Dynamics and non-equilibrium steady state in a system of coupled harmonic oscillators

•A non-equilibrium pre-Lindblad master equation with a linear interaction is analytically solved.•The evolution of the logarithmic negativity is studied.•The approach to non-equilibrium steady state is studied via the entropy.•It is found that there is no steady state entanglement.

A system of two coupled oscillators, each of them coupled to an independent reservoir, is analysed. The analytical solution of the non-rotating wave master equation is obtained in the high-temperature and weak coupling limits. No thermal entanglement is found in the high-temperature limit. In the weak coupling limit the system converges to an entangled non-equilibrium steady state. A critical temperature for the appearance of quantum correlations is found.