Fig. 2: Negative refraction in a 5-layer lattice of two-level atoms with Rb lattice spacing.

Transmission of light through a cubic array of atoms, formed by five stacked infinite planar lattices. Each atom, denoted by a green dot, exhibits an isolated σ⁺-polarised two-level transition, with the quantisation axis along z, and a lattice constant a = 0.68λ, corresponding to Mott insulator-state experiments of Rb atoms^22,23. Negative refraction is demonstrated by normalised light intensity I/I₀ outside the medium at the plane z = 0, where \({I}_{0}=2{\epsilon }_{0}c\,{\max }_{{{{\bf{r}}}}}| {{{{\boldsymbol{{\mathcal{E}}}}}}}^{+}({{{\bf{r}}}}){| }^{2}\) represents the maximum incident field intensity. Within the lattice, delimited by the dashed green box, the atomic polarisation density \(| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }^{2}/| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }_{\max }^{2}\) is visualised and smoothed using convolution with a Gaussian of root-mean-square widths σ_x = 0.25, σ_y = 0.5. The light beam, incident in the xy plane at angle \(\theta=\arcsin (0.2)\) to the lattice normal and with a detuning Δ = −0.1γ from the atomic resonance, shows a lateral displacement along the y-axis of approximately −λ and power transmission T ≃0.95.