Quantum rotor in a two-dimensional mesoscopic Bose gas

1-clear,
2-well organized,
3- a good combination of analytical results and numerics.

1 - it is not clear what aspects of the problem depend on the number of particles in the system and what on the harmonic confinement.

This paper is well-organized and clearly written. While some of the results are quite technical, the authors explain them in simple terms. The description of the angulon quasiparticle in real space in terms of the Gross-Pitaevskii equation (GPe) is new and potentially very useful, given that it allows one to capture the physics of non-uniform condensates in the presence of the strong interactions between the boson and the impurity.

The other strength of the manuscript is that the authors consider the system in a harmonic trap so that it can be realized in the experiment, however, this strength also turns out to be its major weakness, because the effects of the trap on the angulon physics are somewhat obscure. In particular, the authors consider the regime where the central density in the trap remains fixed, so for a small number of particles in the system, the potential has to be more confining, and the effects of the trap are important. At the same time, the effects of the trap should be less important as $N \to \infty$. The main analytical result is Eq.(18) and it is derived under the assumption that $N$ is finite, but the trap is turned off. This analytical result agrees well with numerics for a range of $N\gg1$, but then suddenly breaks down when $N$ becomes larger than some $N^*$. The fact that such behavior does not seem to follow from the formalism used (from what I understand the result in Eq.(18) should hold for arbitrary $N\gg1$), makes the whole approach look questionable. Since it is not clear what triggers such a behavior, it is not clear whether the harmonic trap has to do with it.

Just to sum up the above, when the authors talk about different regimes based on the number of particles in the system, it is hard to tell whether this is something inherent to the angulon problem, or some of the regimes are the artifacts of the harmonic confinement.

I think to make the discussion clearer and make the claims of the authors stronger, they should also consider the problem in the absence of harmonic confinement, for example, by putting the system into a box of finite size. Then one can fix the density and study the problem as a function of $N$ and the box size $L$. If the results in this system agree with the ones in the current manuscript in the regime where $N\gg1$, then I will accept the validity of the presented results and will be happy to recommend this manuscript to be published in SciPost.

Finally, the angulon problem is a close cousin of the Bose polaron problem, as was also pointed out by the authors. For the Bose polarons, there is a thermodynamic relation between the number of particles inside the polaronic cloud and its energy, see https://doi.org/10.1103/PhysRevLett.126.123403. Can one expect something similar to hold for angulons?

1-add results/discussion of the angulon physics in the absence of the harmonic confinement and compare with the current results
2-add radial density profiles for weak and strong interactions and comment on how condensate is distorted.

Ask for major revision

1. clear problem statement
2. simple and powerful method explored
3. clear physical interpretation of the obtained results
4. plenty of new interesting results obtained

I have some tiny remarks (please see the pdf file) but I would have not called them true weaknesses.

I think the journal acceptance criteria are fully met.

1. The physical origin of the impurity-boson interaction is unclear. The manuscript suggests the rotor model for the impurity. Are bosons also rotors?

2. On p.~4 we read: ``The normalized function $\psi$ defines the probability of finding a boson at a given position in a molecular frame of reference. It does not depend explicitly on the angle $\varphi_I$...'', however, coordinates ${\bf r}_i$ explicitly suggest such dependence. I think this issue should be clarified in the revised manuscript.

3. What do the authors mean by the following sentence: ``Indeed, a strong deformation of the density of the Bose gas requires a number of phonons in momentum space for its description.''?

4. The breakdown of the angulon behavior at large angular momenta (see Fig.~6) is signaled as a deviation of the rotation energy from the quadratic law. However, at large $L$ the energy $\Delta E^{num}_L$ seems to support a linear dependence. Is it really linear, and is there some physical reasoning for it?

Publish (easily meets expectations and criteria for this Journal; among top 50%)