Fig. 2: HHG selection rules in a system with broken Ĉ2 symmetry.
From: Selection rules in symmetry-broken systems by symmetries in synthetic dimensions
a For \(\lambda =0\) (top row), the driving field exhibits \({\hat{C}}_{2}\) DS (top row). For \(\lambda \,\ne\, 0\) (bottom row), the Lissajous curve of the driving field is invariant under \({\hat{C}}_{2}\cdot \hat{\zeta }\) where \(\hat{\zeta }\left(\lambda \right)=-\lambda\) b HHG spectrum for \(\lambda =0\) (blue, even harmonic forbidden) and \(\lambda =0.2\) (red, all harmonic orders allowed). The phase of each Lissajous curve is illustrated by a color gradient that transforms under temporal translations. When an \(\hat{x}\)-polarized field of frequency \(4\omega\) and relative amplitude \(\lambda\) perturbs a Ne atom driven by a \({\hat{C}}_{2}\) symmetric field, the DS is reduced to real-synthetic symmetry, which coerces c \(2q\) harmonics of to scale oddly with \(\lambda\) whereas d \(2q+1\) harmonics scale evenly with \(\lambda\). Even (odd) harmonic amplitudes are fit to a linear (quadratic) model in the range \(\left|\lambda \right|\le 0.2\) (green shaded), and the average \({R}^{2}\) value is presented in the top right of each figure. Individual \({R}^{2}\) values are presented above each subfigure. For \(\left|\lambda \right| \, > \, 0.2\), cubic/quadratic contributions deform the lowest order linear/quadratic scaling.
