Self-stabilized Bose polarons

The referee writes:

1. I thank the authors for removing the confusion about "coherent states" in their manuscript.

Our response: We thank the referee again for having pointed this out.

The referee writes:

1. Concerning my suggestion to use a third model potential to confirm the claimed universality, I am surprised that the authors repeatedly declined to do so. It is very easy to do once a numerical code for minimizing Eq. (8) has been written, which the authors have done. I did the numerical calculation myself, and found that model potentials with a van der Waals tail reproduce the results of the manuscript (see attached figure). I am now convinced that the universality claimed by the authors on the basis of only two examples is likely to be correct. Again, I invite the authors to do this calculation themselves and include it in their article to strengthen their claim.

Our response: We recognise the referee’s effort in providing an excellent report on our manuscript and appreciate their concern with regards to testing the claim of universality further. In addition to the data shown by the referee we have thus now also evaluated the case of a physical Lennard-Jones potential given by:

The referee writes:

1. There seems to be a typo in Eq. (8). The $\xi^2$ factor inside the integral should be removed, and a $\xi^3$ factor should be added in front of the integral. Could the authors confirm this point? Also, the notation implies that $E$ in Eq. (7) and (8) are the same, but as far as I understand, the authors have (duly) subtracted the condensate energy $-\mu n_0 /2$, which is divergent for increasing volume. This should be made clear in the notations as well as in the text.

Our response: There was a typo in Eq. (8), and we thank the referee for spotting this. Indeed a bracket was set incorrectly which we have now corrected (in our convention $u$ has units of length). Moreover, we have subtracted the energy $-\mu n_0 /2$ of the unperturbed BEC and we now mention this explicitly in the text above Eq. (8).

The referee writes:

1. I am confused about the units of Eq. (17). According to Eq. (8) (once the typo is corrected), the energy is proportional to $\mu n_0 \xi^3$ and to a dimensionless function of the dimensionless potential parameters $a/\xi$ and $r_{\rm eff}/\xi$. At unitarity, it reduces to a function of only $r_{\rm eff}/\xi$, which for $r_{\rm eff}/\xi \gtrsim 0.2$ has the linear form $A + B\times (r_{\rm eff}/\xi)$, where $A$ and $B$ are found numerically to be about -5 and -9. It follows that Eq. (17) should read:
   $$ E/E_n = A n_0^{1/3}\xi + B n_0^{1/3}r_{\rm eff} $$
   In other words, there should be a factor $n_0^{1/3}\xi = (8\pi n_0^{1/3} a_{BB} m_{\rm red}/m_B)^{-1/2}$ in the first term of the right-hand side. Can the authors confirm whether this is the case?

Our response: The referee is correct about this missing factor which we have now included in the new version of the manuscript. We thank the referee for spotting this!

The referee writes: 5. The comparisons with the works of Refs. [42] and [59] are very confusing.\ 5.1 About the comparison with Ref. [42] (Massignan et al):\ Eq. (11) of Ref. [42] states that at unitarity, the polaron energy for $R \ll \xi$ is:

Our response: Ref. [42] includes a Fig. 1 that shows the polaron energy as function of the effective range. Also in this Figure a nearly linear regime is visible at large range. With the comparison shown in Fig. 5 we wanted to convey that our predictions for the energy agree in this regime (no surprise since the referee states correctly that both we and Ref. [42] apply GPE theory). We did not make or wanted to imply any statements about the scaling predictions in [42], and indeed our comparison was made for a value of the effective range well outside the regime where Ref. 42 predicts their 1/3 scaling. We acknowledge that a comparison might have been more confusing than helpful and thus we now opted to take up the referee’s comment and we removed this explicit comparison altogether from the manuscript (in particular in terms of the green star depicted in Fig. 5).

As a side remark ---also coming back to the referee’s second comment--- one might actually regard the agreement in energy with Ref. [42] as further supporting our claim of universality since Ref. [42] employs yet a different microscopic potential compared to our work.

The referee writes: 5.2 About the comparison with Ref. [59] (Levinsen et al).\ Eq. (6) of Ref. [59] states:

Our response: Here again we only aimed at comparing our energy values to other literature to check for overall consistency. Our comparison to Ref. [59] highlighted that although predicted scalings are very different, actual energies might still be very much comparable for typical experimental parameters. However, as the referee’s comment makes clear a comparison for just a single point (in order to test the energy) causes more confusion than it is helpful and thus we again opted to remove this discussion from the manuscript.\ Concerning the referee’s comments on the $\sqrt{1/x}$ scaling found in Ref. [30]: In this case we refrain from making explicit comments or comparisons since Ref. [30], by applying the Bogoliubov approximation, really represents a result for a fundamentally different model than the one considered in the present work.

Requested changes by the referee: 1. Confirm/Clarify the points mentioned in the report.\ 2. Add the van der Waals potential data to Fig. 5.\ 3. Remove the star in Fig. 5, unless the authors can give a justification. To make a comparison with Ref. [42], I suggest the authors to display the curve corresponding to the formula $E/(\mu n_0 \xi^3) = -3\pi (0.557 y)^{1/3}$ obtained from Ref. [42].

Our response: 1. We thank the referee for bringing up these points. We have addressed these in the report above. \ 2. We have added additional data to Fig. 5 obtained for a Lennard-Jones potential including the van der Waals tail, further supporting the claim of universality.\ 3. Taking into account the referee’s valuable input, in order to unambiguously avoid any confusion or misunderstanding, we have opted to remove the comparison with Ref. [42] and the star from the figure as discussed in our reply above.

1. Included new data for a Lennard-Jones potential in Fig. 5b, further supporting universality. Removed the green star from the figure which lead to confusion.
2. Fixed typos in Eqs. (8) and (17).
3. Stated explicitly that the polaron energy is measured relative to the energy of the unperturbed BEC.