Highly directional photon superbunching from a few-atom chain of emitters

We examine angular distribution of the probability of correlated fluorescence photon emission from a linear chain of identical equidistant two-level atoms. We selectively excite one of the atoms by a resonant laser field. The atoms are coupled to each other via the dipole-dipole interaction and collective spontaneous emission. Our attention is focused on the simultaneous observation of correlated pairs of photons. It is found that the interference between the emitting atoms can result in a highly directional emission of photon pairs. These pairs of photons exhibit strong correlations and their emission is highly concentrated into specific directions. We demonstrate the crucial role of the selective coherent excitation in such a geometrical configuration. Shifting the driving field from an atom located at one end of the chain to the other causes the radiation pattern to flip to the opposite half of the detection plane. Furthermore, we find that atomic systems in which only an atom situated at a particular position within the linear chain is driven by a laser field can radiate correlated twin photons in directions along which the radiation of single photons is significantly reduced. Alternatively, superbunching in the emitted photon statistics preferentially occurs in directions of negligible or vanishing single-photon emission. The effect of superbunching strengthens as more emitters are added to the chain. Depending on the number of atoms and the position of the driven atom within the chain, the strongly correlated pairs of photons can be emitted into few well-defined directions.

Figure 1

A chain of equidistant two-level atoms. An external laser field of Rabi frequency $\mathrm{\Omega }$ drives one of the atoms (black dot). The atoms are coupled to each other through the dipole-dipole interaction and the collective spontaneous emission resulting from the coupling of the atoms to a common vacuum field. The emitted fluorescence field is detected by a single or two photodetectors at distances $R_1$ and $R_2$ and angles ${\theta }_1$ and ${\theta }_2$ from the atomic line, located in the plane normal to the plane defined by the polarization of the atomic dipole moments $\vec{\mu }·{\widehat{r}}_{ij}=0$. Left (right) blob and inset shows a coherently driven (nondriven) atom.

Figure 2

Angular distribution of $g^{\left(2\right)}(\vec{R},\vec{R})$ for a chain composed of two atoms illustrated for $r_{12}=\lambda /2$ and $\mathrm{\Omega }=0.15\gamma$, the parameter values set according to the Rabi frequency $\mathrm{\Omega }/2\pi =4$ MHz and the damping rates of the atoms $\gamma /2\pi =26$ MHz, used in the experiment of Loo et al. [80]. (a) Shows the angular distribution corresponding to the excitation of the left-end atom whereas (b) corresponds to the excitation of the right-end atom with the same Rabi frequency. The black dot indicates the laser-driven atom.

Figure 3

Angular distribution of $g^{\left(2\right)}(\vec{R},\vec{R})$ for a chain composed of two atoms illustrated for two different separations between the atoms and two different excitation configurations. In (a) and (b), $r_{12}=\lambda /4$, and in (c) and (d), $r_{12}=\lambda /2$. (a), (c) Show angular distributions corresponding to the excitation of the left-end atom with a laser field of the Rabi frequency $\mathrm{\Omega }=0.02\gamma$. (b), (d) Correspond to the excitation of the right-end atom with the same Rabi frequency. The black dot indicates the laser-driven atom.

Figure 4

Variation of the degree of the first-order coherence ${\upsilon }_{12}$ with the scaled interatomic separation $r_{12}/\lambda$ for the case of the left-side atom driven by a laser field of Rabi frequency $\mathrm{\Omega }=0.02\gamma$.

Figure 5

Angular distribution of $G^{\left(2\right)}(\vec{R},\vec{R})/{\left[u\left(\widehat{R}\right)\right]}^2$ (solid black line) and $G^{\left(1\right)}\left(\vec{R}\right)/u\left(\widehat{R}\right)$ (dashed blue line) for $N=3$ with $r_{12}=r_{23}=\lambda /4$ plotted against detection angle $\theta$. Different Rabi frequencies of the laser field drive the left-end atom, (a) $\mathrm{\Omega }=0.02\gamma$, (b) $\mathrm{\Omega }=0.2\gamma$, (c) $\mathrm{\Omega }=\gamma$, and (d) $\mathrm{\Omega }=10\gamma$. Also shown is the normalized second-order correlation function $g^{\left(2\right)}(\vec{R},\vec{R})$ (dashed-dotted red line). The curves in (a) and (b) have been scaled with constant factors.

Figure 6

Angular distribution of $g^{\left(2\right)}(\vec{R},\vec{R})$ for a chain composed of three atoms with $r_{12}=r_{23}=\lambda /4$ and for different excitation configurations, (a) left-end atom, (b) middle atom, and (c) right-sided atom driven by a laser field of the Rabi frequency $\mathrm{\Omega }=0.02\gamma$. The black dot indicates the laser-driven atom.

Figure 7

Comparison of the angular distributions of the correlation function $G^{\left(1\right)}\left(\vec{R}\right)/u\left(\widehat{R}\right)$ (solid blue line) with that of $g^{\left(2\right)}(\vec{R},\vec{R})$ (dashed red line) plotted using logarithmic scale against the detection angle $\theta$. The parameters are same as in Fig. 6.

Figure 8

Angular distribution of $g^{\left(2\right)}(\vec{R},\vec{R})$ for a chain composed of (a) $N=4$, (b) $N=5$ equidistant atoms with $r_{ij}=\lambda /4$. The left-end atom is driven by a laser field of the Rabi frequency $\mathrm{\Omega }=0.02\gamma$. The black dot indicates the laser-driven atom.

Figure 9

Angular distribution of $g^{\left(2\right)}(\vec{R},\vec{R})$ for a chain composed of three atoms with $r_{12}=r_{23}=\lambda /4$ when (a) both end atoms are driven by a laser field of the Rabi frequency $\mathrm{\Omega }=0.02\gamma$, (b) all atoms are driven by a weak field of the Rabi frequency ${\mathrm{\Omega }}_p=0.02\gamma$ propagating along the interatomic axis, (c) in addition to the weak probe field interacting with all atoms the left-side atom is driven by another field of the same Rabi frequency. The black dots indicate the laser-driven atoms.

Figure 10

Variation of the correlation functions $[G^{\left(2\right)}(\vec{R},\vec{R})/(u\left(\widehat{R}\right){)}^2]\times {10}^3$ (solid black line), $G^{\left(1\right)}\left(\widehat{R}\right)/u\left(\widehat{R}\right)$ (dashed blue line) and $g^{\left(2\right)}(\vec{R},\vec{R})\times {10}^{-7}$ (dashed-dotted red line) in the vicinity of $\theta ={71}^{\circ }$ at which $g^{\left(2\right)}(\vec{R},\vec{R})$ exhibits a pronounced peak. Also shown is a function $C^{\left(2\right)}(\vec{R},\vec{R})\times {10}^2$ (solid green line). Inset shows the angular distribution of the correlation functions over the entire angle $\theta$. The plots are for a chain composed of three atoms separated by $r_{12}=r_{23}=\lambda /4$ and the left-end atom driven by a laser field of the Rabi frequency $\mathrm{\Omega }=0.02\gamma$.