A real-time approach to frequency-mixing spectroscopies: application to sum and difference frequency generation in two-dimensional crystals

TL;DR

This work develops a real-time framework to compute second- and higher-order nonlinear optical responses under multiple driving fields, enabling accurate SFG/DFG and FI-SHG spectra for two-dimensional crystals. By employing an effective Hamiltonian that can include excitonic effects via TD-aGW (and IPA as a baseline), the authors extract $\boldsymbol{\chi}^{(2)}$ and $\boldsymbol{\chi}^{(3)}$ from time-dependent polarization, using either SVD-based inversion or nonlinear least-squares fitting to handle multi-frequency drives efficiently. The results for h-BN and MoS$_2$ show pronounced excitonic resonances shaping SFG/DFG spectra, with TD-aGW revealing sharp features absent in IPA, confirming the crucial role of electron-hole interactions. Additionally, FI-SHG under THz pumping demonstrates substantial third-order responses in 2D layers, highlighting potential applications in nonlinear THz spectroscopy and sensing, and the method’s extensibility to other nonlinear probes like CARS.

Abstract

We propose a computational framework to extract non-linear response functions from real-time simulations in the presence of more than one external field. We apply this approach to the calculation of sum frequency generation (SFG) and difference frequency generation (DFG). SFG and DFG are second-order nonlinear processes where two lasers with frequencies $ω_1$ and $ω_2$ combine to produce a response at frequency $ω= ω_1 \pm ω_2$. 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. To assess the approach, we calculate the SFG and DFG of two-dimensional crystals, hBN and MoS2 monolayers, from real-time solution of an effective Schrödinger equation. Within the effective Schrödinger equation, one can select from various levels of theory for the effective one-particle Hamiltonian to account for local-field effects and electron-hole interactions. We compare results obtained within the independent-particle picture and including many-body effects. Such comparison allows us to identify and characterize excitonic features in the obtained spectra. 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 the hBN bilayer.

Figures (7)

• Figure 1: A schematic representation of the nonlinear processes studied in this work: (a) sum frequency generation (SFG), (b) difference frequency generation (DFG) and (c) field-induced second-harmonic generation (FI-SHG).
• Figure 2: SFG of h-BN with a pump frequency $\omega_2=3~\mathrm{eV}$ obtained at the IPA level using the full discrete Fourier transformation (FT) (blue solid line), singular value decomposition (SVD) (magenta dashed line), and the least square fit (LSF) (yellow dotted line). Results obtained with different sampling times are shown. The discrete FT needs a sampling time (385 fs) about 26 times larger than the SVD and LSF (15 fs). Below is displayed the difference in logarithmic scales for SVD and LSF, respectively, with 5 fs less sampling time to show the convergence.
• Figure 3: The time-dependent polarization (purple solid line) of h-BN calculated at the independent particle level with two electric fields ($\omega_1=0.2~\mathrm{eV}$, $\omega_2=3~\mathrm{eV}$). The signal can be divided into two regions: an initial "equilibration" region (up to $t\gg\gamma_\mathrm{deph}$, here $t=50~\mathrm{fs}$) during which the system's eigenfrequencies are suppressed by dephasing and a region where Eq. \ref{['eq:FourierSeries2']} holds. In the second region the polarization is logarithmically sampled (teal dots) within a converged time window of 15 fs, smaller than the fundamental period of 20.678 fs of the signal. This sampling time is sufficient to correctly determine the Fourier coefficients by the least square fit (LSF) as verified by reconstructing the polarization (lavender dashed line) within the fundamental period using the Fourier coefficients and the truncated Eq. \ref{['eq:FourierSeries2']}.
• Figure 4: SFG/DFG spectra for h-BN in panels (a), (b) at the independent particle level and in panels (c), (d) at the TD-aGW level. These heatmaps have been generated using a frequency grid of $\omega_1 \times \omega_2 = 96 \times 96$ points. For each frequency pair a real-time simulation was run and the output signal processed.
• Figure 5: Calculated imaginary dielectric function of (a) h-BN and (b) at the IPA (blue dashed line) and TD-aGW (orange solid line) level. The excitonic peaks $E_1$ and $E_2$ of h-BN are located at around 6.1 eV and 7 eV, respectively (panel (a)). The degenerate excitonic peak A/B as well as C can be seen at around 2.2 eV and 3 eV, respectively (panel (b)).