# Zeeman Structure: Nonlinear Magneto-Optical Rotation

An atomic ground state with 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).

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.

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. |

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 ℓ_{0}=-=-, with and Ω_{L} equal to zero, and Ω_{R} and approaching zero.

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).

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 γ, Ω_{L}<<Γ.