SciPost logo

SciPost Submission Page

Self-stabilized Bose polarons

by Richard Schmidt, Tilman Enss

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Tilman Enss · Richard Schmidt
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-11-08 08:38
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.

Author comments upon resubmission

Reply to Report 2

 The authors have gone some way towards addressing the comments of the referees, particularly regarding the regime of validity of their theory. However, there are still some remaining points that I would like to see addressed before publication: - If the regime of validity corresponds to R≫aBB, then presumably this is why the results of the GP theory do not agree with the QMC results in [Phys. Rev. Lett. 127, 033401 (2021)] ? This should be mentioned in the manuscript.

Reply: The comparison with the Monte Carlo results is an interesting point. Contrary to the expectation of the referee, the results show very good agreement for the parameter set for which both works make their predictions, specifically aBB=0.04 n^(-1/3) and r_eff->0. In this caseQMC predicts E=-2.6 n^(2/3)/mB in perfect agreement with the short-range extrapolation of our results in Eq. (17) for the infinite-mass case. We now mention this in the manuscript, along with the reference to the QMC work.

- I do not think that the perturbative limit of weak boson-impurity coupling violates the assumption of R≫aBB, as claimed by the authors. Rather, one requires aIB to be small compared to the BEC healing length, and this can be satisfied by taking the boson density to be sufficiently low, without making any assumptions about the range of the potential. Therefore, if the GP theory does not match the perturbative weak-coupling results, this suggests that the theory cannot accurately capture the low-density limit. The authors should at least comment on this in the manuscript.

Reply: In the low-density limit mentioned by the referee, the impurity potential of range R with xi >> aIB ~ R >> aBB becomes effectively a contact potential on the scale of the healing length. In our manuscript we already discuss below Fig. 5(b) and in Section VI that this limit is not well described by the standard GPE: while for an attractive impurity potential with aIB < 0 a Chevy ansatz might be sufficient, for repulsive aIB > 0 the nonlocal GPE [40] is capable of describing the large polaron cloud around a short-range impurity.

Reply to Report 1

 1. In their response, the authors argue that atomic interactions typically support a large number of bound states, which makes it too difficult to use a van der Waals potential in their work. The authors could have nonetheless considered a shallow van der Waals potential with only one bound state (such as that of helium) to compare with the Gaussian/Exponential potential and check their claimed universality. This would not be particularly more difficult than a Gaussian or exponential potential with one bound state, and would significantly strengthen the manuscript. If the authors do not do this simple check in the present work, it will need to be be done in another publication (if it is ever done). I think this is a good opportunity for the authors, but again I leave it to their discretion.

Reply: We thank the referee for this suggestion; the systematic analysis of the range dependence of the ground-state energy for different model potentials is a worthwhile topic for further research.

 2. I find it regrettable that the authors insist on referring to their ansatz as a "coherent state". Technically, as I pointed out in my first report, they introduce it as a Fock state, not as a coherent state, and it is equivalent with a coherent state only in the limit of large numbers of particles. Moreover, the coherent nature (superposition of states with different numbers of particles) does not play any role in this work. I think terms like "Fock state" or "mean-field" or "Gross-Pitaevskii" would be much more easily understood. Moreover, they oppose their work to the coherent ansatz of Ref. [30,36], which truly is a coherent ansatz where the superposition of number states plays a crucial role (here, the number of excitations). Naming both approaches as coherent ansatz may cause confusion to the readers.

Reply: Our work is concerned with the thermodynamic limit of large particle number where the product and coherent states have indistinguishable density expectation values, and the fluctuations of the particle number become small compared to the total particle number. In this limit, the states used in our work and in Refs. [30,36] become equivalent, with one written in real space and the other in momentum space. Instead, the main difference is that the Hamiltonian is bounded from below in our work. To avoid confusion, we removed references to coherent states.

List of changes

List of changes:
- compared our prediction for the ground-state energy with the recent QMC result in J. Levinsen et al., Phys. Rev. Lett. 127, 033401 (2021), which now appears as reference [59].
- refer to Eq. (6) as product state.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2021-12-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2102.13616v3, delivered 2021-12-01, doi: 10.21468/SciPost.Report.3979


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

2. 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.

3. 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.

4. 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?

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 $R << \xi$ 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:
$E/(\mu n_0 \xi^3) = -3\pi (0.557 y)^{1/3}$ for Ref. [42] ($y\ll1$)
$E/(\mu n_0 \xi^3) = -5 -9 y$ 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.

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.

Requested changes

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].


  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report or comment