Multichannel high-order harmonic generation from solids

We studied the ultrafast dynamics of high-order harmonic generation (HHG) from solids numerically. It is found that a superposition of Bloch oscillation in the same band and Zenner tunneling to its neighboring conduction band (i.e., Bloch-Zener oscillation effect) play significant roles in HHG when the Bloch electrons cross the boundary of the first Brillouin zone. It increases the number of the harmonic emission channels. These multichannel signals extend the cutoff energy of the plateau in the HHG spectra and enhance both the intra- and interband contributions. The interference of different channels makes the structure of the HHG spectra complex. The multichannel dynamics in the monochromatic and two-color laser fields are demonstrated in a periodic potential model and single-crystal MgO, respectively. It provides an alternative way to control the ultrafast electron dynamics and HHG emission processes in solids.

Figure 1

Schematic of Bloch-Zener oscillations. (a),(b) Two situations in which electrons are restricted in the first BZ and extended to the second BZ, respectively. The electron motion in the same band shows the Bloch oscillations (solid arrows), while the tunnel ionization between different bands illustrates the Zener tunneling (dashed arrows) near the edge of the first BZ. The shadow zone denotes the first BZ. The electron dynamics beyond the boundary are illustrated in the extended zone, which is shown as blank in (b). The same dynamics illustrated in the reduced zone scheme are shown in the opposite side of the shadow zone.

Figure 2

(a) The top valence and first two conduction bands. (b) HHG spectra where the electrons do not go across the boundary of the first BZ. (c) HHG spectra where the Bloch electrons cross the edge of the first BZ. $△Z_1$ and $△Z_2$ show the relative intensities between inter- and intraband contributions. Two vertical gray dash-dotted lines indicate the maximum contribution of HHG from each conduction band. $△E_{\mathrm{cutoff}}$ denotes the extension of the cutoff energy in (c).

Figure 3

(a) Time-frequency analysis of the HHG spectrum in Fig. 2. (b) Time-frequency analysis of the HHG spectra in Fig. 2. They are displayed on a logarithmic scale. Violet lines in (a) and (b) show the quasiclassical analysis of the HHG. (c) Trajectory of the electron which is confined to the first BZ (black thick line) and the corresponding electric field of the laser (red thin line) used in (b). The Bloch electron equivalently enters into the first BZ from the opposite side when it goes across the boundary because the quasimomentum $k^{\prime }$ of the Bloch electron satisfies $k^{\prime }=k+m\frac{2\pi }{a}$, where $m$ is an integer.

Figure 4

(a) Harmonic spectrum under the two-color fields. (b) Time-frequency analysis of the HHG (color graph on a logarithmic scale) and quasiclassical predictions of the harmonic emission (violet lines). (c) Energy bands including the multichannel dynamics. (d) Quasiclassical trajectory which is reduced to the first BZ (black thick line) and the temporal profile of two-color fields (red thin line). A circle in (d) marks the multichannel dynamics in (c).

Figure 5

Harmonic spectra in the MgO crystal which are calculated by SBEs. (a) Harmonic spectra where the electrons do not go across the boundary of the first BZ. (b) Time-frequency analysis of the HHG in (a) (logarithmic scale) and quasiclassical predictions of the harmonic emission (violet line). (c) Harmonic spectra where the electrons go across the boundary of the first BZ. (d) Time-frequency analysis of the HHG in (c) and quasiclassical predictions of the harmonic emission. One gray dashed line and two orange dashed lines (from left to right) in (a) and (c) show the positions of the minimal gap and the cutoffs, respectively. $\mathrm{\Delta }{E}_{\mathrm{cutoff}}$ denotes the extension of the cutoff energy in (c). The intensities and wavelength of the laser pulses are 4.64 TW//${\mathrm{cm}}^2$ in (a), 11.89 TW/${\mathrm{cm}}^2$ in (c), and 1.6 $\mu \mathrm{m}$, respectively. The dephasing time $T_2=10.68$ fs (equal to two cycles).