  We propose a novel quantum sensing protocol that leverages the dynamical
response of physical observables to quenches in quantum systems. Specifically,
we use the nitrogen-vacancy (NV) color center in diamond to realize both scalar
and vector magnetometry via quantum response. Furthermore, we suggest a method
for detecting the motion of magnetic nanoparticles, which is challenging with
conventional interference-based sensors. To achieve this, we derive the closed
exact form of the Berry curvature corresponding to NV centers and design
quenching protocols to extract the Berry curvature via dynamical response. By
constructing and solving non-linear equations, the magnetic field and
instantaneous motion velocity of the magnetic nanoparticle can be deduced. We
investigate the feasibility of our sensing scheme in the presence of
decoherence and show through numerical simulations that it is robust to
decoherence. Intriguingly, we have observed that a vanishing nuclear spin
polarization in diamond actually benefits our dynamic sensing scheme, which
stands in contrast to conventional Ramsey-based schemes. In comparison to
Ramsey-based sensing schemes, our proposed scheme can sense an arbitrary
time-dependent magnetic field, as long as its time dependence is nearly
adiabatic.
