SciPost Submission Page
Sum frequency generation from real-time simulations in two-dimensional crystals
by Mike N. Pionteck, Myrta Grüning, Simone Sanna, Claudio Attaccalite
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Claudio Attaccalite |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202505_00003v1 (pdf) |
| Date submitted: | May 2, 2025, 12:05 p.m. |
| Submitted by: | Claudio Attaccalite |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Sum frequency generation (SFG) and difference frequency generation (DFG) are second order nonlinear processes where two lasers with frequencies $\omega_1$ and $\omega_2$ combine to produce a response at frequency $\omega = \omega_1 \pm \omega_2$ . Compared with other nonlinear responses such as second-harmonic generation, SFG and DFG allow for tunability over a larger range. Moreover, the optical response can be enhanced by selecting the two laser frequencies in order to match specific electron-hole transitions. Here, we propose a first-principles framework based on the real-time solution of an effective Schr\"odinger equation to calculate the SFG and DFG in various systems, such as bulk materials, 2D materials, and molecules. Within this framework, one can select from various levels of theory for the effective one-particle Hamiltonian to account for local-field effects and electron-hole interactions. To assess the approach, we calculate the SFG and DFG of two-dimensional crystals, h-BN and MoS$_2$ monolayers, both within the independent-particle picture and including many-body effects. Additionally, we demonstrate that our approach can also extract higher-order response functions, such as field-induced second-harmonic generation. We provide an example using bilayer h-BN.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for sending us the referee's report, which we found very positive.
As suggested, we have responded directly to the referee's comments/observations,
and have resubmitted a revised version of our manuscript, which we now consider more suitable for publication in SciPost.
Below is the list of changes made to the manuscript, also highlighted in red in the text.
Kind regards.
Claudio Attaccalite
List of changes
Here the list of changes in the new version of our manuscript:
1) We updated Ref.31 and Ref. 34 that are now published
2) We added a note to explain that TD-aGW was called TD-HSEX/TD-SEX in previous papers
3) Fixed all abbreviations in references
4) fixed typos
5) We condensed the discussion about problematically long periods at the beginning of Sec. 3.2
6) We changed the introduction of the paragraph about two similar frequencies in Sec. 3.2.1
7) We changed Fig. 3. In the full DFT approach, we used 435 fs sampling time after 40 fs dephasing. For a fair comparison with the other approaches where a 50 fs dephasing is applied, we start also with a 50 fs dephasing in the full DFT approach. Then a simulation time of 385 fs is considered converged. Furthermore, the difference for SVD and LSF in logarithmic scale with 5 fs less sampling time is add to show convergence.
8) We explained why SVD and LSF provide same results and that they are equally efficient according to our tests.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 4) on 2025-8-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00003v1, delivered 2025-08-25, doi: 10.21468/SciPost.Report.11796
Strengths
1- technically very sound, theory is state-of-the-art 2- studied nonlinear processes interesting and relatively uncharted
Weaknesses
1- Maybe a bit too technical for a general audience
Report
Requested changes
1- For clarity, it would be nice if the authors could choose a different letter from “P”, or at least its calligraphy, to denote polarization and momentum, otherwise the derivations are hard to follow at times. This is specially true in sections like 3.1 (page 6); right after Eq. 9, the authors refer to “complex Fourier components \vec{p}_n”, I assume that they want to refer to the quantity defined in Eq. 9 “\vec{p}^(n)”, but the symbol they employ actually corresponds to single-particle momentum introduced above Eq. 6.
2- The description on how the FI-SHG is computed at small finite frequency could be improved. According to Eq. 8, it appears that the response at +-2w is proportional to the third-order susceptibility at frequency set(\nu,\omega\omega). Then, I do not entirely understand the following sentence “summing the χ(3) extracted by P (2ω +) and P (2ω −), one obtains the corresponding FI-SHG for low-frequency time-dependent pump fields.” Do the authors mean that when \nu is positive (negative) it corresponds to 2w+ (2w-)? I believe this should be clarified.
A further minor point: the authors say “Among higher-order responses that can be extracted from Eq. (4), we look at the FI-SHG”, but Eq. 4 explicitly discards higher-than-quadratic contributions. Could they clarify?
3- Why do the authors employ the approximate equality symbol in Eq. 5? To my understanding, this is the exact expression for the second-order susceptibility in perturbation theory.
4- The authors could provide more details on the symmetries of the two studied systems, e.g. their point group, and possibly provide an appropriate reference containing more details on the systems to help the interested reader.
5- The authors rescale the response by a effective thickness d_eff. Is this procedure standard for quadratic responses? Do they expect any non-trivial dependence on this parameter?
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #2 by Anonymous (Referee 3) on 2025-8-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00003v1, delivered 2025-08-20, doi: 10.21468/SciPost.Report.11777
Report
The main contribution of this work is a signal processing strategy that is improved with respect to the previous work Ref. [11] by the same team. Consequently, the core part of the manuscript is its section 3. Given that this is incremental technical progress, I would find the work better placed in SciPost Physics Core.
That being said, there is also the application to monolayer $h$-BN and MoS$_2$ and second-harmonic generation in bilayer $h$-BN. In this context, I unfortunately have an issue with the underlying Eq. (2). According to the Hohenberg-Kohn theorem, ground states for a many-body system can be described by a density functional. While I am aware that it is common practice to apply this also to band structures, this is already starting to stretch things. Once one gets to time-dependent density-functional theory (TDDFT) -which is basically the situation investigated here- things get more complicated. Thus, as someone who is not an expert in the physics of excitons, I cannot help wondering if the additional terms in Eq. (2) really go into the right direction from the independent-particle picture. If the authors could add a few further remarks to justify this approximation, this could make the difference to render the manuscript publishable in SciPost Physics.
There are a number of relatively minor details (some of which were also noted by Referee 2) that I list among "Requested changes".
Requested changes
1- Provide some justification of Eq. (2).
2- The acronym "DFG" carries two meanings in the present manuscript. I thus recommend that the authors check if they cannot find better acronyms than "SFG" and "DFG".
3- Likewise, "DFT" is usually understood to mean "density-functional theory" and this method is actually used in the present manuscript. Again, I recommend looking for a different abbreviation of "discrete Fourier transform" in order to avoid confusion.
4- Fig. 2 is referenced and discussed only after Fig. 3. I thus recommend moving it to a later place (where it belongs).
5- The end of the first paragraph of section 3.2.1 refers to Fig. 3. However, I see no logarithmic sampling in Fig. 3. Do the authors maybe mean Fig. 2?
6- There is a number of minor linguistic and typographic issues. I uploaded an annotated manuscript for the authors to help them with proofreading it.
Recommendation
Accept in alternative Journal (see Report)
see attachment
Attachment:
Report #1 by Anonymous (Referee 2) on 2025-8-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00003v1, delivered 2025-08-14, doi: 10.21468/SciPost.Report.11747
Report
In the present manuscript, the authors adress the question how to efficiently extract the relevant components of the second-order susceptibility tensor from the time-dependent polarization that is obtained from the real-time propagation scheme. The standard option here would be discrete Fourier transform which would, however, require long time propagation in order to reliably capture frequency sums and differences. The authors explore two alternative ways: (i) calculating the Moore-Penrose inverse of the matrix that determines the coefficients (when the polarization is developed into harmonics) using singular value deconposition and (ii) performing a least-square fit of the coefficients, minimizing the deviation of the calculated and approximated polarization on a number of time-steps within the simulated time interval. It turns out that this latter method (least-square fit - LSF) works particularly well and allows to reduce the overall time interval for the simulation considerably.
The computational approach is tested on two prototype 2D materials: hBN and MoS2. Unfortunately, there is no comparison with experimental data due to the absence of experimental data for hBN and the limited availability of data for MoS2 (partially measured on metallic substrate and thus not directly comparable to the simulations of the manuscript). Nevertheless, the simulations constitute a valuable proof of concept, in particular through the exploration and validation of the three computational schemes for the extraction of the frequency information.
The manuscript is very clearly written. The results constitute a valuable contribution to the field of computational non-linear optical spectroscopy. Clearly, this manuscript is suitable for publication in SciPost Physics.
Requested changes
(1) If the authors would like to improve the discussion of their results on hBN and MoS2, they might consider to add results on the band-structures and on the (linear) optical spectra. Even though these are pretty well known in the literature, they could be used to visualize the discussion of the computational results. This might improve the readibility and clarity of the second-order results in this manuscript.
(2) Apart from this, I have only a few typos that should be corrected: - page 3, second line from bottom: "can be accounted for" (insert "be") - page 5, first line: alpha should have two subscripts - page 5, six lines below Eq. (6): "there may also resonance" (a "be" seems to be missing in the sentence) - page 5, bottom: DFT is the standard acronym for "density functional theory". Using it for "discrete Fourier transform" as well, might cause confusion. - page 6, caption of Fig. 2: "logarithmically sampled" - page 7, 5 lines below Eq. (11): "As the frequencies of the external fields are muliples of ..." (no komma here) - page 7, 6 lines below Eq. (11): no "then" at the beginning of the line - page 10, bottom paragraph: The phrase "In Fig. 4, we report ..." is doubled. - page 12, line 4 of section 5.2: "Similar to what was observed ..." (insert "was") line 5: "... while the weake A and B excitons ..." (delete "the")
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
see attachment
Author: Claudio Attaccalite on 2025-09-09 [id 5799]
(in reply to Report 3 on 2025-08-25)see attachment
Attachment:
Answer_to_referee4.pdf