Tunable light sources with sub-femtosecond to picosecond laser pulses offer intriguing opportunities in ultrafast materials science. Contrary to femtochemistry, where chemical processes are watched in real time, the difference is that in solids photodressing changes not only the state of the system, but even affects its effective Hamiltonian. This is due to the much smaller energy scales that govern collective modes in solids vis-a-vis large binding energies in small molecules.

Here we show how this photodressing tool to engineer Hamiltonians can be monitored.

In this paper we demonstrate how the photo-dressing of electrons in a solid leads to a rich non-equilibrium bandstructure and how the formation of quasiparticle bands of the combined photon-electron states can be directly observed in td-ARPES measurements.

We simulate the pump-probe photoemission process and angle reosolved measurement, as depicted in Fig. 1a, where the pump is a circularly polarized monochromatic pulse that is long enough to drive the electronic structure into a non-equilibrium but stationary state. The probe is typically much shorter but with a higher energy, large enough to ionize electrons from the sample (HH: here a bit more on how photoemission gives the electronic structure). The atomic structure of monolayer WSe\({}_{2}\), which crystalises in with hexagonal symmetry, is depicted in Fig. 1b, along with the Brillouin zone and the path across the \(K\) point that we are considering here. The observed photoelectron spectrum depends strongly on the overlap of the pump and probe pulses as shown in Fig. 1d. When the two pulses do not overlap one only measures the equilibrium bandstructure of WSe\({}_{2}\), but in case of overlapping pulses extra features occur in the spectrum. The non-equilibrium state of the driven monolayer consists of quasiparticles that are a combination of electrons from the material and photons from the driving field, so called photon-dressed electronic states. This dressing of the electronic bands leads to replicas of the equilibrium bands shifted by the photon energy. In case this energy is in resonance with the band gap these bands can hybridize with other equilibrium bands, as schematically shown in Fig. 1c. These dressed bands are directly obeservable in the ARPES spectra of Fig. 1d. When the overlap between pump and probe pulse, we can observe how the dressed bands collapse to the equilibrium bands. This means it is possible to directly observe the creating and destruction of the photon-electron quasiparticle by tuning the pump-prpobe overlap: the real-time dressing and undressing of an electron.

The observed quasiparticle structure can be understood in terms of Floquet theory, where a stationary state is expanded into a basis of Fourier components of multiples of the photon frequency \(\Omega\): \(|\psi(t)\rangle=\sum_{m}\exp(-i(\epsilon+m\Omega)t)|u_{m}\rangle\), where \(\epsilon\) is the equilibrium energy of the state. With this ansatz the time-dependent Schrödinger equation becomes an eigenvalue problem \(\sum_{n}\mathcal{H}^{mn}|u_{n}\rangle=\epsilon|u_{m}\rangle\) of the static Floquet Hamiltonian \(\mathcal{H}^{mn}=\frac{\Omega}{2\pi}\int_{2\pi/\Omega}dte^{i(m-n)\Omega t}H(t)+\delta_{mn}m\Omega\mathbf{1}\). The eigenstates of this Hamiltonian span a Hilbert space with the dimension of the original electronic Hilbert space times the multiphoton dimension. The contribution of the latter is in principle infinite, but here can be truncated. The spectrum of this Hamiltonian gives the bandstructure of the dressed quasiparticles. (HH: here more on Flqouet-TDDFT). In Fig. 1d are shown the Floquet bands corresponding to the pump paramters used in the ARPES simulation and the dressed bands observed in the APRES calculation are perfectly reproduced. Thus the dressed states can be interpreted as Floquet-sidebands and the process of measurement in ARPES as an effective averaging of the periodic system. (HH: elaborate this point a bit).

The striking agreement between Floquet theory and ARPES calculation allows to further explore the parameter space of the probe pulse. Fig. 2a shows the dressed bands for the same pump energy but with different amplitudes than in Fig. 1d. The splitting of the bands due to the hybridization can be seen to strongly depend on the strength of the pump. Monolayer WSe\({}_{2}\) is known to have circular pump dichroism, originating from a well defined chiral character of electronic orbitals at the K and K’ points in the Brillouin zone. This pump dichroism also affects the hybridisation of the dressed bands, as shown in Fig. 2b, where only at one of the inequivalent K points the electronic part of the dressed states has the right chirality to interact with the circularly polarized photons in a non-trivial way (HH: this needs to be more precise/better explained). In Fig. 1d and 2a-2b we show results where the energy of the pump is at resonance with the bandgap of the monolayer WSe\({}_{2}\) so that the photon dressing leads to hybridization of the quasiparticle states. The occurance of dressed bands in the phtoto-electron spectrum is, however, not limited to the resonant case. In Fig. 2c we show the effect of off-resonant driving on the electronic quasiparticle structure and observe the same feature of additional bands shifted by the driving frequency.

Now, using results from Floquet theory we can systematically elucidate this behaviour.