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We propose an analytical approach to high-harmonic generation (HHG) fornonperturbative low-frequency and high-intensity fields based on the(Jeffreys-)Wentzel-Kramers-Brillouin (WKB) approximation. By properly takinginto account Stokes phenomena of WKB solutions, we obtain wavefunctions thatsystematically include the repetitive dynamics of production and accelerationof electron-hole pairs and quantum interference due to phase accumulationbetween different pair production times (St\"{u}ckelberg phase). Using theobtained wavefunctions without relying on any phenomenological assumptions, weexplicitly compute electric current (including intra- and inter-bandcontributions) as the source of HHG for a massive Dirac system in(1+1)-dimensions under an ac electric field. We demonstrate that the WKBapproximation agrees well with numerical results obtained by solving thetime-dependent Schr\"{o}dinger equation and point out that the quantuminterference is important in HHG. We also predict in the deep nonperturbativeregime that (1) harmonic intensities oscillate with respect to electric-fieldamplitude $E_0$ and frequency $\Omega$, with a period determined by theSt\"{u}ckelberg phase; (2) the cutoff order of HHG is determined by$2eE_0/\hbar \Omega^2$, with $e$ being the electron charge; and that (3)non-integer harmonics, controlled by the St\"{u}ckelberg phase, appear as atransient effect. Our WKB theory is particularly suited for a parameter regime,where the Keldysh parameter $\gamma=(\Delta/2)\Omega/eE_0$, with $\Delta$ beingthe gap size, is small. This parameter regime corresponds to intense lasers inthe terahertz regime for realistic massive Dirac materials. Our analysisimplies that the so-called HHG plateau can be observed at the terahertzfrequency within the current technology.

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