  We present a formulation for investigating quench dynamics across quantum
phase transitions in the presence of decoherence. We formulate decoherent
dynamics induced by continuous quantum non-demolition measurements of the
instantaneous Hamiltonian. We generalize the well-studied universal
Kibble-Zurek behavior for linear temporal drive across the critical point. We
identify a strong decoherence regime wherein the decoherence time is shorter
than the standard correlation time, which varies as the inverse gap above the
groundstate. In this regime, we find that the freeze-out time
$\bar{t}\sim\tau^{{2\nu z}/({1+2\nu z})}$ for when the system falls out of
equilibrium and the associated freeze-out length
$\bar{\xi}\sim\tau^{\nu/({1+2\nu z})}$ show power-law scaling with respect to
the quench rate $1/\tau$, where the exponents depend on the correlation length
exponent $\nu$ and the dynamical exponent $z$ associated with the transition.
The universal exponents differ from those of standard Kibble-Zurek scaling. We
explicitly demonstrate this scaling behavior in the instance of a topological
transition in a Chern insulator system. We show that the freeze-out time scale
can be probed from the relaxation of the Hall conductivity. Furthermore, on
introducing disorder to break translational invariance, we demonstrate how
quenching results in regions of imbalanced excitation density characterized by
an emergent length scale which also shows universal scaling. We perform
numerical simulations to confirm our analytical predictions and corroborate the
scaling arguments that we postulate as universal to a host of systems.
