Comparison of trapped-ion entangling gate mechanisms for mixed species

Entangling gates are an essential capability of quantum computers. There are different methods for implementing two-qubit gates, with respective advantages and disadvantages. We investigate the experimentally relevant differences and commonalities of laser-based ${\sigma }_z\otimes {\sigma }_z$ light-shift and ${\sigma }_{\phi }\otimes {\sigma }_{\phi }$ Moelmer-Soerensen gates, highlighting the phases of experimental control fields and their long-term stabilities, in the specific case of mixed-species gates. We implement these gates on qubits with very different magnetic field sensitivities, encoded in ${}^{43}{\mathrm{C}\mathrm{a}}^{+}$ and ${}^{88}{\mathrm{S}\mathrm{r}}^{+}$, achieving fidelities of $99.8\mathrm{\%}$ for the ${\sigma }_z\otimes {\sigma }_z$ and $99.6\mathrm{\%}$ for the ${\sigma }_{\phi }\otimes {\sigma }_{\phi }$ gate.