Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment

We would also like to thank the three Referees for their careful examination of our work. Referee 1 and 2 recommended our manuscript for publication in SciPost Physics. We have revised the manuscript and implemented their comments to improve the manuscript. Referee 3 wrote “It could there be published if the authors address my concerns”. We hope that we address in the following his/her concerns satisfactorily.

Particularly, the three Referees have pointed out two main weaknesses of our manuscript: the systematic errors from the experiments are not discussed in a detailed manner; and a more substantial comparison between our numerical methods with existing ones is missing. These two points are both more properly addressed in the new version.

In the new manuscript, the major changes related to the Referee’s comments are marked in blue. Please note that Figs. 4 and 5 in the previous version now become Figs. 5 and 6, respectively.

Reply to comments from Referee 1.

1. (Retardation effects) Indeed the retardation effect of the cavity field would also affect the SSF-SMI boundary. We now discuss it in more detail in Sec. 4.1. As we further discuss in the same section, this retardation effect is inevitable due to the dilemma between the retardation effect and the atom loss.

2 & 3. (Monochromaticity) In experiments, the main loss of the laser pump strength comes from the two side peaks due to servo electronics, which are in the order of MHz away from the central peak. This shift is much larger than the cavity dissipation rate, and thus these two side peaks cannot contribute to the scattering of the cavity field. On the other hand, the linewidth of the central peak cannot be accurately measured because we do not have another stable laser to beat with. However, we estimate that the linewidth is roughly 100Hz, which is much smaller than the cavity dissipation rate. It can thus fully scatter into the cavity. The calibration factor 1.36 seems reasonable to us. We add a few lines discussing this in Sec. 2.2.

4. (Background signal of the cavity field). This large background counts originates from diffuse scattering on the optics implemented in the system, and not from the cavity. It can thus be safely subtracted from the data for the determination of the phase boundary. We add a sentence in the caption of Fig. 6 discussing that. Moreover, as one can see in our recent preprint Ref. 54, the symmetry in our system is very well established and there is no significant breaking of the symmetry due to diffusively scattered light. We clarify this in Sec. 2.1.

5. (Comparison to existing approaches). We thank the Referee for his/her appreciation on this discussion. Inspired by this comment, we now further perform a more quantitative comparison between the MCTDH-X approach and the existing approach based on Wannier function analysis. This is now discussed in the new Appendix F. In the main text, we also move the relevant paragraph to Sec. 4.2. We decide to keep the discussion after the main result instead of moving it to an earlier section, because it requires understanding of the SSF-SMI transition, which is mainly discussed in Sec. 4.1.

Reply to comments from Referee 2.

1, 3 & 4. (Heating and atom loss). Indeed there are multiple effects induced by heating and atom loss which we have not fully discussed in the previous version.

-- (Comment 3): The NP-SSF boundary is slightly shifted to higher pump strengths for the large detunings due to heating and atom loss because we reach the self-ordered phase at larger pump strength and hence later times. Therefore, when we are fitting this boundary, we only consider the data points from low detunings (-2pi5 kHz to -2pi20 kHz). This has already been implemented for the fitting in the last version, and is now explicitly clarified in Sec. 3.2.

-- (Comments 1 & 3): Moreover, the heating and atom loss are also accountable for the plateau seen in $|\alpha|^2$ in Fig. 6(b). This will certainly affect the cavity-induced optical potential. However, for the fast ramp presented in Fig. 6(b), this plateau occurs later than the Mott transition, so it has little effect on the predicted boundary. This is discussed now in Sec. 4.1.

-- (Comment 4): However, inside the SMI phase, indeed the atom loss will change the shape of the central Bragg peak. This indeed partly contributes to the difference between the behavior of W in experiments and simulations. We now also discuss this in Sec. 4.1.

2. (Mean-field density distributions in momentum space). In the mean-field limit, if we blindly use the algorithm to calculate the “momentum-space density distribution”, we would obtain the Fourier transform of the real-space density distribution. Nevertheless, we would say that this result does not have physical significance, because it has not converged numerically. Therefore, we don’t think that it would be useful to present them.

5 & 6. (Terminology and typos) We thank the Referee for pointing out these issues. We have now replaced “superlattice” with “lattice”, and fixed the typos.

Reply to comments from Referee 3.

In the report, the Referee posts two questions regarding to the applicability of the four-well model and the relevance of the beyond-Gross-Pitaevskii mean-field effects, respectively.

1. (Applicability of the four-well model). Indeed, the symmetry of the checkerboard lattice makes us possible to choose a simple model with only four lattice sites, and this is not guaranteed for many other systems. As the number of lattice sites in a unit cell increases, the required computational resources scale up significantly. We now include an extra paragraph in Sec. 3.3 to discuss this.

2. (Beyond mean-field effects). If we constrain ourselves only in the Gross-Pitaevskii mean-field limit, we are able to reproduce the self-organization, but we cannot capture the Mott insulation. This is because the Mott insulation requires at least one single-particle orbital for each lattice site. If we blindly do the simulations in the mean-field limit even in the four-well model, we will consistently obtain “superfluid” states even for very large lattice depth. This has already been discussed in detail in Sec. 3.1, but is now also briefly discussed in the introduction as suggested by the Referee.

The Referee also requests us to clarify two statements.

1. (Quantum-optical Bose-Hubbard model). Here what we have in mind is a Bose-Hubbard model with long-range interaction between lattice sites. We have made the statement more clear now.

2. (Parameters). Indeed our sentence was a bit confusing in the previous version. We want to say that these are the last two parameters which we need to determine for the simulation. We have clarified this now. If we want to use other approaches instead, for example the Wannier-based Bose-Hubbard approach as presented in Sec. 4.2 now, the same parameters are also required.

Sec. 1: We clarify the phrase “quantum-optic version of the Bose-Hubbard model”, and briefly discuss the beyond-mean-field effects in our system.

Sec. 2.2: We further discuss in detail the retardation and heating effects on both the NP-SSF and SSF-SMI boundaries. We also clarify the source of power loss of the pump laser.

Sec. 3.2: We now include a new figure showing the real space density distribution of the full 2D model. We also clarify that the fit for the experimental NP-SSF boundary only takes the data points from small detunings into account, in order to minimize the heating effects.

Sec. 3.3: We discuss how the computational effort scales when the symmetry of the optical lattice becomes weaker.

Sec. 4.1: We now include new panels (s-x) in Fig.5 showing the real space density distribution of the full 2D model. This is for comparison with the four-well model in panels (m-r). We also discuss now the retardation and heating effects observed in Fig. 6 in more detail. We have also fixed a typo in the experimental results in Fig. 6(b). The number of photons is in the order of 10^3.

Sec. 4.2 & Appendix F: We perform a comparison of our MCTDH-X approach with an alternative approach based on Wannier functions and Bose-Hubbard model. The detail of the comparison is shown in the new Appendix F.

We also rewrite several paragraphs to improve the narrative.