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.
Load the package.
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Set graphics options.
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Quadrupole Moment

Here is a density matrix representing an aligned spin-1 system.
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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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These two methods are equivalent.
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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.
The angular-momentum probability surface.
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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 ,,.
Here we rotate by -/4 about the y axis, and then /4 about the z axis.
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And the plot.
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Perform the same rotation on the polarization-moment expansion of the density matrix.
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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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Hexacontatetrapole Moment

We can find a density matrix that represents a hexacontatetrapole (rank 6) moment by converting the components of a =6 irreducible tensor into mm representation.
Here we start with an unpolarized spin-5 density matrix.
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We then add a rank-6 (13 components) irreducible tensor converted to spin-5 mm' 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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Now we have =0 and =6 moments:
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Here is J⁣J(, ):
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and the plot.
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Larmor Precession

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.
The Zeeman sublevels of a J=1 atomic state.
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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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LiouvilleEquation creates the differential equations for the dynamics from the Hamiltonian.
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We choose the initial polarization to be alignment at an angle /4 to the z-axis.
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InitialConditions produces the set of initial conditions corresponding to this density matrix.
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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.
The decomposition of the time-dependent density matrix.
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Here is JJ(,):
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and the plots, corresponding to one Larmor period.
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Stark Beats

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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LiouvilleEquation creates the differential equations for the dynamics from the Hamiltonian:
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Here is a set of initial conditions corresponding to a state stretched along the y axis.
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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.
The decomposition of the density matrix.
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Here is the norm of each polarization moment as a function of time.
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Here is JJ(,):
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and plots corresponding to one quantum-beat period.
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