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Non-Equilibrium Dynamics of a Noisy Quantum Ising Chain: statistics of the work and prethermalization after a sudden quench of the transverse field

We discuss the non-equilibrium dynamics of a Quantum Ising Chain (QIC) following a quantum quench of the transverse field and in the presence of a gaussian time dependent noise. We discuss the probability distribution of the work done on the system both for static and dynamic noise. While the effect of static noise is to smooth the low energy threshold of the statistic of the work, appearing for sudden quenches, a dynamical noise protocol affects also the spectral weight of such features. We also provide a detailed derivation of the kinetic equation for the Green's functions on the Keldysh contour and the time evolution of observables of physical interest, extending previously reported results (J. Marino, A. Silva, Phys. Rev. B 86, 060408 (2012)), and discussing the extension of the concept of prethermalization which can be used to study noisy quantum many body hamiltonians driven out-of-equilibrium.