Numerical modeling for trapped-ion thermometry using dark resonances

The simulation of vibrational energy transport and quantum thermodynamics with trapped ions requires good methods for the estimation of temperatures. One valuable tool for this purpose is based on the fit of dark resonances in the fluorescence spectrum. However, this fit demands numerical simulations of the coupled electronic-motional dynamics which usually involve a trade-off between accuracy and speed. Here, we discuss several techniques with simplified dynamical equations for the simulation of the spectrum of a trapped ion that undergoes thermal motion, identifying the advantages and limitations of each method. We start with a three-level model to provide a better insight into the approximations involved, and then move on to tackle the experimentally relevant case of an eight-level calcium ion. We observe that mimicking the effect of thermal motion by means of additional dephasing is computationally very convenient, but can lead to significant errors in the estimation of the temperature. Nevertheless, this can be counteracted by a proper calibration, supporting the use of dark resonances as a practical thermometer.