Quantum Entanglement of Two Bosonic Modes in Two-reservoir Model∗

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable entanglement for a system consisting of two non-interacting bosonic modes embedded in two independent thermal environments. By using the Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all temperatures of the thermal reservoirs, an initial separable non-squeezed Gaussian state remains separable for all times. In the case of an entangled initial squeezed vacuum state, entanglement suppression (entanglement sudden death) takes place, for all temperatures of the thermal baths. For definite values of temperatures and dissipation constants, one can observe temporary revivals of the entanglement, but the system evolves asymptotically to an equilibrium state which is always separable.