 Rotating half-wave plate frequency shift

A rotating ideal half-wave plate flips circular polarization. The spin change has magnitude $2\hslash$ per photon, so its mechanical energy exchange produces a frequency shift of magnitude $2|\Omega |$. For Jones convention $H(\theta )=R(\theta )\mathrm{diag}(1,-1)R(-\theta )$ and $Q=\mathrm{diag}(1,i)$, the combination $QH(\theta )Q=e^{i2\theta {\sigma }_x}$. Acting on $(1,1{)}^T/\sqrt{2}$ gives $e^{i2\theta }$; with time dependence $e^{-i\omega t}$ and $\theta =\Omega t$, the output frequency is $\omega -2\Omega$. The opposite input eigenpolarization reverses the shift.