been used in experiments at both the NRC ($\Omega_\mathrm{c} \approx 20\Gamma$) [@Madej2004a] and the National Physical Laboratory ($\Omega_\mathrm{c} \approx 7\Gamma$) [@Brownnutt2007].

Types of dark states \[sec:dark\_states\]
=========================================

Ground-level dark states \[sec:g-states\]
-----------------------------------------

Berkeland and Boshier [@Berkeland2002a] state that for a transition between states with half-integer angular momentum and $J_g = J_e$, such as the cooling transition considered here, there is one dark state for circular polarization only. Let us elaborate on this.

Figure \[fig:g-states\] shows the scattering rate as a function of time for different cooling laser polarizations. For a circularly polarized cooling beam in zero magnetic field, we can choose the QA such that the polarization is $\sigma^+$. This polarization only drives the $|1\rangle \rightarrow |8\rangle$ transition and the scattering rate decays rapidly as the population is pumped into the dark state $|2\rangle$ at the optical pumping rate $\Gamma_\mathrm{op} = 0.045 \Gamma$ \[solid (black) curve in Fig. \[fig:g-states\]\]. We have earlier derived an expression for the optical pumping rate in a three-level system with one dark lower level [@Lindvall2009a] $$\label{eq:Gamma_op}
\Gamma_\mathrm{op} = \frac{A_n \Gamma \left(\frac{C\Omega}{2}\right)^2}{\delta^2 + \left(\frac{\Gamma}{2}\right)^2 + \left[ 2+\frac{\delta^2-3 \left(\Gamma/2 \right)^2}{\delta^2 + \left( \Gamma/2 \right)^2} A_n \right] \left(\frac{C\Omega}{2}\right)^2}.$$ Here $A_n$ is the spontaneous decay probability to the noncoupled 