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Quantum Ising model in one dimension is an exactly solvable example of aquantum phase transition. We investigate its behavior during a quench from aparamagnetic to ferromagnetic phase caused by a gradual turning off of thetransverse field at a fixed rate characterized by the quench time $\tau_Q$. Inagreement with Kibble-Zurek mechanism, quantum state of the system after thetransition exhibits a characteristic correlation length $\hat\xi$ proportionalto the square root of the quench time $\tau_Q$. The inverse of this correlationlength determines average density of defects after the transition. In thispaper, we show that $\hat\xi$ also controls the entropy of entanglement of ablock of $L$ spins with the rest of the system. For large $L$, this entropysaturates at $\frac16\log_2\hat\xi$, as might have been expected from theKibble-Zurek mechanism. Close to the critical point, the entropy saturates whenthe block size $L\approx\hat\xi$, but -- in the subsequent evolution in theferromagnetic phase -- a somewhat larger length scale$l=\sqrt{\tau_Q}\ln\tau_Q$ develops as a result of quantum dephasing, and theentropy saturates when $L\approx l$. We also study the spin-spin correlation.We find that close to the critical point ferromagnetic correlations decayexponentially with the dynamical correlation length $\hat\xi$, but in thefollowing evolution this correlation function becomes oscillatory at distancesless than this scale. However, both the wavelength and the correlation lengthof these oscillations are still determined by $\hat\xi$. We also deriveprobability distribution for the number of kinks in a finite spin chain afterthe transition.Comment: 12 pages, 3 figures; updated references and minor cosmetic change

Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model