Bound impurities in a one-dimensional Bose lattice gas: low-energy properties and quench-induced dynamics

1) Comprehensive description
2) Detailed characterization of microscopic properties
3) Exact calculations
4) Indications of persistent behavior from small to larger systems

1) Lack of explanation for certain processes
2) Some of the presented results, such as the formation of impurity dimers, have also been reported elsewhere

In the present work, the authors study the stationary properties and the interaction quench dynamics of two bosonic impurities in a few-body bosonic bath. Both species are trapped in an one-dimensional lattice. The situation of unit filling is considered throughout and exact diagonalization is employed to perform the calculations. Regarding the ground state, the formation of a bound di-impurity dimer caused by the genuine phase-separation process in two-component settings is identified and characterized in detail in terms of the system parameters. Furthermore, following interaction quenches from strong to weak impurity-bath repulsions revealed the regions where the aforementioned dimer persists or can be dynamically destroyed indicating an orthogonality catastrophe.

The results are interesting providing insights into the formation of impurity bound states immersed in bosonic baths under the influence of a lattice geometry. I believe that they will be proven useful for future investigations on impurity and in general quasiparticle states especially concerning their non equilibrium dynamics. This topic is also of interest from an experimental point of view. However, I have several comments and questions regarding the presentation and more importantly the interpretation of the results that should be addressed before final recommendation. A list of suggestions for improvement follows.

1) On page 4, it is stated that “most of the parameters considered here lie around the one-dimensional superfluid-to-insulator transition”. Does this statement refers solely to the majority component? Please clarify.

2) Before Equation (4) the word “stated” should read “state”

3) Equation (5) is the total wave function if I understand correctly and not a general wavevector.

4) Regarding the numerical ED method used, I wonder whether only the lowest band states of the lattice are taken into account or also energetically higher ones in the current study. In this context, what is the relation between the first excited state discussed in section 4 and higher-band states of the lattice? Please clarify. Again here, how numerical convergence is judged in the present work?

5) At the beginning of Section 3, the concept of “di-impurity bound states” is mentioned but not explained. Please provide a brief explanation since it has been studied also elsewhere.

6) In Figure 2 the dashed and dashed-dotted lines and especially the vertical dashed-dotted lines are hardly visible. I suggest to increase their fonts. The same holds for Figures 3, 5, 6, 7 and 8.

7) What is the phase-separation condition in the present setting in terms of the interactions of the bath atoms and the interaction between the impurities and the bath atoms?

8) On page 5, I do not fully understand the comment about the bipolar energy which shows an an abrupt decrease at a fixed ratio and for smaller values a more continuous crossover. I see in all cases a rather continuous curve. Please explain better what is meant here.

9) What is the origin of the energy minimum observed in Figure 2(b) at strong impurity bath interactions? Is it the phase separation?

10) In Equation (7) the index \alpha is referred to as the lattice spacing while in Equation (5) it is the state. This is a bit confusing. I suggest to modify one of these notations.

11) It does not become clear why and how the competition between phase-separation and Mott-insulator or superfluid character of the bath impacts the impurities bound state. It this related to the compressibility of the bath? Please clarify.

12) An interesting possibility would be instead of examining the first excited state of the system to consider only the impurity(ies) in their excited states while the bath in its lowest state. Would then the di-impurity bound state exist? If yes, how its properties will be changed.

13) Are the oscillations observed in the overlap being present for larger systems (in particular lattice sizes and bath atom numbers) or do they become more prominent for smaller system sizes? Is it possible to discern finite effects from genuine ones here? I suspect that there are indeed genuine effects since phase-separation and Mott-insulator to superfluid transition play a decisive role in the behavior of the overlap and in the dynamics of the system. Please clarify.

14) In all dynamical cases considered, if I understand correctly, tunneling of the impurities is prevented due to the large energy gap. What happens for interactions/hoping coefficients where tunneling is favored? Is it possible that the di-impurity tunnels an entity and thus two-body effects are prominent?

Ask for minor revision

1. The authors study a concrete, experimentally relevant setup with exact diagonalization.
2. They look at a number of physical quantities to provide a picture for the nature of binding of the two impurities.
3. They predict collapse-and-revival dynamics of the dimer under an interaction quench, which may motivate experimental observations.

1. The study is not particularly novel as the dimer formation and much of the ground-state properties are already known. In particular, there is significant overlap with a paper involving one of the authors (Ref.33).
2. The study is limited to small system sizes where it is difficult to characterize critical phenomena.
3. The authors have not discussed how their predictions may be detected experimentally.

The authors study the binding of two spin impurities in a 1D Bose-Hubbard model at unit filling using exact diagonalization. In particular, as the bath-impurity repulsion U_{bI} is increased relative to the bath-bath repulsion U_{bb}, the impurities phase separate from the bath, forming a dimer. Although this binding in the ground state was already known from past work, particularly Ref. [33], the authors provide a more detailed characterization and also analyze low-energy excitations and dynamics following an interaction quench, which they show can lead to collapse and revival of the dimer.

The work is thorough and well written, despite being limited to small systems (9 sites), and constitutes a useful addition to the literature. However, it does not meet the acceptance criteria of SciPost Physics for groundbreaking results or novelty. It could be considered for SciPost Physics Core after addressing the following points.

1. The authors assume that the two impurity atoms do not interact with each other. Other than simplicity, is there any motivation to work at this limit? Have the authors considered the effect of nonzero interactions?

2. More generally, it will be useful to discuss in greater depth how realistic the setup is and whether one can observe the results, e.g. collapse and revival, in experiments.

3. I find the description of Fig. 2(b) in the last paragraph of Sec 3.1 somewhat confusing and lacking physical content. Firstly, the cutoff of 0.4 J_b for weak repulsion is quite arbitrary. (Also, “J_b” is missing in the text.) Secondly, it appears from the figure that the binding energy is quadratic for small for U_{bb} and linear for large U_{bb}, which results in the minima. Can the authors explain the physics behind the quadratic growth in the superfluid region?

4. On several occasions the authors use the term “critical” interaction strength for dimer formation. However, all of their plots at nonzero tunneling are (necessarily) smooth, becoming sharper in the Mott regime. Is there any reason to expect a phase transition as opposed to a crossover (in the thermodynamic limit)? If not, the authors should qualify their usage.

5. Can the authors explain why the energy gap in Fig. 6 falls sharply as the impurities become bound? What is the energy scale of dimer tunneling that sets the small gap?

6. In Fig 7(b) what are the almost equally spaced excitations at large U_{bI}? The gap is much smaller than U_{bb} which would be the cost of multiple doublons in the bath (as in Fig. 16).

7. In the quench dynamics do the authors expect the collapse and revival to persist for large systems? For instance, how does the corresponding spectral gaps scale with the number of sites M?

8. Can the authors explain how they get Eq. (9) for the average distance between two free bosons? Is there a similar expression for two free fermions?

Minor points:

1. In Eq. (7) the one of the occupations should have a subscript “\sigma^{\prime}” as opposed to “\sigma”

2. The last column in Table I should probably read “r_s^* / r_0” as opposed to “r_s^* / a” since Eq. (10 ) gives r_s^* > r_0.

3. Below Fig. 4 the authors state that the distance between the impurities vanishes for small U_{bb}. This is only valid if U_{bI} > U_{bb}/2, which would be useful to add here.

4. In Eq. (A.3) should the degeneracy be M choose 2 for small U_{bI}?

5. In labeling the curves in Figs. 2(b), 3(b), and 5(b) one should use “U_{bb}” [not “U_{BB}”] like everywhere else.

1. Discuss experimental realization(s).
2. Address questions in the report on phase transition and spectral gaps.
3. Incorporate the minor corrections listed in the report.