Tripartite nonlocality and continuous-variable entanglement in thermal states of trapped ions

We study a system of three trapped ions in an anisotropic bidimensional trap. By focusing on the transverse modes of the ions, we show that the mutual ion-ion Coulomb interactions set entanglement of a genuine tripartite nature, to some extent persistent to the thermal nature of the vibronic modes. We tackle this issue by addressing a nonlocality test in the phase space of the ionic system and quantifying the genuine residual tripartite entanglement in the continuous variable state of the transverse modes.

Figure 1

Scheme of the physical system at hand. We consider three ions in a parabolic potential along the longitudinal direction (the trapping potential ${\omega }_z$ is the same for all the ions). Ions 1 and 3 have the same mass $m$, while ion 2 (which occupies the position $z=0$ in our reference frame) has mass $M$. The ions are also confined in the transverse direction with the two end-chain ions having trapping frequency ${\omega }_x$. The transverse trapping frequency scales inversely with the mass of the ion, so it will be different for the central one. The equilibrium distance between an ion and its nearest neighbor is ${\delta }_{\text{eq}}$ (see text for details).

Figure 2

(a) The maximum value of $|M_3|$ as a function of $\alpha$ and $\mu$ at zero temperature. The inset shows that the quasiplateau seen in the main panel is always above the local-realistic bound. (b) The maximum value of $|M_3|$ as a function of $\alpha$ and $\mu$ at temperature $T={10}^{-6}$ K, with ${\omega }_z=1$ MHz. (c) We show the maximum of $|M_3|$ against ${\log }_{10}T$ for two different working points. The solid line is for $(\mu ,\alpha )=(1,2.1909)$ while the dashed one is for $(\mu ,\alpha )=(1,2.3)$. The straight line indicates the boundary imposed by local hidden-variable theories. We have assumed the same value of ${\omega }_z$ as in panel (b).

Figure 3

Panels (a), (b), and (c): We show the behavior of $E_{1|23}$ (solid lines) and $E_{2|13}$ (dashed ones) against $\alpha$ for $T=0$ and $\mu =0.5$, 1, and $1.5$, respectively. Panels (d) and (e): The same quantities are plotted against $\alpha$ and $\overline{n}$ for $\mu =1$. Qualitatively similar behaviors hold for other values of $\mu$.

Figure 4

Residual tripartite entanglement in the vibrational state of the transverse modes of our system. We study ${\tau }_{123}$ against $\alpha$ at $T=0$ and for three values of $\mu$. Each curve is given in a range of parameters such that both stability and the validity of the second-order approach are guaranteed. The residual tripartite entanglement is always finite, with a maximum value that depends on $\mu$.