Magnetic-Field-Driven Localization of Light in a Cold-Atom Gas

We discover a transition from extended to localized quasimodes for light in a gas of immobile two-level atoms in a magnetic field. The transition takes place either upon increasing the number density of atoms in a strong field or upon increasing the field at a high enough density. It has many characteristic features of a disorder-driven (Anderson) transition but is strongly influenced by near-field interactions between atoms and the anisotropy of the atomic medium induced by the magnetic field.

Figure 1

Complex eigenvalues $\mathrm{\Lambda }$ of a representative random realization of the Green’s matrix for $N=8\times 10^3$ two-level atoms in a strong magnetic field ($\mathrm{\Delta }=10^3$) at low (a) and high (b) densities of atoms.

Figure 2

Gray-scale maps of the average IPR at low (a) and high (b) densities of atoms in a strong magnetic field $\mathrm{\Delta }=10^3$. Dashed lines show lines along which the eigenvalues of a two-atom system would be situated for atoms placed along the direction of magnetic field ($\theta =0$) or perpendicular to it ($\theta =\pi /2$). For $0<\theta <\pi /2$ the corresponding eigenvalues are in between the two lines. The gray level of each small square in the figure is obtained by averaging over IPR of all eigenvalues that fall inside the square for 16 different realizations of the random Green’s matrix.

Figure 3

Thouless number $g$ as a function of the bare Ioffe-Regel parameter $k_0{\ell }_0=k_0^3/6\pi \rho$ for a strong magnetic field $\mathrm{\Delta }=10^3$. The curves are obtained by averaging over a unit interval of $\mathrm{Re}\mathrm{\Lambda }$ around their positions and over 50, 25, or 16 realizations of random positions of $N$ atoms for $N=2000$, 4000, and 8000, respectively. Different curves at the same value of $\mathrm{Re}\mathrm{\Lambda }$ correspond to different numbers of atoms $N$. The gray plane corresponds to $g=1$.

Figure 4

Same as Fig. 3 but for selected values of $\mathrm{Re}\mathrm{\Lambda }=-2002(\mathrm{a})$, $-2(\mathrm{b})$, and 1998(c). The insets show the scaling function $\beta (g)$ estimated from the numerical data of the main plots.