Enhancing Divalent Optical Atomic Clocks with the $^{1}\mathrm{S}_0$$\leftrightarrow$$^{3}\mathrm{P}_{2}$ Transition

Divalent atoms and ions with a singlet $S$ ground state and triplet $P$ excited state form the basis of many high-precision optical atomic clocks. Along with the metastable $^{3}\mathrm{P}_{0}$ clock state, these atomic systems also have a nearby metastable $^{3}\mathrm{P}_{2}$ state. We investigate the properties of the electric quadrupole $^{1}\mathrm{S}_0$$\leftrightarrow$$^{3}\mathrm{P}_{2}$ transition with a focus on enhancing already existing optical atomic clocks. In particular, we investigate the $^{1}\mathrm{S}_0$$\leftrightarrow$$^{3}\mathrm{P}_{2}$ transition in $^{27}\mathrm{Al}^{+}$ and calculate the differential polarizability, hyperfine effects, and other relevant atomic properties. We also discuss potential applications of this transition, notably that it provides two transitions with different sensitivities to systematic effects in the same species. In addition, we describe how the $^{1}\mathrm{S}_0$$\leftrightarrow$$^{3}\mathrm{P}_{2}$ transition can be used to search for physics beyond the Standard Model and motivate investigation of this transition in other existing optical atomic clocks.