1. Angular correlation in a speckle pattern of cold atomic clouds Eilat 2006 Ohad Assaf and Eric Akkermans Technion – Israel Institute of Technology

2. Outline The scatterers. Correlations in a speckle pattern. Building the multiple scattering. Calculation. Results. Summary.

3. The scatterers Photon-atom interaction: dipolar interaction A degenerate atomic dipole transition allows Rayleigh scattering and Raman scattering Average light propagation in a cold atomic gas: Average over the positions of the atoms Trace over the quantum numbers with a scalar atomic density matrix. Weak disorder of weak and resonant scatterers.

4. Correlations in a speckle pattern We are interested in obtaining the angular correlation function of atomic speckle patterns at the approximation i.e. without quantum crossings. Slab geometry:

5. Intensity Diffuson Paired amplitudes are from the same realization. Correlation Diffuson Paired amplitudes are from distinct realizations.

6. The average transmission coefficient involves the Diffuson The angular correlation function involves the Diffuson with

7. and are obtained from the iteration of single scattering. : Building the multiple scattering

8. Using a standard basis, we decompose into components: Likewise, Diffusons acquire a tensorial structure : Summation over inner photon polarizations

9. Diffusion poles with relaxation times Eigenvalues of Spectral Decomposition: Calculation

10. Memory effect Size of the atomic cloud Elastic mean free path

11. Intensity fluctuations Rayleigh law: Inelastic scattering, Doppler shift, finite size absorption… Results

12. Summary We study the angular correlation in a speckle pattern of a cold atomic cloud. We find two kinds of interaction vertices - for intensity and for correlation - and thus two kinds of Diffusons. The intensity Diffuson gives rise to three “modes” that correspond to energy and angular momentum conservation. The correlation Diffuson for degenerate scatterers gives rise to nine “modes”, one of them negative, which implies a “correlation amplification”. Strong intensity fluctuations for degenerate scatterers.

13. Diffusing photons and superradiance Diffusion coefficient and group velocity

14. We have assumed a model of disorder where scatterers are independent. Edwards model or white noise In atomic gases, there are cooperative effects (superradiance, subradiance) that lead to an interacting potential between pairs of atoms. Dicke states and pairs of degenerate two-level atoms: Pair of two-level atoms in their ground state + absorption of a photon. Unperturbed and degenerate 0-photon states

15. Second order in perturbation theory in the coupling to photons Subradiant state Superradiant state Photon is trapped by The two atoms Characteristics of superradiance Superradiance

16. Scattering properties of Dicke states Scattering amplitudes of a photon by pairs of atoms in superradiant or subradiant states are: (detuning) Photon frequency two-level spacing between At short distance between the two atoms, the subradiant term becomes negligible compared to the superradiant term

17. Multiple scattering and superradiance Consider multiple scattering of a photon by atoms in superradiant states, i.e. coupled by the attractive potential Use Edwards model to calculate the self-energy in the weak disorder limit atomic density Maximum separation between the two atoms.