Four- and six-photon stimulated Raman transitions for coherent qubit and qudit operations

We experimentally demonstrate transitions between electronic angular momentum states with a difference in magnetic quantum numbers $\mathrm{\Delta }{\mathrm{m}}_{\mathrm{J}}=$ 3, 4, and 5 via resonant four- and six-photon stimulated Raman transitions in a single trapped atom. Derivation of the corresponding Rabi frequencies, which are verified experimentally, follows the standard treatment of two-photon transitions including the adiabatic elimination of intermediate states. Finally, we discuss pathways to increase the observed multi-photon transition fidelities to $>99.99\mathrm{\%}$, providing a tool for efficient, high-fidelity control of qu$\mathit{\text{d}}$its and single-atom logical qubits.