Enhancement of High Harmonic Generation in Bulk Floquet Systems

The paper explores various aspects of HHG from a 1D model SSH system, which is simultaneously being driven by a second frequency component that's harmonic to the drive that generates harmonics. This is mainly done by allowing the hoping terms to vary by one frequency in time, while driving with an optical frequency via a Peierls substitution.
There are some nice numerical results, and I appreciate the analytic formulations in the high frequency limit, which might find some potential uses.

As a whole there are some main weaknesses here. Since this is my first time refereeing in SciPost I am not sure which are relevant for acceptance/revision, so I'll just go through things how I see them:

1. I could not understand mainly what is the fundamental difference between utilizing this approach (one color for Floquet physics + one for an HHG drive), or just having a standard two-color driving fields. Both induce a time-periodic Hamiltonian, and both can be analyzed with Floquet theory. In fact, it was difficult for me to follow if the current approach by the authors included also the optical drive in the Floquet theorem, or only the 2nd drive at \Omega for the hopping terms.
In that respect, its not clear if this is just a mathematical choice for analyzing the system for convenience, or if there's some added intuition, because one could just as easily analyze HHG results with multi-colored drives for various solids, as is in fact very commonly done with many other theoretical frameworks. I suppose this version allows extracting some high-frequency limit, but I'm not sure that's the most relevant thing (see point 2).

2. A big portion of the calculations and analysis is performed for high frequencies. This also means the response is not necessarily non-perturbative. Most plots in the paper show quite strong exponential decay of harmonics with harmonic order (e.g. figs. 2,4,9, etc.), and no plateau. Thus, I would make a separation, because the standard notation for HHG is for a truly non-perturbative optical responses. I would either focus the work on perturbative responses, or present spectra that show plateaus.
Also in this regard, a lot of the introduction references are in fact only dealing with SHG and perturbative responses.

3. The analysis and conclusions on the spatio-temporal symmetries and selection rules overlaps with many years of efforts already well-established in the strong-field physics community. It is hard to summarize this effort, but one can look at some main papers, including Phys. Rev. Lett. 80, 3743 (1998), J. Phys. B: At. 34, 5017 (2001), and more recent generalizations that include also time-reversal symmetries and glide symmetries explored by the authors (Nat. Comm. 10, 405 (2019) , Nat. Comm. 13, 1312 (2022)). This approach has already been employed to many HHG setups from atoms and solids, and even recently was used to analyze HHG from topological insulators (Phys. Rev. A 103, 023101 (2021), Nano Lett. 21, 8970–8978 (2021), again, its hard to summarize because its already very well established). It is well known that by employing an additional commensurate field one can control (break/impose) symmetries in the system. This is even the case in EFISH, which is a subcase of Floquet theory, but also its analogue in HHG (Nat. Phot. 12, 465–468 (2018)). Here the authors have to acknowledge previous works, including the most general works and remove claims of novelty. Essentially, I would say the authors re-derive a lot of these results with new terminology, but there is no new physics presented since the selection rules arise from Floquet group theory.

4. The authors claim there is an enhancement of the HHG emission in the driven systems. I suppose this is quite expected. But, I think the enhancement has not been evaluated fairly here - one should compare the HHG yields not between the driven/undriven systems, but between the driven system, and an undriven system where the HHG generating field has a total power that corresponds to the total driving power in the driven system. In other words, the power of the 2nd field at \Omega should be put back into the fundamental field for correct comparison, otherwise the two cases are not on equal footing.

5. Another weakness is that the entire study is done for a 1D model system, such that one cannot make general conclusions for more realistic systems.

Actually my main report was already incorporated in the section above.
Some other points are:

1. The HHG process is itself a Floquet process that can 'drive' a system out of equilibrium. For instance, this was predicted to cause a topological phase transition (Nat. Phot. 14, 728–732 (2020)), a dynamical band dressing (Nat. Phot. 16, 428–432 (2022)), and some other related phenomena, and there's some debate on what picture should be used for interpreting the dynamics (Phys. Rev. Research 4, 033101 (2022)), i.e. the dressed band picture or the field-free bands. I think the authors should comment on this, because it raises the question of which field should be considered the probe and which the pump, and how would one choose which Floquet basis to use if there are two drives, etc.

2. The authors should also address in the introduction Phys. Rev. B 99, 195428 (2019), Phys. Rev. A 99, 053402–(2019), Phys. Rev. A 102, 053112 (2020), which explore similar physics but in non-dressed systems.

3. The notation for curly omega and omega is confusing (e.g. in fig. 2). I would make an effort to change notations and clarify some of the variables.

4. I wonder how can one control the occupations of the Floquet states in a realistic system - would this not be determined by the dephasing properties of the solid?

5. is t_0 averaged over one drive cycle? or more? because it can often take a few driving cycles for the system to enter into a so-called steady-state.

6. When both drives are of the same frequency, labeled \Omega=\omega, what then is the meaning of the 2nd drive? Would it not be equivalent to a case with just a single drive?

Everything is given above.

1 - Valid theory for high-harmonic generation in a Floquet-driven solid
2 - High-harmonic generation in condensed matter is a timely subject
3 - Topological effects are included
4 - Numerical results for a concrete example system (Su-Schrieffer-Heeger chain) are presented
5 - The manuscript is well written

1 - Kind of Floquet drive is unrealistic for real systems in which high-harmonic generation of the emitted radiation occurs
2 - Some of the relevant literature is ignored
3 - Mistake in Fig. 3
4 - Various typos

In their manuscript, Kumar et al. study high-harmonic generation in a bulk Floquet system. A Floquet system is a periodically driven system (by some frequency \Omega) such that the Floquet theorem applies to it, resulting in quasi energies, dressed states and periodic energy zones in analogy to crystal momentum, Bloch states and Brillouin zones in spatially periodic systems. High-harmonic generation (HHG) means that upon irradiation of a system with (laser) light of frequency \omega, multiples of that frequency are emitted. This is almost always the case unless the system is a harmonic oscillator. As an example, they apply their theory to periodically driven Su-Schrieffer-Heeger chain.

Not surprisingly, the emitted harmonics are influenced by the Floquet drive because frequency mixing occurs. The authors develop a theory and derive an expression for the current from which the harmonic emission can be calculated. They discuss selection rules and show results for different initial occupations, relative phases, and laser intensities.

I think the theory outlined by the authors and their calculations are correct, and the results are methodologically interesting for theorists. I recommend publication in SciPost but have some questions the authors might address to improve their manuscript, making it potentially more appealing to a broader audience.

1 - How could HHG in a Floquet system be observed? The kind of driving the authors consider is possible to implement on various platforms that allow to simulate tight-binding hamiltonians, e.g., cold atoms in optical lattices, optically, with written wave guides, or even classically with electric circuits. However, in all such systems no high harmonics of incident light are emitted. HHG occurs in “real” physical systems that sustain the strong incident light. One could drive such a system with a second laser but that would be completely different from the driven hopping amplitudes considered in the manuscript. It would rather be HHG in two-color fields, which has been studied extensively already. The authors might discuss which experimental realisation they envision. 


2 - Concerning selection rules for HHG using Floquet theory, the authors may consult, https://doi.org/10.1103/PhysRevLett.80.3743, https://doi.org/10.1088/0953-4075/34/24/305, https://doi.org/10.1103/PhysRevA.91.053811, for instance.


3 - Figure 3 shows three times the same set of two panels. I wonder how such a mistake remains unnoticed by three authors.


4 - In general, the manuscript is well written but there are still some typos. I leave it as an exercise for the authors to figure them out.