  We present a method that uses dynamic decoupling of a multi-level quantum
probe to distinguish small frequency shifts that depend on $m^2_{j}$, where
$m^2_{j}$ is the angular momentum of level $\left|j\right\rangle$ along the
quantization axis, from large noisy shifts that are linear in $m_{j}$, such as
those due to magnetic field noise. Using this method we measured the electric
quadrupole moment of the $4D_{\frac{5}{2}}$ level in $^{88}Sr^{+}$ to be
$2.973^{+0.026}_{-0.033}\, ea_{0}^{2}$. Our measurement improves the
uncertainty of this value by an order of magnitude and thus helps mitigate an
important systematic uncertainty in $^{88}Sr^{+}$ based optical atomic clocks
and verifies complicated many-body quantum calculations.
