Synthesizing a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>̂</mml:mo></mml:mover><mml:mi>z</mml:mi></mml:msub></mml:math> 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 $\hat{\sigma}_x$ or $\hat{\sigma}_y$ M{\o}lmer-S{\o}rensen type interactions in trapped ion systems. Using this field configuration, it is also possible to synthesize a $\hat{\sigma}_z$ 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.