Self-bound droplets in quasi-two-dimensional dipolar condensates

Summary of changes

• We have removed seceral acronyms such as CS and HFBT.
• We have used Sec. II.C, especially the last two paragraphs, to discuss the properties of the quantum phases.
• In Fig.4(b), we have added the $\sigma_{\rm EGPE}$-$N$ curve obtained by numerically solving EGPE.
• We have added one paragraph at the end of Sec. III.C to discuss the EGPE results for the radial size.
• Reference [49] is added for the validity of the Gaussian density profile.

Response to the comments and questions

1- A first initial comment is that I think the authors use too many acronyms in the text: SVS, SGS, CAN, GST, CS, SCS, HFBT... I understand acronyms are often used in the scientific literature, but I have found this to be a little bit excessive in this work, making the manuscript difficult to read at some points. I think the authors can improve that.

We thank the referee for this advise. In the revised manuscript, we have remove several acronyms and only retained SVS, SCS, CAN, GST, and EGPE.

2- Regarding the Gaussian States theory and its review in section II.B, the authors develop the mathematics of the model but do not discuss the physics behind it, as said above. For that reason it is difficult for a non-specialist in Gaussian State Theory to appreciate most of the results derived on the draft. Since there is no a priori restriction on the length of the works being published in this journal, I urge the authors to at least explain what the physics behind each of the relevant states being discussed is.

We thank the referee for the suggestion. In the revised manuscript, we have used Sec. IIC to explain the physics behind the quantum states. Specifically, we present the criterions for the quantum phases and summarize their properties (e.g., the second-order correlation function and atom number distribution). These discussions can help the readers to capture the physics of the quantum phases.

3- Related to the previous point, for instance in Fig.1 the authors present the dependence of the coherent fraction fc as a function of the number of particles N, where fc = Nc/N with Nc the occupation of the coherent mode. What is the relevance of that plot exactly, what are the physical implications behind that? Actually the most relevant question would be: what does a larger or lower Nc tells us about the physical structure of the Bose condensate? Is it related in any way to the number of particles occupying it?

We thank the referee for the careful reading of our manuscript. Before we explain the physical implication of $f_c$, we would like to point out that the purpose of Fig. 1 is to validate the quasi-2D model, which can be used to dramatically reduce computational complicity. The results regarding the physical implication of $f_c$ are presented in Fig. 2 where $f_c$ acts as the order parameter that determines the quantum phases of the system. For the question about $N_c$, we first note that it is indeed the occupation number in the coherent mode $\bar\phi_c$.

Now, let us explain how $f_c$ reveals the structure of a Bose condensate.

• For a conventional condensate, we have $N_c\approx N$ implying that majority of the atoms are condensed in the coherent mode. The $N_s$ ($\ll N$) uncondensed atoms are depletion described by a multimode squeezed state with mode $\bar\phi_{s,j}$ being occupied by $N_{s,j}$ atoms. Here the total number of squeezed atoms is $N_s=\sum_jN_{s,j}$.
• In the opposite limit with a single squeezed mode being macroscopically occupied, i.e., $N_c/N\approx0$ and $N_{s,1}/N\approx 1$, the condensate is in a macroscopic squeezed vacuum state which has a completely different statistical property compared to the coherent state, e.g., the second-order correlation function and the atom number distribution [1-3].
• For the intermediate case with the coherent and a single squeezed modes being macroscopically occupied (i.e., $N_c/N\neq 0$, $N_{s,1}/N\neq 0$, and $N_{s,j>1}\approx 0$), the condensate is in the squeezed coherent state.

In the revised manuscript, we have summarized these results in Sec. IIC.

4- In the same figure the authors show the radial size \sigma of the system as a function of the particle number. How does that result compare to what one would get with a standard (extended)-Gross Pitaevskii calculation? The later is the reference for most of the calculations on these kind of systems today, so an actual comparison is in order. Not just for this quantity, but also for many others along the paper that can be computed in both ways, for instance the Critical Atom Number, which is known to be a rather delicate quantity to reproduce accurately, see point 6.

We thank the referee for the suggestion. Since the results presented in Fig. 1 are merely the demonstrations for the agreement between the full 3D and reduced 2D calculations, we have added the results for $\sigma_{\rm EGPE}$ (radial size numerically computed using EGPE) versus $N$ in Fig. 4(b) where the physics of the results are also discussed.

From the $\sigma_{\rm EGPE}$-$N$ curve, one may determine the corresponding critical atom number which, as shown in Fig. 4(b), is the same as that obtained from the $\sigma^{(c)}$-$N$ curve. The underlying reason for this is because the many-body wave function associated with EGPE is also a coherent state.

In the revised manuscript, we have added one paragraph at the end of Sec. IIIC to discuss the EGPE results for the radial size.

5- In the same line of point 4, in Eq.(21) the authors provide an expression for the energy per particle of the system, assuming the density profile is a Gaussian. Once again, they do not provide any plot or evidence that this is indeed the case in the range of parameter values where they use it. A plot of the density profiles, compared to the equivalent GPE result, is required here.

We thank the referee for the careful reading of the manuscript. First of all, we would like to clarify that the validity of the Gaussian density profile for dipolar condensates was checked as early as in 2001 (see, e.g., Ref.[4]). And we also used this assumption for binary droplets of K atoms [3]. The general conclusion is that the Gaussian-density-profile assumption holds when interactions are weak, or equivalently, when the number of atoms is small. As a proof for this, we compare the numerically obtained normalized densities $\bar n(\rho)$ (i.e., $2\pi\sqrt{\pi}a_z\int \rho d\rho\bar n(\rho)=1$) with their Gaussian fits in the figure attached to this reply for two different atom numbers. Indeed, nearly perfect agreement is achieved when the number of atoms in the condensate is small. While, for large $N$, obvious deviation is found.

Secondly, in this work, the Gaussian density profile is only introduced to qualitatively understand the W-shape $\sigma$-$N$ curve, which does not require high agreement between the actual and variational density profiles. In fact, for our purpose, we only need to show that the $\sigma$-$N$ curve is of V shape within the range of parameter values for a single quantum phase.

Since the validity the Gaussian-density-profile assumption is rather clear, it is cumbersome to add the comparisons in the present paper. Instead, we cite Ref. [49] in the revised manuscript to avoid any confusion. In addition, we also emphasize before Eq. (21) that we only perform a qualitative analysis.

6- The authors finally use this gaussian ansatz to derive a rather simple expression for the Critical Atom Number (CAN) for droplet formation. This is quite an interesting result because there are actual measurements of the CAN for 162Dy and 164Dy atom droplets, performed by the group of Prof.T.Pfau in Stuttgart (see for instance Ref[23] of the manuscript). This is an interesting benchmark because current theoretical predictions from the GPE show orders of magnitude deviations from the measured data. In this sense, a comparison between the actual data, the author's results and the GPE prediction would be a very interesting result.

We thank the referee for the comment. In the earlier works of ours [2,3], we performed systematical comparisons between the calculated critical atom number with the experimental data for both three-dimensional dipolar droplets of Dy atoms [5] and quasi-two-dimensional binary droplets of K atoms [6]. In both works, reasonable agreements were found.

In the present work, we propose a new configuration to generate quasi-two-dimensional dipolar droplets which has not been experimentally realized yet. Therefore, a direct comparison between our results and the experimental data is not applicable. Nevertheless, it would be very interesting to check our calculations when the experimental data become available.

Reference [1] T. Shi, J. Pan, and S. Yi, arXiv:1909.02432 (2019). [2] Y. Wang, L. Guo, S. Yi, and T. Shi, Phys. Rev. Research 2, 043074 (2020). [3] J. Pan, S. Yi and T. Shi, Phys. Rev. Research 4, 043018 (2022). [4] S. Yi and L. You, Phys. Rev. A 63, 053607 (2001). [5] M. Schmitt et al., Nature 539, 259 (2016). [6] C. R. Cabrera et al., Science 359, 301 (2018).

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