= Rotating half-wave plate frequency shift
{title2=$|\Delta\omega|=2|\Omega|$}
A rotating ideal <half-wave plate> flips circular <polarization>. The spin change has magnitude $2\hbar$ per <photon>, so its mechanical <energy> exchange produces a <frequency> shift of magnitude $2|\Omega|$. For Jones convention $H(\theta)=R(\theta)\operatorname{diag}(1,-1)R(-\theta)$ and $Q=\operatorname{diag}(1,i)$, the combination $QH(\theta)Q=e^{i2\theta\sigma_x}$. Acting on $(1,1)^T/\sqrt2$ 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.
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