Hidden equivalence in the collective emission from a dilute atomic cloud

We investigate the collective spontaneous emission from an ensemble of two-level atoms evenly distributed inside a sphere with a low density. An initial symmetric single-excitation state is considered and polarizations for all atoms are assumed to be aligned in the same direction. We find that the superradiant decay rate of the ensemble exhibits oscillatory features dependent on the radius of the sphere and the atomic radiation wavelength, indicating that the collective emission rate can be less than the single-atom decay rate for certain parameters. Moreover, the system exhibits a hidden equivalence where the collective emission from the ensemble gives the same radiation rate as the case of a two-atom model where one atom is at the center of the sphere and the other effective large atom, composed of all other atoms, is localized at the edge of the sphere along the polarization direction. Our result hence provides a potential method towards exploring complex many-body physics by simplifying the model into a two-body problem.

Figure 1

(a) Equally distributed spherical cloud of atoms where their electric dipoles have the same magnitudes and orientations. (b) The spherical coordinate describes the relativity positions between the $m\mathrm{th}$ atom and the first atom located at the center of the sphere. The $z$ axis is along the polarization direction. (c) Schematic illustration of an equivalent two-atom model in which one atom is at the center and another effective large atom with the dipole $(N-1)\mu$ is set at the edge of the sphere.

Figure 2

Plot of $f\left(\xi \right)\equiv 3(sin\xi -\xi cos\xi )/{\xi }^3$ as a function of the phase difference $\xi$.

Figure 3

(a) Collective spontaneous emission rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}^{11}$ as a function of the phase difference $\xi$ and the atomic density $n$. The black line denotes the constant atomic density $n=3/4\pi {\lambda }^3$ and the red dashed line the constant atomic number $N=1000$. (b) Collective decay rate as a function of the phase difference $\xi$ for constant atomic density $n=3/4\pi {\lambda }^3$ (black solid line) and constant number of atoms $N=1000$ (red dashed line).