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We define an atomic system consisting of two states (a ground state labeled 1 and upper state labeled 2). This is a "toy" system that neglects angular momentum (J and M are not defined). We apply a light field detuned from resonance by a frequency .

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Define the optical field with frequency . This is an *x*-polarized field by default, but for a toy system the polarization does not matter.

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Solve the system of equations with DSolve. We see that the upper state has population _{2, 2}(*t*) given by with generalized Rabi frequency and amplitude A=_{R}^{2}/(_{R}^{2}+^{2}).

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First we remove the time dependence from the density-matrix-element notation. This will have the effect of setting time derivatives of the density-matrix elements to zero.

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There are two forms of relaxation present: intrinsic relaxation of the upper state due to spontaneous decay and "transit" relaxation due to atoms leaving the system. A "relaxation matrix" accounts for these processes.

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The ground state is repopulated by the same two mechanisms: spontaneous decay transfers atoms from the upper state to the ground state, and ground-state atoms enter the system. This is taken into account by the "repopulation matrix":

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Here are the evolution equations, including relaxation and repopulation. Since the density matrix elements are now explicitly time independent, we obtain time-independent equations for the steady state.

Create evolution equations with LiouvilleEquation.

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We can simplify these solutions by assuming that the transit rate is much slower than the natural width of the upper state, and by writing them in terms of the upper-state saturation parameter .

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We see that the upper-state population is a power-broadened Lorentzian in detuning. Here is a plot of the upper-state population as a function of detuning for various values of the saturation parameter.

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Setting the detuning to zero, we see why is called the saturation parameter for this system: when is small, the upper-state population is linear in ; when is increased to unity, the upper-state population begins to saturate, and approaches its limiting value of 1/2.

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