  We present a detailed theoretical and experimental study on the optical
control of a trapped-ion qubit subject to thermally induced fluctuations of the
Rabi frequency. The coupling fluctuations are caused by thermal excitation on
three harmonic oscillator modes. We develop an effective Maxwell-Boltzmann
theory which leads to a replacement of several quantized oscillator modes by an
effective continuous probability distribution function for the Rabi frequency.
The model is experimentally verified for driving the quadrupole transition with
resonant square pulses. This allows for the determination of the ion
temperature with an accuracy of better than 2% of the temperature pertaining to
the Doppler cooling limit TD over a range from 0.5TD to 5TD. The theory is then
applied successfully to model experimental data for rapid adiabatic passage
(RAP) pulses. We apply the model and the obtained experimental parameters to
elu- cidate the robustness and efficiency of the RAP process by means of
numerical simulations.
