Coherent backscattering of inelastic photons from atoms and their mirror images

Coherent backscattering is a coherence effect in the propagation of waves through disordered media involving two or more scattering events. Here we report on the observation of coherent backscattering from individual atoms and their mirror images. This system displays two important advantages: First, the effect can be observed at low optical densities, which allows us to work in very dilute clouds or far from resonance. Second, due to the fact that the radiation of an atom interferes constructively with that of its own image, the phenomenon is much more robust to dephasing induced by strong saturation. In particular, the contribution of inelastically scattered photons to the interference process is demonstrated.

Figure 1

Four processes contributing to m-CBS. In the low intensity limit, reciprocal paths (i) and (ii) contribute to the m-CBS fringes, whereas paths (iii) and (iv) yield a smooth background intensity.

Figure 2

Experimental setup: A laser beam passes through a beamsplitter (BS) before illuminating the atomic cloud a first time. It is then reflected on a virtual mirror (VM), created by two lenses and a physical mirror, at a small angle of ${\theta }_0\sim 1^{\circ }$ with the normal of the mirror. After being reflected, it crosses the cloud again, before being sent by the beamsplitter, to a CCD camera detecting the angular distribution of the light in the focal plane of a lens. The top right picture shows an interference pattern obtained in our experimental setup, exhibiting the predicted circular symmetry. The inset is an azimuthal angular average of the picture.

Figure 3

Mirror position expected from m-CBS theory (i.e., $h_{\text{thr}}=\pi /k{\theta }_0{\mathrm{\Theta }}_f$) as a function of the experimental mirror position. This measurement allowed us to detect an initial experimental misalignment of $x_0=2.1\left(1\right)$ mm, which corresponds to the shifted minimum of the fit $h_{\text{thr}}=|Ah+x_0|$ (dashed line); we obtained $A=0.988\left(12\right)$. The error bars associated with the experimental data are smaller than the symbol size.

Figure 4

Single-atom [colored (shaded) areas] and cloud-averaged [red (gray) thick line] fringe patterns in the linear (a) and saturated (b) regime. The reason for the periodic modulation of the fringes' maxima for different atoms is explained in the main text; the black dashed line denotes the maximum amplitude of individual atomic fluorescence fringes, as a function of the maxima position. The maxima of all single-atom curves are identified by a black dot. In the saturated regime, the saturation of the atoms reduces the amplitude modulation, and the contrast of the averaged fringes decreases.

Figure 5

Experimental m-CBS contrast (blue triangles), compared to the normalized CBS one from a strontium cloud [11] (green circles) and from a rubidium cloud [15] (red stars). The gray areas for the theoretical predictions {for both the elastic scattering only, and the elastic and inelastic scattering [see main text and Eqs. (B11) and (B10) of Appendix pp2)]} account for the precision within which our experimental parameters are known: $h=8\pm 0.5$ mm and ${\sigma }_z=0.9\pm 0.1$ mm. The data are presented as a function of the saturation parameter $s$ at the center of the probe beam.

Figure 6

(a) Angular intensity profile in the presence (middle black curve) and in the absence [top green (gray) curve] of the atomic cloud. The deduced atomic fluorescence after the laser light weighted subtraction (see text) $I_f\left(\theta \right)$ [lower red (gray) curve, right vertical axis] presents a locally isotropic background, plus an interference pattern at the center. (b) Transmission coefficient $T$ and (c) background intensity $I_{\text{fluo}}$ deduced from the fit [see Eq. (A2) and the main text], as a function of the on-axis saturation parameter $s$. The dashed curves are calculated with a simple model for the interaction between our saturated Gaussian laser beam and our Gaussian atomic cloud.

Figure 7

(a) Intensity pattern in the nonlinear regime [Eq. (B10)], with $s=20$. The intensity pattern in the linear regime is virtually identical, up to a scaling factor. (b) Azimuthal average of the intensity pattern normalized to the background intensity, in the linear ($s=0.01$, blue continuous line) and nonlinear ($s=20$, red dashed line) regimes, showing the fringes profile. The figures can be compared to the measurement presented in Fig. 2 of the main text.