  We study the time-dependent fluorescence of an initially hot, multi-level,
single atomic ion trapped in a radio-frequency Paul trap during Doppler
cooling. We have developed an analytical model that describes the fluorescence
dynamics during Doppler cooling which is used to extract the initial energy of
the ion. While previous models of Doppler cooling thermometry were limited to
atoms with a two-level energy structure and neglected the effect of the trap
oscillating electric fields, our model applies to atoms with multi-level energy
structure and takes into account the influence of micromotion on the cooling
dynamics. This thermometry applies to any initial energy distribution. We
experimentally test our model with an ion prepared in a coherent, thermal and
Tsallis energy distributions.
