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 Schrodinger 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(peak) reaches the half width 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'(peak) = 2 pi/a,3 pi/a,.... The multiple cutoff positions are given as functions of A(peak) and the second maximum A(peak), 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.