field or by modulating the laser polarization. Both methods rely on making the dark states evolve in time more rapidly than the optical pumping processes. One of the systems they consider is the $^2S_{1/2} - \,^2P_{1/2} - \,^2D_{3/2}$ system. They recommend Rabi frequencies that give a reasonable scattering rate without excessive power broadening and point out that the repumper should have a positive detuning to tune the system away from CPT resonance (the cooling laser is red-detuned for Doppler cooling). The magnetic field or rate of polarization modulation required to destabilize the dark states is then analyzed.

This paper expands on the results in Ref. [@Berkeland2002a]. We allow the two laser fields to have different polarizations and take into account the effect of the laser linewidths and correlation properties on the CPT resonances. Furthermore, the effect of the motion of the ion, secular and micromotion, is analyzed. Finally, the theoretical results are compared to experimental spectra from the $^{88}\mathrm{Sr}^+$ endcap trap at the National Research Council of Canada (NRC) [@Dube2010a].

Theory
======

The atomic Hamiltonian of the eight-level system shown in Fig. \[fig:8-level\] is

$$H_\mathrm{a} = \hbar\omega_g \sum_{i=1}^2 |i\rangle\langle i| + \hbar\omega_m \sum_{i=3}^6 |i\rangle\langle i| + \hbar\omega_e \sum_{i=7}^8 |i\rangle\langle i|,$$

