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Self-stabilized Bose polarons

by Richard Schmidt, Tilman Enss

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Submission summary

Authors (as registered SciPost users): Tilman Enss · Richard Schmidt
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-07-19
Date submitted: 2022-06-30 21:50
Submitted by: Enss, Tilman
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
Approach: Theoretical


The mobile impurity in a Bose-Einstein condensate (BEC) is a paradigmatic many-body problem. For weak interaction between the impurity and the BEC, the impurity deforms the BEC only slightly and it is well described within the Fr\"ohlich model and the Bogoliubov approximation. For strong local attraction this standard approach, however, fails to balance the local attraction with the weak repulsion between the BEC particles and predicts an instability where an infinite number of bosons is attracted toward the impurity. Here we present a solution of the Bose polaron problem beyond the Bogoliubov approximation which includes the local repulsion between bosons and thereby stabilizes the Bose polaron even near and beyond the scattering resonance. We show that the Bose polaron energy remains bounded from below across the resonance and the size of the polaron dressing cloud stays finite. Our results demonstrate how the dressing cloud replaces the attractive impurity potential with an effective many-body potential that excludes binding. We find that at resonance, including the effects of boson repulsion, the polaron energy depends universally on the effective range. Moreover, while the impurity contact is strongly peaked at positive scattering length, it remains always finite. Our solution highlights how Bose polarons are self-stabilized by repulsion, providing a mechanism to understand quench dynamics and nonequilibrium time evolution at strong coupling.

Published as SciPost Phys. 13, 054 (2022)

Author comments upon resubmission

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:

$$ V_{\rm LJ}(r)=\frac{\lambda R^{10}}{r^{12}} - \frac{R^4}{r^6} . $$

Here, $\lambda$ and $R$ are chosen to produce the desired scattering length $a$ and effective range $r_{\rm eff}$, respectively. In contrast to the cases considered previously, this potential features also a strong repulsive contribution, thus allowing to test the degree of universality of our results further. Again, we find good agreement with our previous results. We added this data now also to Fig. 5 as suggested by the referee. We expect that for very large ranges the repulsive core limits the applicability of universality.

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:

$$ E = - (3\pi n_0 \xi/m_B) (R/\xi)^{1/3} $$

where $R = 0.557 r_{\rm eff}$ for a square well potential. Expressing it in units of $E_n = n_0^{2/3}/(2m_{\rm red}) \approx n_0^{2/3}/(2m_B)$, one finds:

$$ E = - E_n 6 \pi (\frac{R}{8\pi a_{BB}})^{1/3} $$

The authors state that this formula is consistent with their results for the particular value $r_{\rm eff} = 2 a_{BB}$. Where does this value come from? I do not understand why the authors expect an agreement for this particular value. It is also unclear how the value -5 is obtained, as well as how the horizontal coordinate of the green star in Fig. 5b is determined.\ My understanding is that Ref. [42] and the present manuscript essentially use the same GP theory. However, Ref. [42] applies this theory in the limit $r_{\rm eff}/\xi \ll 1$, whereas the present manuscript considers the theory for $r_{\rm eff}/\xi \gtrsim 0.2$. At unitarity, the two papers obtain results in separate regimes, namely:

$$ E/(\mu n_0 \xi^3) = -3\pi (0.557 y)^{1/3} \,\,\textrm{for Ref. [42]}\,\, (y\ll1)\\ E/(\mu n_0 \xi^3) = -5 -9 y \,\,\text{for the present manuscript}\,\, (y\gtrsim0.2) $$

where $y = r_{\rm eff}/\xi$. Plotting these two functions of y (see dotted and dashed curves in the attached figure) shows that they do not cross. One can imagine that they connect to each other in the intermediate regime, but it should be observed from Fig. 1 or Ref. [42] that numerical results match the first function only for $y\lesssim0.002$. Therefore, the mentioned value -5 and star in Fig. 5 look very misleading.

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:

$$ E = -E_n 2 f( n_0^{1/3} a_{BB}) $$

where the function $f$ is found by QMC calculations to follow the law: $f(x) = -0.36 ln( 0.019 x)$. This looks inconsistent with the results of the present manuscript.\ However, the authors state that this formula is consistent with theirs for the particular value $x = n_0^{1/3} a_{BB} = 0.04$. Again, where does this value come from?\ If one considers the extrapolation to zero range of the linear law found in this manuscript, one finds (with the missing factor I mentioned in point 4.): $f(x) = -2.6 /\sqrt{8\pi x}$ which does coincide with the previous formula at $x=0.04$, but this is simply a coincidental point for two completely different laws.\ As an aside, Ref. [42] also mentions that $f(x) = \sqrt{\pi/4x}$ for the coherent ansatz of Ref. [30]. That formula is consistent with the result of the present manuscript, if multiplied factor of about 1.77. Perhaps this is worth mentioning.\ I would be grateful to authors for clarifying these points.

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.

List of changes

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.

Reports on this Submission

Anonymous Report 1 on 2022-7-6 (Invited Report)


I thank the authors for their efforts to improve their manuscript, in particular the calculation with the Lennard-Jones potential which further demonstrates the universality of their results. I am satisfied with all the replies and changes. The manuscript constitutes a nice piece of work and I believe it is now fit for publication in SciPost.

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