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Here we examine the interaction of linearly polarized light and a longitudinal magnetic field with an atomic sample, as in the tutorial "Zeeman Structure: Nonlinear Magneto-Optical Rotation". In this case, we consider the D1 line of the ^{87} Rb atom, and write the input parameters in units appropriate for comparison with experimental results.

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Set options for DensityMatrix and plotting function.

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First we set up the atomic system. We could enter the various atomic-state parameters by hand, but in this case it is convenient to pull them from a database. In this example we will focus on transitions originating from the ground-state F=2 hyperfine sublevel, and so we truncate the system by deleting the ground-state F=1 sublevels.

Choose a transition. AtomicTransition just returns the names of the states associated with the given transition, which we can use to find the associated data in the AtomicData database.

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Get the list of quantum numbers and other information that we will use to set up the atomic system. We could also include the additional numerical atomic parameters in this list, but it can be more convenient to substitute their values in later.

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Create the list of sublevels in the atomic system. We specify BranchingRatio[1]→1, which means that the upper J^{} state decays with 100% probability to the lower J state. The individual F^{}→F branching ratios are calculated automatically.

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DeleteStates removes the states satisfying the given criterion from the list of atomic states.

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Now we form the Hamiltonian, relaxation and repopulation matrices, and Liouville equation assuming the steady-state condition.

Form the Hamiltonian assuming an *x*-polarized optical field with frequency and reduced Rabi frequency _{R}, and a *z*-directed magnetic field with nominal Larmor frequency _{L}.

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Draw the level diagram for the truncated system, showing the optical and magnetic-field interactions.

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Apply the rotating-wave approximation to the Hamiltonian, and write the optical frequency in terms of the detuning .

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Now we find the changes in light polarization parameters in terms of the atomic density-matrix elements.

The absorption, phase-shift, optical rotation, and change of ellipticity per unit path length and unit density.

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ExpandDipoleRME expresses the reduced dipole matrix element in terms of the natural width and energy of the transition.

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Write the total absorption and optical rotation in terms of the path length (in cm), and the atomic density (in cm^{-3}).

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Finally, we plot the fractional electric-field amplitude absorption, and optical rotation (in radians). We can incorporate the effect of Doppler broadening by treating the Doppler shift as a shift in the light frequency. Thus the Doppler-broadened signal is given by the convolution of the Doppler-free signal and the Doppler profile.

Plot the Doppler-free absorption and rotation signals at low magnetic field and fairly high light intensity to see the nonlinear magneto-optical effect. Note that for the chosen set of parameters there is significant power broadening of the Doppler-free signal.

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