$ is not purely of theoretical interest.
3- In many-body systems, the entanglement entropy is more commonly evaluated in real space. Can the results on $S_\mathbf{k}$ be leveraged to infer what happens in real space as well, for instance by considering a half-system bipartition? My expectation for the BEC is that the real-space entropy grows linearly in time in all phases, because even in the stable phase where $S_\mathbf{k}$ oscillates, the different $\mathrm{k}$ mode show revivals at dephased frequencies, and thus no collective revival occurs. What do the Authors think?
4- What are the practical consequences of $S_\mathbf{k}\sim t$? Does this mean that eventually the condensate phase is unstable due to proliferation of excitations? What about the stable phase and the phase transition point? I think further commenting on this can enrich the discussion.
5- Can the Authors clarify whether or not (or to what extent) the phase diagrams they observe in both systems actually depend on the choice of the initial state? I would expect that as long as the system is prepared in any coherent state, the phase diagrams should remain the same, as they are fully determined by the Möbius map.
6- Do the Authors expect that the phase diagram of the paired oscillators is robust against the addition of dissipation? What happens in the realistic situation where the system is held at finite (small) temperature?

Minor comments

Below I list some minor comments and corrections I found while reading the manuscript. I point out that I checked explicitly the correctness of all equations up to Sec.~5.2 included, with the exception of Sec.~3.2
1- At the beginning of Sec.~2, I would make it more clear that Eq.~(1) is just the 2-dimensional representation of the group, but that the operators considered later on are not in that form and are instead unitary. While this is already stated, I think it can be emphasized explicitly to avoid any confusion.
2- Providing an explicit example of $\hat{K}_i$ in terms of Pauli matrices would be useful.
3- I believe that in Eq.~(4) it should be specified that $a_\pm,a_0$ are purely imaginary. Otherwise, the same operator can also be realized using the generators of the $\mathrm{SU}(2)$ group.
4- In Eq.~(6), specify that $|k,A_+\rangle = e^{\xi \hat{K}_+-\xi^*\hat{K}_-}|k,0\rangle$.
5- Can the Authors provide references to the statement ``One can prove that the $\mathrm{SU}(2)$ elements exactly form Möbius transformations'' above Eq.~(8)?
6- Make it clear that $Tr(\mathcal{M})$ coincides with the trace of Eq.~(10).
7- The reader might not be familiar with the notion of the multiplier $\eta$ introduced in Eq.~(12b). I suggest that Eq.~(12c) is used to define it, and after explaining its meaning you later put Eq.~(12b) to provide its explicit form.
8- I feel like Eq.~(18) lacks some context. I would add a comment specifying that you consider a coherent state in momentum space involving modes $\pm\mathrm{k}$, that your goal is to evaluate momentum-space entanglement, and that these states will appear later.
9- I believe Eq.~(19) has two swapped indices. The correct version should have the index $k_1$ at the bra with momentum $\mathrm{k}$, and $j_1$ at the ket with momentum $-\mathrm{k}$. This does not impact later results.
10- I believe there is the same problem in the second line of Eq.~(24). The correct version should have the index $j_1$ at the bra with momentum $\mathrm{k}$, and $k_1$ at the ket with momentum $-\mathrm{k}$.
11- At the beginning of Sec.~4.1, the Authors should provide either an explicit derivation of the effective Hamiltonian, or references that obtain it.
12- Above Eq.~(38), $|\Psi_0\rangle$ is not simply the ground state of Eq.~(37), it is its ground state when assuming a specific parameter choice. I believe its clearer if they simply refer to it as the excitation vacuum state.
13- Referring to Eq.~(40), specify that $k=1/2$.
14- In Eq.~(40), the squared sine should also have a modulus, otherwise it can become negative for $\xi^2<0$.
15- Around Eq.~(41), the Authors should mention that in the unstable mode eventually the approximation used to derive the effective Hamiltonian eventually breaks down because $N_{k\neq 0}\gg 1$.
16- At the start of Sec.~5.2, specify that ``$n$-cycle'' means to apply $\mathcal{M}$ $n$ times.
17- Before Eq.~(48), write the expressions of $\sinh r_i$ and $\cosh r_i$ explicitly in the main text.
18- In the conclusions, I believe the Authors mean ``QED'' rather than ``QCD'' cavity. Some references or discussion on how the Floquet dynamics of the oscillators could be implemented would also be useful.

Ask for major revision

The authors consider the entanglement measures of various systems described by SU(1,1) algebra, driven by quantum quenches or periodic protocols. I think the results are interesting and the paper deserves to be published in Scipost Physics Core after addressing my comments.

1. It it not obvious which quantity is k dependent and which is not. For example, after Eq. (37), the xi_{0,1,2}(k) are defined, by \xi is k independent and later \xi_{0,1,2} appear without the k argument. I find this confusing here and at other places. Similarly, in Fig. 1, n_k and S_k appears without specifying the k values.

2. For Sec. 4, there is a momentum sum for the BEC Hamiltonian, but later this sum is omitted. Is it due to the peculiar initial state?

3. What happens if there are also other terms in the Hamiltonian which do not preserve the SU(1,1) algebraic structures. Maybe distinct the generators of distinct k modes are coupled to each other or some non-linear powers of the generators appear?

Ask for minor revision