  Hybrid traps for the simultaneous confinement of neutrals and ions have
recently emerged as versatile tools for studying interactions between these
species at very low temperatures. Such traps rely on the combination of
different types of external fields for the confinement of either species
raising the question of interactions between the individual traps. Here, the
influence of a strongly inhomogeneous magnetic field used for trapping neutrals
on the trapping and laser cooling of a single Ca$^+$ ion in a radiofrequency
ion trap is studied theoretically using molecular-dynamics simulations based on
multilevel rate equations. The inhomogeneous magnetic field couples the
different components of the ion motion and introduces position-dependent Zeeman
splittings. Nonetheless, laser cooling is still found to work efficiently as
the ion samples different magnetic field strengths and directions along its
trajectory. Offsetting the centres of the two traps generates a linear
magnetic-field gradient so that multiple lasers are required to address the
resulting range of Zeeman splittings in order to ensure efficient cooling. The
present study yields detailed insights into the ion cooling dynamics in
combined magnetic and radiofrequency electric fields relevant for the
characterisation and optimisation of hybrid trapping experiments.
