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A case study of the non-equilibrium dynamics of open quantum systems in themarkovian approximation is presented for two dynamical models based on a singleharmonic oscillator in an external field. Specified through distinct forms ofohmic damping, their quantum Langevin equations are derived from an identicalset of physical criteria, namely the canonical commutator between position andmomentum, the Kubo formula, the virial theorem and the quantum equilibriumvariance. The associated Lindblad equations are derived but only one of them iscompletely positive. Transforming those into Fokker-Planck equations for theWigner functions, both models are shown to evolve towards the same Gibbs state,for a vanishing external field. The phenomenological differences between themodels are illustrated through their quantum relaxations and through the phasediagrammes derived from their re-interpretation as mean-field approximations ofan interacting many-body system.

From Lindblad master equations to Langevin dynamics and back