Angular correlation in a speckle pattern of cold atomic clouds

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Angular correlation in a speckle pattern of cold
atomic clouds

• Eilat 2006
• Ohad Assaf and Eric Akkermans
• Technion Israel Institute of Technology

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Outline

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

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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.

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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
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Correlation Diffuson
Intensity Diffuson
Paired amplitudes are from distinct realizations.
Paired amplitudes are from the same realization.
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The average transmission coefficient
involves the Diffuson
The angular correlation function involves the
Diffuson
with
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Building the multiple scattering
and are obtained from the
iteration of single scattering.

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Using a standard basis
, we decompose into components
Likewise, Diffusons acquire a
tensorial structure
Summation over inner photon polarizations
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Calculation
Spectral
Decomposition
Diffusion poles with relaxation times
Eigenvalues of
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Memory effect
Elastic mean free path
Size of the atomic cloud
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Results
Intensity fluctuations
Rayleigh law
Inelastic scattering, Doppler shift, finite
size absorption
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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.

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Diffusing photons and superradiance

• Diffusion coefficient and group velocity

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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
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Second order in perturbation theory in the
coupling to photons
Superradiance
Superradiant state
Subradiant state
Photon is trapped by The two atoms
Characteristics of superradiance
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Scattering properties of Dicke states
Scattering amplitudes of a photon by pairs of
atoms in superradiant or subradiant
states are
At short distance between the two
atoms, the subradiant term becomes negligible
compared to the superradiant term
(detuning)
Photon frequency two-level
spacing between
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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.