Stark Mixing

Electric fields cause mixing of opposite-parity atomic states, and consequent energy shifts. If all relevant atomic states are included in the system, the mixing and splitting can be found by finding the eigenvalues and eigenvectors of the Hamiltonian.
Here we analyze a J=2 state mixed with a J=1 state by a z-directed electric field.
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Define the atomic system.
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The Hamiltonian for the system subject to a z-directed electric field, in terms of the "reduced" Rabi frequency:
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The level diagram for the system, showing the mixing induced by the electric field.
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Find the eigenvalues and eigenvectors of the Hamiltonian.
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The eigenvalues are the new energies. Here is a plot of them as a function of electric field.
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The eigenvectors are the eigenstates in the presence of the electric field, written as a linear combination of the original sublevels. However, they are not listed in the order corresponding to the original sublevels. Here we reorder the eigenvalues and eigenvectors so that they match the unperturbed quantities when the electric field approaches zero.
Expand the eigenvectors in series for small R.
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Normalize the eigenvectors, and then expand in series again.
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Reorder the eigenvalues and eigenvectors to match the unperturbed eigenvalues and eigenvectors.
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Here we use the correctly ordered energies to plot the level diagram for the energy eigenstates in the presence of the electric field.
We can see that sublevels with the same M are mixed by the electric field, causing them to repel. There are no M=±2 sublevels of the upper state, so the M=±2 sublevels of the lower state do not shift.