Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids
Round 1
Reviewer 1 Report
Victor et al study the interference effects in high harmonics from crystals.
They find that even order harmonics can be produced when only either positive momentum or negative momentum is considered. When they include both directions the even harmonics get interfere destructively.
The results are as one would expect. Authors show details of theirs methods.
Although for now it's not clear what this study presents to the community, I encourage the group very much on considering these problems.
The presentation language and literature study is fine. The new review paper on Nature Physics may be cited. I recommend the publication with the following detail criticisms that authors maybe able to accommodate.
1. Please understand dipole approximation. This is the approximation when wavelength of electromagnetic wave is much longer than bonding distance. The spot size of a laser beam is always bigger than lattice distance, that's not dipole approximation. http://web.ift.uib.no/AMOS/nazila/LaserAndLight/node7.html
2. I assumed that the crystal orientation is such that there is inversion symmetry, i.e. the laser field is perpendicular to the c-axis of the ZnO crystal. What happens when the laser field is along the c-axis of the crystal?
I ask this because in that direction even order harmonics are produced, as shown in the first ZnO results by Stanford group. I think it would be important test to understand the origin of interference. I am not quite convinced with the argument that harmonics are interfered "after they are produced".. It is possible i misunderstood. please explain.
3. Minor comment is what happens when the initial momentum in the conduction band is not zero, i. e. for example in Silicon, which is indirect band-gap material.
Author Response
We thank the referee for his/her careful reading of our manuscript as well as for the useful comments.
1. We completely agree with the referee on this issue. Generally, the condition is that the electromagnetic field should change a little on the length scale defined by the bonding distance (which is characteristically the lattice constant). Although the spatial variation of the field is not independent from its spatial envelope (spot size), much stronger spatial changes result from the periodicity of the radiation, which is characterized by the wavelength. We thank the referee for reminding us this fact. The manuscript has been changed accordingly.
2. This remark has common points with the second critical comment of the third referee, thus we felt it important indeed. A trivial approach would be using effective potentials that do have and don’t have inversion symmetry. However, no qualitative differences were seen, i.e., we observed practically no even harmonics in either case. Most probably, this has to do with the dimensionality of the model, since working in 1D (along the polarization of the excitation) means assuming that only a very narrow set of k vectors from the orthogonal plane plays relevant role in the time evolution, and this may not be the case in reality. However, already this observation is a result (although negative one) worth mentioning, and motivated us to start studying the two-dimensional case.
On the other hand, as mentioned by the third referee, the appearance of the even order harmonics is not determined by the crystal structure alone, the symmetry of the excitation is equally important. Since breaking the (approximate) time-translational symmetry of a long, multicycle excitation is independent from the dimensionality of the target, we expect the appearance of the even order harmonics also in 1D. And indeed, as the newly added Fig. 4 shows, when a weak, few-cycle THz radiation is superimposed on the IR excitation, these harmonics do appear.
Having discussed it, we also feel that the statement that harmonics are interfered "after they are produced” was not formulated precisely enough. It is clear that the dynamics of all initial states together create the source of the HH radiation. And although the time evolution of the states with different k indices can be followed independently, it is only their net effect that can be observed. However, when we calculate this overall effect, we have to add the individual contributions to the net current coherently. Interference takes place when the oscillating current components produce the source of the HH signal. That is, harmonics that do not appear in the spectrum were never created, since they are absent already from the time evolution of the source – as a consequence of the interference described above. We clarified this point in the manuscript.
3. We agree with the referee that it would be interesting to consider various band schemes in the context of HH generation. For a general dispersion relation, e.g., for indirect bandgap materials, the interference described above in point 2 could lead to considerably richer HH structure, since not all k vectors have the counterparts –k with the same energy, i.e., the symmetry in this sense is lower in the system. However, for single electron Bloch states (independently from the dimensionality), the dispersion relation is always an even function of the k vector, which makes predictions for indirect bandgap systems quite speculative. This difficulty is mentioned in the current version of the manuscript.
The review paper Nat. Phys. 15, 10 (2019) were added to the list of references.
New or rewritten parts of the manuscript are marked by blue color in the resubmitted version.
Reviewer 2 Report
The manuscript is devoted to modeling of harmonic generation from solid targets using a quantum mechanical approach.
I have a number of comments and criticisms, however, there is no meaning to list them up in view of a major fault in the present version: the crystal potentials, equations (6) and (7), must be re-written or completely reconsidered.
In particular, in Eq. (6), the sum of two cosine-squared can be replaced with a single cosine of the double period and some phase shift. I made some math and obtained the following mathematically equivalent form: U(a)(x) = u0 + A*cos(2*pi*x/Da + phi), where u0 = -U0(a) = -25 eV, A = U0(a)*cos(pi*(X1-X2)/Da) ~ 24.7318 eV, and phi = - pi*(X1+X2-7*Da)/Da ~ 2.99498 rad. Why do the authors over-complicate the equation, rather than use a simple cosine form?
Further, in Eq. (7), the potential is a non-periodic function of x. I do not understand how it was possible to obtain the Bloch states in that case.
Author Response
We thank the referee for pointing out an issue that really needed clarification.
We forgot mentioning some crucial points in the previous version of the manuscript. For U(b) it is only the negative part of the two summands that are used, i.e., when the formulae produce positive values, they are replaced by zeros. For U(a), additionally, we used only the first cycle of the cos^2 functions, centered around x1 and x2. In this way the potentials are zero both at x=0 and x=a, i.e., they are periodic and usual techniques (eq. 2) can lead to the Bloch-states. Let us also note that the two potentials differ in the sense that the zero-nonzero transition is considerably smoother for U(a), thus they indeed can be used to check to what extent our results depend on the particular choice of the model.
Additionally, it was not our intention to “complicate” the equations, we only wrote what we had in our code (unfortunately, without mentioning the point discussed above). We focused on double-well potentials, the two parts of which were added consecutively. The referee is completely right, using the information we gave in the previous version of the manuscript, U(a) seemed to have a simpler form, but since both cos squared functions are assumed to become constant zero outside a given interval, this does not hold. We clarified this point in the current version of the manuscript.
New or rewritten parts of the manuscript are marked by blue color in the resubmitted version.
Reviewer 3 Report
The manuscript by Szaszko-Bogar et al.presents a theoretical investigation of HHG from bulk solids using a quantum mechanical model. By working in the velocity gauge, the authors avoided coupling of different k-states. As a consequence, the authors found out that the interference of the fields generated from initial states is responsible for the suppression of the even harmonics, as well as controlling the cut-off of the harmonics. The manuscript is well written, the language is very clear and easy to understand. The authors have paid attention into including details of the calculation in the manuscript which is very good. However, the manuscript suffers from conceptual issues hence I would not be able to recommend publication of this manuscript in the current form. I hope the authors will find my critical and minor comments useful in preparing their manuscript for revision and submission elsewhere.
Critical comments:
- Although I understand that the authors might had developed their own formalism and described them already before (for example, PRB, 96, 035112 (2017)), it might be misleading for the unfamiliar readers reading this model description because of not knowing the complete fact. The fact is that the main derivation in this manuscript was carefully discussed and presented in various works before (see for example, Haug, Koch, “Quantum theory of the optical and electronic properties of semiconductors”; or PRB 38, 3342 (1988) for a full description on length gauge; or PRB 87,115201 (2013) for velocity gauge). The authors should give a broad overview of development in this direction, then mention how their approach is different (if there is any) from existing literature. This has to be clearly stated such that readers of this manuscript would get a more “complete” perspective, not confusing that this is the first paper in this direction.
- For any light-matter interaction, the product of the interaction depends on not only matter but also light. For matter: for any quantum system neglecting spin, time reversal symmetry results into bandstructure as an even function of wave vector k. For spatially inversion symmetric system, the transition dipole matrix elements are a real and even function or purely imaginary and odd function of k(PRA96, 053850, (2017)) and non-inversion symmetric systems possess odd function of the transition dipole phase. Therefore, by pure symmetry reason, if the light is inversion symmetric, and the matter is not, then even harmonics are allowed, and vice versa: if the matter is inversion symmetric, but the light (laser pulses) is not (due to short duration, complex, two-color field, for example) then even harmonics are also allowed. This is much more generic than the interference picture the authors discussed and this is already well known in attosecond community. Furthermore, the even harmonics if allowed will be very strong, with comparable intensity as odd harmonics. This is not the case for the results shown in this manuscript.
- The authors used parameters corresponding to a ZnO crystal, yet direct comparison of theoretical investigation and experimental results (Ref. 17) has not been shown. It would be very convincing if the authors show that their model is able to reproduce experimental results. Otherwise, it is hard to judge the importance of this model.
Minor comments:
- It looks like that the numerical implementation in this manuscript suffers from accuracy and precision. In Fig. 2, 3, 4, the difference between peak intensity and noise level is about 10 orders of magnitude or even less. This is surprising considering that the 64 bit representation (default in most computing language) has 15 orders of magnitude in precision. Perhaps the authors used adaptive integration techniques? If so, this can be easily solved to get higher precision.
- It would be good if the authors could investigate the population dynamics, as it may reveal something interesting.
- “SHHG” is used without definition.
Author Response
We thank the referee for carefully reading our manuscript as well as for the suggestions that helped us to improve the content.
Reply to the critical remarks:
1. We thank the referee for this comment. The focus of our manuscript was not on the technique, we wanted to point out a viewpoint that was rarely mentioned in the literature earlier. Neither we wanted to create the impression that the theoretical approach presented in the manuscript is the only possible way of performing similar calculations. And it was far from our intention to exclude any relevant reference. Although the literature is rich in this subject, we did our best. However, the papers mentioned by the referee are indeed closely related and should have been mentioned. Now we added them to the list of references. PRA 91, 013405 (2015), which also describes a model that is close to our approach, is cited in the current version of the manuscript, too.
Besides adding new references, we extended the introductory part of the text and rewrote the beginning of Sec. II to put our model in an appropriate perspective.
2. This remark has common points with the second comment of the first referee, thus we felt it important indeed. We agree with the referee that symmetry-related considerations are more general/fundamental. In the current version of the manuscript we clarify that the intended focus of the manuscript is not whether we have or don’t have even order harmonics (since it can be predicted using the symmetry), but rather about an insight how this effect can be seen from the perspective of interference of the currents (HH sources) created by the different initial states: although the time evolution of the states with different k indices can be followed independently, it is only their net effect that can be observed. However, when we calculate this net effect, we have to add the individual contributions to the total current coherently. Interference takes place when the oscillating current components combine to produce the source of the HH signal. We think that this viewpoint is worth discussing, and we clarified this issue in the manuscript.
Considering our model, a trivial approach to check the role of inversion symmetry would be using effective potentials that do have and don’t have inversion symmetry. However, no qualitative differences were seen, i.e., we observed practically no even harmonics in either case. Most probably, this has to do with the dimensionality of the model, since working in 1D (along the polarization of the excitation) means assuming that only a very narrow set of k vectors from the orthogonal plane plays relevant role in the time evolution, and this may not be the case. However, already this observation is a result (although negative one) worth mentioning, and motivated us to start studying the two-dimensional case.
On the other hand, we thank the remark that the appearance of the even order harmonics is not determined by the crystal structure alone, the symmetry of the excitation is equally important. Since breaking the symmetry of a long, multicycle excitation is independent from the dimensionality of the target, we expect the appearance of the even order harmonics also in 1D. And indeed, as the newly added Fig. 4 shows, when a weak, few-cycle THz radiation is superimposed on the mid-IR excitation, these harmonics do appear, and their amplitudes are comparable to the odd order ones.
3. The 1D model potential we used can only reproduce a limited number of features of the band structure of a ZnO crystal, so no quantitative agreements can be expected. Having compared with the results of Ref. [17], we can conclude that the cutoff scales linearly with peak field strength of the excitation, which is in agreement with [17]. As discussed above, the polarization dependence of the process cannot be investigated using a 1D model. The increase of individual HH peaks as a function of the intensity of the excitation shows qualitative agreement with the experimental results, where, however, because of numerical limitations (see below), we could not reach intensity regions where a new effect (deviation from the power law) was reported in [17].
Note that although it was not the primary goal of our work to reproduce experimental results, the comparison above has been added to the current version of the manuscript.
Responses to the minor comments:
- We used an adaptive stepsize Cash-Karp Runge-Kutta integration routine, where the control parameter was the norm (the trace of the initially normalized density operator) and we kept its deviance from unity below a limit of 10e-7. Motivated by the remark of the referee, by setting more strict conditions, we could improve the accuracy by an order of magnitude, but no observable changes in the spectra were seen. Demanding higher accuracy would easily end up the routine taking “infinitely many infinitely small steps,” especially for higher intensities. We think that the presented results are converged, higher contrast cannot be expected using this model. Let us note, however, that a contrast around 10 orders of magnitude is not unusually low in HHG (neither experimentally nor theoretically).
- The population dynamics for both “pure” and “THz assisted” excitation were calculated, and the result was included in the new version of the manuscript as Fig. 5. It shows that the populations of the conduction bands strongly decrease as the energy increases (i.e., it is the largest in the first conduction band and almost negligible in the third one). The time dependence of the populations shows signatures of the harmonics as well. Additionally, for the multicycle IR excitations we considered, the population in the conduction bands were practically zero after the excitation. Using an additional THz field, these final populations increased orders of magnitude (definitely above the level of the numerical errors), but they are still very small. These results, together with the corresponding figure, were added to the manuscript.
- The term ‘SSHG’ appeared only once in the manuscript. In order to avoid an acronym in a (subsection) title, we reformulated the related part of the text.
New or rewritten parts of the manuscript are marked by blue color in the resubmitted version.
Round 2
Reviewer 2 Report
The authors corrected the description of their potentials, and now the real review is possible.
The authors describe a model of high-order harmonic generation (HHG) in bulk solids. In their model, the authors consider crystals with the band-gap much larger than the photon energy, lattice constant much smaller than the wavelength, and initial population of the valence band only. The authors then describe their 1D formalism based on the single-electron approximation, Bloch states, and density matrix. The authors show representative spectra, (qualitatively) compare them with experimental data from literature, and analyze origin of odd-only harmonic emission and origin of the high-frequency spectrum cut-off. For the latter two, the authors identify the reason: it turned out to be interference of radiation from states with different k vectors.
In short, I found some aspects of the manuscript to be of interest. However, there are several issues that must be addressed before it can be published. Two of these issues are general:
(A) Firstly, the authors should state clearly what are the new method(s) and result(s) that they use/obtain, and distinguish these from the already known methods and results. The authors should also compare their model with already published ones, so that potential readers will be able to knowledgeably pick the authors' model, or otherwise.
(B) Secondly, the authors should state not only the properties which are explained by their model, but also the HHG properties that cannot. How about, for example, even-order harmonics and HHG dependence on the angle between the laser polarization and the crystal axis [Nat. Phys. 7, 138 (2011)] [17]? Or, the appearance of the second plateau [Nature 534, 520 (2016)] [23]? I am sure the authors can additionally list and briefly comment on other properties, as well.
Some particular comments:
1. Lines 72-74: "Let us note here that using Bloch states, the band energies are always the same for +-k, which means a certain limitation for the materials that can be described in this way." Please comment on what kinds of materials can and cannot be described.
2. Eq. (4) and the text before it: "The time evolution of the density operator is governed by the von Neumann equation supplemented by a phenomenological term that takes unavoidable decoherence effects into account:" Please provide references.
3. The meaning of brackets [] in Eq. 4 is different from other equations, why not to mention this or change notation?
4. Lines 91-92: " For the calculations presented here, the diagonal and off-diagonal relaxation rates are gamma_d = 0.1 1/fs and gamma_od = 0.3 1/fs, respectively." Please explain the choice of these parameters and provide a reference. Also, the present notation is confusing, it is very similar to 0.11 (fs^-1) and 0.31 (fs^-1). Why not to change these to 0.1/fs and 0.3/fs, or 0.1 fs^-1 and 0.3 fs^-1?
5. Lines 102-103: " we consider the gauge independent, entire J as the source of the HH radiation." In view of its importance, why not to show the dependence of J(t) in additional frames in Figs. 2-6?
6. Eqs (6), (7): a figure with the model potentials would be useful for potential readers. A figure with the resulting band schemes would be useful as well.
7. Please provide not only the peak field strengths, but the corresponding peak laser intensities [W/cm^2] as well. The readers would also benefit from dimensionless values characterizing the system, such as the dimensionless amplitude a0 = e*F0/(m c omega_L), the ratio of the Bloch and laser frequencies omega_B/omega_L, the ratios of the maximum and minimum bandgaps to the laser photon energy, and one of the dimensionless values characterizing strength of the laser field with respect to the binding potential (the latter one is probably the most important).
8. Lines 143-144: "Furthermore, the increase of individual HH peaks as a function of the intensity of the excitation shows qualitative agreement with the experimental results..." Please show the dependences of harmonics intensity on laser field strengths in the Log-Log scale, such as in Fig. 2 of Ref. [17].
9. Continuing lines 143-145: "Furthermore, the increase of individual HH peaks as a function of the intensity of the excitation shows qualitative agreement with the experimental results, where, because of numerical limitations, we could not reach intensity regions where a new effect (deviation from the power law) was reported in [17]." If an exact numerical reason is known, why not to specify it? There as well may be other reasons, such as model limitations, which prevent this experimentally observed deviation. The authors should also comment on this. By the way, there are results shown for the field of 4 GV/m in Fig. 3, which is 4 times higher than the maximum field in Fig. 2. Does it mean that the numerical limitations do not apply to Fig. 3?
10. Line 147: "Note that for larger excitation intensities, the position of the cutoff is more difficult to identify." In view of this, please show the cutoff frequency at each frame of Fig. 2, and describe the authors' definition of the cutoff. It would also be useful to add an inset with the dependence of the cutoff frequency on the field strength, and compare it with the experimental dependence [Nat. Phys. 7, 138 (2011)] [17]. In particular, in Fig. 2 the authors use the maximum field of 1 GV/m = 1 V/nm = 0.1 V/A, and obtain a large number of harmonics (around 40 to 50); however, the experiment [17] even at two times larger field of 0.2 V/A gave only ~15 harmonics. Please comment on the difference.
11. Continuing Fig. 2: there are two different sets of harmonics with different shapes: quite narrow harmonics of low orders, and quite broad harmonics of higher orders. Can the authors comment on this? How do the authors distinguish the harmonics signal from numerical noise?
12. Figure 4, two insets: why not to use the same time scale?
13. Line 193: " For N CBs and..." Here the meaning of N is different from that of Eq. (3).
14. Figure 6 insets: why not to show axes labels?
15. Lines 219-220: "The crystal was assumed to be initially in thermal equilibrium, with only the valence band being populated." The crystal temperature is not a parameter of the model. Further, the situation of no population in the conduction bands corresponds to a low-temperature limit. Why not to state it as such? In any case, the wording "initially in thermal equilibrium" seems not relevant here.
Author Response
We thank the referee for his/her detailed report, which helped us to improve the manuscript. We took all recommendations into account and modified the text accordingly. New or rewritten parts are denoted by blue color. Figs 2 and 4 are also new, and we added new panels to Figs 3 and 6 as well.
A) We extended the introduction by adding a few sentences in which we discuss the usefulness of our approach.
B) In the revised version of the paper, at the end of in subsection 3.1, we itemize what effects can be described using our model and what are the ones that are beyond the scope of our approach.
1) The symmetry of the energy eigenvalues in k certainly allows the description of direct bandgap semiconductors or dielectrics, where the extrema of the bands are at k=0 and the momentum matrix elements have the maximal magnitude also around here. (Since in this case most of the significant transitions occur in a k-space range where the assumption of symmetry holds.) Although in principle one can imagine indirect bandgaps occur with E(k)=E(-k) for both the conduction and valence bands, but this is quite rare, so practically it is not expected that indirect bandgap materials (like Si) can be described in this way. The discussion above is inserted in the manuscript starting at line 78.
2) We provided references and a short discussion at lines 101-108.
3) The referee is right, [.,.] means a commutator in eq. (4), and a simple bracket everywhere else. In order to avoid confusion, we used the explicit form of the commutator in Eq. (4), so now “[]” simply means bracket all over the manuscript.
4) This point is related to 2), and the discussion (together with references) can be found at the same point, i.e., lines 101-108. The notation has been changed.
5) The time evolution of the current could not have been seen on insets (the plot was too small), thus we decided to add new panels to the figures, as it can be seen in the new part of Fig.3 (current numbering). However, additional plots are quite similar, so in order to keep a normal ratio of figures and text, we added new subfigures only when it contained additional information, like in the case of Fig. 6. (We are not conceptually against adding more figures, but it did not seem necessary.)
6) We added a new figure (Fig. 2) with the two potentials and the band schemes.
7) We inserted the peak intensities at the captions of the figures. The dimensionless ratios are given at lines 149-156.
8) The corresponding graph has been inserted as Fig.4. For numerical issues, see the next point.
9) -10) We were able to improve our numerical method. Now practically the only numerically noisy part is the end of the spectrum. Therefore – although we can figure out a reasonable definition for the cutoff, we are somewhat reluctant to put much focus on it, especially since the cutoff in our case is at higher frequencies than in the experiments. (We discuss the possible reasons in Sec.3, lines 167-169.) If this issue were the only one hindering the publication of the paper, we are not strongly against adding a figure.
11) This point became serious when we draw the new figure, where the maximum of a harmonic peak was plotted versus the exciting filed strength (Fig.4 ). In that case we defined e.g., the height of the 11th harmonic peak as the maximum of the spectrum data between 10 and 12 times the exciting frequency.
12) Now the scales are the same
13) N has been replaced by M.
14) Axis labels were added.
15) For a gap of 3.2 eV, at room temperature, thermal equilibrium is practically the same as the low temperature limit. It is more precise to state the latter one, but the term “thermal equilibrium” implies the independence of the energy eigenstates, even when the temperature is low. Therefore we did not replace this term by something like “low temperature limit” everywhere, but clarified the usage.
Reviewer 3 Report
I do not think the authors fully understand my original comments and also the principle/fundamental of generation of even harmonics that I wrote before. Perhaps it is because I did not write it in details enough. Please find here the more comprehensive/longer explanation: consider that you have an inversion symmetric medium, and a monochromatic light (so that it is inversion symmetric by a 180 degree rotation and retardation). In the first half cycle of the electric field, a total polarization will be induced, and it can possibly contain all harmonics. This total polarization is constructed from coherent superposition of all the coherences as well as interband currents (Eq. 4, 5, 6 of PhysRevB 77, 075330 (2008)) thus it is literally the “interferences of fields emitted by all the initial (valence band) states” as the authors said. Now, this one does not make the net even order harmonics yet, because we need to consider the next half a cycle of the electric field where the same total polarization will be induced, but it is shifted and rotated by 180 degrees. Therefore, this is the reason why even harmonics are not detected because they are destructively interfered in this regime. Similar arguments can be made for other cases (involving non-inversion symmetric medium or electric field). To conclude, the key role in this case is symmetry where it is the symmetry that enable/disable the observation of even harmonics afterward.
Although the authors have improved their manuscript, including more proper introduction, the manuscript still suffers from conceptual problem. The findings are therefore not novel, but they are already well known. Perhaps the authors can improve this point and submit it elsewhere.
Author Response
We do not think that there is conceptual opposition between us and the referee, and we are sorry that he/she found not enough novelty in our work to be published in a special issue of Applied Sciences.
As a final remark, let us emphasize that we completely agree that symmetry based arguments are by far more fundamental than (probably) any other. We also admit that the role of subsequent halfcycles was not discussed in the previous manuscript, simply because that was not our focus. (However, now we added a few sentences.) We investigated the whole time evolution and performed FFT on the complete data file. We were motivated by the observation (FIg. 5 of Phys. Rev. B 96, 035112 (2017)) that a single two-level system emits both odd and even harmonics for many cycle excitation as well. We wanted to know, what happens when we have a collection of few-level systems (with different gaps and coupling matrix elements, as a very simple model for a solid), how the net effect of all these two-level systems builds up the HHG spectra. Not too surprisingly, we observed that the odd-even nature of the spectra is determined by the parameters of both the material and the excitation: adding the effects of different k-s (which, admittedly, involves interferences at the consecutive half-cycles) results in the final spectra.
Round 3
Reviewer 2 Report
The authors improved the manuscript further, and in my opinion, it now represents sufficient interest to potential readers. However, I checked a few calculations and, unfortunately, it seems there are mistakes/misprints in the text. I recommend the authors to check all calculations ones again before submitting the final version.
1. Fig. 2 top: I tried to plot the potentials according to the given equations and parameters. For U(a), I obtained the same (or at least very similar) graph; however, for U(b), my graph was different: it contained two negative regions at the same positions as the authors, but deeper, down to -190 eV, and wider: the negative regions were from ~-0.04 to ~0.41 and from ~0.49 to ~0.92. Of course, this may be just my mistake, but I would ask the authors to check the equation and parameters – there may be a misprint; or there may be a mistake in Fig. 2.
2. Lines 152-153: "the Bloch frequency w_B = aeF_0/h is roughly an order of magnitude larger than w_L at lambda_L = 3 micron." However, actually w_B and w_L are almost equal at these parameters. Please correct the sentence.
Author Response
We thank the referee for his/her careful review.
Answers:
1) The referee is right, there was a typo when we enumerated the parameters of potential b. The figure is correct, that was plotted using direct numerical data. However, we overlooked a division by a factor of 2 for the parameter U_shift. Now its correct value (which is the half of its previous one) is given in the manuscript. We thank the referee for pointing out this issue.
2) We corrected the sentence according to the suggestion of the referee.
Reviewer 3 Report
I think now the authors understand what I meant before. Hence, from the novelty point of view, the manuscript does not satisfy conditions for publication. In order to help the authors improve their work, I would like to suggest few things. If they can do it, and they find anything interesting, I think we could consider it for publication again.
Time-resolved spectroscopy measurements are a powerful technique to study electronic and nuclei dynamics. Now if the authors could use their model to study either of this, it would be nice. For example, the author can put the same beam, delayed, and interact in the collinear configuration, what would happen. Or the authors can use a short wavelength beam, to excite immediately the ZnO and probe the excited dynamics using a longer wavelength beam.
Author Response
We thank for the suggestions of the referee for improving the content of the manuscript.
We were brooding on which direction to follow. Since the electromagnetic pulses we consider have given waveform, i.e., the back-action of the material sample on the pulses is neglected, investigating e.g., fast, light-induced change of the absorption would require strong modifications of the model that are not doable within a couple of weeks.
Finally we decided to consider the dependence of the strength of the harmonics on the delay between a THz and an infrared pulse. The related questions clearly follow from the points that were previously discussed, and they are not too far from the original concept of the manuscript. That is, we felt that these results can be incorporated to the text in a logical way.
We added Fig.7 and the corresponding discussion to the manuscript. Here we can see how the weight of a given even order harmonic increases when the THz and the infrared pulses start to overlap, and results for an odd order harmonic and also for the overall high harmonic gain are shown as well.
