certain transition $|i\rangle \rightarrow |j\rangle$ is equal to $C_{ji} \Omega_k$ ($k=c,r$ depending on which laser field drives the transition). Note that Ref. [@Berkeland2002a] uses rms Rabi frequencies defined as $\Omega_\text{rms}^2 = \sum_{m_J,m_J'} |\Omega_{m_J m_J'}|^2$. From the sum rule [@Sobelman:ASRT] $$\label{eq:matrix_sum}
\sum_{m_J, m_J'} |\langle J' m_J' | T^\kappa_q |J m_J\rangle|^2 = \frac{1}{2\kappa +1} |T^\kappa_{J',J}|^2,$$ we obtain the relation between the rms and the two-level Rabi frequencies: $\Omega_\text{rms} = 3^{-1/2} \Omega$.

The Liouville-von Neumann equation for the density matrix of the eight-level system is $$\label{eq:L-vN}
\frac{\mathrm{d}\rho}{\mathrm{d}t} = \frac{1}{i\hbar}[H_\mathrm{a,m}+V_\mathrm{c}(t)+V_\mathrm{r}(t),\rho] +\frac{\mathrm{d}\rho^\text{relax}}{\mathrm{d}t}.$$ Terms oscillating at optical frequencies are eliminated by the substitutions

$$\begin{aligned}
\tilde{\rho}_{kl} &=& \rho_{kl} e^{-i\omega_\mathrm{c}t}, \quad 1\leq k\leq 2, \;  7\leq l\leq 8, \\
\tilde{\rho}_{kl} &=& \rho_{kl} e^{-i\omega_\mathrm{r}t}, \quad 3\leq k\leq 6, \;  7\leq l\leq 8, \\
\tilde{\rho}_{kl} &=& \rho_{kl} e^{-i(\omega_\mathrm{c}-\omega_\mathrm{r})t}, \quad 1\leq k\leq 2, \;  3\leq l\leq 6.\end{aligned}$$

Relaxation
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The relaxation terms $\mathrm{d}\rho^\text{relax}/\mathrm{d}t$ in Eq. (\[eq:L-vN\]) are added phenomenologically. The excited state populations decay at the rate $\Gamma$. The excited state decays to the ground and metastable states with the probabilities $A_g$ and $A_m=1-A_g$ and the decay probability for a certain magnetic sublevel transition, e.g., $|8\rangle$ to $|6\rangle$ can then be calculated as $$A_{86} = \frac{|C_{86}|^2 A_m}{|C_{86}|^2+|C_{85}|^2+|C_{84}|^2}.$$

The excited state coherence 