Radiation from anN-Atom System. II. Spontaneous Emission from a Pair of Atoms

Abstract

Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms $A_1$, $A_2$, whose separation $r_{21}$ can be comparable to the wavelength $\lambda$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\pi$ pulse, and (c) only $A_1$ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators $O_1$, $O_2$. The atoms exhibit superradiant behavior, whereas $O_1$, $O_2$ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\theta ,t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\to \infty$ is the same as in the independent-atom case $r_{21}\gg \lambda$. For the oscillators, the lobes of $\mathcal{R}(\theta ,t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\Omega }_{12}$ corresponding to the shifts $\pm {\Omega }_{12}$ of the triplet and singlet states due to the $A_1-A_2$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that $A_2$ and $A_2$ radiate simultaneously around the frequencies $\epsilon \pm {\Omega }_{12}$, where $\epsilon$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.