Fermionization of a Few-Body Bose System Immersed into a Bose-Einstein Condensate

The Authors have appropriately addressed many of the 28 questions I raised in my previous report. Still, I believe some questions that might be answered better. The numbering of the comments is the same as in the previous report.

Response: We thank the referee for their time in evaluating our response and updated manuscript, and are happy to see that they agree with many of the points in our reply. We hope that the following responses adequately answer the remaining questions.

1 - The minority particles are confined to a hard-walled box. This typically induces a finite-size effect in the energy of the order of $1/N$. Instead using periodic boundary conditions seems to be much more natural as (a) finite-size effects would scale much more favorably, as $1/N^2$ (b) the geometry will be much more similar to that which appears in the thermodynamic limit, $N\rightarrow\infty$. Is it possible to use periodic boundary conditions for the minority component?

Response: Since in the infinite limit described by periodic boundary conditions the pinned state is only pinned with respect to each particle's neighbour, the appearance of the pinned state requires spontaneous symmetry breaking. This is difficult to simulate and we have therefore opted to rather break the symmetry by hand by introducing a box potential. This also significantly reduces the numerical complexity of the problem and allows to more accurately pinpoint the transition from superfluid to pinned.

If the transition is similar to a phase separation, there should no be problems in using periodic boundary conditions. Note, that periodic boundary conditions do not describe the infinite limit, but rather a system on a ring, which still has finite-size effects. Potentially, the finite-size effects can be smaller in that case. The original question was if it is possible to use periodic boundary conditions for the minority component?

Response: While the referee is correct that periodic boundary conditions for continuous systems have some similarity with the thermodynamic limit, and while they also represent finite ring-geometries, for systems which take on a discrete structure this comes with a number of additional complications. Let us first reiterate that not only in free space, but also on a ring the particles are only pinned with respect to their respective neighbours, and for both situations the appearance of the pinned state requires spontaneous symmetry breaking. As we said before, doing this by introducing a finite size box potential is a very clean and clear way. Furthermore, experiments are inherently of finite size and box potentials are a common experimental tool these days. Secondly, introducing periodic boundary conditions forces a periodicity on the system that is related to the size of the numerical grid and requires an additional step of numerical optimisation of the grid size (i.e. find a grid size that minimises the energy of the system after the transition). Our numerical simulations are already at the boundary of what it possible, and we therefore cannot carry out the additional optimisation of the grid. Our solution to this numerical issue is a finite sized box and we clearly spell out the restrictions this brings with it. Additionally, we are not aware of any unwanted finite size effects appearing in our calculations that would be eliminated when going to a $1/N^2$ scaling. This quadratic nature of the enhanced scaling is much more important when the particle numbers are large, and for our cases of $N=2,3$ it does not play a particularly large role. Furthermore, with the exception of the gray shaded areas in the phase diagrams, generally the immersed component is localized strongly enough either in the superfluid or self-pinned phase that the nature of the boundary conditions does not play a role at all.
If the referee could point out to us any effects related to this we might have missed, we would be happy to discuss them and work on mitigating them. We hope that the above convinces the referee that simulations with periodic boundary conditions would need a very good reason to justify the significant numerical effort needed and also our work would not be as relevant to current experimental setups. The short answer to the referee's question is therefore that it is, of course, possible to use periodic boundary conditions for the minority comment, and we have added a sentence to the conclusions to point this out. For the current work however, we cannot see a reason that justifies the significantly increased numerical effort.

9 - As there are different possible ways to introduce dimensionless parameters in the considered problem (there are three coupling constants under consideration, also it is common to set $2m=1$ in 1D), it would be much clearer to have all reported quantities in the full units. It seems that the used unit of length is directly related to the s-wave scattering length. Maybe it should be used as the unit of length? Also, the phase diagram the way it is presented now seems to depend only on the coupling constant. I do not find the notation in which density is a number without units to be clear enough, ($N/L = 1/4$, etc). The same applies to other physical quantities.

Response: The referee is correct and there are various different ways to scale a system in order to make the underlying physics more clear. The effects we describe in our work mainly depend on the ratios of the different interaction strengths and not on their absolute values. Therefore, using full units or absolute values and fixing a reference scale a priori leads to an unnecessary loss of generality without providing a benefit in terms of clarity. While it is possible using any of the three coupling strengths as the reference scale for introducing dimensionless units, the background BEC intraspecies coupling $g_c$ is the natural choice since it is constant and set to $g_c=1$ throughout the manuscript, whereas both other coupling strengths $g_m$ and $g$ are varied. This rescaling and the units of length $x_0=\hbar^2/mg_c$ are commonly used in the literature [Martin et al., PRL 98, 020402 (2007) and PRA 77, 013620 (2008) or Helm et al., PRL 114, 134101 (2015)]. Furthermore, setting $2m=1$ would also require dimensionless units which we believe contradicts the point the referee is raising. As we have shown in the previous reply and added to the resubmitted version of the manuscript, choosing the reference scale from typical values for cold atom experiments leads to a one-to-one correspondence of the dimensionless lengths appearing in our manuscript and physical lengths in units of $\mu$m, i.e. the position axis shown e.g. in Fig. 2 c) depicts a range of $8$ $\mu$m in total and the dimensionless density of $N/L = 1/4$ used throughout the manuscript corresponds to a line density of $0.25$ $\left(\mu\mathrm{m}\right)^{-1}$ in that case. Giving lengths in units of the scattering length instead and using the same exemplary reference scale would mean plotting e.g. Fig. 2 c) on a position axis spanning $-800$ to $800$ scattering lengths which hardly seems like an improvement in terms of clarity. Therefore it is not a suitable length scale for the physics we discuss. Furthermore, we do not fully understand the referee's comment regarding the phase diagrams only depending on the coupling constants. Apart from the (equal) masses of both species and the intraspecies interaction of the condensate $g_c$, which are all considered constant throughout the manuscript, it seems natural that the system's state and therefore its phase diagram would be a function of the remaining two variables $g$ and $g_m$. We strongly believe that the units we have chosen are the most suitable and intuitive ones for our work, and we have provided context and details for their interpretation. We have furthermore edited the discussion on the units in the manuscript to provide more clarity on this point. Ultimately it is a personal choice which units one prefers, but we do not think that the referee's suggestion of full SI units would be helpful in understanding the underlying principles of the work we discuss.

10 - The Local Density Approximation refers to the assumption of a piece-wise constant potential As far as I understand, the LDA assumes that the chemical potential locally can be approximated as the sum of the external field and the chemical potential of a homogeneous system. There is no requirement for the assumption of a piece-wise constant potential. Please check this point.

Response: The wording piece-wise was chosen to indicate that locally the potential is constant, similarly to e.g. F. Riggio et al. in Phys. Rev. A 106, 053309 (2022). This is equivalent to the definition the referee states, found e.g. in G. E. Astrakharchik, Phys. Rev. A 72, 063620 (2005), i.e. locally having a homogeneous system.

13 - We use this particular notion in order to be consistent with our previous publication The choice of notation used in the study is not clear and intuitive.} We have decided against using a notation like A and B or $\uparrow$ and $\downarrow$ commonly found in descriptions of two-component systems since these notations usually describe two equally sized components. I do not see the point, having labels A and B implies nothing about the number of particles.

Response: As before, choosing labels is partly a personal preference. We have decided to use labels that give information about the different components and have made sure that these are clear and intuitive. One can make an argument that A and B are labels from a set of equal letters and therefore they are not the best suited for describing a system of non-equals. But this is hair splitting and not helpful. The important part is that the labels appropriately describe the respective quantities, are readable and are used consistently. All of these criteria are fulfilled by our choice. Throughout the manuscript we also regularly remind the reader of the labelling, referring to "interpecies interaction $g_m$" and "intraspecies interaction $g$".

21 - Moreover, since we are considering a quasi-one-dimensional system the term condensate fraction might be misleading and also lead to confusion with regards to the background Bose-Einstein condensate. This statement is contradicting itself, if the term "condensate" is already used in one dimension for the background BEC, why should it be misleading applied to the immersed component? If the Authors still want to keep the "coherence" term, please add a note that it corresponds to the definition of the condensate fraction. BTW, the quantity looks more like a "coherence fraction".

Response: Our manuscript shows how a quasi-one-dimensional quantum system immersed into a background BEC fermionizes as its intraspecies repulsion $g$ is increased. As we only consider a small number of immersed particles ($N=2,3$), it would be misleading to suggest that this few-body system ever forms a Bose-Einstein condensate. We are therefore reluctant to refer to the occupation of the lowest natural orbital as the 'condensate fraction', even though it is formally equivalent. To remain consistent with other works on strongly correlated few-body systems [see Sowiński and García-March, Rep. Prog. Phys. 82 104401 (2019) for a recent review] we elect to call this the coherence, but have also added a comment in the manuscript on its correspondence to the condensate fraction as the referee recommends. Furthermore, there is no requirement for the background BEC to also be one-dimensional and as the referee correctly states, its existence already implies that the condensate is quasi-one-dimensional at most. Therefore the strongly-correlated gas could also be immersed into a fully three-dimensional background BEC instead, but we do not treat this case.