Spatial Characterization Of Two-Photon States - GAP-Optique

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