  Universal single-qubit operations based on purely geometric phase factors in
adiabatic processes are demonstrated by utilizing a four-level system in a
trapped single $^{40}$Ca$^+$ ion connected by three oscillating fields.
Robustness against parameter variations is studied. The scheme demonstrated
here can be employed as a building block for large-scale holonomic quantum
computations, which may be useful for large qubit systems with statistical
variations in system parameters.
