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We derive the exact out-of-equilibrium Wigner function of a bosonic modelinearly coupled to a bosonic bath of arbitrary spectral density. Our solutiondoes not rely on any master equation approach and it therefore also correctlydescribes a bosonic mode which is initially entangled with its environment. Ithas been recently suggested that non-Markovian quantum effects lead to dissi-pationless dynamics in the case of a strong coupling to a bath whose spectraldensity has a support bounded from below. We show in this work that such asystem undergoes a quantum phase transi- tion at some critical bath couplingstrength. The apparent dissipationless dynamics then correspond to therelaxation towards the new ground-state.

Exact dynamics and thermalization of an open bosonic quantum system in presence of a quantum phase transition induced by the environment