Interplay of vacuum-mediated inter- and intra-atomic couplings in a pair of atoms

The resonance fluorescence emitted by a system of two dipole-dipole interacting nearby four-level atoms in a $J=1/2\leftrightarrow J=1/2$ configuration is studied. This setup is the simplest realistic model system which provides a complete description of the (inter-atomic) dipole-dipole interaction for arbitrary orientation of the inter-atomic distance vector, and at the same time allows for intra-atomic spontaneously generated coherences. Our main interest is the interplay of both these different coupling mechanisms. We discuss different methods to analyze the contribution of the various vacuum-induced coupling constants to the total resonance fluorescence spectrum. These allow us to find a dressed state interpretation of the contribution of the different inter-atomic dipole-dipole couplings to the total spectrum. We further study the role of the spontaneously generated coherences, and identify two different contributions to the single-particle vacuum-induced couplings. We show that they have a noticeable impact on the total resonance fluorescence spectrum down to small inter-atomic distances, even though the dipole-dipole coupling constants then are much larger in magnitude than the the single-particle coupling constants. Interestingly, we find that the inter-atomic couplings can induce an effect of the intra-atomic spontaneously generated coherences on the observed spectra which is not present in single-atom systems.

Figure 1

Vacuum-mediated couplings between different transition dipole moments. (A–C) illustrate intra-atomic or single-particle vacuum-induced couplings (SPVC) between different transition dipole moments in a single atom. (D–F) show a generalized form of these SPVC in a four-level $J=1/2\leftrightarrow J=1/2$ level scheme. (G–I) depict inter-atomic or two-particle vacuum induced couplings (TPVC) between transition dipole moments in different atoms. In all three cases, the system is initially excited on one of the involved transitions, see (A,D,G). In the second step, a virtual photon is emitted together with the deexcitation of the atom (B,E,H). Finally, the virtual photon is reabsorbed on the second transition (C,F,I).

Figure 2

The analyzed model system consisting of two nearby four-level atoms in $J=1/2\leftrightarrow J=1/2$ atoms. This configuration is the simplest realistic system in which both single-particle and two-particle vacuum-induced couplings occur. (a) Geometrical setup. We assume that atom 1 is located at the origin and atom 2 at the point ${\mathit{r}}_2$. (b) Inner structure of each of our two atoms in the $J=1/2\to J=1/2$ configuration. Note that the energy differences are not to scale.

Figure 3

Energy shifts of the system dressed states, evaluated as the imaginary part ${\upsilon }_j$ of the eigenvalues ${\xi }_j={\chi }_j+i{\upsilon }_j$ of the matrix $\mathcal{M}$ governing the system dynamics. The inter-atomic distance vector has direction given by $\theta =\pi /2$ and $\varphi =\pi /4$. (a) Case of large inter-atomic distance $r_{12}=10{\lambda }_{\pi }$ in dependence of the Rabi frequency $\Omega$. The four branches clearly show the splitting of the spectrum into a central feature and Mollow sidebands due to the ac Stark splitting of the bare states. The horizontal dotted lines mark the values for $\Omega$ chosen for calculations in the later sections. (b) Energy shifts as a function of the inter-atomic distance for fixed Rabi frequency $\Omega =10{\gamma }_{\pi }$. The three different cases of large, intermediate, and small distance can clearly be distinguished. The horizontal dotted lines mark distances for which we will present resonance fluorescence spectra in later sections.

Figure 4

Linearly polarized resonance fluorescence spectra for different inter-atomic distances $r_{12}$. In curve (a) we chose $r_{12}=0.08{\lambda }_{\pi }$, in (b) $r_{12}=0.1{\lambda }_{\pi }$, in (c) $r_{12}=0.2{\lambda }_{\pi }$, and in (d) $r_{12}=10{\lambda }_{\pi }$. In all curves, the Rabi frequency is $\Omega =10{\gamma }_{\pi }$, and the laser field is applied on resonance $\Delta =0$. The inter-atomic distance vector is aligned with $\theta =\pi /2$ and $\varphi =\pi /4$.

Figure 5

Resonance fluorescence spectra at small inter-atomic distances. In (a) $r_{12}=0.04{\lambda }_{\pi }$, in (b) $r_{12}=0.046{\lambda }_{\pi }$ and in (c) $r_{12}=0.06{\lambda }_{\pi }$. All other parameters are chosen as in Fig. 4.

Figure 6

Here, we show the $\pi$ spectrum for the distance $r_{12}=0.04{\lambda }_{\pi }$ while we artificially set some inter-atomic couplings equal to zero. In each curve of the upper subfigure, except curve (d), only the couplings of one ${\mathcal{G}}_a$ are on. In curve (a) the parameters of ${\mathcal{G}}_2$ are turned on, in (b) the ones of ${\mathcal{G}}_3$, in (c) the ones of ${\mathcal{G}}_3$, and in (d) we plot the spectrum with all couplings on. In the lower subfigure, we turn one group of inter-atomic couplings after the other on. In curve (a) only coupling parameters of ${\mathcal{G}}_2$ are on, in (b) the couplings of ${\mathcal{G}}_2$ and ${\mathcal{G}}_3$, in (c) the constants of ${\mathcal{G}}_2$, ${\mathcal{G}}_3$, and ${\mathcal{G}}_4$ are on, and curve (d) depicts the spectrum with all couplings on. All other parameters are chosen as in Fig. 4.

Figure 7

Incoherent resonance fluorescence spectrum emitted by the $\pi$ transitions for the case where the couplings of ${\mathcal{G}}_2$ and ${\mathcal{G}}_3$ are considered, and in addition we progressively switch on the parameters of ${\mathcal{G}}_4$. In curve (a) the coupling constants of ${\mathcal{G}}_4$ are zero, in (b) they are multiplied by $0.1$, in (c) by $0.3$, in (d) by $0.6$, and in (e) by 1. All other parameters are as in Fig. 6.

Figure 8

Splitting of the atomic energy levels due to the inter-atomic coupling parameters for a small distance $r_{12}=0.04{\lambda }_{\pi }$. For this inter-atomic distance, the values of the coupling constants are $\left|{\Omega }_2\right|=286.5{\gamma }_{\pi }$, $\left|{\Omega }_3\right|=\left|{\Omega }_5\right|=91.6{\gamma }_{\pi }$, and $\left|{\Omega }_4\right|=103.2{\gamma }_{\pi }$.

Figure 9

Impact of the intra-atomic couplings on the resonance fluorescence spectrum emitted by the $\pi$ transitions for intermediate distances. The parameters are $r_{12}=0.09{\lambda }_{\pi }$, $\Omega =6{\gamma }_{\pi }$, $\Delta =-14{\gamma }_{\pi }$, $\theta =\pi /2$, and $\varphi =\frac{\pi }{4}$. In curve (i) the complete spectrum with all couplings is shown as it could be observed in an experiment. In (ii), all intra-atomic couplings are artificially turned off. In (iii), all intra-atomic couplings entering the equations of motion are kept, while those entering the expression for the spectrum Eq. (18) are set to zero. In (iv), the intra-atomic couplings in the expression for the spectrum are kept, while those in the equations of motion are dropped.

Figure 10

Impact of the intra-atomic couplings on the resonance fluorescence spectrum emitted by the $\pi$ transitions in the small-distance case. The parameters are as in Fig. 5a. The three subfigures separately show the three spectral features visible in the right half of the spectrum in Fig. 5a. Each subfigure shows four curves (i)–(iv). As in Fig. 9, these correspond to the complete spectrum (i), the spectrum without intra-atomic couplings (ii), and the spectrum with either the intra-atomic coupling entering the expression for the spectrum artificially suppressed (iii) or those entering the equations of motion switched off (iv).

Figure 11

Intra-atomic couplings induced by inter-atomic couplings. (a) Steady-state population of state $|1\rangle$ in atom 1 obtained by tracing over the second atom. (b) Imaginary part of the coherence between states $|1\rangle$ and $|3\rangle$ in atom 1. The different curves are as follows: (i) All couplings included, large-distance case corresponding to no inter-atomic couplings. (ii) Large-distance case, without intra-atomic couplings entering the equations of motion. (iii) and (iv) show the corresponding results for small distance $r_{12}=0.09{\lambda }_{\pi }$. The parameters are as in Fig. 9 except for the variable detuning $\Delta$. Note that in (b), the curves (i) and (ii) are shown multiplied by a factor of $1/2$ for better visibility.