Multiple scattering of photons by atomic hyperfine multiplets

Mesoscopic interference effects in multiple scattering of photons depend crucially on the internal structure of the scatterers. In the present paper, we develop the analytical theory of multiple photon scattering by cold atoms with arbitrary internal hyperfine multiplets. For a specific application, we calculate the enhancement factor of elastic coherent backscattering as a function of detuning from an entire hyperfine multiplet of neighboring resonances that cannot be considered isolated. Our theory permits one to understand why atoms behave differently from classical Rayleigh point-dipole scatterers, and how the classical description is recovered for larger but still microscopic objects like molecules or clusters.

Figure 1

(Color online) Typical level structure of an atomic dipole transition in the $LS$-coupling scheme (energy splittings not drawn to scale). First the electronic spin $\mathit{S}$ ($S=1∕2$ in this example) is coupled to the orbital angular momentum $\mathit{L}$ (here $L=0$ in the ground state and $L=1$ in the excited state) to produce the fine structure angular momentum $\mathit{J}$ (here $J=1∕2$ and $J^{\prime }=3∕2$; $J^{″}=1∕2$ not shown). The frequency of the fine structure resonance line is ${\omega }_0$ (dashed arrow). $\mathit{J}$ is then coupled to the nuclear spin $\mathit{I}$ to give the hyperfine angular momentum $\mathit{F}$. The allowed hyperfine dipole transitions between the levels $F_g$ and $F_e^{\prime }$ with frequencies ${\omega }_{eg}$ are marked with arrows.

Figure 2

(Color online) Level scheme for limiting cases of purely elastic light scattering, requiring a unique ground-state $F_g$. Effectively excited levels are enclosed by a dotted circle. (a) Zero nuclear spin: only the fine structure with a unique excited state is involved. (b) Resonant excitation of a closed transition $F_{\max }\to F_{\max }+1$ or $F_{\min }\to F_{\min }-1$, also with a unique excited level. (c) $J=0$: a unique ground level with $F=I$ is coherently coupled to several excited levels. Cases (a) and (b) reduce to the known results of Ref. 14. Case (c) with a structured excited multiplet is treated analytically in Sec. 4.

Figure 3

(Color online) (a) Total single scattering cross section, Eq. (37), for ${}^{87}\mathrm{Sr}$ as a function of detuning (in units of $\Gamma$). The ground-state angular momentum is $F_g=I=9∕2$. There are three accessible excited states with angular momentum $F_e^{\prime }=9∕2,11∕2,7∕2$ and frequency separation $0.531\Gamma ∕2\pi$ and $1.875\Gamma ∕2\pi$ respectively. The corresponding (overlapping) optical resonances are shown by dashed lines. (b) Frequency dependence of the single scattering bistatic coefficient ${\gamma }_1$ for a semi-infinite medium in the four polarization channels $h\Vert h,h\perp h,l\Vert l$, and $l\perp l$. (c) Frequency dependence of the double scattering CBS interference contrast ${\gamma }_2^{\left(C\right)}∕{\gamma }_2^{\left(L\right)}$ for the same situation. The predictions for isolated resonances (See Ref. 14), indicated by the symbols, are clearly off the lines, indicating the importance of interference. (d) Frequency dependence of the total CBS enhancement factor (40). Monte Carlo calculation including all orders of scattering for a semi-infinite medium.