We study the generation of high harmonic radiation by Bloch electrons in a
model transparent solid driven by a strong mid-infrared laser field. We solve
the single-electron time-dependent Schr\"odinger equation (TDSE) using a
velocity-gauge method [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 [Phys. Rev. B 33, 5494
(1986)], which allows us to separate inter-band and intra-band contributions to
the time-dependent current. We find that the inter-band and intra-band
contributions display very different time-frequency characteristics. We show
that solutions in these two bases are equivalent under an 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
