Precision of single-qubit gates based on Raman transitions

We analyze the achievable precision for single-qubit gates that are based on 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-adabaticity, 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.