Bose polaron in a quantum fluid of light

1-Detailed theoretical description
2-Effective graphics
3-Well written

1-Solution methods are not stated
2-Lacks some discussion

This work reports a study of an impurity in a non-equilibrium exciton-polariton condensate, with the dynamics of both the quantum fluid and the impurity taken into account. A peculiar regime of negative drag on the impurity is identified and characterized. The paper is well written and the theoretical description is clearly presented. The results are interesting, seem sound and are nicely illustrated by the figures. I think however that some additions are needed to make the paper clearer and more relevant.

First of all I think that the means (numerical, analytical...) with which the results are obtained should be stated more clearly, for example at the beginning of Section 4.
Another point that in my opinion is important to discuss is the experimental relevance of these calculations. In particular: how would a moving impurity be realized in an exciton-polariton condensate?
Moreover I think that the results of Ref. [22] are not quoted with sufficient clarity, since some of the results obtained here regarding the negative drag were anticipated in that paper, albeit with a much simpler model of a fixed impurity. A discussion of those results should be included in the Introduction.
Finally the exposition may benefit from adding some explanations to the theoretical part of Section 3, as I indicate in the "Requested changes", where I also indicate some minor points.

Even if I think that (after these changes) this work is well worth publishing, I am not sure that it meets one of the very selective expectations of SciPost Physics. While the paper contains interesting results obtained in an original way, I must admit that I do not see a groundbreaking discovery, a novel link between different fields or much follow-up work. For these reasons in my opinion the paper would fit better in SciPost Physics Core.

In order of appearance
1-I think the predictions of Ref.[22] should be reported in the introduction
2-A discussion of the experimental relevance of the calculations should be included
3-Operators $\hat{a}_\mathbf{k}$ in Eq. (2) and $\hat{\alpha}_\mathbf{k}$ in Eq. (4) are not defined
4-Maybe the $\hat{p}$ operator in equation (3) (and also in (10)) should be bold
5-I think the density n appearing after equation (8) is not defined
6-At the end of Sec. 2.1 and at the beginning of Sec. 2.2 it would be nice to have references for the Bogoliubov-Frolich Hamiltonian and the Lee Low Pines transformation
7-When $\Pi$ is an operator it should have the hat
8-After Eq. (22) a coherent state of excitations is considered. When is this a valid procedure? I think some comment is worth adding.
9-At the end Sec. 3.2 I think the definition of the effective mass and of $\eta$ and how they follow from Eq. (26) should be explained in more detail
10-Also the definition of the drag force in Eq. (28) would benefit from some more details. Also I think $n_{imp}$ is not defined.
11-Eq. (28) and (29) should have only one number
12-In the line after (29) ie should be i.e.
13-A discussion of the solution methods should be added in Section 4
14-At the end of the second paragraph of Sec. 4.1 the value $\Delta/gn=2$ is used to distinguish the different regimes, but in Fig. 2 $\Delta/gn=1$ appears as a threshold
15-I could not find a definition of the $\Gamma$ independent of $\mathbf{k}$ that is used in Section 4
16-Is "decreasing $\Gamma$" in at the end of Sec. 4 supposed to be "increasing $\Gamma$"
17-In the conclusions it is stated that for $\Delta>0$ the impurity is subjected to negative drag force, but this does not always happen for $\Delta>0$
18-In the caption of Fig. 6, in the list of values of $\Delta$, there is a cursive "and"

The manuscript by A. Vashisht et al. discusses the properties of a single impurity in a driven-dissipative polariton bath. The paper is very well written, the problem is experimentally interesting, the formalism is clearly explained, and the equations and the results appear correct. As such, I can surely recommend publication of this manuscript in SciPost.

Upon resubmission, I think the Authors may extend the discussion on the key result presented here, i.e., the presence of a region of negative effective mass. In particular, is there an intuitive understanding of why such effect appears when the Bogolubov dispersion becomes complex? If I understood correctly, such effect was already noticed in earlier works on static impurities (Refs. [22,23]). In this case, this discussion may be placed already in the Introduction.

Here below follow further minor remarks and suggestions, presented in chronological order, that the authors may want to take into account.

# caption of Fig. 1: "Central panel as a" ---> "Central panel: as a " (":" missing). Moreover, define what is an "LP fluid".

# Sec. 2, line 3: "by its coupling" ---> "by their coupling"

# one line below Eq. (1): explain the adjective "lower" used in front of "polariton"

# Eqs. (2) and (4): define the operators $a_k$ and $\alpha_k$

# Eq. (5): in the moving reference frame, shouldn't the two operators $a_{k_p}$ and $a^\dagger_{k_p}$ in the last line of the equation actually read $a_0$ and $a^\dagger_0$?

# one line below Eq. (9): "all the the" ---> "all the". Further, the discussion in this paragraph is unclear. It is first said that $u_k = u^*_k$, but then it is specified that when $\Delta>0$ this is not the case. To avoid ambiguities, I suggest then to say "if $\Delta>0$ then $u_k = u^*_k$. Else, ..."

# to simplify notation, I suggest to introduce the operator $\Pi$ already around Eq. (13).

# Eqs. (17) and (20): a factor $1/M$ is missing in the term $\hbar k \cdot p$

# two lines above Eq. (22): $2\pi/\Gamma_k$ ---> $2\pi/\Gamma_{\bf k}$

# Eq. (28): $\langle \int$ ---> $\int \langle$. Moreover, I have the impression that $n_{imp}$ is not defined.

# line following Eq. (29): "ie" ---> "i.e."

# Eq. (34): I have the impression that $\Gamma$ (without pedix k) is not defined, please check. It is also unclear how such quantity emerges from the previous equations. A few more lines of explanation may be spent to explain this key equation.

# end of page 9: The two sentences "In all the above cases... . This is possible ..." contain important information, which I think deserves further clarification for the reader unfamiliar with polaritons. The authors may want to extend slightly the discussion here.

# Fig.2b: inside the various panels, $\Delta = ...$ should be $\Delta/gn = ...$

# Fig. 3: what is the dashed part of the red line?

# last paragraph of Sec. 4: "the region ... disappear" ---> "the region ... disappearS"