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 />
According to equation 1.6, the electric field operators Ên in equation 5.4,<br />
are given by<br />
Ê (+)<br />
<br />
n (rn, t) = dkn exp [ikn · rn − iωnt]â(kn) (5.6)<br />
where, as in equation 1.23, we are using a more convenient set of transverse<br />
wave vector coordinates given by<br />
ˆxs,as =ˆx<br />
ˆys,as =ˆy cos ϕs,as + ˆz sin ϕs,as<br />
ˆzs,as = − ˆy sin ϕs,as + ˆz cos ϕs,as.<br />
(5.7)<br />
Under these conditions, at first order perturbation theory, the spatial quantum<br />
state of the generated pair of photons is<br />
<br />
|Ψ〉 =<br />
(5.8)<br />
dqsdqasΦ (qs, qas) |qs〉s|qas〉s<br />
where the mode function Φ (qs, qas) of the two-photon state is<br />
<br />
<br />
Φ (qs, ωs, qas, ωas) = dqpdqcEp (qp) Ec (qc) exp − ∆20R 2<br />
4 − ∆21R 2<br />
4 − ∆22L 2 <br />
4<br />
(5.9)<br />
with the delta factors defined as<br />
∆0 = q x s + q x as<br />
∆1 = (ks − kas) sin ϕs + (q y s − q y as) cos ϕs<br />
∆2 = kp − kc − (ks + kas) cos ϕ + (q y s − q y as) sin ϕs<br />
and the longitudinal wave vector of the pump beam given by<br />
ωpnp kp =<br />
c<br />
2<br />
− ∆ 2 0 − ∆ 2 1<br />
1/2<br />
(5.10)<br />
. (5.11)<br />
In the case of the Stokes and anti-Stokes two-photon state, there are no correlations<br />
between space and frequency due to the narrow bandwidth (∼ GHz)<br />
of the generated photons [60]. Therefore, to analyze the spatial shape of the<br />
mode function Φq (qs, qas), we can consider ωs = ω 0 s and ωas = ω 0 as.<br />
The effect of the unavoidable spatial filtering produced by the specific op-<br />
tical detection system used is described by Gaussian filters. The angular acceptance<br />
of the single photon detection system is 1/ k0 <br />
sws,as . In most experimental<br />
configurations, ws ≈ 50-150 µm and the length of the cloud is a few<br />
millimeters or less. For simplicity, we will assume that ws = was.<br />
In the calculations, the pump and the control are Gaussian beams with<br />
the same waist wp at the center of the cloud. As the waist is typically about<br />
200-500 µm, the Rayleigh range of the pump, Stokes and anti-Stokes modes<br />
Lp = πw2 p/λp and Ls,as = πw2 s/λs,as satisfy L ≪ Lp, Ls,as. This condition<br />
allows us to neglect the transverse wavenumber dependence of all longitudinal<br />
wave vectors in equations 5.9 and 5.10.<br />
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