Geometry-dependent dynamics of two $\Lambda$-type atoms via vacuum-induced coherences

The dynamics of a pair of atoms can significantly differ from the single-atom dynamics if the distance of the two atoms is small on a scale given by the relevant transition wavelengths. Here, we discuss two nearby three-level atoms in $\Lambda$ configuration, and focus on the dependence of the optical properties on the geometry of the setup. We find that in general transitions in the two atoms can be dipole-dipole coupled by interactions via the vacuum field even if their transition dipole moments are orthogonal. We give an interpretation of this effect and show that it may crucially influence the system dynamics. In particular, for a fixed setup of driving fields and detectors, the spatial orientation of the two-atom pair decides if the system reaches a true constant steady state or if it exhibits periodic oscillations in the long-time limit. As an example observable, we study the resonance fluorescence intensity, which is either constant or is modulated periodically in the long-time limit.

Figure 1

(Color online) The system setup. Atom A is located in the coordinate origin, atom B at ${\mathit{r}}_{12}$, as shown in the left part of the figure. In our coordinate system, the interatomic distance vector is parametrized by the length $r_{12}$ and the angles $\theta ,\varphi$. Internally, both atoms (A, B) are three-level systems in $\Lambda$ configuration as shown in the right subfigure. ${\Omega }_1$ $\left({\Omega }_2\right)$ is the Rabi frequency of the driving laser field coupling to transition $1\leftrightarrow 3$ $(2\leftrightarrow 3)$ with detuning ${\Delta }_1$ $\left({\Delta }_2\right)$. The two lower states have frequency difference $\delta$. The spontaneous decay rates are ${\gamma }_1$, ${\gamma }_2$.

Figure 2

(Color online) The time-dependent fluorescence intensity $I_y$. The parameters chosen are ${\Omega }_1=3\gamma$, ${\Omega }_2=5\gamma$, $\delta =0$, ${\Delta }_1=0$, ${\Delta }_2=2\gamma$, $r_{12}=0.1\lambda$, and $\varphi =\pi ∕4$. The solid line corresponds to $\theta =\pi ∕2$, the dashed line is for $\theta =0$. (a) Short-time evolution, (b) long-time limit. The oscillatory behavior of the intensity for $\theta =\pi ∕2$ remains undamped in the long-time limit.

Figure 3

(Color online) Same as in Fig. 2, but with $\delta =-2\gamma$ and thus $\Delta =0$. In this case, both for $\theta =0$ and $\theta =\pi ∕2$, the long-time limit intensity is time independent.

Figure 4

(Color online) Dependence of the modulation of the fluorescence in the long-time limit on the angle $\varphi$. (a) $\varphi =0.25\pi$, (b) $\varphi =0.1\pi$, (c) $\varphi =0.4\pi$, (d) $\varphi =0$, (e) $\varphi =0.5\pi$. The other parameters are as in Fig. 2 with $\theta =\pi ∕2$.

Figure 5

(Color online) Dependence of the vacuum-induced coupling constants ${\Gamma }_{\mathit{vc}}^{\mathit{dd}}$ and ${\Omega }_{\mathit{vc}}^{\mathit{dd}}$ on the angle $\varphi$. The parameters are as in Fig. 4. The solid curve shows ${\Omega }_{\mathit{vc}}^{\mathit{dd}}$, the dashed curve is for $250\times {\Gamma }_{\mathit{vc}}^{\mathit{dd}}$. The vertical lines indicate the phase values shown in Fig. 4.

Figure 6

(Color online) Dependence of the modulation of the fluorescence in the long-time limit on the angle $\theta$. (a) $\theta =0.25\pi$, (b) $\theta =0.1\pi$, (c) $\theta =0.4\pi$, (d) $\theta =0$, (e) $\theta =0.5\pi$. The other parameters are as in Fig. 2 with $\varphi =\pi ∕4$.

Figure 7

(Color online) Dependence of the vacuum-induced coupling constants ${\Gamma }_{\mathit{vc}}^{\mathit{dd}}$ and ${\Omega }_{\mathit{vc}}^{\mathit{dd}}$ on the angle $\theta$. The parameters are as in Fig. 6. The solid curve shows ${\Omega }_{\mathit{vc}}^{\mathit{dd}}$, the dashed curve is for $250\times {\Gamma }_{\mathit{vc}}^{\mathit{dd}}$. The vertical lines indicate the phase values shown in Fig. 6.

Figure 8

(Color online) Dependence of the modulation of the fluorescence in the long-time limit on the lower-state splitting $\delta$. Parameters are as in Fig. 2 with $\theta =\pi ∕2$, but with ${\Delta }_1=-\gamma$ and ${\Delta }_2=\gamma$. (a) $\delta =-2\gamma$, (b) $\delta =-\gamma$, (c) $\delta =0$, (d) $\delta =\gamma$, (e) $\delta =2\gamma$.