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

TL;DR

This paper develops an analytical WKB-based theory for high-harmonic generation in massive Dirac systems under intense low-frequency fields, revealing quantum interference effects, oscillatory harmonic intensities, and a terahertz regime HHG plateau.

Contribution

It introduces a systematic WKB approach incorporating Stokes phenomena to analyze HHG in massive Dirac electrons without phenomenological assumptions.

Findings

WKB approximation matches numerical solutions of Schrödinger equation.

Harmonic intensities oscillate with electric field parameters.

Non-integer harmonics emerge transiently due to St"{u}ckelberg phase.

Abstract

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 the repetitive dynamics of production and acceleration of electron-hole pairs and quantum interference due to phase accumulation between different pair production times (St\"{u}ckelberg phase). Using the obtained wavefunctions without relying on any phenomenological assumptions, we explicitly compute electric current (including intra- and inter-band 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\"{o}dinger 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 $\Omega$, with a period determined by the St\"{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 a transient effect. Our WKB theory is particularly suited for a parameter regime, where the Keldysh parameter $\gamma=(\Delta/2)\Omega/eE_0$, with $\Delta$ 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.