 Atomic Density Matrix Tutorial

# Zeeman Structure: Nonlinear Magneto-Optical Rotation

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).
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.
Define the atomic system.
 Out= Remove explicit time dependence from the notation for the density-matrix elements.
 Out= Here is what the density matrix looks like.
 Out//MatrixForm= The Hamiltonian for the system subject to the optical field, in terms of the "reduced" Rabi frequency R and Larmor frequency L.
 Out//MatrixForm= The level diagram for the system, showing the Zeeman splitting and the light coupling.
 Out= Apply the rotating-wave approximation to the Hamiltonian.
 Out//MatrixForm= IntrinsicRelaxation and TransitRelaxation supply the relaxation matrices.
 Out//MatrixForm= OpticalRepopulation and TransitRepopulation supply the repopulation matrices.
 Out//MatrixForm= Here are the evolution equations.
 Out//TableForm= 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.
 Out= 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.
 Out= The values of the observables per absorption length.
 Out= 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.
 Out= The absorption, phase shift, polarization rotation, and change of ellipticity as a function of light detuning.
 Out= 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<< .
The density-matrix equations with and L neglected compared to .
 Out//TableForm= The solution of the approximate equations substituted into the expression for optical rotation.
 Out= Optical rotation written in terms of the ground-state saturation parameter , the dimensionless Larmor frequency x= L/ , and the dimensionless detuning y= / :
 Out= Again neglect compared to .
 Out= 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.
Expand the optical rotation signal to first order in saturation parameter.
 Out= 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.
Set detuning to zero.
 Out= 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.
Solve for the magnetic fields corresponding to the maxima in the optical rotation signal.
 Out= The full width of the dispersive resonance.
 Out= The width takes a simple form when the branching ratio from the upper state goes to zero.
 Out= The peak-to-peak height of the dispersive resonance.
 Out= The small-field slope.
 Out= Plot the width, height, and slope for two values of the branching ratio.
 Out= 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.
Find the maximum of the slope with respect to .
 Out= Plot as a function of branching ratio.
 Out=   