Dynamics of bipartite and tripartite entanglement in a dissipative system of continuous variables

With the help of the characteristic function and the covariance matrix, we study the effect of a dissipative bath on bipartite entanglement in a system of continuous variables with no direct inter-mode coupling. Under the dissipation induced by an Agarwal bath, the time evolution of entanglement between two oscillators initially in a squeezed state has been examined, and an analytical criterion, as a function of the squeezing parameter s and the bath temperature n̄, has been derived for bipartite entanglement at long times, resulting in an entanglement phase diagram comprising of a sudden death region and a no-sudden death region. If (s, n̄) lies in the no-sudden death region, there will be sustained bipartite entanglement in the long-time limit. The analysis has been extended to three-mode systems including fully symmetric tripartite states and bisymmetric states, which are invariant with the exchange of two modes (out of the three). If two symmetric modes are propagated in separate baths, the bipartite entanglement between one of the two and the third mode will vanish in a finite time. The disappearance of the entanglement leads to a decrease in cloning fidelities to below the classical value of 0.5. For fully symmetric tripartite states, both the separate- and common-bath cases have been considered. It is found that entanglement vanishes in a finite time in the presence of separate baths, while it persists for a long time in the presence of a common bath.