Determining stationary-state quantum properties directly from system-environment interactions

Considering stationary states of continuous-variable systems undergoing an open dynamics, we unveil the connection between properties and symmetries of the latter and the dynamical parameters. In particular, we explore the relation between the Lyapunov equation for dynamical systems and the steady-state solutions of a time-independent Lindblad master equation for bosonic modes. Exploiting bona fide relations that characterize some genuine quantum properties (entanglement, classicality, and steerability), we obtain conditions on the dynamical parameters for which the system is driven to a steady state possessing such properties. We also develop a method to capture the symmetries of a steady state based on symmetries of the Lyapunov equation. All the results and examples can be useful for steady-state engineering processes.

Figure 1

Separability surface $\mathcal{S}$ (bottom) and classicality surface $\mathcal{P}$ (top). States either on or above $\mathcal{S}$ are separable while states either on or above $\mathcal{P}$ are classical. Between $\mathcal{S}$ and $\mathcal{P}$ there are classical and nonclassical separable states. Below $\mathcal{S}$, there are separable and entangled states. The divergence of both functions for ${\overline{N}}_1\to 0$ and for ${\overline{N}}_2\to 0$ are not shown in this plot.

Figure 2

Necessary and sufficient conditions for classicality and for separability of the steady state. We show the functions ${\mathcal{P}}^{\prime }\left(\overline{N}\right)$ in Eq. (49) and ${\mathcal{S}}^{\prime }\left(\overline{N}\right)$ in Eq. (50), as indicated in the graph. The regions limited by these two functions indicates the nature of the steady state. We also plot the functions $\mathcal{S}(\overline{N},\overline{N})$ in Eq. (45) and $\mathcal{P}(\overline{N},\overline{N})$ in Eq. (47) as dashed curves, see Fig. 1. In this plot $\omega /\kappa =0.5$.