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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].

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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.

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Generate the Hamiltonian for a z-polarized light field with frequency and amplitude corresponding to a Rabi frequency _{R}, a z-directed static magnetic field with strength corresponding to a Larmor frequency _{L} , and a y-directed rf field with frequency _{rf} and amplitude corresponding to a Larmor frequency of _{rf}.

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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 from resonance.

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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.

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The Hamiltonian in the rotating frame, with far-off-resonant terms neglected, and the rf frequency written in terms of the detuning _{rf} from resonance. The Hamiltonian is now time-independent.

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Matrix describing relaxation due to spontaneous emission at a rate and atom transit through the light beam at a rate _{t}.

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The steady-state equations in the rotating frame. We set the optical detuning to zero for simplicity, write the Rabi frequency in terms of the saturation parameter , and scale all frequencies to the transit width by setting _{t} equal to one.

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The density-matrix equations can be solved numerically to all orders. For illustrative purposes, we will instead solve them analytically to third order in the optical field. For simplicity, we assume that the light field is on resonance, and that the natural width is much greater than all of the other rates in the problem.

To solve the equations perturbatively in , we first find the zeroth-order solution by solving them with set to zero. We then use this solution by substituting it back into the equations, but only for the particular density-matrix variables that appear multiplied by a factor of . Solving this new set of equations gives the first-order solution. This procedure is iterated to solve the system to successive orders.

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Solve the evolution equations to third order in the optical field. At each iteration we simplify under the assumption that is large.

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We now have the steady-state solution for the density matrix in the frame rotating with the rf field. We can find the steady-state optical-rotation signal for a fictitious light field propagating along the y-axis in this frame. This is essentially what a lock-in amplifier in the laboratory frame would report as the in-phase optical-rotation signal of the actual light field. (The quadrature component of the signal would be given by the optical rotation of a fictitious field propagating along the x-axis in the rotating frame.)

Find the expression for the optical rotation of a z-polarized field propagating along y in the rotating frame, in terms of the rotating-frame density-matrix elements.

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Substitute in the third-order solution for the density matrix, and simplify the presentation by dividing out a multiplicative factor.

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For small _{rf} (_{rf}<1 in the dimensionless units), the in-phase signal has a dispersive-Lorentzian dependence on rf frequency.

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For larger _{rf} (_{rf}>1 in the dimensionless units), an additional feature appears in the resonance line shape.

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We can get a better idea of what is going on by plotting the angular-momentum probability surface for the ground-state density matrix. We only briefly explore this topic here—for a more thorough discussion see [Zigdon2010].

Define a function to plot the AMPS, the field configuration, and the in-phase rotation signal for a given set of parameters.

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At low rf-field strengths, the optical rotation in the rotating frame results from precession of the atomic polarization about the resultant static-magnetic-field direction (plotted as a dotted line). An animation could also be generated as the rf detuning is varied. Here is a link to an animation showing the rf detuning being varied in the low-rf-field regime.

Plot the AMPS at a low rf-field strength. We exaggerate to emphasize the nonlinear contribution to the angular-momentum distribution.

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At higher field strengths, the atomic polarization is averaged about the magnetic-field direction. Depending on the rf detuning—i.e., the angle of the resultant magnetic-field direction—this can result in a "peanut" shape, a "donut" shape, or, at the *magic angle*, the complete averaging out of all atomic polarization. This is the source of the additional feature in the resonance line shape. Here is a link to an animation showing the rf detuning being varied in the high-rf-field regime.

Plot the AMPS at a higher rf-field strength, for five values of the rf detuning. The magic angle is attained at the third value.

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To find the time-dependent optical-rotation signal in the laboratory frame, we use the unitary transformation defined above to write the laboratory-frame density matrix in terms of the density matrix in the rotating frame, and then use this density matrix in the expression for the optical rotation.

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