metastable levels depends on the laser linewidths [@Arimondo1996a] $$\label{eq:g-m_dephasing}
\gamma_{g,m} = \frac{1}{2}\left( \gamma_m + \Delta\omega_\mathrm{c}+\Delta\omega_\mathrm{r} - 2\Delta\omega_\mathrm{c,r} \right),$$ where $\Delta\omega_\mathrm{c}$ and $\Delta\omega_\mathrm{r}$ are the FWHM linewidths of the cooling and repumper lasers and $\Delta\omega_\mathrm{c,r}$ is the cross-correlated linewidth. For perfectly cross-correlated lasers the linewidths cancel, whereas for completely uncorrelated lasers $\Delta\omega_\mathrm{c,r} = 0$ and the linewidths add up. In the low-intensity limit, the linewidth of the CPT resonances is $2\gamma_{g,m}$.

Solution of density matrix equations
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The time-dependent density matrix is solved by direct numerical integration of Eq. (\[eq:L-vN\]) with the initial condition $\rho_{11}=\rho_{22}=1/2$, i.e., a thermally populated ground state. This describes the situation well when the cooling and repumping beams are turned on after a probe-laser interrogation on the $^2S_{1/2} - \,^2D_{5/2}$ quadrupole transition. When the ion is first trapped, the situation is more complicated, as it is hot and there might be another far-detuned cooling field present [@Brownnutt2007a]. However, also in this case the treatment should be valid for the “final” state, i.e., when the ion has cooled down and reached quasi-steady state (when the only time dependence is due to the possible polarization modulation of the repumper). As the density matrix describes ensemble averages, the time-dependent solution 