\nonumber \\*
&+& f^\mathrm{r}_{+1}(t) (C_{73}|7\rangle\langle 3| + C_{84}|8\rangle\langle 4|) \nonumber \\*
&+& f^\mathrm{r}_{-1}(t) (C_{75}|7\rangle\langle 5|+C_{86}|8\rangle\langle 6|)\big] \nonumber \\*
&\times& e^{-i\omega_\mathrm{r}t} + \text{H.c.},\end{aligned}$$ where H.c. stands for Hermitian conjugate. The coefficients $C_{ji}$ are the relative transition amplitudes of the electric dipole transitions and can be obtained using the Wigner-Eckart theorem, which states how a matrix element depends on the $m_J$ quantum numbers [@Sobelman:ASRT] $$\begin{aligned}
 \label{eq:W-E_theorem}
\langle J' m_J' | T^\kappa_q |J m_J\rangle &=& (-1)^{J'-m_J'} \langle J'||T^\kappa ||J\rangle
\begin{pmatrix} J' & \kappa & J \\ -m_J' & q & m_J \end{pmatrix} \nonumber \\
&=& C_{J'm_J',Jm_J} T^\kappa_{J',J}.\end{aligned}$$ Here $T^\kappa_q$ is a set of tensor operators of rank $\kappa$ and $T^\kappa_{J',J} = \langle J'||T^\kappa ||J\rangle$ is the reduced matrix element that does not depend on $m_J$, $m_J'$, or $q$. Using Eq. (\[eq:W-E\_theorem\]) for the electric dipole operator ($\kappa=1$; $q = 0$ for $\pi$ and $\pm1$ for $\sigma^\pm$ transitions), one obtains: $C_{82}=-C_{71}=6^{-1/2}$, $C_{72}=-C_{81}=3^{-1/2}$, $C_{73}=C_{86}=2^{-1}$, $C_{74}=C_{85}=-6^{-1/2}$, and $C_{75}=C_{84}=2^{-1}3^{-1/2}$.

The “two-level” Rabi frequencies corresponding to the two laser fields are $\Omega_\mathrm{c} = \mu_{P_{1/2},S_{1/2}} \mathcal{E}_\mathrm{c} /\hbar$ and $\Omega_\mathrm{r} = \mu_{P_{1/2},D_{3/2}} \mathcal{E}_\mathrm{r} /\hbar$, where $\mu_{L'_{J'},L_J} = \langle L'_{J'}||-e\mathbf{r} ||L_J\rangle$ is the reduced dipole moment. The Rabi frequencies can be assumed to be real. The true Rabi frequency for a 