Spatial Characterization Of Two-Photon States - GAP-Optique

5. Spatial correlations in Raman transitions
pump
g
e
s
anti
Stokes Stokes
control
Figure 5.1: One atom with a Λ-type energy level configuration, can produce Stokes
and anti-Stokes photons by the interaction with the pump and control beams.
5.1 The quantum state of Stokes and anti-Stokes photon
pairs
This section describes the Stokes and anti-Stokes state generated by Raman
transitions in cold atomic ensembles in an analogous way to the description of
the two-photon state generated via spdc in chapter 1. The section discusses the
general characteristics of the nonlinear process, and introduces the two-photon
mode function.
Consider as a nonlinear medium an ensemble of n identical Λ−type cold
atoms trapped in a magneto-optical trap (mot). The atoms have an energy
level configuration with one excited state: |e〉 and two hyperfine ground states:
|s〉, and |g〉. This is the case, for example, in the d2 hyperfine transition of
87rb. In the initial state of the cloud all atoms are in the ground state |g〉, and
after emission all atoms return to their initial state, as figure figure 5.1 shows.
The two-photon generation results from the interaction of a single atom of
the cloud with two counter-propagating classical beams in a four step process.
In the first step, the atom gets excited by the interaction with the pump beam
far detuned from the |g〉 → |e〉 transition. In the second step, the excited atom
decays into the |s〉 state by emitting one Stokes photon in the direction zs as
shown in figure 5.2. In the third step, the atom is re-excited by the interaction
with the control beam far detuned from the |s〉 → |e〉 transition. In the last
step, the atom decays to the ground state by emitting an anti-Stokes photon
in the zas direction.
If ω 0 i
is the central angular frequency for the photons involved in the pro-
cess (i = p, c, s, as), and k 0 i is the corresponding wave number at the central
frequencies, energy and momentum conservation implies
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and
g
e
ω 0 p + ω 0 c = ω 0 s + ω 0 as, (5.1)
k 0 p − k 0 c = k 0 s cos ϕs − k 0 as cos ϕas, (5.2)
k 0 s sin ϕs = k 0 as sin ϕas. (5.3)
s