= Solution
The <Pauli matrices> obey $\sigma_x^2=\sigma_y^2=I$ and $\sigma_x\sigma_y+\sigma_y\sigma_x=0$, so $A=\cos\phi\,\sigma_x+\sin\phi\,\sigma_y$ has $A^2=I$. Its entries are $A_{12}=\cos\phi-i\sin\phi=e^{-i\phi}$ and $A_{21}=e^{i\phi}$, with zero diagonal. Substitute this matrix into the <involution exponential formula>:
$$
\boxed{e^{-i\theta A}=\begin{pmatrix}
\cos\theta&-ie^{-i\phi}\sin\theta\\
-ie^{i\phi}\sin\theta&\cos\theta
\end{pmatrix}.}
$$
The drive phase selects the transverse rotation axis, while the <quantum pulse area> selects its rotation angle.
Back to article page
