Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 2 / b / Solution

The Pauli matrices obey ${\sigma }_x^2={\sigma }_y^2=I$ and ${\sigma }_x{\sigma }_y+{\sigma }_y{\sigma }_x=0$, so $A=\cos \varphi {\sigma }_x+\sin \varphi {\sigma }_y$ has $A^2=I$. Its entries are $A_{12}=\cos \varphi -i\sin \varphi =e^{-i\varphi }$ and $A_{21}=e^{i\varphi }$, with zero diagonal. Substitute this matrix into the involution exponential formula:The drive phase selects the transverse rotation axis, while the quantum pulse area selects its rotation angle.