Photon scattering from strongly driven atomic ensembles

The second-order correlation function for light emitted from a strongly and near-resonantly driven dilute cloud of atoms is discussed. Because of the strong driving, the fluorescence spectrum separates into distinct peaks, for which the spectral properties can be defined individually. It is shown that the second-order correlations for various combinations of photons from different spectral lines exhibit bunching together with super-Poissonian or sub-Poissonian photon statistics, tunable by the choice of the detector positions. Additionally, a Cauchy-Schwarz inequality is violated for photons emitted from particular spectral bands. The emitted light intensity is proportional to the square of the number of particles, and thus can potentially be intense. Three different averaging procedures to model ensemble disorder are compared.

Figure 1

Schematic setup of a dilute atomic ensemble pumped with a coherent field with wave vector ${\vec{k}}_L$. (a) The energy levels of each of the ensemble particles, and the interaction with the strong coherent light with coupling strength $\Omega$ is shown. (b) The ensemble with typical interparticle distance length scale denoted by $l$ with $l\gg 2\pi /k_L$ is depicted. We consider the case of photon pair emission in the forward direction and denote the angle between the two emitted photons with wave vectors ${\vec{k}}_1$ and ${\vec{k}}_2$ as ${\phi }_0$, and ${\vec{k}}_1+{\vec{k}}_2\approx 2{\vec{k}}_L$. The direction of the emission cone defined by ${\vec{k}}_1$ and ${\vec{k}}_2$ is characterized by the angle $\phi$ between ${\vec{k}}_1+{\vec{k}}_2$ and ${\vec{k}}_L$.

Figure 2

Normalized and configuration averaged second-order correlation function $g_{CC}^{\left(2\right)}\left(0\right)$ between two photons emitted from the central spectral band. In (a), the correlation function is shown for pairs of photons emitted in the same direction (${\phi }_0=0$, see Fig. 1), and plotted as a function of the emission direction $\phi$. In (b), the correlation function is plotted as a function of the opening angle ${\phi }_0$ between the two photons, which corresponds to the case of two distinct detectors. The two photons are measured at positions symmetric with respect to the incident laser field direction (i.e., $\phi =0$). The (blue) solid line shows averaging over a spherical shell, the dashed (red) line averaging over a spherical volume, and the (green) dotted line the numerical sampling.

Figure 3

Same as Fig. 2, except that the normalized and configuration averaged second-order correlation functions $g_{LR\left(RL\right)}^{\left(2\right)}\left(0\right)$ between one photon emitted in the left and one photon emitted in the right sideband is shown.

Figure 4

The Cauchy-Schwarz inequality $\chi$ characterizing the cross correlations of photon pairs with one photon emitted from each sideband. The photons are detected symmetrically around the incident laser field direction ($\phi =0$), and the result is shown as a function of the angle between the wave vectors of the two emitted photons ${\phi }_0$. The Cauchy-Schwarz inequality is violated for $\chi <1$. The curves are as in Fig. 2.