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 (St\"uckelberg 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 Schr\"odinger 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 $\mathrm{\ensuremath{\Omega}}$, with a period determined by the St\"uckelberg phase, (2) the cutoff order of HHG is determined by $2e{E}_{0}/\ensuremath{\hbar}{\mathrm{\ensuremath{\Omega}}}^{2}$, with $e$ being the electron charge, and that (3) noninteger harmonics, controlled by the St\"uckelberg phase, appear as a transient effect. Our WKB theory is particularly suited for a parameter regime, where the Keldysh parameter $\ensuremath{\gamma}=(\mathrm{\ensuremath{\Delta}}/2)\mathrm{\ensuremath{\Omega}}/e{E}_{0}$, with $\mathrm{\ensuremath{\Delta}}$ 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.