Coherence and entanglement of mechanical oscillators mediated by coupling to different baths

We study the non-equilibrium dynamics of two coupled mechanical oscillators with general linear couplings to two uncorrelated thermal baths at temperatures $T_1$ and $T_2$, respectively. We obtain the complete solution of the Heisenberg-Langevin equations, which reveal a coherent mixing among the normal modes of the oscillators as a consequence of their off-diagonal couplings to the baths. Unique renormalization aspects resulting from this mixing are discussed. Diagonal and off-diagonal (coherence) correlation functions are obtained analytically in the case of strictly Ohmic baths with different couplings in the strong and weak coupling regimes. An asymptotic non-equilibrium stationary state emerges for which we obtain the complete expressions for the correlations and coherence. Remarkably the coherence survives in the high temperature, classical limit for $T_1 \neq T_2$. In the case of vanishing detuning between the oscillator normal modes both coupling to one and the same bath the coherence retains memory of the initial conditions at long time. A perturbative expansion of the early time evolution reveals that the emergence of coherence is a consequence of the entanglement between the normal modes of the oscillators \emph{mediated} by their couplings to the baths. This \emph{suggests} the survival of entanglement in the high temperature limit for different temperatures of the baths which is essentially a consequence of the non-equilibrium nature of the asymptotic stationary state. An out of equilibrium setup with small detuning and large $|T_1- T_2|$ produces non-vanishing steady-state coherence and entanglement in the high temperature limit of the baths.