# Strong-field response of solids and molecules

### Prof. Dr. Stefanie Gräfe, Friedrich-Schiller-Universität Jena

The interaction of light with matter covers a large number of physical phenomena that we literally *see* in our everyday life, as it is responsible (amongst other things) for the vision process. Early scientists mostly focused on investigations of electromagnetic radiation in the visible range („Zur Farbenlehre“ J. W. v. Goethe, 1810) and at low intensities, where material polarization responds linearly to incident electromagnetic fields. Although unknown at that time, quantum mechanical effects already resulted in measurable effects such as color alterations of stained glass (e.g. in church windows) when including gold nanoparticles of varying size.

With the advent of tunable, high-intensity light sources, the investigation of nonlinear effects, where the material polarization changes in a nonlinear fashion with electric field of incident light, became accessible.

The interactions of intense light and matter give rise to a plethora of interesting phenomena such as multiphoton absorption/ionization, four-wave mixing, sum- and difference frequency generation, optical parametric amplification, laser induced electron diffraction [1,2] or second-, third or higher-order harmonic generation.

Spectral signatures however, become increasingly complex when investigating quantum systems and their dynamics in strong fields since the increasing importance of non-linear effects requires careful theoretical description and examination. So far, most of our understanding of strong-field effects is based on the simplest atoms and molecules such as H_{2} and H_{2}^{+} or simple model systems. [3,4]

This project aims at simulating and interpreting the strong-field dynamics of real molecules and larger systems in a rigorous real-space real-time approach including non-linear strong-field effects such as photoionization and high-order harmonic generation (HHG) of systems ranging from small (chiral) molecules over nano-systems to the condensed phase. The project aims at advancing the theoretical description of light-driven dynamics of multi-electron systems. To this end, we will employ a state-of-the-art numerical description of the strong-field response of solids and molecules based on the real-time real-space time-dependent density functional theory (rtTDDFT) as realized in the Octopus program package. The numerical propagation of time-dependent Kohn-Sham orbitals from which the time-dependent electron density is constructed, is done in short (attosecond) time steps, typically requiring several ten-thousand consecutive propagation steps for covering a short excitation pulse of about 50-100 fs duration. This comprehensive microscopic description of light-matter interaction allows for a detailed investigation of non-linear effects such as high harmonic generation (HHG) or photo electron emission.

Exemplary we present first results of simulations of HHG from CdSe nanoparticles performed at Noctua PC2 computer cluster. (Higher) harmonic generation is a process, in which atoms and molecules interacting with high intensity laser pulses emit radiation at frequencies that are multiples of the incident laser frequency. For small gas-phase systems, HHG is usually explained via the so-called three-step model, that includes (1.) tunneling ionization of an electron out of the potential well of the atom/molecule and (2.) acceleration of the electron due to interaction with the external field and. For an intense, time-dependent incident laser field, the electron may return to the parent ion, where it has a possibility to (3.) recombine and emit. the accumulated kinetic energy in a coherent short burst of electromagnetic radiation – the HHG. Due to the coherent nature of the high energetic radiation, HHG is the routine method to generate attosecond pulses.

In solids, the HHG processes are much more complex since electrons do not freely propagate through space but propagate in the conduction band(s), giving rise to inter- and intra-band currents, forming the basis of HHG. Although much more challenging for theoretical scientific efforts, solids, however, provide a promising route towards bright and compact HHG sources since their electron density is much higher than that of gas phase systems.

On Noctua we have so far performed simulations of CdSe nanoparticles with different particle sizes (4-64 atoms, corresponding to about 0.5-1.5 nm diameter). HHG spectra are simulated in a rigorous real-time real-space approach. The incident laser field parameters are chosen corresponding to recent experiments of our experimental collaborators. The field strength (intensity) of the field is such that most likely, ionization is suppressed, thereby restricting HHG to effects of electron and hole dynamics within the nanoparticles.

Figure 1 shows the different employed structures including spherical boxes of varying size that were discretized with an equidistant grid of 0.3 a.u. resolution, resulting in more than 7,000,000 mesh points. Electron dynamics were propagated for >100 fs (17600 steps of about 6 as). A complex absorbing boundary at the periphery of the simulation box monitored electronic density leaving the nanoparticle. As the integrated electronic charge (i.e. number of electrons) of the system did not significantly change during simulation runs, we assured that the resulting HHG spectra originate from the current dynamics within the nanoparticles only. Simulations were performed including rotational averaging over multiple orientations of the nanoparticle. The resulting HHG spectrum of a 64-atom nanoparticle is shown in comparison to a bulk simulation of CdSe using periodic boundary conditions in Figure 2.

The simulation of bulk CdSe clearly shows peaks low orders (1^{st} – the fundamental – and the 3^{rd} harmonic order). From 5th to 9th order the signal is rather noisy, which might be due to a high joint density of states in this energy window. For higher energies, clear peaks of odd harmonics are visible up to the 17th harmonic.

In contrast to this, the HHG spectrum of 1.5 nm (64 atom) CdSe nanoparticles looks very different.

Clearly, the confinement significantly reduces the contribution of high orders (>5) to the spectrum.

Our goal is a systematic investigation of the effect of different laser and material parameters on the resulting HHG spectra using a rigorous state-of-the-art ab-initio approach based on the rtTDDFT. As computation of the spectra is extremely demanding (up to 100,000 CPU hours per simulation run), HPC facilities such as the PC² computer cluster are of utmost importance for our work.

**References:**

[1] Kasra Amini, Michele Sclafani, Tobias Steinle, Anh-Thu Le, Aurelien Sanchez, Carolin Müller, Johannes Steinmetzer, Lun Yue, José Ramón Martínez Saavedra, Michaël Hemmer, Maciej Lewenstein, Robert Moshammer, Thomas Pfeifer, Michael G. Pullen, Joachim Ullrich, Benjamin Wolter, Robert Moszynski, F. Javier García de Abajo, C. D. Lin, Stefanie Gräfe, and Jens Biegert. Imaging the Renner–Teller effect using laser-induced electron diffraction, *PNAS* 116 (17) 8173-8177 (2019), 10.1073/pnas.1817465116.

[2] B. Wolter, M. G. Pullen, A.-T. Le, M. Baudisch, K. Doblhoff-Dier, A. Senftleben, M. Hemmer, C. D. Schröter, J. Ullrich, T. Pfeifer, R. Moshammer, S. Gräfe, O. Vendrell, C. D. Lin, J. Biegert, "Ultrafast electron diffraction imaging of bond breaking in acetylene", *Science*, 354, 308 – 312 (2016).

[3] Lun Yue, Philipp Wustelt, A. Max Sayler, Florian Oppermann, Manfred Lein, Gerhard G. Paulus and Stefanie Gräfe, Strong-field polarizability-enhanced dissociative ionization. *Phys. Rev. A*, **98(4)**, 043418 (2018)

[4] Matthias Paul and Stefanie Gräfe. Strong-field ionization dynamics of asymmetric equilateral triatomic model molecules in circularly polarized laser fields, *Phys. Rev. A* 99(5), 053414, (2019), 10.1103/PhysRevA.99.053414