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 $^{43}\mathrm{Ca}^+$ and $^{88}\mathrm{Sr}^+$, achieving fidelities of $99.8\%$ for the $\sigma_z\otimes\sigma_z$ and $99.6\%$ for the $\sigma_\phi\otimes\sigma_\phi$ gate.