Spatial Characterization Of Two-Photon States - GAP-Optique

2.3. Correlations between signal and idler
Table 2.1: The parameters used in figures 2.3 and 2.4
Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c)
Crystal liio3 liio3 bbo bbo
L 1 mm 1 mm 1.5 mm 2 mm
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦
T0 → ∞ → ∞ → ∞ → ∞
wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm
λp 405 nm 405 nm 405 nm 351.1 nm
∆λp 0 0.4 nm
λ 0 s 810 nm 810 nm 810 nm 702 nm
∆λs 0.2 nm 10 nm 10 nm
ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦
2.3 Correlations between signal and idler
In contrast to the previous section, here the two-photon system is considered
as composed by two photons, each described by space and time variables, as
figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity
of the signal photon calculated in this section is a measurement of the degree of
spatiotemporal entanglement between signal and idler photons, as in references
[5, 10, 11, 47, 48].
spdc as a source of two-photon states can be used to generate pairs of
photons maximally entangled or two single photons [5, 49]. The purity of the
signal photon in the first case should be equal to 0 and in the second case equal
to 1. This section explores the necessary conditions to be in each regime.
This section is divided in three parts, the first one discusses the origin of
the correlations and the strategies to suppress them. The second part describes
how to obtain an analytical expression for the signal photon purity. And the
third part shows the results of numerical calculation of that purity.
2.3.1 Origin of the correlations between photons
The presence of cross terms in variables of both signal and idler in equation
1.34 indicates correlations between the generated photons. Figure 2.5 shows
those terms: i, j, l, m, n, r, t, v, and z.
In contrast to the last section, filtering one degree of freedom is not enough
to suppress the correlations between signal and idler. For example, by using a
frequency filter, the photons can be still correlated in space. The total suppression
of the correlation requires infinite narrow filters in both degrees of freedom
for one of the photons, as shown experimentally in reference [50]. But that kind
of filtering reduces the number of available pairs of photons substantially.
The other possibility to generate uncorrelated photons is to tailor the parameters
of the spdc process to make the cross terms negligible simultaneously.
According with appendix A, when α = 0 ◦ , this condition is equivalent to mak-
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