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 x N coupling matrix Uij = cos xij, where xij 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.
Photon propagation in a gas of N atoms is studied using an effective Hamiltonian describing photon-mediated atomic dipolar interactions. The density P(Gamma) of photon escape rates is determined from the spectrum of the NxN random matrix Gamma_{ij}=sin(x_{ij})/x_{ij}, where x_{ij} is the dimensionless random distance between any two atoms. Varying disorder and system size, a scaling behavior is observed for the escape rates. It is explained using microscopic calculations and a stochastic model which emphasizes the role of cooperative effects in photon localization and provides an interesting relation with statistical properties of "small world networks."
We study the influence of cooperative effects such as superradiance and subradiance on the scattering properties of dilute atomic gases. We show that cooperative effects lead to an effective potential between two atoms that decays as
1∕r. In the case of superradiance, this potential is attractive for close enough atoms and can be interpreted as a coherent mesoscopic effect. We consider a model of multiple scattering of a photon among superradiant pairs and calculate the elastic mean free path and the group velocity. We study first the case of a scalar wave which allows us to obtain and to understand basic features of cooperative effects and multiple scattering. We then turn to the general problem of a vector wave. In both cases, we obtain qualitatively similar results and derive, for the case of a scalar wave, analytic expressions for the elastic mean free path and for the group velocity for an arbitrary detuning (near resonance).
We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\ensuremath{\gg}1$ atoms with the radiation field. We analyze the interplay between cooperative effects and disorder for both scalar and vectorial radiation fields. We show that for dense gases, the resonance width distribution follows, both in the scalar and vectorial cases, a power law $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}1}$ behavior, which has been considered as a signature of Anderson localization of light in random systems. We show that in dilute clouds, the center of the energy distribution is described by Wigner's semicircle law in the scalar and vectorial cases. For dense gases, this law is replaced in the vectorial case by the Laplace distribution. Finally, we show that in the scalar case the degree of resonance overlap increases as a power law of the system size for dilute gases, but decays exponentially with the system size for dense clouds.
We present a framework to investigate the collective properties of atomic lattices in one, two, and three dimensions. We analyze the single-photon band structure and associated atomic decay rates, revealing a fundamental dependence on dimensionality. One- and two-dimensional arrays are shown to be inherently radiative, exhibiting band gaps and decay rates that oscillate between superradiant and subradiant regimes, as a function of lattice spacing. In contrast, three-dimensional lattices are found to be fundamentally non-radiative due to the inhibition of spontaneous emission, with decay only at discrete Bragg resonances. Furthermore, we demonstrate that this structural difference dictates the system dynamics, which crosses over from dissipative decay in lower dimensions to coherent transport in three dimensions. Our results provide insight into cooperative effects in atomic arrays at the single-photon level.
We investigate the properties of the cooperative decay modes of a cold atomic cloud, characterized by a Gaussian distribution in three dimensions, initially excited by a laser in the linear regime. We study the properties of the decay rate matrix
S, whose dimension coincides with the number of atoms in the cloud, in order to get a deeper insight into properties of cooperative photon emission. Since the atomic positions are random,
S is a Euclidean random matrix whose entries are a function of the atom distances. We show that in the limit of a large number of atoms in the cloud, the eigenvalue distribution of
S depends on a single parameter
b0, called the cooperativeness parameter, which can be viewed as a quantifier of the number of atoms that are coherently involved in an emission process. For very small values of
b0, we find that the limit eigenvalue density is approximately triangular. We also study the nearest-neighbor spacing distribution and the eigenvector statistics, finding that although the decay rate matrices are Euclidean, the bulk of their spectra mostly behaves according to the expectations of classical random matrix theory. In particular, in the bulk, there is level repulsion and the eigenvectors are delocalized, therefore exhibiting the universal behavior of chaotic quantum systems.