cooling field is slightly red-detuned from the $|g\rangle \rightarrow |e\rangle$ transition (for Doppler cooling) and induces the transitions shown as black arrows in Fig. \[fig:8-level\], while the repumping field is near resonant with the $|m\rangle \rightarrow |e\rangle$ transition and induces the transitions shown as gray (red) arrows in Fig. \[fig:8-level\]. The electric field of the laser field $i=\mathrm{c}, \mathrm{r}$ is written as $$\label{eq:E_laser}
\mathbf{E}_i(t) = \frac{\mathcal{E}_i}{2} \left[ f_0^i(t) \mathbf{u}_0 + f_{+1}^i(t) \mathbf{u}_{+1} + f_{-1}^i(t) \mathbf{u}_{-1}\right] e^{-i\omega_i t} + \text{c.c.},$$ where $\mathcal{E}_i$ is the electric field amplitude, $\omega_i$ is the frequency, c.c. stands for complex conjugate and $f_q^i(t)$ is the time-dependent complex amplitude of the polarization component $q$, obtained as the projection of the unit polarization vector of the field onto the spherical unit vector $\mathbf{u}_q$. These are $$\mathbf{u}_{\pm1} = \mp \frac{1}{\sqrt{2}} (\mathbf{u}_x \pm i\mathbf{u}_y), \quad \mathbf{u}_{0} = \mathbf{u}_z,$$ and describe $\sigma^\pm$ and $\pi$ polarized light, respectively.

In the electric dipole and rotating wave approximations, the interaction Hamiltonians for the two fields become $$\begin{aligned}
V_\mathrm{c}(t) = -\frac{\hbar \Omega_\mathrm{c}}{2} \big[&f^\mathrm{c}_0(t)& (C_{71}|7\rangle\langle 1| + C_{82}|8\rangle\langle 2|) \nonumber \\*
&+& f^\mathrm{c}_{+1}(t) C_{81}|8\rangle\langle 1| \nonumber \\*
&+& f^\mathrm{c}_{-1}(t) C_{72}|7\rangle\langle 2|\big] e^{-i\omega_\mathrm{c}t} + \text{H.c.}\end{aligned}$$ and $$\begin{aligned}
V_\mathrm{r}(t) = -\frac{\hbar \Omega_\mathrm{r}}{2} \big[&f^\mathrm{r}_0(t)& (C_{74}|7\rangle\langle 4| + C_{85}|8\rangle\langle 5|) 