Collective Lamb Shift in Single Photon Dicke Superradiance

The collective Lamb shift and associated radiative decay of a large cloud of radius $R$ containing $N$ atoms uniformly excited by one photon of wavelength $\lambda$ is analyzed. It is shown that the time evolution of the symmetric state prepared by single photon absorption in the limit $R\gg \lambda$ is similar to that encountered in the Dicke limit of small sample ($R\ll \lambda$) superradiance. The theory includes virtual (counterrotating) terms naturally and thus provides a simple calculation of the collective Lamb shift of a single Dicke state.

Figure 1

The timed Dicke states corresponding to the first three energy levels of a uniformly excited cloud of $N$ atoms. The important $B_l$ states are given in Table . The solid lines indicate the Fano-Agarwal coupling from the initially prepared $B_0$ state. The dashed lines indicate virtual transitions in which an atom jumps from the ground state to an excited state while emitting a photon. The $B_l$ states have one excited atom and the $A_l$ states have two excited atoms. The $A_{\ell }$ state can be computed from the $B_{\ell }$ states by applying $\sum _j{e}^{i{\overrightarrow{k}}_0·{\overrightarrow{r}}_j}{\sigma }_j^{+}$.