A comparative study of perturbative and nonequilibrium Green's function approaches for Floquet sidebands in periodically driven quantum systems

TL;DR

This work benchmarks two complementary theoretical frameworks for Floquet-sideband physics in periodically driven graphene: PB1, a $1^{st}$-order perturbative approach, and tdNEGF, a dynamical nonequilibrium Green’s function method. It disentangles Floquet-dressed initial states, Volkov-dressed final states (LAPE), and their interference in pump–probe ARPES, highlighting the crucial roles of photoemission matrix elements and interface screening modeled by $f_V$ and $f_F$. PB1 yields analytical, momentum-resolved sideband amplitudes and captures symmetry trends, while tdNEGF delivers full energy–momentum spectra with higher-order processes, spectral weight redistribution, and hybridization gaps. Qualitative agreement between the two methods emerges when matrix elements are included, but quantitative differences appear near hybridization regions and at certain incidence/angle conditions, underscoring their complementary use for interpreting tr-ARPES in driven quantum materials.

Abstract

We compare two complementary theoretical approaches to compute and interpret Floquet sidebands in periodically driven quantum materials: a first-order perturbative approach (first-order perturbative Born approximation, PB1) and time-dependent nonequilibrium Green's functions (tdNEGF). Using graphene as a model Dirac system, we disentangle in pump-probe setups Floquet-dressed initial states, Volkov-dressed final states (also known as laser-assisted photoelectric effect, LAPE), and their interference. We quantify how photoemission matrix elements, polarization, incidence angle, and near-surface screening shape the momentum-resolved sideband intensity observed in tr-ARPES. PB1 yields an analytical expression for the momentum-dependent sideband intensity, and for graphene it captures the correct symmetry trends, such as the magnitude of the intensities when considering the interference between the Floquet and Volkov states and photoemission matrix elements. tdNEGF reproduces the full energy-momentum-resolved spectra, including hybridization gaps and spectral-weight redistribution. We find qualitative agreement between PB1 and tdNEGF once matrix elements are included; quantitative differences arise near hybridization regions and at specific angles where higher-order processes and self-energies are essential. Thus, for systems with simple band structures and away from these regions, the two approaches can be used in a complementary way.

Figures (15)

• Figure 1: Setup geometry and relevant vectors: (a) The scattering plane is along the $\Gamma-K_1$ direction. Both IR drive and EUV probe impinge under $\theta_{\mathrm{in}}$. In this coordinate frame, the in-plane field is parametrized as $(\textbf{E}_{xy},\theta_E)$ and the momenta as $(\textbf{k}_{xy},\theta_k)$; similarly $\textbf{E}_z$ and $\textbf{k}_z$ are out of plane components. The polarization is defined by the angle $\phi$. (b) Sketch of the $(k_x,k_y)$-plane parametrized around the $K_1-$ point using the vector $\mathbf{k_D}$.
• Figure 2: First Floquet sideband intensity in the $(k_x,k_y)$ plane as obtained from the perturbative approach (PB1) and from the tdNEGF, respectively. Columns (a,b) show PB1 results; columns (c,d) show tdNEGF results extracted from constant-energy cuts ($E–E_F=0.63eV$) that isolate the first sideband. Top, middle, and bottom rows correspond to Floquet–Volkov (FV), Volkov (V), and Floquet (F) contributions, respectively. Columns (a,c) are without photoemission matrix elements (no-ME); columns (b,d) include matrix elements (ME). Parameters used $\theta_{\mathrm{in}}=68^{\circ}$, $\phi=85^{\circ}$, and field strength $4~\mathrm{MV/cm}$. All intensities are normalized to unity.
• Figure 3: Azimuthal ($\theta$) dependence of the first sideband intensity for Floquet–Volkov (FV), Volkov (V), and Floquet (F) contributions, corresponding to the momentum maps in Fig. \ref{['fig:Park_NEGF_overview_WME']}. The blue lines represent tdNEGF results, and the red lines the PB1 results. Each curve is normalized to its own maximum. Left panel: results without photoemission matrix elements (no-ME). Right panel: results including matrix elements (ME). The inset in (e) illustrates the way the azimuthal data is obtained.
• Figure 4: Comparison between theoretical photoemission intensities obtained by tdNEGF and PB1, respectively, and experimental tr-ARPES data for monolayer graphene as published in Ref. Merboldt2025. The plots show the angular dependence of the normalized photoemission intensity for the first sideband, shown for two different values of the polarization of the incident light: near p-polarized light ($\phi=10^{\circ}$, left column) and near s-polarized light ($\phi=76^{\circ}$, right column). Black crosses represent the experimental data, the blue lines represent tdNEGF results, and the red lines the PB1 results. Theoretical results include contributions from Floquet-Volkov (FV) in (a) and (d), Volkov only (b) and (d), and Floquet data in (c) and (f). Parameters are $\theta_{\mathrm{in}}=68^{\circ}$, $(f_F,\,f_V)=(0.5,\,0.5)$, and ME are included.
• Figure 5: Sensitivity of the first sideband angular intensity to screening. Top: tdNEGF; bottom: PB1. Matrix elements are included throughout. The incidence and polarization are fixed to $\theta_{\mathrm{in}}=68^{\circ}$ and $\phi=85^{\circ}$. Screening pairs $(f_F,f_V)$, defined in Sec. \ref{['subsec:comparison_with_experiment']} [Eqs. \ref{['eq:eff_out']}, \ref{['eq:eff_xy']}], are varied from the “perfect-surface’’ limit $(0.0,1.0)$ (black) to $(0.7,0.3)$ (dark red) and encoded by the color bar; all curves are normalized to unity.