should be interpreted as the average over an infinite number of experimental cycles for a single trapped ion.

If the polarization amplitudes $f_q^i$ are time independent, the steady-state solution can be solved by setting the time derivative to 0 in Eq. (\[eq:L-vN\]). If they are time-dependent, we must integrate Eq. (\[eq:L-vN\]) until quasi-steady state is reached and average the density matrix over one polarization modulation period, which is considerably more computer time consuming.

The (quasi-)steady-state value of the scattering rate is the most relevant parameter for comparison with experiments. To collect the fluorescence of the single ion, it is customary to use a bandpass filter at the cooling wavelength to minimize problems with background light and scattering. We therefore define the scattering rate as $\Gamma_\mathrm{sc} = A_g \Gamma (\rho_{77}+\rho_{88})$.

Experimental parameters for $^{88}$S$^+$ \[sec:exppar\]
=======================================================

Table \[tab:exp\_par\] summarizes the numerical parameters used for the $^{88}$Sr$^+$ ion in this paper. We use the dipole matrix elements, spontaneous decay rate, and branching ratio calculated by Jiang *et al.* [@JiangD2009a]. Their values are in good agreement with the experimental $^2 P_{1/2}$ lifetime in [@Pinnington1995a] and with the calculated lifetime and oscillator strengths in [@Brage1998a]. Gallagher’s [@Gallagher1967a] experimental oscillator strengths and branching ratio are 