Multichannel High-Order Harmonic Generation from Fractal Bands in Fibonacci Quasicrystals

High-order harmonic generation (HHG) in solids was expected to be efficient due to their high density. However, the strict transition laws in crystals restrict the number of HHG channels. Quasicrystals with fractal band structures could solve this problem and produce multichannel HHG emissions, which has been rarely studied. We simulate the Fibonacci quasicrystal (FQ) HHG for the first time and investigate the electron dynamics on the attosecond timescale. Our results reveal that (i) the acceleration theorem is approximately applicable in FQ, which provides us a valuable tool to analyze the electron trajectories. (ii) Fractal bands lead to more excitation channels, analogous to the forbidden nonvertical electron transitions in crystals. (iii) The broken symmetry results in more frequent backscattering of electrons. (iv) Compared with crystals, multichannel HHG in FQ has a higher yield and wider spectral range. Our results pave the way to understand and control the HHG in quasicrystals and shed light on a potential shorter and stronger attosecond light source based on compact solids.

Figure 1

(a) Schematic diagram of HDPM to obtain a 1D quasicrystal. $\theta$ is the angle between the vectors $x_{\parallel }$ and $\mathbf{X}$, controlling the sequence of atoms. (b) Schematic diagram of the 1D atomic chain based on HDPM. (c) Energy spectrum of FQ ($m=1$, black circles) and the crystal ($m=100$, magenta circles). The dashed gray line separates the valence band (VB) and conduction bands. (d) Modulus of the 150th eigenstate in FQ and the crystal.

Figure 2

Numerical results of HHG in FQ ($m=1$) and crystal ($m=100$). The wavelength $\lambda$ is 0.5 and $1\mu \mathrm{m}$ in (a) and (b), respectively. In (b), the black arrows are the cutoff positions predicted by our theory. Other laser parameters are shown in the text.

Figure 3

Band structures of (a) FQ ($m=1$) and (b) crystal ($m=100$) in $k$ space. (c) EC band (solid black line, $q=0$) and (d) crystal band are calculated by PWEM. In (c), we translate the EC band by Eq. (3) with $p\in [-40,40]$ and $q=\pm 1,\pm 2,\pm 3$ to reconstruct the FQ band.

Figure 4

(a) and (b) Schematic diagrams of electron excitation channels and backscattering of FQ and crystals in $k$ space, respectively. The bands come from Figs. 3 and 3. The energy-resolved TDEP is displayed for (c) $\lambda =0.5\mu \mathrm{m}$ and (e) $\lambda =1\mu \mathrm{m}$, respectively, taking the highest valance state as the initial state. (d) The band is the same as it is in Fig. 3, and the brown dotted lines mark the vertical transition of the highest valence electron. (f) EC band and its band gap between all bands and VB are drawn in solid black and red lines, respectively.