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arXiv:1805.04314 · Physical Review Letters

Localization Transition for Light Scattering by Cold Atoms in an External Magnetic Field

S. E. Skipetrov

Published Aug 29, 2018• 33 citations
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Abstract

We establish a localization phase diagram for light in a random three-dimensional (3D) ensemble of motionless two-level atoms with a threefold degenerate upper level, in a strong static magnetic field. Localized modes appear in a narrow spectral band when the number density of atoms $\ensuremath{\rho}$ exceeds a critical value ${\ensuremath{\rho}}_{c}\ensuremath{\simeq}0.1{k}_{0}^{3}$, where ${k}_{0}$ is the wave number of light in the free space. A critical exponent of the localization transition taking place upon varying the frequency of light at a constant $\ensuremath{\rho}>{\ensuremath{\rho}}_{c}$ is estimated to be $\ensuremath{\nu}=1.57\ifmmode\pm\else\textpm\fi{}0.07$. This classifies the transition as an Anderson localization transition of 3D orthogonal universality class.

DOI
10.1103/physrevlett.121.093601

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We establish a localization phase diagram for light in a random three-dimensional (3D) ensemble of motionless two-level atoms with a threefold degenerate upper level, in a strong static magnetic field. Localized modes appear in a narrow spectral band when the number density of atoms \ensuremathρ\ensuremath{\rho}\ensuremathρ exceeds a critical value \ensuremathρc\ensuremath≃0.1k03{\ensuremath{\rho}}_{c}\ensuremath{\simeq}0.1{k}_{0}^{3}\ensuremathρc​\ensuremath≃0.1k03​, where k0{k}_{0}k0​ is the wave number of light in the free space. A critical exponent of the localization transition taking place upon varying the frequency of light at a constant \ensuremathρ>\ensuremathρc\ensuremath{\rho}>{\ensuremath{\rho}}_{c}\ensuremathρ>\ensuremathρc​ is estimated to be \ensuremathν=1.57\ifmmode±\else\textpm\fi0.07\ensuremath{\nu}=1.57\ifmmode\pm\else\textpm\fi{}0.07\ensuremathν=1.57\ifmmode±\else\textpm\fi0.07. This classifies the transition as an Anderson localization transition of 3D orthogonal universality class.
We explore the potential of a static electric field to induce Anderson localization of light in a large three-dimensional cloud of randomly distributed, immobile atoms with a nondegenerate ground state (total angular momentum Jg=0{J}_{g}=0Jg​=0) and a threefold-degenerate excited state (Je=1{J}_{e}=1Je​=1). We study both the spatial structure of quasimodes of the atomic cloud and the scaling of the Thouless number with the size of the cloud. Our results indicate that unlike the static magnetic field, the electric field does not induce Anderson localization of light by atoms. We explain this conclusion by the incomplete removal of degeneracy of the excited atomic state by the field and the relatively strong residual dipole-dipole coupling between atoms which is weaker than in the absence of external fields but stronger than in the presence of a static magnetic field. A joint analysis of these results together with our previous results concerning Anderson localization of scalar waves and light suggests the existence of a critical strength of dipole-dipole interactions that should not be surpassed for Anderson localization to be possible in three dimensions.
Common belief, confirmed by existing experiments, is that arbitrarily weak disorder should lead to spatial localization of eigenmodes of scalar wave equations when wave propagation is two-dimensional (2D). We predict that contrary to this belief, a localization-delocalization transition can take place for light scattered by two-level atoms placed at random positions in the middle plane of a parallel-plate 2D waveguide fed by its fundamental transverse-magnetic (TM) mode (electric field polarized perpendicular to the waveguide and atomic planes). This transition, driven by near-field dipole-dipole interactions between atoms, occurs upon increasing the areal number density of atoms beyond some critical value. A finite-size scaling analysis of the transition yields an estimate of its critical exponent ννν = 1.4 ±\pm± 0.2.
Large ensembles of cold, immobile atoms might serve as ideal model systems to study Anderson localization of light. In contrast to dielectric systems, atomic clouds allow for a fine experimental control and admit a precise theoretical description. In this talk, we will review the recent theoretical developments aimed at paving a path towards an experimental realization of Anderson localization of light in three-dimensional (3D) atomic systems. Because atoms are the simplest point scatterers, considering light scattering on them is a good opportunity to illustrate several basic concepts and methods of the localization theory (Thouless conductance and Ioffe-Regel criterion, the mobility edge and the critical behavior, the scaling and self-consistent theories, the finite-size scaling procedure, etc.) on a concrete example. We will show that, surprisingly, Anderson localization of light by atoms is impossible in 3D if no special efforts are made to suppress near-field interactions between atoms. The necessary suppression may be achieved by subjecting the atoms to a constant external magnetic field or by arranging them in a crystalline structure in which a small amount of disorder is introduced. However, even when spatially localized states appear in a system of immobile atoms, their experimental observation remains challenging because of the residual thermal motion of atoms.
We demonstrate that a weak disorder in atomic positions introduces spatially localized optical modes in a dense three-dimensional ensemble of immobile two-level atoms arranged in a diamond lattice and coupled by the electromagnetic field. The frequencies of the localized modes concentrate near band edges of the unperturbed lattice. Finite-size scaling analysis of the percentiles of Thouless conductance reveals two mobility edges and yields an estimation $\ensuremath{\nu}=0.8--1.1$ for the critical exponent of the localization length. The localized modes disappear when the disorder becomes too strong and the system starts to resemble a fully disordered one where all modes are extended.
We demonstrate that the transport of coherent quasiresonant light through a dense cloud of immobile two-level atoms subjected to a static external electric field can be described by a simple diffusion process up to atomic number densities of the order of at least 102{10}^{2}102 atoms per wavelength cubed. Transport mean-free paths well below the wavelength of light in the free space can be reached without inducing any sign of Anderson localization of light or of any other mechanism of breakdown of diffusion.