Cooperative light scattering in any dimension

We present a theory of cooperative light scattering valid in any dimension: connecting theories for an open line, open plane, and open space in the nonrelativistic regime. This theory includes near-field and dipole-orientation effects, highlighting how field-mode confinement controls the phenomena. We present an experimental implementation for planar collective effects.

Figure 1

(a) Schematic showing a pair of emitters embedded in a 2D slab extending in the $x_1{x}_2$ plane. The emitters are separated a distance ${\mathit{r}}_{ıȷ}$ apart in the ${\widehat{\mathit{x}}}_2$ direction. Emission is detected by a detector D. (b) Energy diagram for two-atom superradiance, with $|0\rangle ={|g\rangle }_1{|g\rangle }_2, |E\rangle ={|e\rangle }_1{|e\rangle }_2$, and the superradiant and subradiant states $|\pm \rangle =\frac{1}{2}{\left(\right|e\rangle }_1{|g\rangle }_2\pm {|g\rangle }_1{|e\rangle }_2)$; $|\pm \rangle$ have transition energies ${\omega }_0\mp {\omega }_{12}$ and rates $\gamma \pm {\gamma }_{12}$, as labeled in the diagram.

Figure 2

Spherical polar plots of (a)–(c) dimensionless ${\tilde{\omega }}_{12}={\omega }_{12}/{\gamma }_{11}$ and (d)–(f) ${\tilde{\gamma }}_{12}={\gamma }_{12}/{\gamma }_{11}$ up to a multiplicative constant for parallel dipoles ${\mathbf{\mu }}_1={\mathbf{\mu }}_2={\sum }_{l=1}^3{x}_l{\widehat{\mathit{x}}}_l, r_{12}\ll \lambda$, and ${\widehat{\mathit{r}}}_{12}={\widehat{\mathit{x}}}_1$.

Figure 3

Dimensional and separation dependence of (a) and (b) dimensionless ${\tilde{\omega }}_{ıȷ}:={\omega }_{ıȷ}/{\gamma }_{ıı}$ and (c) and (d) ${\tilde{\gamma }}_{ıȷ}:={\gamma }_{ıȷ}/{\gamma }_{ıı}$ vs dimensionless separation ${\tilde{r}}_{ıȷ}=2\pi \frac{r_{ıȷ}}{\lambda }$ for identical parallel dipoles ${\widehat{\mathbf{\mu }}}_{ı}={\widehat{\mathbf{\mu }}}_{ȷ}={\widehat{\mathit{x}}}_3$. (a) and (c) show results interpolated for real-valued dimensions $1\le d\le 3$. (b) and (d) compare $d=1$ (dotted blue line), $d=2$ (solid red line), $d=3$ (dot-dashed green line).