Phases and dynamics of quantum droplets in the crossover to two-dimensions

The manuscript {\it ``Phases and dynamics of quantum droplets in the crossover to two-dimensions''} describes a theoretical analysis of quantum droplets formed within a bosonic binary mixture of equal mass atoms, in 2D and quasi 2D configurations realized by imposing a tight box potential along the $z$-direction. The Authors recall earlier results from literature to write the energy functionals and the wavefunction equations, which are solved numerically and by means of a variational ansatz for the wavefunction in the form of a generalized Gaussian function.

The Authors compare the 2D and quasi-2D configurations through the energy per particle and the droplet size and show that the results are in agreement when the ratio of the intra-component mean-field energy to the transverse box energy is sufficiently small. Quite reasonably, such agreement occurs for low values of $\delta a < 0$, i.e. close to the mean-field ``collapse'', and for all values $\delta a>0$, and for small values of the transverse box length. The agreement is traced to the behaviour of the interaction terms in the energy functionals.

Subsequently, the Authors also numerically calculate the dynamics of the droplet after an interaction quench, whereby the $\delta a$ parameter is changed abruptly. The spectrum of the ensuing collective oscillations is extracted, focusing on the main components, i.e. the breathing mode in the isotropic 2D and quasi-2D configuration. Finally, the collective oscillations are numerically investigated also in the case where a weak harmonic confinement is introduced with a tunable anisotropy.

Generally, I find the analysis interesting with a good balance of analytical results, mainly from literature, and numerical simulations. The former are used to illustrate the latter and help developing insight. The proposed variational approach/ansatz is very pertinent for this purpose.
These strengths are evident in the analysis of the ground state properties, less so in the results about dynamics.

I think the paper is worth of publication, provided the Authors address the following remarks.

1- At beginning of Sec. 3, the Authors give some numbers, e.g. $\chi_{q2D}<0.3$ and trace some boundaries of the regions where the agreement between 2D and quasi-2D occurs, e.g. in Fig. 1. It is unclear how this number is obtained and what is the level of agreement. Some clarification occurs later with the plots of the energy and the droplet size, but I recommend the Authors to state earlier on what they consider an agreement, e.g. in terms of the fractional difference of energy.

2- What is the energy scale of the collective oscillation frequencies in the absence of an harmonic potential, like in the isotropic case? The interaction energy? The frequencies should be compared to this scale.

3- Eq. (8) is dimensionally wrong, the LHS being an energy, the RHS a volume;

4- Eq. (9): it is stated that ``{\it Notice that the first LHY contribution is negative for} $\delta a \gtrsim 0$'': this must be explained since the first LHY term does not contain $\delta a$;

5- Eq. (14) contains $\Gamma(\dots)$ that, presumably, is the Gamma function: I think the Gamma function should not be taken for granted and must be named explicitly;

6- Fig. (1): for clarity, I recommend stating that these are contour plots of $\chi_{q2D}$ and $|\chi_{2D} - \chi{q2D}|$, otherwise they might be confused for ordinary plots of $L_z$ vs $\delta a$ at different values of $\chi_{q2D}$;

7- Fig. (2a): in the inset the three lines touch at $L_z=0.1, \mu$m implying that the three energy per particle are equal, and so also the lines in the main plot should touch: why not?

8- Page 9: it is stated that ``{\it one can see from Fig. 2 \dots that for $\delta a < 0$ the mean-field (logarithmic LHY) interaction energy is negative (positive), while for $\delta a > 0$ this is reversed}. I suggest to explicitly state that it is seen in the insets, the main plots having the magnitudes of the various terms;

9- Figure (5): there is some confusion about colors, some lines are orange color whereas they should be green, I think.

Ask for minor revision

1-Authors study both two-dimensional and quasi-two-dimensional geometries
2-ALternative approaches are used, such as doing numerics and using the variational procedure
3-The topic is relevant to the present experiments

1-It might be not clear why this specific geometry (hard-walls) is used

The Authors study both static and dynamic properties of self-bound droplets in two-dimensional and quasi-two-dimensional geometry. The Gross-Pitaevskii equation for the droplets is solved both numerically and by using the variational technique.

Overall the paper is well-written and the obtained results seem to be credible. The considered problem is relevant to experiments with ultracold atomic gases. The level of the research is appropriate for the Sci Post journal. Nevertheless, I do not find that in the present form, the presentation is clear enough to allow the publication in its present form.

1- It is said in the Abstract that "One of our main findings is that droplets can become substantially extended when their binding energies become small ...". Phrased this way this result seems trivial, as this is a rather general property of bound states, even not necessary droplets. Actually, one might rather check if the binding energy is inversely proportional to the square of the droplet's size.
2-The Authors opted for using a micrometer scale for the distances, instead of reporting the results in dimensionless quantities. This makes sense when specific experimental realization is kept in mind. Thus I would prefer to have a detailed explanation of which parameters are used and why.
3-I am a little bit puzzled by the geometry used by the Authors. It is said that hard-wall boundary conditions are used. There are two issues with that. In the 2D plane it is more common to use periodic boundary conditions for describing the gas-like unbound state. For the finite-size bound state, the boundary conditions are not important if the size of the bound state is small compared to the box size. This should be mentioned. Instead it is not obvious why the hard-wall condition is used in the transverse direction. Using a harmonic oscillator would sound to be more natural to me. So I would prefer to have a stronger argumentation of that particular choice.
4-Section 2 starts with an overview of 3D experiments. Immediately after an "experimentally verified condition" is given, which involves $na^2$, which seems to be more typical to 2D. Please clarify the dimensionality.
5- Section 2.1, I think that it would be beneficial for the clarity of the presentation to have two subsections, one on 2D, the other on the quasi-2D
6-Equation (7) should be introduced better, with detailed explanations of what are the main assumptions which are needed for obtaining this equation
7-I am puzzled by the statement that the relative error during the time evolution is $10^{-14}$. Firstly, the way it is written it is not clear if the whole time evolution is intended, or a single time step. Secondly, this number is pretty close to machine epsilon for a double and is smaller than that for a float.
8-Beginning of Section 3, I would prefer here and in other places to see also the physical terms added to $\delta a$, $L$, $\chi$, etc
9- "is small ... namely $\leq$0.2", it would be more natural to have $\lesssim0.2$
10-Figures 3a and 3b, the blue line is non-monotonous, why is that? Does that mean that at the maximum the flat-top regime disappears or that is a different effect?

To summarize, I believe that the mentioned commenters must be addressed before the article can be published.

Ask for minor revision