Entanglement of a pair of atomic qubits near a carbon nanotube

We investigate the entanglement of quasi-one-dimensional atomic polariton states formed by atoms (ions) located close to or encapsulated inside carbon nanotubes and strongly coupled to the nanotube virtual photonic modes. Small-diameter metallic nanotubes are shown to result in sizable amounts of two-qubit atomic entanglement, which persists with no damping for very long times.

Figure 1

(Color online) (a) Transverse local photonic DOS functions ${\xi }^{\perp }\left(x\right)$ for the two-level atom in the centers of the four “zigzag” nanotubes. (b) Normalized two-particle local photonic DOS functions ${\xi }^{+}$ (solid lines) and ${\xi }^{-}$ (dashed lines) taken at the peak frequencies of ${\xi }^{\perp }\left(x\right)$ [see (a)], as functions of the distances between the two atoms on the axes of the (10,0) (lines 1, $x=0.29$), (11,0) (lines 2, $x=0.25$), and (12,0) (lines 3, $x=0.24$) CN’s. (c) Same as in (b) for the two atoms on the axis of the (9,0) CN $(x=0.32)$. (d) Two-particle DOS functions ${\xi }^{+}\left(x\right)$ (solid lines) and ${\xi }^{-}\left(x\right)$ (dotted lines) for the two atoms located in the center of the (9,0) CN and separated from each other by the distances of $2.1R_{cn}\approx 7.4\mathrm{\AA }$ (lines 1), $4.2R_{cn}\approx 14.8\mathrm{\AA }$ (lines 2), and $6.3R_{cn}\approx 22.2\mathrm{\AA }$ (lines 3) [shown by the vertical lines in (c)]; the dashed line shows ${\xi }^{\perp }\left(x\right)$ for the atom in the center of the (9,0) CN.

Figure 2

(Color online) (a), (b), (c) Upper decay of initially excited atom $A$ (lines 1) and initially unexcited atom $B$ (lines 2), and the entanglement (lines 3), as functions of dimensionless time for two atoms in the center of the (9,0) CN separated from each other by the distances of $2.1R_{cn}\approx 7.4\mathrm{\AA }$, $4.2R_{cn}\approx 14.8\mathrm{\AA }$, and $6.3R_{cn}\approx 22.2\mathrm{\AA }$, respectively [the situation shown in terms of the local photonic DOS in Figs. 1c, 1d]. (d) Short-time and long-time entanglement evolution of the initially fully entangled atoms separated by the distance of $7.4\mathrm{\AA }$ on the axis of the (9,0) CN; lines 1 and 2 are for $C_{+}\left(0\right)=1$, $C_{-}\left(0\right)=0$ and $C_{+}\left(0\right)=0$, $C_{-}\left(0\right)=1$, respectively.