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 wavefunctions that systematically include repetitive dynamics production acceleration electron-hole pairs quantum interference due phase accumulation between different pair times (St\"{u}ckelberg phase). Using obtained without relying any phenomenological assumptions, explicitly compute electric current (including intra- inter-band contributions) as source HHG a massive Dirac system in (1+1)-dimensions under ac field. demonstrate approximation agrees well with numerical results by solving time-dependent Schr\"{o}dinger equation point out is important HHG. also predict deep regime (1) harmonic intensities oscillate respect electric-field amplitude $E_0$ frequency $\Omega$, period determined St\"{u}ckelberg phase; (2) cutoff order $2eE_0/\hbar \Omega^2$, $e$ being electron charge; (3) non-integer harmonics, controlled phase, appear transient effect. Our theory particularly suited parameter regime, where Keldysh $\gamma=(\Delta/2)\Omega/eE_0$, $\Delta$ gap size, small. This corresponds intense lasers terahertz realistic materials. analysis implies so-called plateau can be observed at within technology.