  We consider a linear quench from the paramagnetic to ferromagnetic phase in
the quantum Ising chain interacting with a static spin environment. Both
decoherence from the environment and non-adiabaticity of the evolution near a
critical point excite the system from the final ferromagnetic ground state. For
weak decoherence and relatively fast quenches the excitation energy,
proportional to the number of kinks in the final state, decays like an inverse
square root of a quench time, but slow transitions or strong decoherence make
it decay in a much slower logarithmic way. We also find that fidelity between
the final ferromagnetic ground state and a final state after a quench decays
exponentially with a size of a chain, with a decay rate proportional to average
density of excited kinks, and a proportionality factor evolving from 1.3 for
weak decoherence and fast quenches to approximately 1 for slow transitions or
strong decoherence. Simultaneously, correlations between kinks randomly
distributed along the chain evolve from a near-crystalline anti-bunching to a
Poissonian distribution of kinks in a number of isolated Anderson localization
centers randomly scattered along the chain.
