= Pulse-and-precession spin one-half map
{title2=$U_F=e^{ia\sigma_z}e^{-ib\sigma_x}$}
Multiplying the two <matrix exponentials> with the <Pauli matrix multiplication law> gives
$$
U_F=CI-i\mathbf v\cdot\boldsymbol\sigma,\qquad C=\cos a\cos b,\qquad
\mathbf v=(\cos a\sin b,-\sin a\sin b,-\sin a\cos b).
$$
One has $C^2+|\mathbf v|^2=1$. If $|\mathbf v|\ne0$, put $\theta=2\arccos C$ and $\mathbf n=\mathbf v/|\mathbf v|$. The <exponential of a Pauli vector> gives $U_F^N=I\cos(N\theta/2)-i\mathbf n\cdot\boldsymbol\sigma\sin(N\theta/2)$. Its action on a <Bloch vector> is a rotation through $N\theta$ about $\mathbf n$, and its <quasienergies> are $\pm\hbar\theta/(2T)$ modulo $2\pi\hbar/T$. When $U_F=\pm I$, its axis is arbitrary and the <Bloch vector> is unchanged.
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