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Linear quench of the transverse field drives the quantum Ising chain across aquantum critical point from the paramagnetic to the ferromagnetic phase. Wefocus on normal and anomalous quadratic correlators between fermionic kinkcreation and annihilation operators. They depend not only on the Kibble-Zurek(KZ) correlation length but also on a dephasing length scale, which differsfrom the KZ length by a logarithmic correction. Additional slowing down of theramp in the ferromagnetic phase further increases the dephasing length andsuppresses the anomalous correlator. The quadratic correlators enter Pfaffiansthat yield experimentally relevant kink correlation functions, the probabilitydistribution of ferromagnetic domain sizes, and, closely related, emptinessformation probability. The latter takes the form of a Pfaffian of a blockToeplitz matrix that allows for some analytic asymptotes. Finally, we obtainfurther insight into the structure of the state at the end of the ramp byinterpreting it as a paired state of fermionic kinks characterized by its pairwave function. All those quantities are sensitive to quantum coherence betweeneigenstates with different numbers of kinks, thus making them a convenientprobe of the quantumness of a quantum simulator platform.

Kink correlations, domain size distribution, and the emptiness formation probability after the Kibble-Zurek quench in the quantum Ising chain