Visualizing Polarization Moments and Quantum Beats

Here we graphically illustrate the angular-momentum polarization of static and dynamic atomic systems using functions from the AtomicDensityMatrix package.

Quadrupole Moment

Here is a density matrix representing an aligned spin-1 system.

We can see that the density matrix contains only population and alignment by decomposing it into its irreducible tensor components using Decomposition. A list containing one rank-0 (one component), one rank-1 (three components), and one rank-2 (five components) irreducible tensor is produced. Irreducible tensors are indicated by double sets of curly brackets.

The angular-momentum probability surface is the plot of ρ_JJ(θ,ϕ), where θ and ϕ represent the direction of the quantization axis. This is the matrix element 〈JJ_(θ,ϕ)|ρ|JJ_(θ,ϕ)〉, where |JM_(θ,ϕ)〉 are the angular-momentum eigenstates with quantization axis (θ,ϕ). The surface shows the probability of measuring the maximum projection of angular momentum in the (θ,ϕ) direction.

To plot the probability surface, we can use SphericalPlot3D. Evaluate is used so that the expression for is found prior to plotting, instead of for each plot point. Axes3D adds the axes and labels.

WignerRotate can be used on either the density matrix or the tensor decomposition to perform rotations by Euler angles α,β,γ.