Dynamical paths and universality in continuous-variable open systems

We address the dynamics of quantum correlations in continuous-variable open systems and analyze the evolution of bipartite Gaussian states in independent noisy channels. In particular, we introduce the notion of dynamical path through a suitable parametrization for symmetric states and focus attention on phenomena that are common to Markovian and non-Markovian Gaussian maps under the assumptions of weak coupling and the secular approximation. We find that the dynamical paths in the parameter space are universal, that is, they depend only on the initial state and on the effective temperature of the environment, with non-Markovianity that manifests itself in the velocity of running over a given path. This phenomenon allows one to map non-Markovian processes onto Markovian ones and may reduce the number of parameters needed to study a dynamical process, e.g., it may be exploited to build constants of motions valid for both Markovian and non-Markovian maps. Universality is also observed in the value of Gaussian discord at the separability threshold, which itself is function of the initial conditions only, in the limit of high temperature. We also prove the existence of excluded regions in the parameter space, i.e., sets of states that cannot be linked by any Gaussian dynamical map.

Figure 1

On the left are Markovian thermalization paths at different temperatures. The initial state of the two oscillators is a two-mode squeezed vacuum with $r_0=1.2$; the different paths (black lines) correspond to different numbers of thermal photons in the environment. From top to bottom we have paths for $n_T=0,0.1,0.5,1.0,10$. The solid blue line corresponds to thermal product states $\nu \otimes \nu$ with zero discord. The solid red line denotes the separability threshold. The high-temperature limit is already achieved for $n_T\gtrsim 3$. On the right are dynamical paths in the high-temperature limit for the two oscillators initially prepared in a two-mode squeezed vacuum state with different values of initial entanglement. The curves correspond to (from bottom to top) $r_0=0.5,0.7,1.0,1.5,2.0$.

Figure 2

Discord at the separability threshold as a function of the initial squeezing. The solid black line denotes the universal function $D_{\mathrm{sep}}\left(r_0\right)$ of Eq. (19) obtained in the high-temperature limit, whereas the colored symbols are the solutions of the full non-Markovian dynamics in Eq. (8) for $n_T=10$ and for three different environment spectra corresponding to Ohmic, super-Ohmic, and white-noise spectral density, respectively (red circles, green diamonds, and blue squares). The horizontal dashed line is the high-temperature high-squeezing limiting value $d_{*}\simeq 0.3863$. The dashed gray lines denote the low-temperature Markovian curves $D(\frac{1}{2}+c\left(t_{\mathrm{sep}}\right),c\left(t_{\mathrm{sep}}\right))$ for $n_T=0.5,0.1,{10}^{-2},{10}^{-3}$ respectively. We also report the solutions of the full non-Markovian dynamics for $n_T=0.5,0.1$ and the same three spectra, whereas for $n_T={10}^{-2},{10}^{-3}$ the separability threshold is definitely in the Markovian regime.