Entanglement within the Quantum Trajectory Description of Open Quantum Systems

The degree of entanglement in an open quantum system varies according to how information in the environment is read. A measure of this contextual entanglement is introduced based on quantum trajectory unravelings of the open system dynamics. It is used to characterize the entanglement in a driven quantum system of dimension $2\times \infty$ where the entanglement is induced by the environmental interaction. A detailed mechanism for the environment-induced entanglement is given.

Figure 1

Entanglement in direct detection, for $\overline{\Omega }=1$: typical temporal behavior of the von Neumann entropy for ${\overline{\Gamma }}_b=0.5$ and ${\overline{\Gamma }}_a=0.3$ (a) , $2.0$ (b) , and $20.0$ (c) ; (d) entanglement [Eq. (2)] as a function of ${\overline{\Gamma }}_a$ (for ${\overline{\Gamma }}_b=0.5$); (e) entanglement as a function of ${\overline{\Gamma }}_a$ and ${\overline{\Gamma }}_b$.

Figure 2

Behavior of the semiclassical steady-state amplitudes $\alpha /\overline{\Omega }$ (a) and $\beta$ (b) [Eq. (9)] with increasing ${\overline{\Gamma }}_a$; the labeled points are (1) ${\overline{\Gamma }}_a=0$, (2) ${\overline{\Gamma }}_a=1/2\overline{\Omega }$, (3) ${\overline{\Gamma }}_a=1/\sqrt{2}\overline{\Omega }$, and (4) ${\overline{\Gamma }}_a\gg 1$. (c) Steady-state Wigner function for ${\overline{\Gamma }}_a=1/\sqrt{2}\overline{\Omega }$ (maximum peak separation) for $\overline{\Omega }=3$.

Figure 3

Entanglement in homodyne detection: (a) as a function of ${\overline{\Gamma }}_a$ and $\overline{\Omega }$ for $\theta =0$ (upper surface) and $\theta =\pi /2$ (lower surface); (b) for fixed $\overline{\Omega }=3$ and (i) $\theta =0$, (ii) $\theta =\pi /40$, (iii) $\theta =\pi /10$, and (iv) $\theta =\pi /2$, the dashed line is the entanglement in ensemble (11). All results are for ${\overline{\Gamma }}_b=0$.