Unambiguous discrimination of high harmonic generation mechanisms in solids

Abstract

Using real-space view of high harmonic generation (HHG) in solids, we develop a physically transparent and gauge-invariant approach for distinguishing intraband and interband HHG mechanisms. Our approach relies on resolving the harmonic emission according to the separation between Wannier states involved in radiative transitions. We show that the intra- and inter-band HHG emission exhibit striking qualitative differences in their dependence on this separation and can be clearly distinguished using the Wannier basis.

Figures (3)

• Figure 1: (a) In the Wannier basis, the photon energy $\Omega$ of intraband HHG emission is constrained by the product of the peak electric field amplitude $F_0$ and the separation $\Delta R$ between the Wannier states involved in the transition. The range of photon energies related to separations of three and six lattice sites are overlaid. (b) The transition cross-section for interband (circles) rapidly decays with electron-hole separation WQC and is greatest for small separations, whereas the intraband transition cross-section (triangles) is much greater for large separations. (c) Thus, interband emission is dominated by small electron-hole separations, shown by fading opacity of conduction band Wannier functions. (d) These mechanisms result in a distinct structure when HHG spectra are resolved according to separation, wherein intraband (interband) emission occurs in the red (purple) regions. The oblique and vertical lines denote the intraband described by Eq. (\ref{['eq:intrabandDR']}) and the harmonic order of the minimum band gap.
• Figure 2: The lattice site separation-resolved total HHG spectra generated using driving lasers with wavelengths (a) 600 nm, 1600 nm, (c) 2400 nm, and (d) 3200 nm. All spectra are calculated with a dephasing time of 10 fs using the formalism presented in PhysRevB.97.144302, one valence and seven conduction bands. The driving laser is a 5 cycle Gaussian pulse with an intensity set such that all simulations have an equal peak vector potential amplitude, such that the peak intensity for the 3200 nm case is $3.1 \times 10^{11}$ W/cm$^{2}$. In all figures, the vertical green line and oblique white line denote the harmonic order of the minimum band gap and the intraband cutoff for each separation, respectively.
• Figure 3: The (dashed red) interband and (dashed purple) intraband HHG spectra calculated from the length gauge simulation presented in Fig. \ref{['fig:2']} (a) and the (solid blue) interband and (solid green) intraband HHG spectra obtained from the discrimination according to the separation between Wannier states obtained by coherently summing the spectra in regions ($i$) and ($ii$), respectively.