Overlapping Resonances in Open Quantum Systems

An $N$-level quantum system is coupled to a bosonic heat reservoir at positive temperature. We analyze the system-reservoir dynamics in the following regime: The strength $\lambda$ of the system-reservoir coupling is fixed and small, but larger than the spacing $\sigma$ of system energy levels. For vanishing $\sigma$ there is a manifold of invariant system-reservoir states and for $\sigma>0$ the only invariant state is the joint equilibrium. The manifold is invariant for $\sigma=0$ but becomes quasi-invariant for $\sigma>0$. Namely, on a first time-scale of the order $1/\lambda^2$, initial states approach the manifold. Then they converge to the joint equilibrium state on a much larger time-scale of the order $\lambda^2/\sigma^2$. We give a detailed expansion of the system-reservoir evolution showing the above scenario.