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