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.