On the Theory of Solid-State Harmonic Generation Governed by Crystal Symmetry

The solid-state harmonic generation (SHG) derives from photocurrent coherence. The crystal symmetry, including spatial-inversion, mirror, rotational symmetries and time-reversal symmetry, constrains the amplitude and phase of the photocurrent, thus manipulates the coherent processes in SHG. We revisit the expression of photocurrent under the electric dipole approximation and give a picture of non-equilibrium dynamical process of photoelectron on laser-dressed effective bands. We reveal the indispensable role of shift vector and transition dipole phase in the photocurrent coherence in addition to the dynamical phase. Microscopic mechanism of the selection rule, orientation dependence, polarization characteristics, time-frequency analysis and ellipticity dependence of harmonics governed by crystal symmetry is clarified analytically and numerically. The theory in this paper integrates non-equilibrium electronic dynamics of condensed matter in strong laser fields, and paves a way to explore more nonlinear optical phenomena induced by the crystal symmetry.