We study the non-equilibrium dynamics of two coupled mechanical oscillatorswith general linear couplings to two uncorrelated thermal baths at temperatures$T_1$ and $T_2$, respectively. We obtain the complete solution of theHeisenberg-Langevin equations, which reveal a coherent mixing among the normalmodes of the oscillators as a consequence of their off-diagonal couplings tothe baths. Unique renormalization aspects resulting from this mixing arediscussed. Diagonal and off-diagonal (coherence) correlation functions areobtained analytically in the case of strictly Ohmic baths with differentcouplings in the strong and weak coupling regimes. An asymptoticnon-equilibrium stationary state emerges for which we obtain the completeexpressions for the correlations and coherence. Remarkably the coherencesurvives in the high temperature, classical limit for $T_1 \neq T_2$. In thecase of vanishing detuning between the oscillator normal modes both coupling toone and the same bath the coherence retains memory of the initial conditions atlong time. A perturbative expansion of the early time evolution reveals thatthe emergence of coherence is a consequence of the entanglement between thenormal modes of the oscillators \emph{mediated} by their couplings to thebaths. This \emph{suggests} the survival of entanglement in the hightemperature limit for different temperatures of the baths which is essentiallya consequence of the non-equilibrium nature of the asymptotic stationary state.An out of equilibrium setup with small detuning and large $|T_1- T_2|$ producesnon-vanishing steady-state coherence and entanglement in the high temperaturelimit of the baths.
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