Spatial Characterization Of Two-Photon States - GAP-Optique

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