system [@Zurek_2005; @Dziarmaga_2005] and for the disordered system [@Dziarmaga_2006; @Caneva_2007]. Here we are going to add the effect of the coupling to an external environment, modelled through a master equation of the form in Eq. .

To retain analytic solvability of the full open-system problem [@Prosen_2008; @Eisler_2011], we will however start from a mapped version of the Ising chain  into a free-fermion model. The latter can be achieved by employing a Jordan-Wigner transformation (JWT), which maps the spin operators in terms of spinless fermions: $$\sigma_n^- = \exp \Big( i \pi \sum_{m<n} c^\dagger_m c_m \Big) c_n ,
  \label{eq:JWT}$$ where $\sigma_n^\pm = \tfrac12 (\sigma_n^x \pm i \sigma_n^y)$, while $c_n$ ($c_n^\dagger$) denotes the fermionic annihilation (creation) operator on site $n$, obeying the anticommutation relations $\{ c^\dagger_m ,c_n \} = \delta_{m,n}$, $\{ c_m, c_n \} = 0$. The resulting Hamiltonian for an Ising chain of length $L$ is quadratic in such operators and reads: $$\label{eq:Ham_quadratic}
  H(t) = \sum_{m,n} \Big[ c^{\dagger}_{m} \, A_{m,n}(t) \, c_{n}
  + \tfrac{1}{2} (c^{\dagger}_{m} \, B_{m,n} \, c^\dagger_{n} + \text{H.c.}) \Big],$$ where $A, B$ respectively are a symmetric and an antisymmetric $L \times L$ matrix whose sole non-zero elements are $A_{n,n}(t) = - \Gamma(t), A_{n,n+1} = A_{n+1,n} = - J/2 ,
