  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)%.
