We demonstrate quantum control techniques for a single trapped ion in a cryogenic, surface-electrode trap. A narrow optical transition of Sr+ along with the ground and first excited motional states of the harmonic trapping potential form a two-qubit
system. The optical qubit transition is susceptible to magnetic field fluctuations, which we stabilize with a simple and compact method using superconducting rings. Decoherence of the motional qubit is suppressed by the cryogenic environment. AC Stark shift correction is accomplished by controlling the laser phase in the pulse sequencer, eliminating the need for an additional laser. Quantum process tomography is implemented on atomic and motional states using conditional pulse sequences. With these techniques we demonstrate a Cirac-Zoller Controlled-NOT gate in a single ion with a mean fidelity of 91(1)%.
