Three-step approach to Fokker-Planck equation for an interacting open waveguide system and localizable entanglement dynamics

We consider a system of three coupled single-mode waveguides each locally interacting with its own Gaussian environment and present a general solution for this coupled system initially in any Gaussian state using the symplectic operations. We investigate the dynamics of two-mode localizable entanglement contained in the evolved state when the system is initially in three-mode bisymmetric Gaussian state in contact with the independent decoherence. We show that such an entanglement exhibits a damped oscillation in a regime of weak waveguide-waveguide coupling and small mean photon numbers of the bath. Remarkably, we find that the entanglement can reappear after the long-time death and arrives at a steady-state oscillation, whose maximum depends strongly on both the squeezing of the bath and the coupling strength between these waveguides. Finally, we generalize the approach to a common squeezed environment case.