Incoherent repumping scheme in the $^{88}$Sr$^{+}$ five-level manifold
Laser-cooled trapped ions are at the heart of modern quantum technologies and their cooling dynamics often deviate from the simplified two-level atom model. Doppler cooling of the $^{88}$Sr$^{+}$ ion involves several electronic levels and repumping channels that strongly influence fluorescence. In this work, we study a repumping scheme for the $^{88}$Sr$^{+}$ ion by combining precision single-ion spectroscopy with comprehensive numerical modeling based on optical Bloch equations including 18 Zeeman sublevels. We show that, although the observed fluorescence spectra retain a Lorentzian lineshape, their width and amplitude cannot be explained by a two-level atom description. Moreover, we find the optimal repumping conditions for maximizing the photon scattering rate.