$\rho_{78}$ decays at the rate $\Gamma$ and can be partially transferred to the ground state coherences in the spontaneous emission process [@Cohen-Tannoudji1977a] $$\begin{aligned}
\frac{\mathrm{d}\rho_{m_g,m_g'}^\text{relax}}{\mathrm{d}t} = \Gamma_{e\rightarrow g} (2J_e+1)
\sum_{q=-1}^1 C_{J_e m_e, J_g m_g} C_{J_e m_e', J_g m_g'} \nonumber \\
\times \rho_{m_e=m_g+q,m_e'=m_g'+q}, \nonumber \\
 { }\end{aligned}$$ where the transition amplitudes $C_{J_e m_e, J_g m_g}$ are as defined above. This adds the following relaxation terms to the evolution equations

$$\begin{aligned}
\frac{\mathrm{d}\rho_{i,i+1}^\text{relax}}{\mathrm{d}t} &=& 2 C_{7i} C_{8,i+1} A_m \Gamma \rho_{78}, \quad i = 3,4,5, \\
\frac{\mathrm{d}\rho_{12}^\text{relax}}{\mathrm{d}t} &=& 2 C_{71} C_{82} A_g \Gamma \rho_{78}.\end{aligned}$$

The optical coherences decay at the rate $\Gamma/2$ (we assume the laser linewidths to be $\ll\Gamma$) and the metastable state populations and coherences decay at the rate $\gamma_m \ll \Gamma$. The transition probabilities for the quadrupole transitions can be obtained from Eq. \[eq:W-E\_theorem\] with $\kappa = 2$ and the branching ratios for decay from the metastable state are then $A_{mg} = |C_{mg}|^2/\sum_g  |C_{mg}|^2$: $A_{62}=A_{31}=1/5$, $A_{61}=A_{32}=4/5$, $A_{52}=A_{41}=2/5$, and $A_{51}=A_{42}=3/5$.

We have assumed that there are no external processes, such as background gas collisions, that cause transitions between the ground or metastable $m_J$ levels or dephasing of the ground and metastable coherences. The dephasing rate of the coherences between the ground and the 