Nonlinear Magneto-Optical Rotation in a Radio-Frequency Field

In this example we study atoms subject to linearly polarized light, a static bias magnetic field and a transverse oscillating (rf) magnetic field. The atoms are initially polarized by the resonant light. If the Larmor frequency associated with the bias field and the oscillation frequency of the rf field satisfy a resonance condition, the atoms are put into a polarization state that precesses about the bias field direction. The atoms then induce oscillating polarization rotation in the light field, which can be detected with a lock-in amplifier. We illustrate the mechanism leading to the observed line shape using angular-momentum probability surfaces. For further discussion see [Zigdon2010].

Load the package.
Draw the experimental diagram.

We first generate the evolution equations for the system. Under the approximations that we will employ, all explicit time dependence can be removed from the equations. We anticipate this by writing the density-matrix variables without the time variable as an argument.

Remove explicit time dependence from the density-matrix variables.
Generate a 10 atomic system.
Generate the Hamiltonian for a z-polarized light field with frequency ω and amplitude corresponding to a Rabi frequency ΩR, a z-directed static magnetic field with strength corresponding to a Larmor frequency ΩL , and a y-directed rf field with frequency ωrf and amplitude corresponding to a Larmor frequency of Ωrf.
Energy-level diagram for the system, showing the optical and magnetic couplings.
Apply the rotating-wave approximation for the optical field (this is almost always an excellent approximation), and write the optical frequency in terms of the detuning Δ from resonance.

We will assume that the rf amplitude is low enough that the rotating-wave approximation can also be applied to the rf-field coupling. This is equivalent to decomposing the rf field into two counter-rotating components, transforming into the frame that is rotating with the component that is resonant with the rf transition, and neglecting the other component. The resulting set of density-matrix evolution equations will be time independent. This approximation is not necessary, but it will simplify the system enough to allow analytic solutions to be obtained, while retaining a number of interesting effects.

Unitary matrix for transformation into the frame rotating with the rf field.