Induced dipole-dipole interactions in light diffusion from point dipoles

We develop a perturbative treatment of induced dipole-dipole interactions in the diffusive transport of electromagnetic waves through disordered atomic clouds. The approach is exact at order 2 in the atomic density and accounts for the vector character of light. It is applied to the calculations of the electromagnetic energy stored in the atomic cloud, which modifies the energy transport velocity, and of the light scattering and transport mean free paths. Results are compared to those obtained from a purely scalar model for light.

Figure 1

Sketch of light propagation in a dilute atomic cloud. Left: multiple scattering from independent atoms $\mathbf{\Sigma }={\mathbf{\Sigma }}^{\left(1\right)}$]; the propagating wave is never scattered more than one time by the same atom. Right: multiple scattering involving the possibility of repeated scattering (IDDC) between pairs of atoms $[\mathbf{\Sigma }={\mathbf{\Sigma }}^{\left(1\right)}+{\mathbf{\Sigma }}^{\left(2\right)}]$.

Figure 2

First- [(i), (ii)] and second-order [(iii), (iv), (v), (vi)] diagrams involved in the calculation of ${\ell }_s, a$, and ${\ell }^{*}$. Dotted arcs connect identical atoms. Solid lines refer to the free-space Green's tensor ${\mathit{G}}_0$. Circled crosses denote the atomic $t$ matrix.

Figure 3

Stored electromagnetic energy per atom $a/\eta$ [Eq. (14)] in units of the quality factor ${\omega }_0/\mathrm{\Gamma }$ for $\eta =0.4$ (solid blue curve). The dashed red curve is the independent-scattering approximation, Eq. (15). Inset: Second-order contribution $\delta a$, Eq. (16).

Figure 4

Induced dipole-dipole interaction potential $V_{\text{DD}}\left(r\right)$ between two atoms as a function of the interatomic distance $r$ for three values of $\mathrm{\Delta }$. When $\mathrm{\Delta }\gtrsim 1$, the curve displays a narrow subradiance peak. When $\mathrm{\Delta }\to 0$, this peak is smoothed out, and the potential becomes essentially attractive.

Figure 5

Scattering cross section ${\sigma }_s$ [Eq. (24)] in units of $6\pi /k_0^2 \left(k_0={\omega }_0/c\right)$ for $\eta =0.4$ (solid blue curve). The dashed red curve is the independent-scattering approximation, Eq. (25). Inset: second-order contribution $\delta {\sigma }_s$, Eq. (26).

Figure 6

Transport cross section ${\sigma }^{*}$ [Eq. (30)] in units of $6\pi /k_0^2$ for $\eta =0.4$ (solid blue curve). The dashed red curve is the independent-scattering approximation, Eq. (25). Inset: second-order contribution $\delta {\sigma }^{*}$, Eq. (31).

Figure 7

Relative correction $\delta a/\left(\eta a_{\text{ISA}}\right)$ as a function of $\mathrm{\Delta }$. The blue curve is the result for vector waves, Eq. (16), and the orange curve is the result for scalar waves, Ref. [8].

Figure 8

Relative correction $\delta {\sigma }^{*}/\left(\eta {\sigma }_{\text{ISA}}\right)$ as a function of $\mathrm{\Delta }$. The blue curve is the result for vector waves, Eq. (31), and the orange curve is the result for scalar waves, Ref. [8]. The dashed curve shows the contribution of only the lowest-order crossed diagram, calculated for scalar waves.

Figure 9

Schematic representation of Eq. (A3), indicating the conventions for tensor indices and momenta. The upper line symbolizes $G_{ik}$, and the lower line symbolizes $G_{jl}^{*}$.