Entanglement dynamics of two two-level atoms in the vicinity of an invisibility cloak

We study the entanglement between two identical two-level atoms located near an ideal model of invisibility cloaks by monitoring the time evolution of the concurrence measure. We obtain the reduced density operator of the atomic subsystem based on a canonical quantization scheme presented for the electromagnetic field interacting with atomic systems in the presence of an anisotropic, inhomogeneous, and absorbing magnetodielectric medium. It is shown that two atoms, which are prepared initially in an unentangled state, are correlated in the weak coupling regime via the spontaneous emission and the dipole-dipole interaction of two atoms mediated by the invisibility cloak. We therefore find that the invisibility cloak, independent of the hidden object, works fairly well at frequencies far from the resonance frequency of the object and the cloak, whereas near the resonance frequency the hidden object becomes detectable due to a sharp reduction of the concurrence.

Figure 1

The geometry of the system. Two two-level atoms are placed at fixed points ${\mathbf{r}}_A$ and ${\mathbf{r}}_B$ near a spherical invisibility cloak where the material parameters $\overline{\overline{\mathbf{\varepsilon }}}$ and $\overline{\overline{\mathbf{\mu }}}$ are given by Eq. (1). The clock has an inner and outer radius of $b$ and $a$, respectively, and a homogeneous and isotropic material is inserted to the central hollow region of the cloak, $r<b$, with the electric primitivity and the magnetic permeability function ${\varepsilon }_c$ and ${\mu }_c$.

Figure 2

The time evolution of the concurrence $\mathcal{C}$ as a function of a dimensionless parameter ${\mathrm{\Gamma }}_{0}t$ for the case that the two-atom system is placed in free space (solid blue line), in the vicinity of the cloaked object (dash-dotted green line) and near the object alone (dashed red line). Time evolution of the population $|C_{+}{|}^2+{\left|C_{-}\right|}^2$ is represented by the solid cyan curve (free space), the dashed black curve (with the cloaking shell), and the dot-dashed purple line (without cloaking shell). The two identical atoms are located at diametrically opposite positions $r{\omega }_0/c=4.7$. The material absorbtion of the central hidden object and the cloak are described by the Lorentz model with parameters $\alpha =1.3$, ${\omega }_p/{\omega }_0=0.1$. In panels (a) and (b) $\gamma /{\omega }_0=0.01$, and in panel (c) $\gamma /{\omega }_0=0.1$. The inner and outer radii of the cloaking shell [panels (a) and (c)] are $b{\omega }_0/c=3$ and $a{\omega }_0/c=4.5$, respectively, and the radius of the object without cloak in panel (b) is $b{\omega }_0/c=4.5$. The atoms are nonresonant with the cloak $\omega /{\omega }_0=0.1$, and initially prepared in the state $|u_A,l_B\rangle$. The inset shows the time evolution of $\mathcal{C}$ and the populations of states $|+\rangle$ (solid yellow line) and $|-\rangle$ (solid brown line) in the presence of the cloaked object.

Figure 3

The time evolution of the concurrence $\mathcal{C}$ as a function of a dimensionless parameter ${\mathrm{\Gamma }}_{0}t$. In panel (a) the inner and outer radii of the cloaking shell are $b{\omega }_0/c=3$ and $a{\omega }_0/c=4.5$, respectively, and in panel (b) the radius of the object without cloak is $b{\omega }_0/c=4.5$. The material parameters of the cloaking shell and the hidden object are identical to those used in Fig. 2.

Figure 4

The time evolution of the concurrence $\mathcal{C}$ as a function of a dimensionless parameter ${\mathrm{\Gamma }}_{0}t$ for (a) $\omega /{\omega }_0=0.1$ and (b) $\omega ={\omega }_0$. The atoms are resonant with the cloak $\omega /{\omega }_0=1$, and initially prepared in the state $|u_A,l_B\rangle$. The geometrical and the material parameters of the cloaking shell and the hidden object are identical to those used in Fig. 2.