I am curious about how different rotations on the Bloch sphere are done in practice. For example, assuming we start in the lower energy state of the $z$-axis (call it $|0\rangle$), a resonant rotation on the Bloch sphere by around the $x$-axis will take you to $\frac{|0\rangle-i|1\rangle}{\sqrt{2}}$ (where $|0\rangle$ is the excited state in the $z$ direction). If we do the same thing around the $y$-axis we end up with $\frac{|0\rangle-|1\rangle}{\sqrt{2}}$. This phase difference matters in practice in various scenarios (e.g. when doing a spin echo). But how do you change the rotation axis in practice i.e. experimentally (e.g. using a microwave pulse)? The field applied in the lab frame is $E\cos{(\omega t + \phi)}$. You can make $\omega$ resonant and $E$ such that you get a $\pi/2$ pulse for the right time, but if you solve the Schrödinger equation in the rotating wave approximation, the $\phi$ actually cancels in the final formula, so I am not sure what other degrees of freedom one has in order to achieve this.
EDIT: For the 2 level Hamiltonian:
$$\begin{pmatrix} 0 & \Omega\cos{(\omega t + \phi)} \\ \Omega\cos{(\omega t + \phi)} & \omega_0 \end{pmatrix},$$ in RWA, the equations for the coefficients of the 2 level system are:
$$i\dot{c_1}=c_2e^{i(\omega-\omega_0)t+i\phi}\frac{\Omega}{2}$$ $$i\dot{c_2}=c_1e^{-i(\omega-\omega_0)t-i\phi}\frac{\Omega}{2}$$
From which one gets, for example:
$$\frac{d^2c_2}{dt^2}+i(\omega-\omega_0)\frac{dc_2}{dt}+\frac{\Omega^2}{4}c_2 = 0$$
and hence no dependence on $\phi$
