Adiabatic transitions in a two-level system coupled to a free Boson reservoir

We consider a time-dependent two-level quantum system interacting with a free Boson reservoir. The coupling is energy conserving and depends slowly on time, as does the system Hamiltonian, with a common adiabatic parameter $\varepsilon$. Assuming that the system and reservoir are initially decoupled, with the reservoir in equilibrium at temperature $T\ge 0$, we compute the transition probability from one eigenstate of the two-level system to the other eigenstate as a function of time, in the regime of small $\varepsilon$ and small coupling constant $\lambda$. We analyse the deviation from the adiabatic transition probability obtained in absence of the reservoir.