# Rotations on the Bloch sphere

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 $$\phi$$

• It really depends on the physical system you are considering. For example, for the system you described, I believe the resting Hamiltonian will apply a phase rotation around $Z$ axis.
– Ali
Commented Mar 20 at 5:42
• What makes you think that $\phi$ drops out? (it doesn't - it's precisely the control knob you're after.) Commented Mar 20 at 6:50
• I tried to do the derivation using RWA (not perturbatively) and I don't end up with a phi in the final formula. I am not sure how to put the derivation here, but basically, when you take derivatives with respect to time, phi disappears, except from the exponentials. But in the end all the exponentials cancel out when you write down the second order ODE for one of the coefficients of the 2 level system. Commented Mar 20 at 13:20
• @EmilioPisanty I managed to add what I meant as an edit to the original post. Commented Mar 20 at 13:48
• Firstly, you have a clear typo - there is no way that it's $c_2$ in the RHS on both equations. Commented Mar 22 at 3:10

If your dipole is in an external magnetic field, when you apply a $$\pi/n$$ radio pulse, it will spiral down in the Bloch sphere's lab frame because of precession. It would continue to rotate at the elevation angle the pulse left it at if it weren't for relaxation mechanisms. So, it seems irrelevant to specify or label the directions you mention. Many images of Bloch spheres lack these labels probably for this reason.