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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4. OAM transfer in noncollinear configurations<br />
ing induced by the spatial walk-off. The symmetry breaking implies that the<br />
correlations between oam modes do not follow the relationship lp = ls + li.<br />
Figure 4.7 also shows that for larger pump beams the azimuthal changes are<br />
smoothened out.<br />
The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at<br />
α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect,<br />
and the weight of the ls = 0 mode is larger than the weight of any other<br />
mode. This angle is optimal for the generation of heralded single photons with<br />
a Gaussian shape.<br />
The degree of spatial entanglement between the photons also exhibits az-<br />
imuthal variations depending on their emission direction. Figure 4.8 shows the<br />
] as a function of the azimuthal angle α with and with-<br />
signal purity T r[ρ2 signal<br />
out walk-off, for a pump beam waist wp = 100 µm with collection modes of<br />
ws = wi = 50 µm. When considering the walk-off, the degree of entanglement<br />
increases and becomes a function of the azimuthal angle. The purity has a<br />
minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ .<br />
The spatial azimuthal dependence affects especially those experimental configurations<br />
where photons from different parts of the cone are used, as in the<br />
case in the experiment reported in reference [12] for the generation of photons<br />
entangled in polarization. Using two identical type-i spdc crystals with the<br />
optical axes rotated 90◦ with respect to each other, the authors generated a<br />
space-frequency quantum state given by<br />
|Ψ〉 = 1<br />
<br />
√ dqsdqi Φ<br />
2<br />
1 q(qs, qi)|H〉s|H〉i + Φ 2 <br />
q(qs, qi)|V 〉s|V 〉i<br />
(4.8)<br />
where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the<br />
first crystal<br />
Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y<br />
i )], (4.9)<br />
and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the<br />
second crystal<br />
Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10)<br />
Since the photons generated in the first crystal are affected by the walk-off as<br />
they pass by the second crystal, the mode functions are different.<br />
Following chapter 2, the polarization state of the generated photons is calculated<br />
by tracing out the spatial variables from equation 4.8. The resulting<br />
state is described by the density matrix<br />
where<br />
46<br />
ˆρp = 1<br />
<br />
|H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i<br />
2<br />
<br />
+c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i<br />
(4.11)<br />
<br />
c = dqsdqiΦ 1 q(qs, qi)Φ 2 q(qs, qi). (4.12)