Solid high-order harmonic generation: Review of the semiconductor Bloch equations

High-order harmonic generation (HHG) is an important nonperturbative and nonlinear optical phenomenon in the interaction between intense lasers and matter. In addition to opening up new ways of generating extreme ultraviolet light sources, it has also been used to synthesise attosecond pulses, thus opening the door to attosecond physics. Solids have higher atomic density than gases, leading to more intricate mechanisms for HHG. The semiconductor Bloch equations (SBEs) are an essential theoretical approach to study solid HHG, playing an important role in revealing the physical mechanisms, interpreting and predicting experimental phenomena. This work reviews the theoretical framework and recent progress of the SBEs. First, we summarize the SBEs in the Bloch basis and the adiabatic Houston basis, and analyze their respective advantages and limitations in numerical calculations. To address the random phase problem of wave functions, the parallel and twisted parallel transport gauges are also introduced. Finally, we present two sets of SBEs, one in gauge-invariant form and the other in the Wannier gauge. This not only perfectly solves the random phase problem but also offers a new perspective for studying the mechanism of HHG.