Analytical WKB theory for high-harmonic generation and its application to massive Dirac electrons

We propose an analytical approach to high-harmonic generation (HHG) for nonperturbative low-frequency and high-intensity fields based on the (Jeffreys-)Wentzel-Kramers-Brillouin (WKB) approximation. By properly taking into account Stokes phenomena of WKB solutions, we obtain wave functions that systematically include the repetitive dynamics of production and acceleration of electron-hole pairs and quantum interference due to phase accumulation between different pair production times (Stückelberg phase). Using the obtained wave functions without relying on any phenomenological assumptions, we explicitly compute electric current (including intra- and interband contributions) as the source of HHG for a massive Dirac system in $(1+1)$ dimensions under an ac electric field. We demonstrate that the WKB approximation agrees well with numerical results obtained by solving the time-dependent Schrödinger equation and point out that the quantum interference is important in HHG. We also predict in the deep nonperturbative regime that (1) harmonic intensities oscillate with respect to electric-field amplitude $E_0$ and frequency $\mathrm{\Omega }$, with a period determined by the Stückelberg phase, (2) the cutoff order of HHG is determined by $2e{E}_0/\hslash {\mathrm{\Omega }}^2$, with $e$ being the electron charge, and that (3) noninteger harmonics, controlled by the Stückelberg phase, appear as a transient effect. Our WKB theory is particularly suited for a parameter regime, where the Keldysh parameter $\gamma =(\mathrm{\Delta }/2)\mathrm{\Omega }/e{E}_0$, with $\mathrm{\Delta }$ being the gap size, is small. This parameter regime corresponds to intense lasers in the terahertz regime for realistic massive Dirac materials. Our analysis implies that the so-called HHG plateau can be observed at the terahertz frequency within the current technology.

Figure 1

(a) A typical Stokes graph, composed of Stokes lines (blue lines) and turning points (red points), and (b) the corresponding physical processes during the real-time evolution.

Figure 2

HHG spectrum for the oscillating field (10), with $\mathrm{\Omega }/(\mathrm{\Delta }/2)=1/4$, $e{E}_0/{(\mathrm{\Delta }/2)}^2=1$ (i.e., $\gamma =1/4$), and $\mathrm{\Omega }{t}_{\mathrm{in}}=-17\pi /3$, $T_w=2\left|t_{\mathrm{in}}\right|$. The parameter set corresponds to, e.g., $\mathrm{\Omega }/2\pi =1\mathrm{THz}$ and $E_0=4.2\mathrm{kV}/\mathrm{cm}$ for a Dirac material with Fermi velocity $v_{\mathrm{F}}={10}^6\mathrm{m}/\mathrm{s}$ and mass $\mathrm{\Delta }=33\mathrm{meV}$.

Figure 3

The magnitude of $N=3$ (left), 9 (middle), and 15 (right) harmonic peaks plotted against amplitude $e{E}_0$ with fixed frequency $\mathrm{\Omega }/(\mathrm{\Delta }/2)=1/4$ (a) and inverse frequency ${\mathrm{\Omega }}^{-1}$ with fixed amplitude $e{E}_0/{(\mathrm{\Delta }/2)}^2=1$ (b). The other parameters are chosen as $\mathrm{\Omega }{t}_{\mathrm{in}}=-17\pi /3$, $T_w=2\left|t_{\mathrm{in}}\right|$.