Noisy propagation of Gaussian states in optical media with finite bandwidth

We address propagation and entanglement of Gaussian states in optical media characterised by nontrivial spectral densities. In particular, we consider environments with a finite bandwidth J(ω ) = J_0 [ θ (ω -Ω ) - θ (ω - Ω - δ )] , and show that in the low temperature regime T≪Ω ^-1 : (i) secular terms in the master equation may be neglected; (ii) attenuation (damping) is strongly suppressed; (iii) the overall diffusion process may be described as a Gaussian noise channel with variance depending only on the bandwidth. We find several regimes where propagation is not much detrimental and entanglement may be protected form decoherence.