Fig. 1: Modulated Floquet parametric driving, exceptional lines and non-equilibrium steady states.
a Illustration of the quasi-energy bands of electrons in a gapped system under MFPD. The dashed gray lines represent the undriven band structure. A Floquet signal with the frequency ΩF, where ℏΩF > Eg, is applied. The quasi-energy bands in the vicinity of the Fermi surface for a non-zero driving amplitude are shown in red. By modulating the amplitude of the fast Floquet drive with a frequency 2ω1, an oscillation of the dispersion and the effective mass of the electrons at the Fermi level is induced (see Eq. (6)). b The periodic modulation of the Fermi velocity induces two lines of exceptional points (shown as red circles) in the rotationally symmetric dispersion of the soft THz plasmon mode (green surfaces)—see Fig. 2 for an in-detail cross-sectional view. c) Above a critical value, the mass oscillation parametrically excites THz plasmons. Due to nonlinear and interaction effects, the plasmons arrange themselves in a crystal-like structure whose periodicity is determined by the condition \({\omega }_{1}={\omega }_{{{{{\rm{pl}}}}}}\left({q}^{*}\right)\), where \({\omega }_{{{{{\rm{pl}}}}}}\left(q\right)\) is the plasmon dispersion. The crystalline state breaks the rotational and translational symmetries of the undriven system and supports soft phonon-like Goldstone modes.
