Quantum-interference-initiated superradiant and subradiant emission from entangled atoms

We calculate the radiative characteristics of emission from a system of entangled atoms which can have a relative distance larger than the emission wavelength. We develop a quantum multipath interference approach which explains both super- and subradiance though the entangled states have zero dipole moment. We derive a formula for the radiated intensity in terms of different interfering pathways. We further show how the interferences lead to directional emission from atoms prepared in symmetric $W$ states. As a byproduct of our work we show how Dicke’s classic result can be understood in terms of interfering pathways. In contrast to the previous works on ensembles of atoms, we focus on finite numbers of atoms prepared in well characterized states.

Figure 1

Scheme of the investigated system: $N$ equidistant two-level atoms are localized along a chain at positions ${\vec{R}}_1$ to ${\vec{R}}_N$. A detector at position $\vec{r}$ registers a photon, scattered by the atoms, in the far field.

Figure 2

Possible quantum paths of the initially separable state $|S_{2,0}\rangle$. Black circles denote atoms in the excited state and white circles denote atoms in the ground state. The middle row depicts the different quantum paths. The lower row displays the final states of the atoms and the phases accumulated by the photon along the different quantum paths. See text for details.

Figure 3

Possible quantum paths of the initial $W$ state $|W_{2,1}\rangle$.

Figure 4

Point plot of the maximum of $I_{|W_{n_e,N-n_e}\rangle }$ as a function of $n_e$ and $N=n_e+n_g$ (the surface serves only to guide the eye). ${\left[I_{|W_{N/2,N/2}\rangle }\right]}^{\mathit{max}}$, given in units of the intensity produced by a one-atom system, is highlighted by the solid line.

Figure 5

Intensity distribution $I_{|W_{1,N-1}\rangle }\left(\theta \right)$ of the state $|W_{1,N-1}\rangle$ as a function of the detector position $\theta$ (cf. Fig. 1) for different numbers of atoms $N$ ($N=2$ dotted, $N=4$ striped, $N=8$ solid) and with $kd=\frac{3}{2}\pi$. $I_{|W_{1,N-1}\rangle }$ is given in units of the intensity produced by a one-atom system. A strong directionality in the emission of radiation by the symmetric $W$ states can be seen.

Figure 6

Intensity distribution of the state $|W_{n_e,10-n_e}\rangle$ for $N=10$ and different numbers of excited atoms $n_e$ ($n_e=3$ striped, $n_e=5$ solid, $n_e=7$ dotted); $kd=\frac{3}{2}\pi$ and $I_{|W_{n_e,10-n_e}\rangle }$ is given in units of the intensity produced by a one-atom system.

Figure 7

Contour plots of the intensity distribution $I_{|W_{1,4}\rangle }$ of the state $|W_{1,4}\rangle$ dependent on the observation angle $\theta$ and the interatomic spacing $kd$ in units of $\pi$. The numbers 1 and 3 on the left-hand side in each plot indicate the two contours $I_{|W_{1,4}\rangle }=1$ and $I_{|W_{1,4}\rangle }=3$, i.e., dark areas correspond to a low intensity, bright areas to a high intensity. $I_{|W_{1,4}\rangle }$ is given in units of the intensity produced by a one-atom system.

Figure 8

Quantum paths of the antisymmetric state $|W_{2,1}^{-}\rangle$. Black circles denote atoms in the excited state and white circles denote atoms in the ground state. The middle row depicts the different quantum paths. The lower row displays the final states of the atoms and the phases accumulated by the photon due to different quantum paths. Quantum paths leading to constructive interference are denoted by solid arrows and quantum paths leading to destructive interference are depicted by dashed arrows. See text for details.

Figure 9

Intensity of the initial anti-symmetric state $|W_{2,1}^{-}\rangle$ (striped), the anti-symmetric state $|{\tilde{W}}_{2,1}^{-}\rangle$ (solid) and the symmetric $W$-state $|W_{2,1}\rangle$ (dotted) as a function of the observation angle $\theta$; $kd=\frac{3}{2}\pi$ and $I$ is given in units of the intensity produced by a one-atom system.