Weak-value magnetometry for precision tests of fundamental physics

Progress in testing fundamental physics relies on our ability to measure exceedingly small physical quantities. Using a $^{40}$Ca$^{+}$ trapped ion system as an example we show that an exceedingly weak synthetic magnetic field (at the scale of $10^{-19}$ T) can be measured with current technology. This improved sensitivity can be used to test the effects of spin coupling that affect the equivalence principle and, if present, may impact the performance of the proposed entangled optical clocks arrays.