Analytic solutions for the spatial character and coherence properties of light scattered from two dipole-coupled atoms

Analytic solutions for steady-state expectation values of atomic quantities and second-order correlations are obtained for a fully quantum treatment of two stationary dipole-coupled atoms driven in a standard geometric configuration by a near-resonant laser. Explicit expressions for the spatial and coherence properties of the far-field scattered light intensity are derived, valid for the full range of system parameters. A comprehensive survey of the steady-state scattering behavior is given, with key features precisely characterized, including suppression of scattering, and the regime in which the dipole-dipole coupling has significant effect. A regime is also found where the incoherent scattered light develops spatial interference fringes. We examine in detail a decorrelation approximation that has potential application for larger systems of atoms that are intractable in a full quantum treatment. Finally, we introduce the concept of an effective driving field and show that it can provide a direct and intuitive physical interpretation of key aspects of the system behavior.

Figure 1

System diagram indicating the directions of the wave vector and polarization of the laser field and the perpendicular and parallel atomic configurations.

Figure 2

Polar plot of the intensity $\mathcal{I}\left(\widehat{\mathbf{r}}\right)$ (solid line) in the $x-y$ plane for (a) $R=0.25{\lambda }_L$, (b) $R=0.5{\lambda }_L$, and (c) $R=0.75{\lambda }_L$. Other parameters are $\mathrm{\Delta }=0,\mathrm{\Omega }=0.5\gamma$. The dotted line is the incoherent component of the intensity. The directions of the laser field wave vector and polarization are shown in (a) and are the same for all of the subfigures.

Figure 3

Forward-scattered intensity ${\mathcal{I}}_{\mathrm{fwd}}$ as a function of $R, \mathrm{\Omega }$, and $\mathrm{\Delta }$, for the perpendicular configuration. The intensity is scaled as in Eq. (33). The dashed red line indicates the uncoupled result ${\mathcal{I}}_{\mathrm{fwd}}^{\mathrm{uc}}$.

Figure 4

Incoherent fraction of forward-scattered intensity as a function of $R, \mathrm{\Omega }$, and $\mathrm{\Delta }$, for the perpendicular configuration. The dashed red line is $f_{\mathrm{fwd},\mathrm{inc}}^{\mathrm{uc}}$.

Figure 5

Comparison of ${\mathcal{I}}_{\mathrm{fwd},D}$ (dashed red line) and ${\mathcal{I}}_{\mathrm{fwd}}$ (solid line) for the case of $\mathrm{\Delta }=0$. The dotted line is the solution for the forward intensity obtained using the linear approximation.

Figure 6

Relative errors arising from the decorrelation approximation. (a),(b) ${\mathcal{E}}_{\mathcal{I}}$; (c),(d) ${\mathcal{E}}_d$ . Yellow regions: $0.01<\mathcal{E}<0.05$. Orange regions: $0.05<\mathcal{E}<0.1$. Red regions: $0.1<\mathcal{E}$. The black solid (dashed) lines give the boundary of the $\mathcal{E}>0.1$ (0.01) regions as approximated by Eqs. (57) and (59).

Figure 7

Effective field for two atoms in the perpendicular configuration, as a function of interatomic distance $R$. Parameters are the same as Fig. 3.

Figure 8

Polar plot of the intensity $\mathcal{I}\left(\widehat{\mathbf{r}}\right)$ (solid line) in the $x-y$ plane scattered from two atoms in the parallel configuration. The dotted line is the incoherent component of the intensity. The directions of the laser field wave vector and polarization are indicated in green. Parameters are $R=0.75{\lambda }_L, \mathrm{\Delta }=0$, and $\mathrm{\Omega }=0.5\gamma$.

Figure 9

(a) Comparison of ${\mathcal{I}}_{\mathrm{fwd}}$ solutions for perpendicular and parallel configurations. Parameters are $\mathrm{\Omega }=0.1\gamma , \mathrm{\Delta }=0.$ (b) Effective field magnitude as a function of $R$ for the parallel configuration. Parameters are the same as (a).