Spontaneous scattering errors from Raman transitions in metastable trapped-ion qubits

Trapped-ion quantum information processing (QIP) often makes use of coherent stimulated-Raman transitions to perform logic gates on the qubits. Spontaneous photon scattering during this process is an important source of errors; therefore, characterization of such errors is important for calculating limits to logic gate fidelity. These errors have been well-studied for qubits encoded in the S_1/2 ground state; here, we present calculations of scattering probabilities for hyperfine qubits stored in the metastable D_5/2 manifold of Ca⁺, Sr⁺, or Ba⁺. Additionally, we consider some effects which have been (reasonably) assumed to be negligible in studies of ground state qubits but cannot be ignored here: the detuning dependence of the scattered photon frequency and the Lamb-Dicke parameter, scattering from higher levels, and contributions from a second Feynman diagram. We conclude that detuning and power requirements for a given error are generally higher in the metastable qubits than in ground state qubits, but the ultimate limits on achievable error may be lower in some cases.  Additionally, the required wavelengths are convenient for high-power solid-state lasers and offer better compatibility with integrated optics approaches.