Trajectory analysis of high-order-harmonic generation from periodic crystals

We theoretically study high-order-harmonic generation (HHG) from solids driven by intense laser pulses using a one-dimensional model periodic crystal. By numerically solving the time-dependent Schr\"odinger equation directly on a real-space grid, we successfully reproduce experimentally observed unique features of solid-state HHG such as the linear cutoff-energy scaling and the sudden transition from a single- to multiple-plateau structure. Based on the simulation results, we propose a simple model that incorporates vector-potential-induced intraband displacement, interband tunneling, and recombination with the valence-band hole. One key parameter is the peak-to-valley amplitude of the pulse vector potential, which determines the crystal momentum displacement during the half cycle. When the maximum peak-to-valley amplitude ${A}_{\mathrm{peak}}$ reaches the half width $\frac{\ensuremath{\pi}}{a}$ of the Brillouin zone with $a$ being the lattice constant, the HHG spectrum exhibits a transition from a single- to multiple-plateau structure, and even further plateaus appear at ${A}_{\mathrm{peak}}=\frac{2\ensuremath{\pi}}{a},\frac{3\ensuremath{\pi}}{a},\ensuremath{\cdots}$. The multiple cutoff positions are given as functions of ${A}_{\mathrm{peak}}$ and the second maximum ${A}_{\mathrm{peak}}^{\ensuremath{'}}$, in terms of the energy difference between different bands. Using our recipe, one can draw electron trajectories in the momentum space, from which one can deduce, for example, the time-frequency structure of HHG without elaborate quantum-mechanical calculations. Finally, we reveal that the cutoff positions depend on not only the intensity and wavelength of the pulse, but also its duration, in marked contrast to the gas-phase case. Our model can be viewed as a solid-state and momentum-space counterpart of the familiar three-step model, highly successful for gas-phase HHG, and provide a unified basis to understand HHG from solid-state materials and gaseous media.