Multicomponent one-dimensional quantum droplets across the mean-field stability regime

Dear Editor-in-charge of SciPost Physics,

Thank you very much for your communication on the above manuscript submitted to SciPost Physics. We are pleased with the positive receptions of our work by the referee. For instance, the referee remarks: "I see that this work has a clear potential for multi-pronged follow-up work. I recommend publication." Also, the recommendation of the referee is: "Publish (easily meets expectations and criteria for this Journal; among top 50%)".

Yet, the reviewer offers some helpful remarks towards the improvement of the manuscript. In the presently resubmitted revised version of our work, we have taken all of these comments into consideration and performed the corresponding changes. In the following, you can find our detailed reply to all questions and suggestions raised by the referee. A list of changes is appended after the point by point reply to the comments of the referee.

Given the positive outlook of the review, we hope that you will find the revised manuscript suitable for publication in SciPost Physics. We will, thus, be looking forward to your final editorial decision on the manuscript.

On behalf of all the authors, Ilias Englezos

• The referee writes: "The manuscript reports an in-depth analysis of BMF quantum mixtures in 1D. The main merit is the derivation of LHY term valid for a large range of interspecies interactions, not only close to the critical points of MF collapse and immiscibility. The extension to three and four components is valuable, although inevitably limited to a few specific configurations, due to the large number of parameters (masses, interaction strength). In terms of usability for experiments, a limitation could be traced in the restriction to the homogenous case. However, in my opinion this is a necessary and justified choice to keep the number of external parameters at a manageable level. I see that this work has a clear potential for multi-pronged follow-up work. I recommend publication."

Our response is: We thank the referee for their positive evaluation of our work and useful comments which helped us to improve and further clarify our presentation. Below, we address in detail all the issues raised. Corresponding modifications have been performed in the revised manuscript and for convenience a list of changes is also appended at the end of this letter.

• The referee writes: "1- Page 2, in the sentence The emergence of the self-bound liquid-type configurations known as quantum droplets in contact interacting three-dimensional (3D) bosonic mixtures'', the reference [24] should be complemented with [44], as the latter was actually published earlier."

Our response is: We agree with the referee. In the revised version of the manuscript, we have also placed this reference at the suggested location (see also the list of changes).

• The referee writes: "2- Page 4, sec. 2.1: it is stated that 1D configurations with periodic boundary conditions, i.e. a 1D ring,\dots can be experimentally reached with digital micromirror devices''. This is not correct, DMD can make rings, but these are hardly 1D, as the transverse confinement is insufficient. Also, Ref. [69] describing a disk-shaped potential seems quite not appropriate here."

Our response is: We agree with the referee that these DMD setups do not necessarily reach the 1D geometry considered in our work. In the revised manuscript, we have removed this sentence and added a comment about the persistence of our findings with box potentials (see also the list of changes).

• The referee writes: "3- Page 4, sec 2.1: dimensionally, $\hbar^3/(m_Ag_A)$ cannot be a unit of time."

Our response is: We thank the referee for pointing out this typo. We have corrected it to $\hbar^3/(m_A g_A^2)$, which is indeed a unit of time (see also the list the changes).

• The referee writes: " 4- Page 5, below eq. 3: LaTeX, \verb|diag| should be typed as text, I think.

Our response is: We agree with the referee and we have changed ``diag" to text format in the revised manuscript (see the list of changes).

• Finally the referee points out the weakness: "In the three component mixture, the MF stability region is asymmetric: $-g/2\leq G\leq g$. No insight is given about this asymmetry."

Our response is: We thank the referee for their suggestion. We have included a new discussion in Appendix A, to provide an intuitive explanation for this asymmetry. An additional comment has been incorporated in the main text, see also the list of changes.

List of changes in the revised manuscript:

1) In the second paragraph of page 2, Section 1, the reference Science 359(6373), 301 (2018) has been placed earlier. 2) In the first paragraph of page 4, Section 2.1, a sentence regarding the possible DMD experimental implementation has been removed and replaced with a comment on the persistence of our results with a box potential. 3) In the second paragraph of page 4, Section 2.1, a typo in the time units used throughout the text has been fixed. 4) In the first paragraph of page 5, Section 2.2, the term "diag" has been changed to text format as suggested. 5) Second paragraph of page 16, Section 4.1, a comment pointing to Appendix A has been added. 6) In the last paragraph of pages 23-24, Appendix A, a short discussion about the asymmetry of the mean-field stability regime in higher-component mixtures has been included. 7) All references have been updated whenever necessary. 8) On page 9, we fixed a typo in the legend of Figure 1. 9) On page 10, a typo has been corrected in the legend of Figure 2. 10) On page 12, a typo has been amended in the legend of Figure 3. 11) On page 18, a typo has been updated in the title of Figure 5.