# Open orbits on the Bloch sphere of two level atom

I am studying the Bloch sphere representation of a two level atom in a classical electric field, starting in the ground state at t=0. The Bloch equations for the time evolution are

\begin{align} \langle \sigma _x\rangle & = \sin(\Omega_r t)\cos\left(\omega t -\frac{\pi} {2}\right)\\ \langle \sigma _y\rangle & = \sin(\Omega_r t)\sin\left(\omega t -\frac{\pi} {2}\right)\\ \langle \sigma _z\rangle & = -\cos(\Omega_r t) \end{align} where $\Omega_r$ is the Rabi frequency and $\omega$ is the laser frequency.

By numerically plotting the evolution of the system, I get this feeling that if $\Omega_r \, / \, \omega$ is rational that the orbits are closed, and if its irrational they are open. Is this true, and if so, how would I go about mathematically proving this? How do you work out the orbital period of the system?

• You are correct. If the orbit is closed, let $T$ be its period. Then $\langle \sigma_z(t) \rangle = \langle \sigma_z(t+T) \rangle$ means $\Omega_r T = 2n\pi$, while from $\langle \sigma_x (t) \rangle = \langle \sigma_x(t+T) \rangle$ or $\langle \sigma_y(t) \rangle = \langle \sigma_y(t+T) \rangle$ we also have $\omega T = 2m\pi$. Wherefrom $$T = \frac{2n\pi}{\Omega_r} = \frac{2m\pi}{\omega}$$ and $$\frac{\omega}{\Omega_r} = \frac{m}{n}$$ – udrv May 6 '17 at 1:30