to present some simple, yet non-trivial, examples where the adiabatic dynamics can be analysed in full details. Most importantly, the phenomenology that we will extract is related to the dynamics of certain spin systems that are very close to the relevant implementations of AQC. This means that the exactly solvable models that we consider here may be used as a very useful benchmark to test important approximations in more complex cases.

The bath we will deal with is Markovian. This means that the dissipative term in Eq.  can be written in the Lindblad form $$\mathbb{D} [\rho] = \sum_n \kappa_n \Big( L_n \rho L^\dagger_n - \tfrac12 \{ \rho, L^\dagger_n L_n \} \Big) ,
  \label{eq:Lindbladian}$$ where $L_n$ are suitable local Lindblad operators that describe the environment (to be defined later) and $\kappa_n$ are the corresponding couplings, which have to be positive for a Markovian Lindblad master equation. The choice of local Lindblad operators that we are going to study does not lead to a thermal state in the steady state. We will dwell in more details on this point later.

We will consider quadratic fermionic models whose Lindblad dynamics can be worked out analytically [@Prosen_2008; @Eisler_2011; @Horstmann_2013], for different types of local 