High-harmonic generation from Bloch electrons in solids

We study the generation of high-harmonic radiation by Bloch electrons in a model transparent solid driven by a strong midinfrared laser field. We solve the single-electron time-dependent Schr\"odinger equation (TDSE) using a velocity-gauge method [M. Korbman et al., New J. Phys. 15, 013006 (2013)] that is numerically stable as the laser intensity and number of energy bands are increased. The resulting harmonic spectrum exhibits a primary plateau due to the coupling of the valence band to the first conduction band, with a cutoff energy that scales linearly with field strength and laser wavelength. We also find a weaker second plateau due to coupling to higher-lying conduction bands, with a cutoff that is also approximately linear in the field strength. To facilitate the analysis of the time-frequency characteristics of the emitted harmonics, we also solve the TDSE in a time-dependent basis set, the Houston states [J. B. Krieger and G. J. Iafrate, Phys. Rev. B 33, 5494 (1986)], which allows us to separate interband and intraband contributions to the time-dependent current. We find that the interband and intraband contributions display very different time-frequency characteristics. We show that solutions in these two bases are equivalent under a unitary transformation but that, unlike the velocity-gauge method, the Houston state treatment is numerically unstable when more than a few low-lying energy bands are used.