Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian

TL;DR

This paper investigates how cooperative effects and disorder influence the spectral properties of a large atomic gas's effective Hamiltonian, revealing distinct statistical behaviors in different density regimes and field types.

Contribution

It provides a numerical analysis of the spectrum of the non-Hermitian effective Hamiltonian, highlighting the impact of density and field type on spectral distributions and resonance overlaps.

Findings

Resonance width distribution follows a power law $P( ext{Γ}) \,\sim\, \text{Γ}^{-4/3}$ in dense gases.

Energy distribution follows Wigner's semicircle law in dilute gases.

In dense gases, the scalar resonance overlap decays exponentially with system size.

Abstract

We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\gg 1$ 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 $P(\Gamma) \sim \Gamma^{-4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $ P(\Gamma) \sim \Gamma^{-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.