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Now we treat specifically the double quantum dot system in which one part of the system plays the role of "detector" whereas the other dot is the so-called "device". By doing this we model the particular $P(E)$ function \cite{Aguado_2000}. Let us assume the following Hamiltonian   \begin{multline}  H=\sum_{k\alpha\sigma} \epsilon_{\alpha k\sigma} c^\dagger_{\alpha k\sigma} c_{\alpha k\sigma} + \epsilon_{d\sigma} n_{d\sigma}+ n_{d\sigma}  \\  +  E_C [n_d-n}]^2 (n_{d\sigma}-n})^2  \end{multline}