Magnetic Feshbach resonances between atoms in $^2$S and $^3$P$_0$ states: mechanisms and dependence on atomic properties

Magnetically tunable Feshbach resonances exist in ultracold collisions between atoms in $^2$S and $^3$P$_0$ states, such as an alkali-metal atom colliding with Yb or Sr in a clock state. We investigate the mechanisms of these resonances and identify the terms in the collision Hamiltonian responsible for them. They involve indirect coupling between the open and closed channels, via intermediate channels involving atoms in $^3$P$_1$ states. The resonance widths are generally proportional to the square of the magnetic field and are strongly enhanced when the magnitude of the background scattering length is large. For any given pair of atoms, the scattering length can be tuned discretely by choosing different isotopes of the $^3$P$_0$ atom. For each combination of an alkali-metal atom and either Yb or Sr, we consider the prospects of finding an isotopic combination that has both a large background scattering length and resonances at high but experimentally accessible field. We conclude that $^{87}$Rb+Yb, Cs+Yb and $^{85}$Rb+Sr are particularly promising combinations.