controlling and measuring ever faster electron dynamics with ever better accuracy is going to have applications across multiple disciplines

Made popular in the terahertz spectral domain, electro-optical sampling provides an alternative route to electric-field information, making use of nonlinear optical crystals to measure a laser field by its influence on the polarization of another laser pulse that simultaneously passes through the detection medium. Using ultrashort optical pulses, we are able to directly access the field oscillations of even near-infrared light.

Using extremely short laser pulses to excite the active region of a small metal-dielectric-metal junction with extreme temporal precision, we can use solid-state optoelectronic devices to measure electric field transients based on the laser-field-induced generation of electric current. Driving these circuits with a longer-wavelength field provides access to the electric field waveform and gives new insight into the dynamics of charge currents switched at petahertz repetition rates.

Adopting a versatile scheme from the fields of microwave and terahertz spectroscopy, we can measure the laser-field waveform after passage through a nonlinear medium. This measurement is done twice. Once with the peak intensity of the laser set high enough to accumulate significant nonlinear modifications, and once at an intensity level low enough to ensure that the light-matter interaction is virtually free of nonlinearities. By comparison of the two recorded waveforms normalized for their difference in peak amplitude, we can extract the temporal evolution of the nonlinear polarization wave inside the sample. This allows for a sensitive quantitative investigation of the energy exchange dynamics between a laser field and a sample, the response time of the underlying nonlinearities and the electron kinetics responsible for the optical response.

Of course, different experiments demand different observations to provide a full physical picture. For this reason, we utilize three different attosecond beamlines for advanced experiments, including time-resolved photoelectron spectroscopy, x-ray and extreme ultraviolet time-resolved spectroscopy and more. Our toolbox includes electro-optical sampling and photoconductive switching at frequencies ranging from the microwave to near-infrared, photoelectron time-of-flight spectroscopy from gases and surfaces and attosecond transient absorption measurements in gaseous and solid systems. Nonlinear polarization sampling relies on the capability of a modified gas-phase photoelectron streaking set-up to sample the electric field of light.

Theoretical research in our group has several purposes: explaining measured data, providing a theoretical basis for planning future measurement campaigns, as well as identifying new promising research directions. For all these purposes, our main tool is a computational experiment, where we use various models and approximations to deepen our understanding of processes that accompany the interaction of extremely short light pulses with condensed matter. There is no single theoretical model that would adequately describe all the relevant processes, and we often learn important lessons by comparing outcomes from different models.

In this approximation, every electron interacts with other electrons via a time-dependent mean- field potential, which depends on the electron density. Even though this approach is limited in its ability to account for electron-electron collisions and correlation, it is currently the most advanced approach that can be used to model highly nonlinear interaction of intense laser pulses with solids. Employing more sophisticated methods is prohibited by the fact that such a pulse typically populates tens of energy bands in the entire Brillouin zone with a final state that requires thousands of crystal momenta to discretize reciprocal space properly. Even though very good agreement between experiment and time-dependent density functional theory was demonstrated, this approach has drawbacks: calculations require significant computational resources (hundreds of core hours), results are sensitive to the choice of an exchange-correlation potential, and accounting for decoherence is not straightforward within this approach.

In its general formulation, this is a many-body approach that is capable of describing electron- electron and electron-phonon interactions from first principles. Such a general description, however, would require the employment of many-particle density matrices, rendering simulations infeasible in the context of strong-field light-matter interaction. As long as the state of a many- electron quantum system is approximated with single-particle density matrices, the capability of this approach to account for correlation effects is rather limited. Nevertheless, semiconductor Bloch equations have an advantage over methods that operate with single-electron wave functions: density matrices make it easy to account for electron-phonon interaction and electron-electron correlation via semi-phenomenological relaxation terms.

In those cases where a laser pulse is so short that relaxation plays no significant role during light-matter interaction, relaxation terms in semiconductor Bloch equations can be dropped. If, at the same time, an external electric field negligibly changes the average Coulomb interaction between electrons, a system of time-dependent Schrödinger equations adequately describes electron dynamics. In this independent-particle approach, electrons do not explicitly interact with each other, except via a macroscopic electromagnetic field that their motion may induce. In spite of this, this method does not neglect the electron-electron interaction entirely. This interaction enters the model via a stationary basis of Bloch states that we obtain from ab initio calculations, which do account for the interaction between electrons. By comparing outcomes of such simulations to those of the time-dependent density functional theory, we often see a very good agreement. The main advantage of working in a stationary basis of Bloch states is the flexibility of this approach: input data (band energies and matrix elements) can be obtained with any DFT code or even with post-DFT methods, such as quasiparticle self-consistent GW. Also, solving a system of time-dependent Schrödinger equations in a stationary basis is much faster than using time-dependent density functional theory or solving semiconductor Bloch equations. For this reason, we use this method often, and we have developed a python code for such calculations. In this code, the most time-consuming functions are parallelized using Numba, and the code is publicly available via GitHub