Cooperative effects in one-dimensional random atomic gases: Absence of single-atom limit

We study superradiance in a one-dimensional geometry, where N ≫ 1 atoms are randomly distributed along a line. We present an analytic calculation of the photon escape rates based on the diagonalization of the N × N coupling matrix U_ij = cos x_ij, where x_ij is the dimensionless random distance between any two atoms. We show that unlike a three-dimensional geometry, for a one-dimensional atomic gas the single-atom limit is never reached and the photon is always localized within the atomic ensemble. This localization originates from long-range cooperative effects and not from disorder as expected on the basis of the theory of Anderson localization.