  Linear quench of the transverse field drives the quantum Ising chain across a
quantum critical point from the paramagnetic to the ferromagnetic phase. We
focus on normal and anomalous quadratic correlators between fermionic kink
creation and annihilation operators. They depend not only on the Kibble-Zurek
(KZ) correlation length but also on a dephasing length scale, which differs
from the KZ length by a logarithmic correction. Additional slowing down of the
ramp in the ferromagnetic phase further increases the dephasing length and
suppresses the anomalous correlator. The quadratic correlators enter Pfaffians
that yield experimentally relevant kink correlation functions, the probability
distribution of ferromagnetic domain sizes, and, closely related, emptiness
formation probability. The latter takes the form of a Pfaffian of a block
Toeplitz matrix that allows for some analytic asymptotes. Finally, we obtain
further insight into the structure of the state at the end of the ramp by
interpreting it as a paired state of fermionic kinks characterized by its pair
wave function. All those quantities are sensitive to quantum coherence between
eigenstates with different numbers of kinks, thus making them a convenient
probe of the quantumness of a quantum simulator platform.
