Wavelength dependence of high-order harmonic yields in solids

Abstract

We theoretically investigate the wavelength dependence of high-order harmonic yields in solids driven by a mid-infrared laser field. By solving the three-dimensional two-band density matrix equations in the wavelength range of $2.0--7.5\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$, it is shown that, in the limit of slow dephasing (dephasing time ${T}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$), the high-order harmonic yield from a crystal follows a scaling of ${\ensuremath{\lambda}}^{\ensuremath{-}4}$ for a fixed energy interval. The ${\ensuremath{\lambda}}^{\ensuremath{-}4}$ scaling is attributed to the wave-packet spreading (${\ensuremath{\lambda}}^{\ensuremath{-}3}$) for the overall yield and the energy distribution effect (${\ensuremath{\lambda}}^{\ensuremath{-}1}$) due to the increase of the cutoff. For the crystal with a finite dephasing time ${T}_{2}$, we find that the exponential factor $x$ of ${\ensuremath{\lambda}}^{\ensuremath{-}x}$ increases with a decay of ${T}_{2}$. An apparent and rapid fluctuation on a fine wavelength mesh is also observed in the harmonic yields. The fine-scale oscillation originates from the quantum interference effect, and the corresponding modulation period $\ensuremath{\delta}\ensuremath{\lambda}$ scales as ${\ensuremath{\lambda}}^{\ensuremath{-}1}$.