Effects of quantum interferences among crystal-momentum-resolved electrons in solid high-order harmonic generation

We theoretically demonstrate the effects of quantum interferences among valence electrons with various initial crystal momenta in the high-order harmonic generation (HHG) in solids. By investigating crystal-momentum-dependent harmonics from solids in linearly polarized laser fields, some unique radiation characteristics of the observed overall harmonics are attributed to the consequences of interferences among valence electrons with different crystal momenta. It is shown that the electron pairs with opposite crystal momenta result in the destructive interferences of even-order harmonics for a crystal with the inversion symmetry, which eventually leads to the only odd orders in observed overall harmonic spectra for a semiconductor material because of the complete pairing of all valence electrons. Additionally, each of the harmonic plateaus in the overall multiple-plateau harmonic spectrum is identified to be contributed by the valence electrons within different crystal momentum zones. We also find that the solid-phase HHG in the below-band-gap region is substantially suppressed due to the collective responses of multiple valence electrons. This work sheds light on the essential impacts of interferences among crystal-momentum-resolved electrons on the HHG in solids and is helpful to explore ultrafast coherent processes in condensed matter.

Figure 1

Band structures of the highest valence band VB ($n=2$) and first three conduction bands CB1, CB2, and CB3 ($n=3$, 4, 5). The lowest valence band VB0 ($n=1$) is not shown. The valence band VB is fully occupied with electrons initially for a band-gap semiconductor.

Figure 2

High harmonic spectra contributed by valence electrons with various crystal momenta: (a) $k=\pm \pi /\left(2a_0\right)$, (b) $\left|k\right|=\pi /\left(2a_0\right)$, and (c) $k\in \left[-\pi /a_0,\pi /a_0\right]$.

Figure 3

High harmonic spectra contributed by the valence electrons at high-symmetry points: (a) $\mathrm{\Gamma }$ point $k=0$, (b) Bragg plane $k=\pm \pi /a_0$.

Figure 4

(a) $k$-space semiclassical picture for the electron pair with arbitrary crystal momenta $\left|k_0\right|$. (b) Harmonic spectrum contributed by the electron pair with $\left|k\right|=\pi /\left(2a_0\right)$. (c) Harmonic spectra from the electron pair with $\left|k\right|=\pi /\left(2a_0\right)$ for the vector potentials $A_0=0.25$ a.u., 0.26 a.u., and 0.27 a.u. The relation of $A_0+\left|k\right|=\pi /a_0$ exactly holds for $A_0=0.26$ a.u. in plane (c).

Figure 5

(a) The sectionalization of crystal momenta for the electrons in VB. ${\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2$, and ${\mathrm{\Omega }}_3$ correspond to the crystal momentum intervals of $\left[-\pi /\left(4a_0\right),\pi /\left(4a_0\right)\right], \left[-\pi /\left(2a_0\right),\pi /\left(2a_0\right)\right],$ and $\left[-3\pi /\left(4a_0\right),3\pi /\left(4a_0\right)\right]$, respectively. (b) Harmonic spectra contributed by the electrons at or within only $\mathrm{\Gamma }$ point, ${\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2, {\mathrm{\Omega }}_3$ and the whole BZ. (c) Harmonic spectra from a single electron at $\mathrm{\Gamma }$ point and the electron pairs with $\left|k\right|=\pi /\left(4a_0\right), \pi /\left(2a_0\right)$, and $3\pi /\left(4a_0\right)$. The laser intensity is $4\times {10}^{11}$ W/${\mathrm{cm}}^2$ in panels (b) and (c).

Figure 6

(a) Harmonic spectra contributed by the electrons at or within only the $\mathrm{\Gamma }$ point, ${\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2, {\mathrm{\Omega }}_3$ and the whole BZ. (b) Harmonic spectra from a single electron at $\mathrm{\Gamma }$ point and the electron pairs with $\left|k\right|=\pi /\left(4a_0\right), \pi /\left(2a_0\right)$, and $3\pi /\left(4a_0\right)$. The laser intensity is $6\times {10}^{11}$ W/${\mathrm{cm}}^2$.

Figure 7

(a) Harmonic spectra contributed by valence electrons at or within the only $\mathrm{\Gamma }$ point and the whole BZ. (b) Harmonic yield $S$ as the function of $\left|k\right|$ for third, fifth, seventh, and the first plateau harmonics. (c) Harmonic intensity ratio $\eta$ as the function of $\left|k\right|$ for the third-, fifth-, and seventh-order harmonics. $\eta$ represents the intensity ratio between $n\mathrm{th}$-order harmonic and the first harmonic plateau. In plane (a), the vertical gray dashed line indicates the positions of minimum band gap between CB1 and VB, i.e., $E_g$. In plane (b), the horizontal gray dashed line indicates the value of $S_{\mathrm{plateau}}$ when $\left|k\right|=1$.