Precision of single-qubit gates based on Raman transitions

We analyze the achievable precision for single-qubit gates that are based on off-resonant Raman transitions between two near-degenerate ground states via a virtually excited state. In particular, we study the errors due to non-perfect adiabaticity and due to spontaneous emission from the excited state. For the case of non-adiabaticity, we calculate the error as a function of the dimensionless parameter chi=Delta tau, where Delta is the detuning of the Raman beams and tau is the gate time. For the case of spontaneous emission, we give an analytical argument that the gate errors are approximately equal to Lambda gamma/Delta, where Lambda is the rotation angle of the one-qubit gate and gamma is the spontaneous decay rate, and we show numerically that this estimate holds to good approximation.