Two-atom system as a nanoantenna for mode switching and light routing

We determine how a system composed of two nonidentical two-level atoms with different resonance frequencies and different damping rates could work as a nanoantenna for controlled mode switching and light routing. We calculate the angular distribution of the emitted field detected in a far-field zone of the system including the direct interatomic interactions and arbitrary linear dimensions of the system. The calculation is carried out in terms of the symmetric and antisymmetric modes of the two-atom system. We find that as long as the atoms are identical, the emission cannot be switched between the symmetric and antisymmetric modes. The switching may occur when the atoms are nonidentical and the emission can then be routed to different modes by changing the relative ratio of the atomic frequencies, or damping rates, or by a proper tuning of the laser frequency to the atomic resonance frequencies. It is shown that in the case of atoms of different resonance frequencies but equal damping rates, the light routing is independent of the frequency of the driving laser field. It depends only on the sign of the detuning between the atomic resonance frequencies. In the case of atoms of different damping rates, the emission can be switched between different modes by changing the laser frequency from the blue to red detuned from the atomic resonance. The effect of the interatomic interactions is also considered, and it is found that in the case of unequal resonance frequencies of the atoms, the interactions slightly modify the visibility of the intensity pattern. The case of unequal damping rates of the atoms is affected rather more drastically, the light routing becoming asymmetric under the dipole-dipole interaction with the enhanced intensities of the modes turned towards the atom of smaller damping rate.

Figure 1

Geometry of the system. Two nonidentical two-level atoms, labeled $\left(1\right)$ and $\left(2\right)$, are located at fixed positions a distance $r_{12}$ from each other. The atoms are driven by a cw laser field propagating in a direction that is taken to form an angle $\varphi$ with the $z$ axis. The emitted field is detected at point $A$, located on the $xy$ plane at a distance $R$ from the atoms. The position of the detector on the $xy$ plane is determined by the angle $\theta$, the direction towards the detecting point relative to the direction of the interatomic axis.

Figure 2

Angular distribution of (a) symmetric and (b) antisymmetric modes for different interatomic separations, $r_{12}=\lambda /4$, $r_{12}=\lambda /2$, and $r_{12}=\lambda$. The solid lines show the directions of the modes to which the interference term contributes positively to produce maxima in the radiation intensity, and the dashed lines show the directions of the modes to which the interference term contributes negatively resulting in minima of the intensity.

Figure 3

Collective energy states of the system. The dashed lines indicate the possible spontaneous transitions between the states. The presence of the coherence ${\rho }_{sa}$ between the intermediate states $|s\rangle$ and $|a\rangle$ is essential for mode switching and directional emission.

Figure 4

The radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda /4$, $\Omega =0.2{\Gamma }_0$, ${\Gamma }_1={\Gamma }_2$, ${\Delta }_L=0$, and different values of $\Delta$: $\Delta =0$ (solid blue line), $\Delta =0.5{\Gamma }_0$ (red dashed line), $\Delta =-0.5{\Gamma }_0$ (green dashed-dotted line), and $\Delta =20{\Gamma }_0$ (solid black line, $\times {10}^2$).

Figure 5

The angular distribution of the steady-state radiation intensity and its variation with the laser detuning ${\Delta }_L$ for $\Omega =0.2{\Gamma }_0$, $r_{12}=\lambda /2$, ${\Gamma }_1={\Gamma }_2$, and $\Delta =2{\Gamma }_0$.

Figure 6

Angular distribution of the radiation intensity for the atomic separation $r_{12}=\lambda /2$, $\Omega =0.2{\Gamma }_0$, ${\Delta }_L=-0.75{\Gamma }_0$, ${\Gamma }_1={\Gamma }_2$, and different values of $\Delta$: $\Delta =2{\Gamma }_0$ (dashed blue line), $\Delta =-2{\Gamma }_0$ (dashed-dotted green line), and $\Delta =0$ (red solid line).

Figure 7

The variation of the steady-state radiation intensity with the laser detuning ${\Delta }_L$ for $\Omega =0.2{\Gamma }_0$, $r_{12}=\lambda /2$, $\Delta =0$, ${\Gamma }_2/{\Gamma }_1=10$, and two different directions of observation, $\theta =\pi /3$ (dashed-dotted blue line) and $\theta =2\pi /3$ (dashed green line). The red solid line is the sum of the two.

Figure 8

The steady-state intensity contrast factor $C$ as a function of the laser detuning ${\Delta }_L$ for $\Omega =0.2{\Gamma }_0$ and $r_{12}=\lambda /2$. The solid blue line is for $\Delta ={\Gamma }_0,{\Gamma }_2/{\Gamma }_1=10$; the dashed red line is for $\Delta =0,{\Gamma }_2/{\Gamma }_1=10$; and the dashed-dotted green line is for $\Delta ={\Gamma }_0,{\Gamma }_2/{\Gamma }_1=1$.

Figure 9

Angular distribution of the emitted field intensity for the atomic separation $r_{12}=\lambda$, $\Omega =0.2{\Gamma }_0$, ${\Gamma }_1={\Gamma }_2$, ${\Delta }_L=-0.75{\Gamma }_0$, and different values of $\Delta$: $\Delta =2{\Gamma }_0$ (solid blue line), $\Delta =-2{\Gamma }_0$ (dashed green line), and $\Delta =0$ (red dashed-dotted line).

Figure 10

Angular distribution of the radiation intensity for the atomic separation $r_{12}=\lambda /2$ at which ${\Gamma }_{12}=-0.152{\Gamma }_0$, ${\Omega }_{12}=0.215{\Gamma }_0$, for $\Omega =0.2{\Gamma }_0$, ${\Gamma }_1={\Gamma }_2$, ${\Delta }_L=-0.75{\Gamma }_0$, and different values of $\Delta$: $\Delta =2{\Gamma }_0$ (solid blue line), $\Delta =-2{\Gamma }_0$ (dashed green line), and $\Delta =0$ (dashed-dotted red line).

Figure 11

The steady-state intensity contrast factor $C$ as a function of the laser detuning ${\Delta }_L$ for $\Omega =0.2{\Gamma }_0$, $r_{12}=\lambda /2$, ${\Gamma }_2={\Gamma }_1$, and $\Delta =2{\Gamma }_0$. The dashed red line is for independent atoms $({\Omega }_{12}={\Gamma }_{12}=0)$, and the solid blue line is for interacting atoms with numerical values of the collective parameters ${\Gamma }_{12}=-0.152{\Gamma }_0$ and ${\Omega }_{12}=0.215{\Gamma }_0$, evaluated for $r_{12}=\lambda /2$ and ${\widehat{\mu }}_{1\left(2\right)}\perp {\vec{r}}_{12}$.

Figure 12

The variation of the steady-state radiation intensity with the laser detuning ${\Delta }_L$ in the presence of the direct interaction between the atoms for $r_{12}=\lambda /2$, $\Omega =0.2{\Gamma }_0$, $\Delta =0$, ${\Gamma }_2/{\Gamma }_1=10$, and the propagation directions of the two antisymmetric modes, $\theta =\pi /3$ (solid blue line) and $\theta =2\pi /3$ (dashed green line). The red dashed-dotted line is the sum of the two.

Figure 13

The variation of the steady-state radiation intensity with the laser detuning ${\Delta }_L$ in the presence of the direct interaction between the atoms for $r_{12}=\lambda$, $\Omega =0.2{\Gamma }_0$, $\Delta =0$, ${\Gamma }_2/{\Gamma }_1=10$, and the two pairs of modes propagating at angles $(0.41\pi ,0.77\pi )$ (solid blue line) and $(0.23\pi ,0.58\pi )$ (dashed green line). The red dashed-dotted line is the sum of the two.

Figure 14

The contrast factor $C$ of the symmetric modes, Eq. (52), plotted as a function of the laser detuning ${\Delta }_L$ for $\Omega =0.2{\Gamma }_0$, $r_{12}=\lambda /4$, ${\Gamma }_2={\Gamma }_1$, and different $\Delta$: $\Delta =0$ (dashed blue line), $\Delta =0.5{\Gamma }_0$ (solid green line), and $\Delta =20{\Gamma }_0$ (dashed-dotted black line).