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arXiv:0806.0941 · Physical Review Letters

Photon Localization and Dicke Superradiance in Atomic Gases

Éric Akkermans, Aharon Gero, Robin Kaiser

Published Sep 5, 2008• 152 citations
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Abstract

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."

DOI
10.1103/physrevlett.101.103602

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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 photon escape rates from two-dimensional atomic gases while taking into account cooperative effects between the scatterers. Based on the spectrum of the random Euclidean matrix Uij=J0(xij){U}_{ij}={J}_{0}({x}_{ij})Uij​=J0​(xij​), where xij{x}_{ij}xij​ is the dimensionless random distance between any two atoms, a scaling behavior for the escape rates is obtained. This behavior is similar to the one obtained in the three-dimensional case, but qualitatively different from the result obtained in one dimension.
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.
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.
Many-body superradiance in ordered atomic arrays is a phenomenon where atomic synchronization gives rise to a burst in photon emission. This superradiant burst only occurs if there is one -- or just a few -- dominant decay channels. We demonstrate that superradiance exists in arrays of any dimensionality, as interference in photon emission leads to the preeminence of certain channels over others. In atomic chains superradiance occurs only below a critical interatomic distance, which we derive analytically. Increasing the array dimensionality leads to a critical distance that scales with system size: sub-logarithmically for 2D, and seemingly faster than that for 3D lattices. We perform calculations for both infinite and finite systems, the latter by employing a highly-efficient algorithm with a computational complexity that scales only linearly with system size, which enables us to study very large arrays. Our results provide a guide to explore this many-body phenomenon in state-of-the art experimental setups.
We show that in atomic gases cooperative effects like superradiance and subradiance lead to a potential between two atoms that decays like 1/r. In the case of superradiance, this potential is attractive for close enough atoms and can be interpreted as a coherent mesoscopic effect. The contribution of superradiant pairs to multiple scattering properties of a dilute gas, such as photon elastic mean free path and group velocity, is significantly different from that of independent atoms. We discuss the conditions under which these effects may be observed and compare our results to recent experiments on photon transport in cold atomic gases.
Coherence and interference play crucial roles in emission and absorption of photons to and from large systems with many atoms. Confusion has arisen because nuclear X-ray physicists and atomic quantum-optics physicists do not understand one another's individual descriptions of related phenomena. Basic physics same for all wave lengths from optical to nuclear gamma ray photons. But different languages are used to describe this physics in different domains. Crucial parameters vary over many orders of magnitude and what is intuitive or counterintuitive varies widely. Differences in parameters arising from differences between coherent emission effects in different domains produce very different results. Unified general treatment of the entire photon spectrum makes basic physics intelligible to all. In the optical region the mean distance between the scattering atoms is much longer than the photon wave length, Dicke superradiant scattering is isotropic and multiple scattering, Fano couplings are important and the lifetimes of intermediate stats are sufficiently short to be negligible. In X-ray scattering the mean distance between atoms is comparable to the photon wave length, Dicke superradiance is concentrated in a forward peak, multiple scattering and Fano effects are negligible, lifetimes are measurably long, speedup shortening the lifetime is important and most of the radiation is not elastically scattered but lost to absorption. Explicit calculations for a one-dimensional array shows the great difference between the case where the photon wave length is much shorter or comparable to the distance between nearest neighbors. A full investigation of the angular distribution and speedup of the intensity for two and three-dimensional scatterers give very different results for the two cases.