  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.
