  Considering ultracold atoms in optical lattices, we propose experimental
protocols to study many-body localization (MBL) length and criticality in
quench dynamics. Through numerical simulations with exact diagonalization, we
show that in the MBL phase the perturbed density profile following a local
quench remains exponentially localized in post-quench dynamics. The size of
this density profile after long-time-dynamics defines a localization length,
which tends to diverge at the MBL-to-ergodic transition as we increase the
system size. The determined localization transition point agrees with previous
exact diagonalization calculations using other diagnostics. Our numerical
results provide evidence for violation of Harris-Chayes bound for the MBL
criticality. The critical exponent $\nu$ can be extracted from our proposed
dynamical procedure, which can then be used directly in experiments to
determine whether the Harris-Chayes-bound holds for the MBL transition. These
proposed protocols to detect localization criticality are justified by
benchmarking to the well-established results for the non-interacting 3D
Anderson localization.
