A dark state of Chern bands: Designing flat bands with higher Chern number

-new and interesting result
-simulation of the center-of-mass response relevant to the experimental realization

-the comparison of the related works should be improved
-very little intuition behind the main result of CD=C1+C2-C3
-numerics justifying the expression for CD is lacking--it is only stated in the appendix that the authors checked the expression for CD in a number of scenarios. Given the lack of analytical results and intuition, this should be improved by presenting their numerics explicitly; and probably also showing when this expression stops to work (when?).

The authors propose how to engineer the high Chern number bands by coupling multiple topologically nontrivial bands using external light fields. They discuss how to prepare and measure the system in the state of interest.

The work seems to be very similar to Panas2020. There, three levels are studied, the dark state considered, but the coupling is different.
-The discussion relating the two should be extended.
-What are the strengths of this approach compared with Panas2020? What are the weaknesses? How does it compare with other approaches to (flexibly?) engineering C>1?
-Why CD=C1+C2-C3 here, but there is simply CD=C1+C2+C3? What makes the difference?
-If there is no intuitive explanation, could the authors explain why it is hard to come up with an explanation? It seems that Panas2020 hasn't given too much intuition as well, but could one cook up a simple toy model to illustrate the physics?
-Dark state has a negligible contribution from |3> state, so why does it contribute to the Chern number.
-Why the phase between states 0 and 1 does not contribute (or does it?) to the expression of CD?
-What if the light coupling between the bands has an angular momentum? Would it lead to different results?
-Why do we need three levels? Is dark state considered so that we have flatter bands? Or we want to simply avoid lossy state |3> and still have strong and tunable coupling between |1> and |2>. Could authors shed more light on that?
-What if we consider coupling between two long-lived bands: do we have C=C1\pm C2 for some circumstances? Which sign should it be?
-For the M-scheme with states 1-2-3-4-5 (the dash indicates which re coupled) with the dark state being a superposition of 1,3, and 5, what is the expression for CD? Is it CD=C1+C3+C5-C2-C4?

-I agree with Referee 2 when it comes to the experimental parameters.

I believe that the work is interesting and worth publishing in the end in some journal. Before however making my final decision, I think that presentation could be improved for the profit of a reader.

Minor:
-what makes HH bands reminiscent of the Landau levels?

Please, address my question in the Report above.

Minor:
Though the draft is in general well written the Authors could make sure that there are no more shortcoming similar to which I bought from my quick read:
-Eq 4, why not add subscript \nu to F in Eq 4, I think it would make the notation clearer.
-Abstract:
--remove comma in 'band", by coupling'
--replace 'very flat' by e.g. 'nearly flat'
-remove comma in 'approach, in view'

1) A new idea to generate a topologically nontrivial dark band in cold atom systems is introduced.
2) A simple sum rule is presented substantiated by detailed numerical simulations.
3) Experimental protocols to verify their findings are clearly demonstrated.

1) Many important details are left in Appendices, which could be integrated into the main texts for better readability.

This work introduces an interesting idea to generate a topologically nontrivial dark band in cold atom systems. The sum rule of Chern numbers is unique and extensively verified by the numerical simulations. In particular, the authors proposed a setup to generate $C_D = 2$ band, which is shown to be experimentally detectable by the center-of-mass responses.

The findings are important for further investigation of exotic topological phases in cold atom systems, and this work meets the criteria for publication. Therefore, I recommend the publication of the manuscript.

I appreciate it if the authors can clarify the following points to improve the presentation.

In Sec. 4, a new setup is introduced to generate a $C_D = 2$ band. I have several questions on this:

1) While it is understandable that Eq. (6) produces one dark band and two bright bands, its connection to the previous $\Lambda$ system is not clear. Here all the three bands are overlapping without the coupling $A_s$, thus it doesn't look like a $\Lambda$-type.
2) What is the value of $A_s$ in Fig. 2(c), and its relation to $\Omega_s(\mathbf{k})$ (in the caption of Fig. 2)?
3) Finally, why is the trivial band so flat? If naively $\hat{H}_\text{HH}$ with $\phi = 0$ is taken, the bandwidth is $\sim J$. Is it due to the fact that only every $q$ sites are used along the $x$-axis, which implies $J_x \ll J_y$?

1) The article proposes a way of producing novel topological band structures that have a flat band characterised by a non-zero integer Chern number. The authors present a particular realisation of a flat band with a Chern number equal to 2, and describe how one could generalise this to arbitrary integers.
2) This proposal is feasible and sensible with an experimental realisation with ultra-cold atoms in optical lattices.
3) The authors provide proof-of-principle simulations for detecting the topological Chern number in current cold atom experiments.
4) This work is novel and interesting and provides a way of preparing and detecting novel topological features directly in an experimental setting. This has the potential to have a high impact in the field.

1) When the authors discuss experimental preparation and detection, specifically the timescales that are required experimentally, they do not discuss the experimental limitations. For example, some of the timescales that they quote in section 5 – for typical experimental parameters – would correspond to many seconds or even minutes, which are out with the capabilities of current experiments due to decoherence effects. It is a little unclear how their conclusions would change if more realistic values were used in their simulations.

This is an interesting and well written article. The results are presented clearly, the experimental proposals are for the most part sensible and the authors conclusions are backed up by numerical simulations.
The ability to experimentally prepare novel topological band structures opens up new avenues for exploring the behaviour of topological systems. In particular, being able to realise these systems in experiments with ultra-cold atoms offers a feasible way of including interactions between atoms allowing for investigations into interacting topological systems such as fractional quantum hall phases.

In this case I would recommend publication in this journal. However, I suggest that the authors address the below points, as I believe that this will increase the impact of their article.

1) In the figure 3 caption, the authors state that for (a) they use a value of $Fa=5\times 10^{-4} J$ and a timescale of $t=15 \hbar/J$. In typical cold atom experiments, $J$ usually takes values around $J/h \approx 100 \rightarrow 1000 ~{\rm Hz}$. If the larger values are used then this will then correspond to a timescale of $t\approx 5~{\rm s}$.
These values for the timescales seem large compared to typical experimental coherence times of $< 1 {\rm s}$. Can the authors discuss how important these timescales are for detecting the Chern number in the way that they are proposing.
For (b) they use a value of $Fa=5\times 10^{-3} J$, which then corresponds to $t\approx 0.5~{\rm s}$ - which seems more experimentally feasible. But if smaller tunnelling amplitudes are realised (see Ref.[9] in the article for example) then even the timescales here may be too large.

2) Similarly, on page 9, when discussing preparing atoms in the flat band, they quote timescales for their linear adiabatic ramp of $T=10^6 \hbar/J = 160~{\rm s}$. The authors do state that this can be improved for different ramping procedures, but they do not present specific simulations or estimates. Can the authors discuss the errors in this preparation scheme for more experimentally realistic timescales, such as $T<1~{\rm s}$.