 Three-mode revival of a planar quantum rotor

For a quantum rotor with angle period $2\pi$, let $u_m=e^{im\varphi }/\sqrt{2\pi }$ and prepare $\psi (\varphi ,0)=\sqrt{4/(3\pi )}{\cos }^2\varphi$. Its only nonzero angular-momentum amplitudes are $c_0=\sqrt{2/3}$ and $c_{\pm 2}=1/\sqrt{6}$. Their energies are $0$ and $2{\hslash }^2/I$, respectively, soThe state revives after $T=\pi I/\hslash$ and its density at $T/2$ is the initial density rotated by $\pi /2$. At $T/4$ there are four equal angular maxima. This is interference between a stationary component and a degenerate pair of counter-rotating components; it is not a uniformly rotating rigid profile. Angular-momentum and energy probabilities stay constant.