Extraction of genuine tripartite entanglement from the vacuum

We demonstrate and characterize the extraction of genuine tripartite entanglement from the vacuum of a periodic cavity field. That is, three probe quantum systems (detectors) can become both bipartitely and tripartitely entangled without coming into causal contact, by means of interaction with a common quantum field. We do this by using an oscillator-detector model that forgoes the need for perturbation theory and which instead is solved exactly. We find that the extraction of tripartite entanglement is considerably easier than that of bipartite. As a secondary result, we also compare a periodic cavity with one that has Dirichlet boundary conditions. We find that the extraction of both bipartite and tripartite entanglement is more easily achieved using the former case.

Figure 1

Convergence of the value of entanglement $E_{ms}$ (notation explained in Sec. 3) under increasing number of cavity field modes, $N$. The parameter values are $\lambda =0.01,L=10,T=r$, and $\Omega =0.4\pi$ in natural units and the boundary conditions are Dirichlet.

Figure 2

Entanglements $E_{ss},{\overline{E}}_{sss}$ in the system subject to periodic boundary conditions, as a function of $T$ and $\Omega$ at $L=10$ and $\lambda =0.01$. Right: regions of existence of $E_{ss}$ and ${\overline{E}}_{sss}$ plotted together. All quantities are given in natural units.

Figure 3

Entanglements $E_{ss},{\overline{E}}_{sss}$ in the system subject to periodic boundary conditions, as a function of $L$ and $\Omega$ at $T=0.4r$ and $\lambda =0.01$. Right: regions of existence of $E_{ss}$ and ${\overline{E}}_{sss}$ plotted together. All quantities are given in natural units.

Figure 4

Entanglements $E_{ms},E_{ss}$, and $E_{m|ss}$ in the system subject to hard-wall boundary conditions, as a function of $T$ and $\Omega$ at $L=10$ and $\lambda =0.01$. Bottom right: regions of existence of $E_{ms}$ and $E_{m|ss}$ plotted together. All quantities are given in natural units.

Figure 5

A comparison, between the cases of periodic and Dirichlet boundary conditions, of the regions in which bipartite entanglement is harvested between neighboring detectors. Here $L=10$ and $\lambda =0.01$. All quantities are given in natural units.