Synthesizing a ${\widehat{\sigma }}_z$ spin-dependent force for optical, metastable, and ground state trapped-ion qubits

A single bichromatic field near-resonant to a qubit transition is typically used for ${\widehat{\sigma }}_x$ or ${\widehat{\sigma }}_y$ Mølmer-Sørensen type interactions in trapped ion systems. Using this field configuration, it is also possible to synthesize a ${\widehat{\sigma }}_z$ spin-dependent force by merely adjusting the beat-note frequency. Here, we expand on previous work and present a comprehensive theoretical and experimental investigation of this scheme with a laser near-resonant to a quadrupole transition in ${}^{88}$Sr${}^{+}$. Further, we characterise its robustness to optical phase and qubit frequency offsets, and demonstrate its versatility by entangling optical, metastable, and ground state qubits.