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An atomic ground state with J>0 can be polarized by the application of light resonant with a transition to an upper state. This polarization will precess about the direction of an applied magnetic field. The atomic polarization can then act back on the light to change the light's polarization, for example by rotating the polarization angle of the light, in a process known as nonlinear magneto-optical rotation (NMOR).

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We define an atomic system consisting of a J=1 ground state and a J=0 upper state. We apply an *x*-polarized light field and a *z*-directed magnetic field.

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The Hamiltonian for the system subject to the optical field, in terms of the "reduced" Rabi frequency _{R} and Larmor frequency _{L}.

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Observables supplies the fractional electric-field absorption, phase shift, polarization rotation, and change of ellipticity experienced by the light in terms of the density-matrix elements.

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It is convenient to write the observables in terms of the absorption length in the atomic medium. The on-resonance, unsaturated absorption length is given by , with and _{L} equal to zero, and _{R} and approaching zero.

Find the absorption length by solving for the differential fractional change in electric field amplitude with zero detuning and magnetic field, and then letting the electric field amplitude and transit rate go to zero.

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We can solve the equations numerically and plot the values of the observables as a function of various parameters. As a function of magnetic field, we see "nested features" in the line shapes: a narrow central resonance of width ~ (the "transit effect"), and a wider resonance of width ~ (the linear effect).

Here are the absorption, phase shift, polarization rotation, and change of ellipticity as a function of Larmor frequency for particular values of the parameters.

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The absorption, phase shift, polarization rotation, and change of ellipticity as a function of light detuning.

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We can also solve the equations analytically. The general solution is complicated, so it is helpful to look at specific cases. For example, if we are interested in the transit effect and wish to neglect the linear effect, we can make the assumptions , _{L}<<.

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Optical rotation written in terms of the ground-state saturation parameter , the dimensionless Larmor frequency x=_{L}/, and the dimensionless detuning y=/:

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The analytic result, even neglecting the linear effect, is still fairly complicated. We can examine this result in several ways. First, we look at the limit of low light power.

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We see that the optical rotation is linear in the saturation parameter to lowest order. (We have neglected the linear magneto-optical rotation, which is zeroth order in the saturation parameter.) The magnetic-field dependence is a dispersive Lorentzian of width , and the optical detuning dependence is the square of a Lorentzian of width . The latter dependence results from the fact that both optical pumping and probing have Lorentzian dependencies on light detuning.

We can also set the detuning to zero and analyze the dependence of the magnetic-field resonance on the saturation parameter and branching ratio.

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We see that the magnetic-field dependence remains a dispersive Lorentzian for any value of the saturation parameter and branching ratio (note that this approximation breaks down when the saturation parameter is large enough that the power-broadened width approaches ). We can find the width, height, and small-field slope of this Lorentzian.

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For the best magnetometric sensitivity, one needs to maximize the slope of the signal. We find that the saturation parameter should be on the order of unity for the best sensitivity.

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