Isotope Shifts and Dark Matter Searches with Trapped Atomic Ions*

Atomic & molecular spectroscopy and the interplay between theory and experiment has long played a central role in the development of new theories for fundamental physics.  Landmark experiments, such as measurments of parity violation and limits on the electron electric dipole moment, have combined with ever-improving theory to set limits on ``new" physics which complement collider experiments.  Many of the today's most stringent limits on possible extensions to the Standard Model arise from AMO experiments.

One area in which AMO experiments are especially useful is in searches for dark matter and any associated ``dark forces" which may weakly couple dark matter to ordinary matter.  Such couplings would give rise only to exceptionally rare events which would struggle to reach statistical significance in collider experiments, despite the fact that they might occur at energies well within a collider's reach.  And yet, even weak couplings could give rise to effects which can be accessed via precision measurements of atomic structure.  One recent proposal suggests searching for nonlinear relationships between the scaled isotope shifts in two different electronic transitions using a King Plot.  Violations of "King's Linearity" might indicate the signature of electron-neutron couplings arising from intermediate-mass gauge bosons in the dark sector, or provide an avenue into measuring higher-order effects arising within the Standard Model, such as nuclear polarizabilities, which have previously been inaccessible to experiment.  This talk will introduce ongoing work in this area while also describing experiments at Williams College searching for nonlinear isotope shifts using co-trapped Ca$^+$ ions.