High harmonic generation from Bloch electrons in solids

TL;DR

This paper investigates high harmonic generation in solids driven by strong laser fields, analyzing the spectral features and time-frequency characteristics using different computational methods.

Contribution

It introduces a stable velocity-gauge method for modeling harmonic generation in solids and compares it with Houston state basis analysis, revealing differences in numerical stability.

Findings

Harmonic spectrum shows a primary plateau with a cutoff linearly scaling with field strength.

A secondary weaker plateau arises from higher conduction bands with similar linear cutoff scaling.

Inter-band and intra-band contributions have distinct time-frequency signatures.

Abstract

We study the generation of high harmonic radiation by Bloch electrons in a model transparent solid driven by a strong mid-infrared laser field. We solve the single-electron time-dependent Schr\"odinger equation (TDSE) using a velocity-gauge method [New J. Phys. 15, 013006 (2013)] that is numerically stable as the laser intensity and number of energy bands are increased. The resulting harmonic spectrum exhibits a primary plateau due to the coupling of the valence band to the first conduction band, with a cutoff energy that scales linearly with field strength and laser wavelength. We also find a weaker second plateau due to coupling to higher-lying conduction bands, with a cutoff that is also approximately linear in the field strength. To facilitate the analysis of the time-frequency characteristics of the emitted harmonics, we also solve the TDSE in a time-dependent basis set, the Houston states [Phys. Rev. B 33, 5494 (1986)], which allows us to separate inter-band and intra-band contributions to the time-dependent current. We find that the inter-band and intra-band contributions display very different time-frequency characteristics. We show that solutions in these two bases are equivalent under an unitary transformation but that, unlike the velocity gauge method, the Houston state treatment is numerically unstable when more than a few low lying energy bands are used.