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

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We can see that the density matrix contains only population and *z* 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.

Display the decomposition of the density matrix as a list of column vectors using TensorForm.

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The angular-momentum probability surface is the plot of _{JJ}(,), where and represent the direction of the quantization axis. This is the matrix element *JJ*_{(,)}||*JJ*_{(,)}, where |*JM*_{(,)} are the angular-momentum eigenstates with quantization axis (,). The surface shows the probability of measuring the maximum projection of angular momentum in the (,) direction.

The function MaxProjectionProbability generates the formula for the probability surface.

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MaxProjectionProbability can also be applied to the tensor decomposition. Then _{JJ}(,) is found as the sum of spherical harmonics.

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

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The AMPSPlot function is a shorthand to produce an angular-momentum probability surface plot directly from the density matrix or tensor decomposition.

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

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Plot the AMPS using the rotated polarization moments. The surface is the same as the one obtained directly from the density matrix.

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We can find a density matrix that represents a hexacontatetrapole (rank 6) moment by converting the components of a =6 irreducible tensor into *m**m*^{} representation.

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We set *a* as large as possible subject to the condition that the eigenvalues of the matrix are nonnegative, so that the density matrix is physical:

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Here we will find the dynamics of a polarized atomic system subject to a magnetic field. We first create a list of the sublevels of a J=1 state. This list will be fed to the functions that create the Hamiltonian and the evolution equations.

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Hamiltonian creates the Hamiltonian for the system, here including the effect of a *z*-directed magnetic field.

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DSolve is used to solve for the dynamics directly, and the results are put into the density matrix generated with DensityMatrix.

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We can see the effect on the polarization moments by decomposing the result into its irreducible tensor components. As expected, the dynamic behavior is rotation, which does not mix polarization moments.

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Now let's look at the dynamics of system subject to an electric field. We first create a list of the sublevels of a J=1 state and assign the state zero scalar polarizability and nonzero tensor polarizability. The polarizability is a parameter that accounts for level shifts due to Stark mixing with other atomic states not included in the system under consideration.

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Hamiltonian creates the Hamiltonian for the system, in this case including the effect of a *z*-directed electric field. The "Polarizability" interaction puts in terms accounting for the polarizability of this state.

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DSolve is used to solve for the dynamics directly, and the results are put into the density matrix generated with DensityMatrix.

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We can see the effect on the polarization moments by decomposing the result into its irreducible tensor components. The dynamic behavior here is not just rotation: There is initially alignment (rank-two polarization moment) but no orientation (rank-one polarization moment) at time zero, but orientation appears at *t*>0. This has been described as *alignment-to-orientation conversion*.

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