Crossover from attractive to repulsive induced interactions and bound states of two distinguishable Bose polarons

1. Clear description of a very wide collection of phenomena.

2. Simple models are presented which capture the relevant physics.

3. Powerful numerical methods which help ellucidate relevant physics and sustain the simple models.

1. Too much topics addressed, which makes difficult the revision of corresponding previous literature, which also is extensive; some from the same group.

2. The different topics can be addressed with more detail and focus, and I hope it could be done in the future in other papers.

The authors study a system of two distinguishable particles interacting with a small number of bosons of a third distinguishable type, with contact interactions in the ultracold regime, in one dimension, in the presence of a harmonic trap. Particularly, they consider NA=15 atoms in the largest component. Their main tool, but not the only one, is the numerical method coined as ML-MCTDHX. Their study focuses on the ground state, which they find via imaginary time evolution method. They study both attractive and repulsive intracomponent and various  intercomponent interactions. Their findings reveal a collection of effects, from Bipolaron, bunching or antibunching,   trimer state, etc. The paper deals with a lot of different aspects, which one may think that should deserve a bit more attention. Nevertheless, the basic aspects are well captured by numerics and with simple models. So I think the paper should be accepted in Scipost.

1. The main comment arises from the fact that the paper tackles so many aspects. I think they miss the context of previous research as a direct consequence. The most prominent lack I think is that classical papers from the open quantum system community should be cited and commented (such as PRL 97, 25060, PRA 80, 032110, PRA 77, 042305) and others maybe PRL 102, 160501 or JPHYSA 45 065301 - this is not a comprehensive list, it should be researchd ).

2. The second comment is that I find it a bit too much to call NA=15 atoms a bath, though for most of the physics discussed here it has little practical implication. I leave on authors hands a comment on this, if they will.

3. This is a technical comment about convergence: at the end of section 3. More details on this may be helpful for anyone to reproduce results.

4. The last paragraph of section 6.1. about larger number of atoms, referring to appendix D is an example of the paper tackling too many aspects: there, some results with a few more atoms are found, but just some; I don’t know if everything what happens with more particles can be deduced from here as the paragraph seems to indicate.

5. I find the simple model in 6.2 very nice and appealing, catching the relevant physics in a simple way. No change is required here, of course.

6. Section 8 stands out over the rest of the paper for being confusing, as compared to the rest of the paper which is very clear. I assume they have calculated with 17 atoms, and traced out to obtain three particle correlations. This is what indicates Eq. 15 and caption of fig. 6. But I find that there is confusion introduced with sentences like “The three particles are correlated” before eq. 15; “demonstrating a bunching behavior of the three particles. ”; or in the very abstract "trimer state in the strongly attractive regime, where the latter consists of two impurities and a medium atom”. Also, the Jacobi coordinates, if one has 15  A atoms, should go over all centers of mass. That is the distance of the third particle to COM of 1 and 2; then distance of 4th to COM of 1,2,3, etc, I believe. So I misunderstand probably eqs. 16 and following, being possible that there is some assumption or evident fact that I miss here.

1. Timely and sound topic of research.

2. Established state-of-the-art ab initio method.

1. Similar system and methodology presented in previous publications of the author. The new elements present in the current work are not clear enough.

2. Experimental realization seems unviable and poorly described.

3. Range of validity of the method and results are not discussed.

The authors investigate the structural properties and the ground-state energy of a system that consists of two distinguishable impurities immersed in a harmonically trapped quasi-1D Bose gas at zero temperature. The highly imbalanced three-component system is characterized by coupling strengths $g_{\sigma,\sigma'}$, where $\sigma=1,2,3$. By tuning different coupling strengths, particularly the impurity-bath ones, the two impurities can correlate and they experience either an attractive or repulsive induced interaction between them mediated by the weakly interacting bath. In particular, a bipolaron state is formed in the strongly interacting regime. The authors also predict the formation of a trimer state in the highly imbalanced three-component mixture.

To quantify these correlations, the authors present a calculation of observables, including the two-body coherence, defined in terms of the one and two-body reduced density matrix. The authors employ the multi-configurational time-dependent Hartree method to compute these observables. The methodology has been extensively used in other systems, such as degenerate quantum mixtures(see references [53-56]). Additionally, the bipolaron is further characterized by measuring the binding energy and the size of the two-body bound state. Finally, the trimer state is characterized using a three-body correlator inspired by the previously defined two-body coherence.

The authors emphasize the high controllability of a highly imbalanced mixture. They present an example using two isotopes of Rubidium. However, the systematic experimental implementation is poorly discussed thoroughly in the text, rendering the work primarily theoretical without clear prospects for possible observation. For example, in the calculations, the impurity-impurity coupling strength is set to zero, which is reasonable since the focus is on observing induced interactions rather than direct ones. However, to achieve this, it is necessary to identify a suitable range of magnetic fields with a constant scattering length (zero for impurity-impurity interactions), two adjustable impurity-boson scattering lengths, and a constant and positive boson-boson scattering length. In the particular case of the current draft, the authors propose a $^{85}\mathrm{Rb}$ condensate. The background scattering length is negative (arxiv.org/pdf/1003.4819.pdf) and this may be problematic. The choice of atomic species is quite limiting in experiments and should be discussed carefully.

On the one hand, the work provides an interesting calculation concerning the correlation between impurities and the formation of few-body bound states like bipolarons and trimers, using a state-of-the-art method. On the other hand, I find it challenging to distinguish the current work from previous publications by the authors, specifically PRA 104, L031301 (2021) and PRA 105, 053314 (2022), where a very similar physical system and methodology are employed. Thus, this work represents a new but incremental calculation compared to the previous works, and I cannot identify any significant evidence of groundbreaking theoretical results, apart from the crossover that is not well discussed.

The system may undergo a miscible-immiscible transition when considering a highly imbalanced triple mixture and for a certain range of parameters. Yet, the authors state that the mediated interaction of polarons causes an effective repulsion or attraction. The question is: How can the authors differentiate between these two scenarios?

In this work, typical values for the impurity-boson coupling strength typically range between -3 and 3. Why are the authors constrained within this range? For instance, in PRL 127,103401 (2021), the strongly interacting regime for the double imbalanced mixture is attained with larger values of the impurity-boson coupling strength.

Given the previous remarks, especially the close connection with the aforementioned works (PRA 104, L031301 (2021) and PRA 105, 053314 (2022)), which limits the originality of the research, and the absence of a proper experimental protocol, the current work falls short of meeting all four Scipost acceptance criteria. Nevertheless, the methodology is deemed reliable, and the results could be suitable for publication in a more specialized journal. For example, physical review A or B.

Other less strong concerns may also be addressed prior to publication in any journal.

The introduction contains the sentence, “In particular, induced interactions are solely attractive as long as the impurities are indistinguishable and thus couple in the same way to their medium.” Please provide a reference for such a claim.

If I understood well, in Fig.1, the authors plot the density of each component and the effective potential as well. The density is directly related to the potential via Eq. 6. In order to check the quality of the effective potential model; I encourage the authors to plug the effective potential into a simple Schrödinger equation and obtain its respective wave function. The wave function squared may be compared to the density obtained from the many-body calculation.

Linked to the experimental realization, could the author discuss losses?

What about the formation of high few-body states on top of bipolarons and trimmers, for instance, tetramers, pentamers, etc?

In Fig. 2, the panels a1,b1 and c1 show $\mathcal{G}_{AB}^{(2)}\left(x_{1}^{A},x_{2}^{B}\right)$, however in the horizontal axes reads $x_{1}^{B}$. Vertical units are wrong. Same for of $g_{AB}$ in Fig.3.

The coherence, as defined in the article, should contain dimensions.

1. Relevant and timely topic
2. Solid numerical framework
3. Well-detailed methodologies

1. Explanation of the limits of the numerical methods could be improved. (See report)
2. Direct potential experimental consequence and/or challenges could be explained in some more detail.

The manuscript titled: "Crossover from attractive to repulsive induced interactions and bound states of two distinguishable Bose polarons," presents a study of a population-imbalanced three-component mixture. The authors focus on a Bose gas coupled to two different species of minority atoms, which are considered single impurities. The different impurity-boson interactions allow for tuning the sign of the mediated impurity-impurity interactions, resulting in a wide range of two-body impurity states. These states are characterized by their energy, as well as two-body and three-body correlations and spatial distribution. The authors employed a recently developed variational multilayer multiconfiguration time-dependent Hartree method for atomic mixtures (ML-MCTDHX) approach in their studies.

The manuscript presents original and interesting results. The numerical approach, based on a method recently developed by the authors for 1D systems, is solid. Finally, the manuscript is clearly written, with well-explained sections and some Appendices which help the understanding of technical points of the manuscript.

In my opinion, the manuscript fulfills at least one of the expected acceptance criteria, in particular: a) detailing a groundbreaking theoretical-computational discovery and b) opening a new pathway in an existing or new research direction, with clear potential for multipronged follow-up work.

After some minor requests, I recommend its publication in Sci. Post. Phys.

1. Please provide details regarding the restrictions of the approach in terms of the strength of impurity-boson and boson-boson interactions for their studies.

2. Concerning the repulsive impurity-boson interaction, is there an underlying bound state? Can the physics of the attractive branch be captured for repulsive interactions? Please provide your comments on this.

3. In Section 6.2, if I understood correctly, this approach directly provides the spatial distribution of the impurities, and the authors manage to derive an effective mass. Can the coherence of the impurity (residue) be extracted in a similar manner?

4. In principle, in a homogeneous system as given in Equation 21, bipolarons are expected to form for both attractive and repulsive interactions (when g_AB and g_AC have the same sign). Is it because the trap allows for the phase separation of the impurities, preventing the formation of a bound state?

5. Could you comment on the feasibility and challenges of measuring in the experiment the correlation regimes and the integrated correlation function?

6. Is it realistic to extend their approach to 2D or 3D systems?

7. The authors mention that a non-local mediated interaction may emerge for strong boson-impurity interactions with a different sign. Do the authors expect Equation 12 to hold in this case as well?