state. If we neglect the metastable state, Eq. (\[eq:Gamma\_op\]) can be applied to the effective three-level system consisting of $|1\rangle$, $|2\rangle$, and $|8\rangle$ ($A_n=A_{82}$, $C=C_{81}$), giving $\Gamma_\mathrm{op} = 0.041 \Gamma$ in reasonable agreement with the numerical result above. If we apply a magnetic field in the direction of the cooling beam, $|2\rangle$ is still a dark eigenstate and the Zeeman shift only changes the detuning.

![\[fig:g-states\](Color online) Time-dependent scattering rates for different cooling polarizations. A $\sigma^+$ polarized cooling beam rapidly pumps the population into the dark state $|2\rangle$ \[$B=0$; solid (black) curve\]. In a transversal magnetic field, the dark state is a superposition state, which is destabilized by the field \[$\mathbf{B} = 350\;\mu\mathrm{T} \mathbf{u}_x$; dashed (red) curve\]. A combination of $\pi$ and $\sigma^+$ polarization creates a partially dark state \[$B=0$; dash-dotted (green) curve\]. For a linearly polarized cooling beam, there are no ground-level dark states \[$B=0$; dotted (blue) curve\]. For all curves, $\Omega_\mathrm{c} = \Omega_\mathrm{r} = \Gamma$, $\delta_\mathrm{c} = -\Gamma/2$, $\delta_\mathrm{r} = \Gamma/2$, $\gamma_{g,m}=\Gamma/10$, and repumper polarized along $\mathbf{u}_y$.](Fig03-g-states){width=".85\columnwidth"}

If we choose the QA in another direction, e.g., in the $x$ direction when the beam is in the $z$ direction, the polarization in the QA frame is $-2^{-1/2}[\mathbf{u}_0-i2^{-1/2}(\mathbf{u}_{+1} - \mathbf{u}_{-1})]$, 